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Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences
13-2 | 2009 Varia
Édition électronique URL : http://journals.openedition.org/philosophiascientiae/224 DOI : 10.4000/philosophiascientiae.224 ISSN : 1775-4283
Éditeur Éditions Kimé
Édition imprimée Date de publication : 1 octobre 2009 ISBN : 978-2-84174-504-3 ISSN : 1281-2463
Référence électronique Philosophia Scientiæ, 13-2 | 2009 [En ligne], mis en ligne le 01 octobre 2009, consulté le 15 janvier 2021. URL : http://journals.openedition.org/philosophiascientiae/224 ; DOI : https://doi.org/10.4000/ philosophiascientiae.224
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SOMMAIRE
Actes de la 17e Novembertagung d'histoire des mathématiques (2006) 3-5 novembre 2006 (University of Edinburgh, Royaume-Uni)
An Examination of Counterexamples in Proofs and Refutations Samet Bağçe et Can Başkent
Formalizability and Knowledge Ascriptions in Mathematical Practice Eva Müller-Hill
Conceptions of Continuity: William Kingdon Clifford’s Empirical Conception of Continuity in Mathematics (1868-1879) Josipa Gordana Petrunić
Husserlian and Fichtean Leanings: Weyl on Logicism, Intuitionism, and Formalism Norman Sieroka
Les journaux de mathématiques dans la première moitié du XIXe siècle en Europe Norbert Verdier
Varia
Le concept d’espace chez Veronese Une comparaison avec la conception de Helmholtz et Poincaré Paola Cantù
Sur le statut des diagrammes de Feynman en théorie quantique des champs Alexis Rosenbaum
Why Quarks Are Unobservable Tobias Fox
Philosophia Scientiæ, 13-2 | 2009 2
Actes de la 17e Novembertagung d'histoire des mathématiques (2006) 3-5 novembre 2006 (University of Edinburgh, Royaume-Uni)
Philosophia Scientiæ, 13-2 | 2009 3
An Examination of Counterexamples in Proofs and Refutations
Samet Bağçe and Can Başkent
Acknowledgements Partially based on a talk given in 17th Novembertagung in Edinburgh, Scotland in November 2006 by the second author. Authors acknowledge the helpful feedback of the two anonymous referees. Eric M. Hielcher and David Chabot helped improving the readability of the text. …the first important notions in topology were acquired in the course of the study of polyhedra. H. Lebesgue
1 Introduction
1 Lakatos’s influential work Proofs and Refutations (PR, afterwards), first published in the British Journal of Philosophy of Science in four parts between 1963 and 1964, introduced many new concepts to both philosophy and the methodology of mathematics. From these we focus here on the concept of heuristics. We analyze the heuristics employed in PR by focusing on the counterexamples discussed in the original work.
2 PR introduced the methods of proofs and refutations by discussing the history and methodological development of Euler’s formula V — E + F = 2 for three dimensional polyhedra, where V, E and F are the number of vertices, edges and faces respectively. Lakatos considered the history of polyhedra to be a good example for illustrating his philosophy and methodology of mathematics and geometry. There are several reasons for this. The first is the fact that the history of Euler’s formula spans the paradigm shift from the Cauchyian analytic school to the Poincareian topological school, and thus is a natural illustration for theory change. The second such reason is that Lakatos’s exposition of Euler’s formula is a “rationally reconstructed” account of the subject matter, therefore diverging from the actual history in certain respects, as was pointed out by Lakatos in the Introduction to PR. However, this drawback does not detract from
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Lakatos’s position as his focus was on theory change rather than precise historiography, evidenced by his remark that “informal, quasi-empirical mathematics does not grow through a monotonous increase of the number of indubitably established theorems, but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations” [Lakatos 1976, 5]. However, as Steiner emphasized, “this conclusion tends to undermine ‘formalism’ (...)” [Steiner 1983, 503]. In the present paper, our principal goal is to fill in the gap between Lakatosian heuristics and the formalism present in the counterexamples given in PR. Thus, the aforementioned divergence from the actual history of the formula is not relevant to the topic at hand.
3 Before beginning our exposition, we give a very short summary of the methods of proofs and refutations. However, we do not go into a detailed account of Lakatos’s methodology; the interested reader is advised to refer to [Koetsier 1991] and [Kampis, Kvasz, Stöltzner 2002] for a more sophisticated treatment of Lakatos’s methodology of proofs and refutations. Here, we use the exposition given by Corfield in [Corfield 2002]. The method of proofs and refutations is comprised of the following methodological steps: 1. Primitive conjecture. 2. Proof (a rough thought experiment or argument, decomposing the primitive conjecture into subconjectures and lemmas). 3. Global counterexamples. 4. Proof re-examined. The guilty lemma is spotted. The guilty lemma may have previously been hidden or misidentified. 5. Proofs of the other theorems are examined to see if the newly found lemma occurs in them. 6. Hitherto accepted consequences of the original and now refuted conjecture are checked. 7. Counterexamples are turned into new examples, and new fields of inquiry open up.
4 In this work, our aim is to show the heuristic role of the counterexamples. We identify how particular counterexamples helped to improve conjectures and how these counterexamples are identified. We employ certain geometrical and topological notions such as genus, connected component etc., in order to be as mathematically precise as possible.
5 The present paper is organized as follows. First, we briefly revisit the formula in question and its Cauchy proof. Then, in the second part, by following the train of thought followed by Lakatos himself in PR, we discuss the counterexamples one by one and point out their heuristic functions. We conclude by pointing out some possibilities for future work.
2 The Conjecture and the Proof
6 The main conjecture discussed in PR is the following: V – E + F = 2,
7 where V, E and F denote the number of vertices, edges and faces respectively for all regular polyhedra. This conjecture is often called the Descartes-Euler conjecture for historical reasons. The integer which is the solution to the equation V — E + F for some polyhedron P is called the Euler characteristic of P.
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8 The proof given in PR was due to Cauchy. Let us briefly recall it step by step. Step 1 : Imagine that the polyhedron is hollow and made of a rubber sheet. Cut out one of the faces and stretch the remaining faces onto a flat surface (or board) without tearing. In this process, V and E will not alter. Thus, we have V — E + F = 1, since we have removed a face. Step 2: The remaining map to be triangulated. Drawing diagonals for those curvilinear polygons which we obtained after stretching, will not alter V — E + F since E and F increase simultaneously. Step 3: Remove the triangles. This can be done in one of the two ways: either one edge and one face are removed simultaneously; or one face, one vertex and two edges are removed simultaneously.
9 At the end of this process, we end up with an ordinary triangle for which V — E + F = 1 holds trivially.
10 However, observe that there are three lemmas that have been used implicitly throughout the proof. Lemma (i) Any polyhedron, from which a face has been removed, can be stretched lat on a lat surface. Lemma (ii) In triangulating the map, a new face is obtained for every new edge. Lemma (iii) There are only two alternatives for removing a triangle out of the triangulating map: the removal of one edge, or the removal of two edges and a vertex. Furthermore, the final result of this process will be a single triangle.
11 These three lemmas play a rather significant role in the proof. Thus, we focus more on them now.
3 Counterexamples and Their Heuristic Patterns
12 The first counterexamples are directed to the three lemmas which were seen to have been used in the above proof. We will categorize these counterexamples by following the classification which was employed in PR. The heuristic patterns of each counterexample and refutation played a significant role in PR, and we clarify them here from a topological/mathematical perspective.
3.1 Local but not Global Counterexamples
13 Local counterexamples contradict the specific lemmas or constructions which were used in the proof without being relevant to the main conjecture. We now consider the method of proof and the lemmas used in the proof.
14 Removing a triangle from the ‘inside’ of the triangulized map of the cube In removing a triangle from the ‘inside’ of the triangulized map, one is able to remove a triangle without removing any single edge or vertex. This, however, is not one of the two ways of removing a triangle given in Lemma (iii), and since this lemma claimed these were the only ways to remove a triangle, this proves the lemma to be false. But, the Euler conjecture clearly does hold for the cube. Thus, we see that this counterexample does not refute the main conjecture but only a lemma which comprised part of its proof. In order to handle this counterexample, a modification of Lemma (iii) is considered.
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15 Modified lemma (iii) The new idea is to remove any boundary triangle by exactly following the pattern described in the third lemma. Yet, another counterexample was given which contradicts the modified Lemma (iii).
16 Counterexamples by disconnecting the network One can easily remove the triangles by disconnecting the network. In this way, V — E + F is reduced by 1 by the removal of two edges and no vertices. So, Lemma (iii) again needs to be modifed to save the proof.
17 Second Modification of Lemma (iii) Remove the triangles in such a way that V — E + F is not changed. Alternatively, we can modify the terminology: we call boundary triangles those whose removal does not disconnect the network. Equivalently, the triangles in our network can be numbered that in removing them in the right order V — E + F will not change until we reach the last triangle.
18 The role of this counterexample was to describe a way to remove the triangles from the triangulized map of the polyhedron in such a way that the proof will succeed. As previously observed, one cannot remove the triangles from the inside of the lat mapping. Thus, the role of this counterexample is that of a positive heuristic as it helped us to prove what we set out to prove. The positive heuristic role of this particular counterexample played a significant role in emphasizing the topological character of Cauchy’s proof. The underlying reason for this is the fact that the polyhedron was manipulated adopting a topological point of view, disregarding its metric qualities and quantities. Whatever the size of an edge or a face of the polyhedron, the proof still applies. But, what makes this specific proof applicable? The present counterexample underlines the topological “simple connectedness” property of the proof since, as soon as the planar network is simply connected, the removal of the triangles from outside does not alter the simply connected character of the network. Thus, the topological invariants of the network remain the same. Note that the “removal” operation which makes the proof go smoothly is a homeomorphism. Recall that a homeomorphism (or topological isomorphism) is an isomorphism which preserves topological properties. A homeomorphism, thus, is a bijective and bicontinuous function. As a result, homeomorphic topological objects have the same topological properties. For instance, a sphere and a torus are not homeomorphic, thus do not enjoy the same topological properties. In conclusion, this lemma shows that the simple topological object is preserved under homeomorphism, and in this particular case, the homeomorphism is the function of removing a triangle from outside of the planar network. Further, this mapping does not violate the simpleness of the topological object.
3.2 Global Counterexamples
19 Global counterexamples contradict the main conjecture. Essentially, global counterexamples, outside of any reference to the proof of the conjecture, demonstrate that it is false. However, the first global counterexample which was given in PR was also local.
20 A pair of nested cubes A nested cube is a complex object which is composed of two cubes one inside the other. Hence, it is a counterexample for the first lemma, since the nested cube cannot be stretched onto a plane after a face has been removed from the inner polyhedron. Also, for the nested cube, we have V — E + F = 4.
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21 At this stage, the students in the fictional class of Lakatos, disagreed on whether a pair of nested cubes is a genuine polyhedron at all. Thus, they altered some definitions of polyhedra in order to save the conjecture, and suggested the following definitions. Definition 1 A polyhedron is a solid whose surface consists of polygonal faces. Definition 2 A polyhedron is a surface consisting of a system of polygons.
22 Let us now see how these attacks on the conjecture can be evaluated from a heuristic point of view. First, note that in order to deal with the case of the nested cube, the protective belt of the hard core of the theory of polyhedra was employed. This opened up a discussion on the definitions of polyhedron in order to save the conjecture. The underlying motivation was the following. In order to exclude the nested cube as a freak, or monster as Lakatos himself called it, it must be proven that the nested cube is not a polyhedron at all. To accomplish this, one can define the polyhedron as a proof generated concept. As long as the students stuck to the Cauchy proof, a rigorous and precise definition of polyhedra was needed and thus this definition was derived from the proof in such a way that the definition would not conflict with the proof and any of its lemmas. Essentially, this is a very important notion in Lakatos’s methodology of scientific research programs (MSRP, henceforth), and this example is one of the examples Lakatos himself used in order to reject the deductivist and Euclidean methodology of mathematics which he attacked in Appendix 2 of PR, entitled “Deductivist Approach vs. Heuristic Approach”. As Lakatos clearly and radically considered these concepts, we will not repeat them here. However, we note that, for Lakatos, the proof was in fact needed in order to shape the notion (or the definition) of the polyhedra.
23 Heuristically speaking, these definitions must be classified as negative heuristics, since they exclude some forms of geometric objects which lie beyond the given two definitions and thus restricted the class of objects to which they apply. Notice that this revision of definitions seems to be an ad hoc. After each counterexample, one might need to revise the definitions in order to exclude the monsters. As might be expected, new counterexamples emerged immediately after the new definitions were introduced.
24 Counterexamples to Definitions 1 and 2 A pair of two tetrahedra which have an edge in common and another pair of two tetrahedra which have a vertex in common serve as counterexamples to Definitions 1 and 2. In PR, they were labeled as Counterexamples 2a and 2b. Observe that both counterexamples fit both definitions. Furthermore, for both counterexamples, we observe that V — E + F = 3. In order to exclude these counterexamples, an immediate revision of the definitions took place in a rather ad hoc manner.
25 Definitions 1 and 2 were very intuitive, so it is not surprising that counterexamples to them were found. Counterexamples 2a and 2b were presented in a way as to show that those definitions were much too restrictive, i.e. their degree of negativity in terms of heuristic was too high. Being rather intuitive but very weak notions, Definitions 1 and 2 included too many objects as being regular polyhedra allowing some to serve as counterexamples.
26 We now try a new definition. Definition 3 A polyhedron is a system of polygons arranged in such a way that (1) exactly two polygons meet at every edge and (2) it is possible to go from the inside of any polygon to the inside of any other polygon via a route which never crosses any edge at a vertex.
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27 It is easy to see that in the first twin tetrahedron in our last counterexample, there was an edge at which four polygons meet and in the second twin tetrahedron it was impossible to get from the inside of a polygon of the upper tetrahedron to the inside of another polygon of the lower tetrahedron without traveling via a route which crosses some edge at a vertex.
28 As a consequence, Counterexamples 2a and 2b provided cause for a revision of the proof generated concept of polyhedra and Definition 3 arose as a result of the aforementioned revision. This was the protective belt of the theory which intervened and protected the hard core of the theory by introducing the Definition 3. The result was the Perfect Definition, which we denote by Definition P. Definition P. A polyhedron is a system of polygons for which the equation V — E + F = 2 holds.
29 Definition P. is a rather deductivist definition which Lakatos would never agree. We believe that this is the reason why PR contained further material.
30 Even though, it was the perfect definition, the discussions on polyhedra did not finish with its treatment. The question then becomes, why did Lakatos include this ad hoc definition which was indeed very narrow, restricted and heuristically weak? In our opinion, the underlying reason for the introduction of Definition P. was for showing the fallacies of the Euclidean mathematical tradition which Lakatos attacked in the Appendices of PR. Obviously, in the very narrow domain described by Definition P., the Descartes-Euler conjecture was trivially satisfied as the Definition P. reduced the Eulerness of the polyhedra to a single formula, i.e. V — E + F = 2, which did nothing to explain the global properties of Eulerness. Moreover, it should also be underlined that this definition was proof-generated, and thus came after the proof. At the first glance, Definition P. seems to be a deductive concept. However, when considering the fact that the Definition P. is proof generated, it is easy to see that it cannot be a deductive notion. We believe that this move was a very significant accomplishment of Lakatos: he turned a deductive-looking concept into a proof generated concept by revising the heuristic presentation.
31 The Urchin of Kepler The Urchin of Kepler is the original name for the small stellated dodecahedron, and it was a counterexample due to the fact that for this “polyhedron” V — E + F = —6. Yet, it satisfied Definition 3.
32 One of the most interesting counterexamples of PR was definitely the urchin. It was proposed to refute Definition 3. So, by the rules of negative heuristics, Definition 3 was revised to exclude the urchin.
33 It was clear that Definition 3 needed to be revised after the urchin was introduced. In order to exclude the urchin, another new definition, Definition 4, is proposed. Definition 4 A polygon is a system of edges arranged in such a way that (1) exactly two edges meet at every vertex, and (2) the edges have no points in common except the vertices.
34 With Definition 4, the problem of the urchin was avoided since, in the urchin, the edges have common points beyond the vertices. Consequently, Definition 4 was modified again in order to “save” the urchin. Definition 4’ A polygon is a system of edges arranged in such a way that exactly two edges meet at every vertex.
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35 In order to save the urchin, a new version of the definition of the polyhedra appeared, which was Definition 4’. This definition was again a product of negative heuristics, since it excluded a clause of the previous definition and restricted the concept of a polygon even further. The proponent of Definition 4’ gave a brilliant explanation in order to justify Definition 4, viz. that one should consider the polygons in space not in the plane. In that way, he thought, the second clause of Definition 4 becomes useless, since “what you think to be a point in common is not really one point, but two different points lying one above the other” [Lakatos 1976, 17].
36 Clearly, the increase in the spatial dimensions reflects a topological intuition. The lines which intersect in two dimensions may not intersect in three dimensions. One can approach this naive observation from low-dimensional topology or even from linear algebra of vector spaces.
37 Area of the urchin An attempt at solving the problem of the urchin was made by pointing out the null areas of some polygons inside it. Urchin was tried to be refuted by indicating the null areas of the some polygons in it. However, this attack was parried by a disregard for the connection between the “idea of area” and the “idea of polygon”.
38 This refutation was again via a negative heuristic following the Popperian tradition. However, mathematically speaking this metric approach to geometric objects is far from being a topological approach. As soon as the concept of area is introduced as a quantity, one might ignore, for instance, the degenerate polygons. In other words, two objects might be homeomorphic but might have different areas. For example, consider two cubes, one having area a, the other having area 2a. Clearly, these two cubes are homeomorphic, but their areas are different.
39 The problem of null areas can also be considered from a topological point of view. Null areas appear when we have a dimension lacking such that some regions are still covered. It is essentially similar to the crossing lines problem which we discussed above.
40 The next counterexample we will discuss is the picture frame.
41 The Picture frame The picture frame was a counterexample to Definition 4 and Definition 4’. It satisfies all the definitions but for the picture frame we have V — E + F = 0. Immediately after this observation, the picture frame was handled by a new definition of polyhedra. Definition 5 In the case of a genuine polyhedron, through any arbitrary points in space, there is at least one plane whose cross-section with the polyhedron consists of a single polygon.
42 The picture frame satisfied all of the definitions hitherto proposed. However, it had an Euler characteristic of 0. It should be recalled that the picture frame had a “hole”, and can only be inflated onto a torus. The reason for this is that a torus has only one hole, or more properly speaking, a torus is of genus one. More precisely, one can always find a homeomorphism between a picture frame and a torus. Thus, the topological properties of a torus and a picture frame are identical. Therefore, it would not be too strange to expect that the picture frame might be handled in some way via these properties. Definition 5 served for this purpose. It emphasized the simple connectedness requirement for Descartes-Euler polyhedra. Observe that if a point is taken in the picture frame, a plane whose cross-section with the picture frame consists of a single polygon does not exist.
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43 Nonetheless, Definition 5 is not the final definition. The following counterexamples challenge Definition 5.
44 Cylinder The cylinder was a counterexample to Definition 5 and all other definitions presented hitherto. The cylinder was labeled Counterexample 5 since V — E + F = 1 for it. The “unusual structure” of the cylinder was given attention in the following discussion on the concept of an edge. It was then claimed that the cylinder cannot be a counterexample to the definitions of a polyhedron, since it was inconsistent with the proper definition of an edge. As a result, a new definition of an edge was proposed. Definition 6 An edge has two vertices.
45 The cylinder was addressed in the discussion of the proof-generated concept of an edge. We observe that by defining an edge, the cylinder was excluded from being a polyhedron by the methods of negative heuristics. Nevertheless, this is not the final step. The consequent definition of an edge in this respect can be considered a product of positive heuristics as well. By subsequently discussing the definition of an edge, Lakatos demonstrated precisely how his MSRP worked. One negative heuristic can indeed be followed by a positive heuristic move in such a way that, similar to the Hegelian dialectics, a new concept would be generated.
46 Having discussed the defining terms such as an edge, the discourse now proceeded onto one of the most significant aspects of Lakatosian heuristics. Let us now briefly review one such aspect, monster-barring.
47 The Method of Monster-Barring “Using this method one can eliminate any counterexample to the original conjecture by a sometimes deft but always ad hoc redefinition of the polyhedron, [or] of its defining terms, or of the defining terms of its defining terms” [Lakatos 1976, 23].
48 As Lakatos indicated in PR, the method of monster-barring is “always ad hoc”. In monster-barring, we may theoretically redefine the terms recursively, i.e. we first start with the terms and define them, and then define the terms that defined the original term and so on. Therefore, the process of definining becomes ad hoc, and we may need another tool to get ourselves out of this vicious circle of ad hocness. This is precisely what Lakatos wanted us to do.
49 Nonetheless, the method of monster-barring does not necessarily yield positive heuristics. This is because, it can be used to avoid the counterexamples or similarly to improve the conjecture by polishing the defining terms.
50 A New Statement of the Theorem For all polyhedra that have no cavities, no tunnels and no “multiple structures”, we have V — E + F = 2.
51 This new statement excludes all monsters, i.e. exceptions. Yet, this process, as we pointed out, is ad hoc. Although it did exclude all the exceptions, we might not be able to determine whether this is true. In fact, as was remarked in PR, the urchin and the cylinder are counterexamples to this new statement of the theorem.
52 It is also worthwhile to underline that the method of monster-barring forgets about the proof. Instead, it focuses on the definitions, its domain and its truth sets—the objects that satisfy the conjecture. As was said in PR, in monster-barring, “by suitably restricting both conjecture and the proof to a proper domain, the conjecture, which is now true, will be perfected, and the basically sound proof, which is now rigorous, will be perfected and obviously will contain no more false lemmas” [Lakatos 1976, 29].
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53 Let us now briefly review an application of monster-adjustment which “adjusted” the urchin. In other words, the defining terms of the monster (urchin) were redefined in such a way that the monster became a regular.
54 Method of Monster-Adjustment on the Urchin It was claimed that there were no star-polygons in the urchin, but only triangular faces. 60 faces, 90 edges and 32 vertices give an Euler characteristic of 2. It was then concluded that the urchin is a polyhedron and is Eulerian as well. However, its star-polyhedral interpretation was faulty.
55 Recall that on the first interpretation of the urchin, it was claimed that its faces were star-polygon. In order to be able to utilize positive heuristics, it was then claimed that the urchin had triangular faces. Thus, this interpretation yielded the observation that the urchin was in fact Eulerian with the appropriate calculations. Yet, some objections were given against the monster-barring interpretation of the urchin. Since all of its triangles lie on the same plane in the groups of five and thus surround a regular pentagon behind the solid angle, it was claimed that the triangular interpretation was false. In this case, these objections played the role of positive heuristics as all of the interpretations or misinterpretations exposed what the urchin really was and how it should have been truly understand. Positive heuristics again helped us how and in which direction we should proceed. In this specific example, we were told how to analyse the urchin, and how to apply it to the Euler conjecture by these positive heuristics.
56 Discussions on Geometrical Topology Having discussed the monster adjustment, a naive geometrical topological approach is laid out. The picture frame was considered first in this regard. It was concluded that the picture frame cannot be inflated into a sphere or a plane. The reason for this is that the genus of the sphere is zero, and this is true even under the planar interpretation of stretching. Namely, it was topologically the “same” to stretch the polyhedra onto the [Euclidean] plane or onto the sphere. In other words, a homeomorphism between these two manifolds exists. From this, it was concluded that the picture frame could be inflated into a torus. It should now be recalled that the torus has genus one. Therefore, the general formula of Euler characteristics in manifolds is: v - e + f = 2.g(S)
57 where S is the surface onto which the polyhedron is being inflated.
58 For a torus S, we have the Euler characteristic V — E + F = 2 — 2 x 1 = 0 as mentioned earlier. Hence, it can be concluded that the picture frame is not planar. This is the exact reason which made the picture frame a global counterexample. Since the torus is not simply connected, it was argued that the original conjecture holds only for simple polyhedra, namely for those which, once one of their faces has been removed can be stretched onto a plane.
59 In this way, the domain of the conjecture and Lemma (i) gets restricted. However, the proof still remains the same.
60 This interpretation of the picture frame in this context is a positive heuristic. Apparently, by the introduction of the concepts of torus and that of genus, we were shown how to interpret the picture frame correctly. Thus, we observe that the positive heuristics introduced us to a new concept of genus in order to allow us to interpret the non-simple polyhedra precisely. The conjecture was improved and thus we ended up with a new formulation.
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61 Yet, this does not mark the end of the counterexamples. The next such counterexample is called a crested cube.
62 Crested Cube Counterexample 6 was the crested cube. It agreed with all the definitions hitherto proposed, in that it had no cavities, tunnels or multiple structures. Yet, the Euler characteristic of the crested cube was 3.
63 The crested cube was handled by modifying Lemma (ii). Consequently, the original conjecture was modified by incorporating the new version of Lemma (ii) as follows: For a simple polyhedron, with all its faces simply connected, V — E + F = 2. Thus, in this case, “even though [proofs] may not prove, [they] help [us] to improve the conjecture” [Lakatos 1976, 37]. From this observation of Lakatos, we conclude that the restriction on Lemma (ii) was a positive heuristic since this restriction let us understand the proof better, and thus improve it with a modified lemma. In this specific case, the sides of the crested cube were interpreted as ring-shaped faces. Therefore, the simply- connectedness condition excluded the crested cube as it had sides which were not simply-connected. However, it was then claimed that the ring-shaped face interpretation was misleading. So, we were told by negative heuristic that, this was a misinterpretation and we should reject it, and that the crested cube is a genuine counterexample to the Euler conjecture. “The misinterpretation interpretation” again utilizes the observation that in lower dimensional manifolds, faces which would not intersect in higher dimensions might intersect.
64 All these discussions resulted in the following formulation of the theorem.
65 A New Formulation of the Theorem All polyhedra are Eulerian, which (a) simple, (b) have each face simply connected, and (c) are such that the triangles in the planar triangular network, which results from the processes of stretching and triangulating, can be so numbered that, in removing them in the right order, V — E + F will not change until we reach the last triangle. In this formulation the lemmas have been turned into conditions.
66 The ultimate application of the method of lemma incorporation yielded a new formulation of the theorem. This formulation was, without doubt, a positive heuristic, since it explicitly and clearly revealed what a polyhedron was and to which objects the Euler conjecture applied. Note that, in this new formulation, all the lemmas we discussed previously were turned into conditions. In other words, the “facts” we used in the proof were discussed and found suspicious, and then, in order to alleviate the suspicion, they were pushed to the meta-level as conditions of the theorem.
67 However, the method of the lemma incorporation suffers from an old problem, that is the problem of induction. How can one ensure that we have incorporated all the lemmas into the formula as conditions?
68 Hidden Lemmas It was claimed that there were some hidden lemmas in some of the mathematical formulations such as “all triangles have three edges and vertices”. In the method of lemma incorporation, one must include all lemmas. However, in this way, the problem of infinite regress appears: how can one determine where to stop turning lemmas into conditions?
69 From the heuristic point of view, we observe that positive heuristics told us that we must include and incorporate the lemmas into the formula in order to properly apply this method. However, in practice, we included only the lemmas for which we had
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counterexamples thus far. Ad hocness thus still remains although we adopted a new method.
3.3 Global but not Local Counterexamples
70 Cylinder Again The cylinder, it was observed, served as a counterexample for both the naive conjecture and the theorem. Therefore, it is a global but not a local counterexample since it refutes the theorem, but not the lemmas. The problem appears when one tries to inflate the cylinder onto a plane by removing the side face (the jacket). At this point, it seems that the cylinder should satisfy another lemma: the resulting planar network needs to be connected. But it turned out that this was a hidden assumption.
71 Moreover, it was claimed that the cylinder did not satisfy lemma (ii), viz. that is “any face dissected by a diagonal falls into two pieces”. The counter argument that followed immediately was the observation that, since one cannot draw a diagonal on circular faces, the cylinder should satisfy the lemma. It was then observed that this interpretation of diagonalization did not follow from the proof due to the fact that it was impossible to arrive at a triangular network and conclude the proof.
72 We first observe that the problem of cylinder was avoided by a positive heuristic rule which says that, in the proof, the resulting network should be connected. On the other hand, a negative heuristic told that we should not remove the side face to obtain a planar network. So, in the case of the cylinder, both heuristic patterns are used. A similar procedure goes for the applicability of Lemma (ii) since we cannot draw a diagonal inside a circle. This discussion might lead to the proof generated concept of diagonal addressing the question of whether it can be drawn on a circle or not.
3.4 Returning to the Local but not Global Counterexamples
73 Problem of Content “Proof-analysis, when increasing certainty, decreases content. Each new lemma in the proof analysis, corresponding to a new condition in the theorem, reduces its domain.” [Lakatos 1976, 57] Another student then introduced the following terminology: Quasi-convex polyhedra are poly-hedra which have at least one face from which we can photograph inside of the polyhedron. One should notice that this interpretation is quite similar to the notion of inflating the polyhedra onto the sphere. In response to this, the theorem next suggested by the student was “All quasi- convex polyhedra with simply-connected faces are Eulerian”. Consider the great stellated dodecahedron. For this polyhedron, we have V — E + F = 2. It was then proposed that a proof must explain the phenomenon of Eulerianness in its entirety. This is because it was not possible to imagine a proof that would explain the Eulerian character of both convex and concave polyhedra, e.g. the cube and the great stellated dodecahedron respectively, by one single idea. This is to say, the conditions of the theorem were not only sufficient, but also necessary. There must not be any counterexamples in its domain and likewise there must not be any examples outside its domain.
74 One student thought that the problem was to discover the domain of truth of V — E + F = 2.. But another disagreed, as for him the problem was to discover the relations between V, E, and F; not only in the case where V — E + F = 2., but also when
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V — E + F = 0., V — E + F = —6, etc. They then observed that the picture frame with ring- shaped faces both in the front and in the back is Eulerian, but is not a Cauchy polyhedron.
75 Therefore, in discussing both the inductive and the deductive basis of the method of proofs and refutations, they set out to discuss as well the relations between edges and vertices in polygons.
76 The idea of deductive reasoning was considered next. The student with this idea performed a pasting on polyhedra which is called Connected Sum in topology. He explains [Lakatos 1976, 76]: Take two closed normal polyhedra and paste them together along a polygonal circuit so that two faces that meet disappear. Since for the two polyhedra V — E + F = 4, the disappearance of two faces in the united poyhedron will just restore the Euler formula. (...) [Then], let us now try a double-pasting test: let us paste the two polyhedra together along two polygonal circuits. Now 4 faces will disappear and for the new polyhedron V — E + F = 0.
77 Going on in this way he concluded that:1 For a monospheroid polyhedron V — E + F = 2, for a dispheroid polyhedron V — E + F = 0, (...) for a n-spheroid polyhedron V — E + F = 2. — 2 x (n — 1).
78 The immediate counterexample is the crested cube with V — E + F = 1, recalled by Omega. But Zeta thinks that his method may not be applied to all poyhedra, but only to all n-spheroidal polyhedra built up according to his construction. Now, Sigma argues about the polyhedra with ringshaped faces. For him, it is possible to construct a ringshaped polygon by deleting in a suitable proof-generated system of polygons an edge without reducing the number of faces. Then, he concludes with the formula:
79 From a heuristic point of view, we remark that the distinctio
80 ns and the interaction between positive and negative heuristics are used to determine the necessary and sufficient conditions of a mathematical theorem, the Descartes-Euler conjecture. However it would be too hasty to draw the conclusion that positive heuristics corresponds to necessary conditions whereas negative heuristics correspond to sufficient conditions, or vice versa. Thus, we avoid making this inference for the time being.
81 The problem of content is concerned with the task of containing every genuine polyhedron in its domain and excluding every counterexample. As we already pointed out, the tools that were used were primarily positive and negative heuristics. Occasionally, they helped to enlarge and restrict the domain. It is not clear for Lakatos whether they have pre-defined and predetermined tasks.
82 However, since Lakatos disagrees with Euclidean methodology considerably, we can not expect Lakatos to give definite tasks for both heuristic rules.
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83 This is precisely the reason why we avoid drawing conclusions too quickly about the correspondence between “sufficient and necessary conditions” and “negative and positive heuristics”.
84 One of the significant ideas that Lakatos used in PR was the interpretation of the Euler characteristic as a function. This idea was stated by one of the students in Lakatos’s imaginary classroom. This positive heuristic approach enabled us to understand the domain of the Euler characteristic better. In this way, not only some specific integers that the Euler characteristic function returned are considered, but in addition, any integer that can be the output of the function is considered. Hence, in hindsight, it is not surprising that such long and sophisticated formulas were present in the preceding pages. They were the products of positive heuristics and led us to all polyhedral objects with our formulation.
85 Lakatos concluded PR by a rather optimistic motto which we treat in the following section.
3.5 How Criticism may Turn Mathematical Truth into Logical Truth
86 In the last part of PR, Lakatos’s optimism reveals itself via his conclusion with an observation about “stretching”.
87 New Version of the Theorem At this stage, a new version of the theorem was put forward: “For all simple objects with simply-connected faces such that the edges of the faces terminate in vertices V — E + F = 2”. However, an immediate refutation, namely the twin tetrahedron, was given. In inflating the twin tetrahedron, the critical edge is split into two edges. Definition 7 Stretching is a bicontinuous one-to-one mapping. Disagreements about its definition were centered on stretching a square along its boundaries. After a few discussions on the concept of simply connectedness the session ended. One student remarked “(...) now I have nothing but problems” [Lakatos 1976, 80].
3.6 Heuristics and Necessitation
88 Thus far, we have not given a precise definition of the notions of positive and negative heuristics. We will briefly recall their definitions now in order to confirm that our exposition agreed with them.
89 Lakatos informally defined negative heuristics as “the methodological rule (...) [which] tells us what paths of research to avoid”; and similarly stated that positive heuristics “tells us what paths [of research] to pursue” [Lakatos 1978, 47]. However, in the present paper, our goal is not to elaborate on the concepts of negative and positive heuristics nor on their relations with the hard core of the theory. Thus, we refer the interested reader to the original paper of Lakatos for detailed discussion and exposition [Lakatos 1978].
90 Kiss remarked that “the aim of heuristic investigations is to find the method of thinking, the rules by which one can receive results more easily and surely” [emphasis is hers] [Kiss 2002, 243]. Our goal in this respect is not much different from that of Kiss. However, based on our present case study, we refrain ourselves from giving a formal account of positive and negative heuristics which depends on the mathematical necessitation. As was noted by one of Lakatos’s colleagues, PR “removes the last
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Aristotelian element, the element of necessitation, from modern science” [Feyerabend 1975, 14]. In this paper, we have attempted to document why and how PR removed necessitation from topological reasoning.
91 One of the reasons for this is the fact that some thought experiments and counterexamples had the function both of negative and of positive heuristics which prevented us from concluding the role of these thought experiments and counterexamples with respect to the Aristotelian notion of necessitation. This point also agrees with Lakatos’s thesis that mathematics is a quasi-empirical science. In other words “nothing in mathematics is self-evident. Self-evidence in mathematics is an illusion” [emphasis is his] [Koetsier 1991, 24].
92 However, we can pursue Lakatosian methodology further in order to formalize the logic of heuristics by diverging a bit from the actual philosophical notions. Recent work in epistemic logic seems to give hints regarding the logic of heuristics by utilizing modal operators for necessitation [Ba§kent 2007]. However, it is far from being precise in terms of Lakatosian MSRP. Therefore, the modal logical tool at hand—assuming that modal logic is an appropriate tool for formalizing necessitation—is still far from giving a precise and formal account of Lakatosian heuristics.
4 Conclusion and Future Work
93 The main aim of this paper was to identify the counterexamples in PR and to discuss whether they play the role of positive heuristics or negative heuristics or both by using some simple and intuitive ideas of geometric topology.
94 While we were treating Lakatos’s philosophy and methodology of mathematics (as he outlined in the Appendices of PR), we refrained from making clear-cut distinctions between the two types of heuristics. This is because, we observed that this distinction would mistakenly lead us to make a similar type of distinction between the correspondence of positive heuristic to the necessary conditions of a theorem and the correspondence of negative heuristic to the sufficient conditions of a theorem. The main reason in our view that Lakatos did not make that kind of distinction lies in the fact that Lakatos did not disagree with the rule of double negation and hence the method of Reductio
95 Ad Absordum [Lakatos 1976]. Therefore, any utilization of intuitionistic and thus modal logical tools to formalize Lakatosian heuristics will be doomed to failure since they cannot be perfectly consistent with these heuristic notions.
BIBLIOGRAPHY
BAS◌̧KENT, CAN — 2007 Topics in Subset Space Logic, Institute for Logic, Language and Computation, Universiteit van Amsterdam, Technical Report.
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CORFIELD, DAVID — 2002 Argumentation and the Mathematical Process, Appraising Lakatos: Mathematics, Methodology, and the Man, G. Kampis, L. Kvasz and M. Stoltzner (eds.), Dordrecht: Kluwer Academic Publishing.
FEYERABEND, PAUL — 1975 Imre Lakatos, British Journal of Philosophy of Science, vol. 26 (1), 1-18.
KAMPIS, G, KVASZ, L., STÖLTZNER, M. — 2002 Appraising Lakatos: Mathematics, Methodology, and the Man, Dordrecht: Kluwer Academic Publishing.
KISS, OLGA — 2002 Mathematical Heuristics—Lakatos and Polya, Appraising Lakatos: Mathematics, Methodology, and the Man, G. Kampis, L. Kvasz and M. Stoltzner (eds.), Dordrecht: Kluwer Academic Publishing.
KOETSIER, TEUN — 1991 Lakatos’ Philosophy of Mathematics, Studies in the History and Philosophy of Mathematics, Amsterdam: Elsevier Science Publishers.
LAKATOS, IMRE — 1976 Proofs and Refutations, Cambridge: Cambridge University Press. — 1978 The Methodology of Scientific Research Programmes, vol. 1, Cambridge: Cambridge University Press.
STEINER, MARK — 1983 The Philosophy of Mathematics of Imre Lakatos, The Journal of Philosophy, vol. 80 (9), 502-521.
NOTES
1. Omega, Sigma and Zeta are the names of the students in Lakatos’s imaginary classroom.
ABSTRACTS
Abstract: Lakatos’s seminal work Proofs and Refutations introduced the methods of proofs and refutations by discussing the history and methodological development of Euler’s formula V — E+F = 2 for three dimensional polyhedra. Lakatos considered the history of polyhedra illustrating a good example for his philosophy and methodology of mathematics and geometry. In this study, we focus on the mathematical and topological properties which play a role in Lakatos’s methodological approach. For each example and counterexample given by Lakatos, we briefly outline its topological counterpart. We thus present the mathematical background and basis of Lakatos’s philosophy of mathematical methodology in the case of Euler’s formula, and thereby develop some intuitions about the function of his notions of positive and negative heuristics.
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Dans son influent Proofs and Refutations (Preuves et Réfutations), Lakatos introduit les méthodes de preuves et de réfutations en discutant l’histoire et le développement de la formule V — E+F = 2 d’Euler pour les polyèdres en 3 dimensions. Lakatos croyait, en effet, que l’histoire du polyèdre présentait un bon exemple pour sa philosophie et sa méthodologie des mathématiques, incluant la géométrie. Le présent travail met l’accent sur les propriétés mathématiques et topologiques qui sont incorporées dans l’approche méthodologique de Lakatos. Pour chaque exemple et contre-exemple utilisé par Lakatos, nous présenterons brièvement sa contrepartie topologique, ce qui nous permettra de présenter les fondations et les motivations mathématiques derrière sa philosophie de la méthodologie des mathématiques et finalement, par ce fait même, nous développerons certaines intuitions sur le fonctionnement de ses notions d’heuristique négative et d’heuristique positive.
AUTHORS
SAMET BAĞÇE Department of Philosophy, Middle East Technical University, Ankara
CAN BAŞKENT Department of Computer Science, The Graduate Center, The City University of New York
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Formalizability and Knowledge Ascriptions in Mathematical Practice
Eva Müller-Hill
AUTHOR'S NOTE
We thank the audience at the 17th Novembertagung on the History and Philosophy of Mathematics in Edinburgh and one of the anonymous referees of Philosophia Scientiae for helpful comments on earlier versions of this paper.
1 Introduction
1 In this paper, we will be concerned with the role of formalizability in an epis-temology of mathematics.
2 Formalizability is a feature of informal mathematical proofs: A formaliz-able proof is a proof that can be transformed into a formal proof, that means it can be transformed into a formal derivation with respect to a formal axiomatic system with consistent axioms. However, the notion of formalizability is not a fixed notion like the notion of formal proof. What has still to be specified is the meaning of “can be transformed” in this context. We may choose the semantics of the phrase “formalizable proof” from a spectrum spread between two extremes. One extreme in the definition spectrum would be the weak reading “a proof of p is formalizable iff the informally proven mathematical theorem is also formally derivable in a consistent formal axiomatic system”; the other extreme is the strong reading “a proof of p is formalizable iff it can be translated step by step into a formal proof”, which may refer to, for example, proofs that are written in some semi-formal language.1
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3 Formalizability is an important feature of mathematical proofs, regarding foundational issues in the philosophy of mathematics: [Rav 1999, 11] suggested referring to the thesis that every informal mathematical proof can be transformed into a formal proof as Hilbert’s thesis, as it is linked to the so-called Hilbert programme2 We will not be concerned with Hilbert’s thesis generally, but with an epistemological application of the thesis: Is formalizability also essential for a philosophical understanding of mathematical knowledge?
4 This question is open, and several answers have been proposed. For one, Rav emphasizes that according to the popular opinion about mathematics (which he argues against), a mathematician’s job consists more or less in manipulating formulas to render a decision “true” or “false” about certain theorems. Facing the increasing use of computer tools, such as computer algebra systems, simulation software or proof checkers, up to the famous examples of computer proofs (e.g., the proof of the Four Colour Theorem), the question arises whether mathematicians, in so far as they are working on proofs, might even be completely replaceable by computers without loss.
5 Consider taking Hilbert’s thesis as an exhaustive characterization of the epistemic role of mathematical proof. What effect would this assumption have on an epistemology of mathematics? As one aspect of this question, we will examine the epistemological impact of criteria for “X knows that the mathematical statement p is true” that are based on the availability of formalizable proof.
6 The paper is also a first report on work in progress concerning a socio-empirical investigation of actual mathematical practice. Besides a purely analytical treatment of the epistemic role of formalizability in mathematics, we will take into account first results of our recently conducted empirical research on formalizability and knowledge ascriptions in mathematical practice.
2 Preliminaries
7 A formalizability criterion for “X knows that p is true” is a criterion of the following form: (*) X has available a formalizable proof of p, where the notion of availability may be weaker than the notion of current cognitive access.
8 Our main research question is: Is there any specification of ”formalizable“ and ”available“ such that a formalizability criterion provides an adequate criterion for the truth of” X knows that the mathematical statement p is true“?
9 As we will discuss below (Section 3.2), prima facie there seems to be a natural way to answer ‘yes’ to this question from the philosophical point of view.
10 However, it turns out that in actual mathematical practice, formalizability seems to play no major role. Employing the results of an empirical study, we will suggest that, at least for common readings of (*), mathematicians do not necessarily use formalizability criteria when ascribing knowledge. Knowledge ascriptions in mathematical practice work in a way that differs systematically from how they should work if they were based on these readings of formalizability criteria. This points to the conclusion that, in spite of their prima facie attractiveness, formalizability criteria may be inadequate for
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capturing conceptions of mathematical knowledge as employed in actual mathematical practice.
11 We hold the view that for an adequate investigation of the epistemic role of formalizability, one needs to distinguish between a purely analytical account of the concept of mathematical knowledge, and an approach towards conceptions of knowledge that are actually employed in mathematical practice. Both should contribute to an adequate philosophical understanding of mathematical knowledge. In our paper, we will focus on the following aspect of this requested mutual contribution: The truth conditions of ”X knows that the mathematical statement p is true“ which a philosophical theory of mathematical knowledge yields should somehow fit, or should at least not systematically contradict, the meaning of knowledge ascriptions employed in actual mathematical practice. Our investigation will proceed (iteratively) in three main steps: - Identify those philosophical conceptions of knowledge that are consistent with a formalizability criterion—this presupposes an appropriate classification of knowledge conceptions, due to the different possible specifications of ”formalizable“ and ”available“. - Examine and analyze conceptions ofmathematical knowledge, especially the meaning of knowledge ascriptions, employed in actual mathematical practice. Can they be captured by particular philosophical conceptions of knowledge? - Combine the results of both analyses. Are there differences between the results of Step 1 and Step 2? Are these differences due to different aspects of knowledge? Can we develop a conceptual epistemological framework which is able to cope with these different aspects?
12 The approach is also meant to shed light on the nature and epistemic role of mathematical proof, which is of great interest for modern philosophy of mathematics (see for example [Detlefsen 1992]).
13 The paper will sketch a showcase part of the analysis from Step 1 of our agenda. We will then turn to Step 2 with two main concerns: On the one hand, we will present first results of our recently conducted empirical study about the conception of knowledge in mathematical practice. On the other hand, we will discuss a reasonable embedding of these empirical results into the overall philosophical framework. This will provide an orientation for further elaboration of Step 1, and is also necessary for developing Step 3, a long term goal. While we will give a quite detailed examination of a part of our empirical results (Step 2), we will not be able to perform an equally detailed discussion of Step 1 in full generality here.
3 Step 1: Knowledge conceptions behind formalizability criteria—a showcase analysis
14 Mathematical knowledge is alleged to have a special epistemic status. On the one hand, this is due to the fact that its theorems can be proven, which is exceptional among the sciences. On the other hand, it is due to the historical stability of the body of mathematical knowledge, and the high degree of systematic unity and uniformity even across different branches of mathematics.
15 We stated above that, prima facie, there seems to be an obvious specification of an adequate formalizability criterion, as there seems to be a particular reading of formalizability criteria that can easily account for the exceptional epistemic status of
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mathematical knowledge. In this section, we will first examine this claim by discussing the specification of formalizability criteria involved and the way it accounts for the exceptional epistemic status. We will then investigate the question whether this specification really yields an adequate semantics for ”X knows that the mathematical statement p is true“ in view of Step 2 and Step 3 of our agenda. This question will lead us to the empirical part of the paper.
3.1 The classical conception of knowledge
16 Classical epistemology defines knowledge as justified true belief via the following three individually necessary and jointly sufficient conditions for ”X knows that p is true“: - X believes that p is true - p is true - X has good reasons to believe that p is true with respect to some fixed epistemic standards.
17 This definition of knowledge has high intuitive appeal. Yet, it has been seriously challenged by the famous Gettier examples, and by scepticism.3 The Gettier examples, as well as the sceptical challenge, are essentially based on the fact that there is a gap between justification and truth. As the classical conception of knowledge cannot fill this gap for the general case, this conception has to be regarded as inadequate for a general theory of knowledge, despite its intuitive appeal.
18 By employing the distinction between internalistic and externalistic conceptions, and the distinction between strict invariantist and context sensitive conceptions of knowledge, the classical conception of knowledge finds a systematical place among the different conceptions of knowledge that have been proposed as a reaction to Gettier and scepticism. We will understand strict invariantism and context sensitivism as two competing views, concerning the epistemic standards a subject X must meet in order to know that p is true: A strict invariantist demands that these epistemic standards are fixed, whereas a context sensitivist allows the epistemic standards to vary situationally. 4 Each of these views is compatible with both internalistic and externalistic theories of knowledge. An internalist claims that X has to meet the relevant epistemic standards and has to have (actual) cognitive access to this fact, whereas an externalist claims that X has to meet the standards objectively, but does not need to have access to that fact. Accordingly, the classical conception of knowledge falls into the category of internalistic strict invariantism.
3.2 Internalistic strict invariantism and formaliz-ability criteria
19 A criterion for ”X knows that p is true“ that satisfies the following conditions: (Int) It yields an internalistic and (Inv) strict invariantist conception of knowledge, and (T) links epistemic justification reliably to truth apparently characterizes some exceptional epistemic status, following the line of thought developed in Section 3.1.
20 The following specification of (*) satisfies the conditions (Int) and (Inv), due to the employed adjustments of the two parameters ”available“ and ”for-malizable“: (*’) X has current cognitive access to a formal proof of p.
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21 Moreover, having access to a formal proof guarantees the existence of a formal derivation in a consistent axiomatic system, and thus the truth of the derived theorem relative to the truth of the axioms. In that sense, one might say that (*’) exemplifies a perfectly reliable relation between truth and justification, and therefore fulfills (T). So, ”X has current cognitive access to a formal proof of p“ as a criterion for ”X knows that the mathematical statement p is true“ may seem to account for the desired exceptional epistemic status of mathematical knowledge. But is (*’) also adequate with regard to our agenda, respecting both the alleged exceptional nature of mathematical knowledge and actual mathematical practice?
22 In a recent paper on a context sensitive account of mathematical knowledge [Löwe & Müller 2008], Benedikt Löwe and Thomas Müller argue that any attempt to specify the notion of availability in ”X has available a formal proof of p“ as a criterion for knowing that p is true without violating (Inv) or (Int) either leads to the conclusion that mathematicians actually have nearly zero non-trivial mathematical knowledge, or leads to counter-intuitive examples of true knowledge ascriptions. In the first case, ”available“ is understood in the narrowest sense as current cognitive access, such as in (*’). Here, Löwe and Müller argue that facing the empirical fact that a formal derivation of some non-trivial mathematical statement can easily take up to thousands of logical steps, current cognitive access to formal proofs of theorems of higher mathematics is not a realistic option for a working mathematician. In the second case, the reading of ”available“ is severely weakened. As a paradigmatic example, Löwe and Müller discuss the possibility of introducing a certain time frame to distinguish availability from actual possession. We may formulate the corresponding specification of (*) as follows: (*”) Within a time period of length t, X will have current cognitive access to a formal proof of p.
23 Here, their objections are based on more philosophical grounds: If we take (*“) as a criterion for knowing that p is true, which amount of time is reasonable, regarding the enormous complexity of formal proofs of the theorems of higher mathematics?
24 Following the argumentation of Löwe and Müller, and somehow in anticipation of Step 2 and 3 for the particular case, (*’) (and (*”) as well) should thus be rejected as candidates for an adequate criterion for “X knows that p is true” in the spirit of our agenda.
25 Still, there may be other specifications of (*) that satisfy (Inv), (Int), and (T), and there may also be different ways to account for the exceptional epis-temic status of mathematical knowledge instead of an analysis via (Inv), (Int) and (T). A question of special interest in this context is whether a strictly invariantist and at the same time internalist reading of formalizability criteria could be adequate for the purpose at hand at all, or whether an externalist reading could be better suited. Yet, we will have to leave this as an open question for now, as it is a question of a much more general kind than questioning the adequateness of special readings of formalizability criteria which we discuss in the scope of this paper. Anyway, abstracting from the special case discussed in [Löwe & Müller 2008], we may state as a conjecture that an adequate reading of (*) that strictly fulfills (Inv) and (T) cannot fulfill (Int), as it is hard to imagine how “available” could be appropriately specified under any strong (due to (T)), invariantist reading of “formalizable”.
26 For the purpose of this paper, we will postpone further philosophical analysis of possible readings of (*) to Section 5, and will now turn to first results of a socio-
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empirical study on the conditions for ascribing knowledge in mathematical practice. The results already point to the conclusion that a particular family of possible readings of (*) is incompatible with conceptions of knowledge that appear to be employed in actual mathematical practice. On the other hand, the results also suggest an alternative way of specifying (*) that appears to be a promising candidate for Step 3 of our agenda.
4 Step 2: A socio-empirical study on the meaning of knowledge ascriptions in mathematical practice
27 How can we investigate the meaning of knowledge ascriptions employed in actual mathematical practice? Emphasizing the fact that the mathematical community is, like any other scientific community, first and foremost a social collective, we make the following explicit assumptions regarding our approach: - An important part of the meaning (including truth-conditional semantics) of knowledge ascriptions in mathematical practice is determined by the actual use of knowledge ascriptions among working mathematicians. - A working mathematician affirms a knowledge ascription if and only if she takes it to be true.5
28 If we are interested in the meaning of knowledge ascriptions in mathematical practice, we should thus shift our attention to how mathematicians use knowledge ascriptions in practice. We should investigate the conditions under which working mathematicians use sentences of the form “X knows that p is true”.6
29 This shift of attention provides a natural starting point for an empirical investigation of mathematical practice, particularly by means of empirical sociology. While it might not be reasonable to investigate empirically whether some particular knowledge ascription is actually true or not, the use of knowledge ascriptions seems to be a natural domain for empirical research on mathematical practice. In focussing on the latter, we regard knowledge ascriptions as actions that can be more or less appropriate relative to the mathematical community. Some ascriber Y acts in a certain way by saying “X knows that the mathematical statement p is true” [Kompa 2001, 16-17]. Whether an action counts as appropriate (or justified) in a certain community is determined by social mechanisms rather than by explicit verbal rules. Tools for an empirical investigation of social actions and social mechanisms are provided by sociology in the form of quantitative and qualitative research techniques. Under the methodological assumption that mathematicians ascribe knowledge systematically, the results of a socio-empirical investigation of knowledge ascriptions in mathematical practice will point to certain standards for knowledge ascriptions that are taken to be appropriate by mathematicians.
30 In what follows, we will present a mainly quantitative web-based survey on standards for knowledge ascriptions in actual mathematical practice. Recalling that our overall interest is in the role of available formalizable proof for mathematical knowledge, we have restricted the range of situational, concrete case studies to knowledge ascriptions based on claimed proof.7
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4.1 What is out there?
31 Many authors emphasize the role of sociological considerations for an epistemology of mathematics, for instance the later Wittgenstein, Philip Kitcher [Kitcher 1984], Paul Ernest [Ernest 1998], or David Bloor [Bloor 1996, 2004].
32 Yet, there has hardly been done any socio-empirical research about actual mathematical research practice: The first socio-empirical study published was Bettina Heintz’s work about the culture and practice of mathematics as a scientific discipline [Heintz 2000]. In her study, Heintz used qualitative methods. Her work is based on a detailed field study (at the Max-Planck-Institute for Mathematics in Bonn, Germany) and a series of qualitative interviews with mathematicians.8
33 In our survey on the conditions for knowledge ascriptions in mathematical practice, we employed mainly quantitative methods. We used an online questionnaire which contained mostly multiple choice questions, and some space for free-text comments. The survey was announced together with a link to the online questionnaire via postings in different scientific newsgroups. It was opened in August 2006 and closed in October 2006.
4.2 Methodology and project data
4.2.1 Quantitative vs. qualitative methods
34 Our study may be seen as a methodological attempt to employ quantitative methods for a socio-empirical investigation of actual mathematical research practice, though we do not aim at using all state-of-the-art statistical evaluation tools from empirical sociology. In the first place, we try to explore if and how the empirical results may generally be brought to bear on an epistemology of mathematics. The use of quantitative methods is due to the assumption that the way mathematicians think about mathematics and mathematical knowledge is heavily influenced by individual factors and might even be a matter of personal style. In contrast, mathematical practice appears to be very homogenous and uniform. It is a well-known phenomenon in sociology that the way people think about certain issues does not necessarily coincide with how they act, it may only point into the right direction [Klammer 2005, 220]. By the use of quantitative methods, we hope to get more significant results about the conditions for appropriate knowledge ascriptions on which mathematicians agree.
35 The qualitative results from the free-text part are supposed to deepen the hypotheses that were tested in the quantitative part. These results were also used to develop an interview guideline for a follow-up qualitative interview study with international research mathematicians.
4.2.2 Target group and adequacy of the sample
36 A great methodological obstacle for anyone who wants to investigate mathematical practice by surveys is the apparent unwillingness of its protagonists to participate. To circumvent this obstacle, we used an online questionnaire in our project which was posted in three scientific Internet newsgroups, as newsgroup readers are supposed to be less averse to surveys. We do not regard the resulting restriction of the sample as a
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serious limitation: A significant correlation between the habit of reading newsgroups and a certain attitude towards formalizable proof does not seem very likely.9
37 The link to the online questionnaire was posted two times. The first posting was a test- run in the newsgroup sci.math. For the second time, the link was posted in sci.math.research and de.sci.mathematik, with 214 responses in total.
38 The target group of our survey consisted of participants with research or teaching experience in any branch of mathematics. A preliminary personal data section of the questionnaire served to decide whether a participant belonged to the target group or not. A response was considered as valid if the personal data part and at least one question in one of the three main parts of the questionnaire (cf. Section 4.3) was completed. We received 76 valid responses from the target group.10 13.2% of these 76 participants received a B.A. (or an equivalent degree), 19.7% a M.Sc. (or an equivalent degree), and 46.1% a Ph.D. (or an equivalent degree) in mathematics. Most of the valid responses from the target group came from the United States of America, Germany, and the Netherlands, from mathematicians with both research and teaching experiences at university level for at least one year.
39 What we will present below are the results from the second posting, exclusively based on valid answers from the target group.
4.3 Empirical hypotheses
40 The questionnaire we used fell into three main parts: • Part I dealt with the abstract concept of knowledge and proof mathematicians develop, • Part II focussed on knowledge ascriptions “in action”, and • Part III was on mathematical beauty.
41 In what follows, we will only consider results from the first and the second part, because the results from Part III do not bear on the hypotheses we want to discuss within the scope of this paper.
42 The results from Part I and II were supposed to complement each other. The questions in Part I also served as control questions for Part II.
43 The empirical hypotheses that were tested in Part II derive from two sources. The first source is the analysis of the concept of knowledge underlying formalizability criteria, which means, the results of Step 1. The other source are experiences from mathematical practice itself. We will discuss three of these hypotheses here:
44 (H1) Whether mathematicians ascribe knowledge based on claimed proof depends essentially and systematically on contextual factors. The claim that in mathematical practice, knowledge ascriptions based on claimed proof are context sensitive has been made in [Löwe & Müller 2008]. As mentioned in Section 3.2, we believe that this claim might be true due to an essential incompatibility of an account of the epis-temic role of mathematical proof that strictly preserves (T) and (Inv), and an internalist view of epistemic justification. Hence, it must not be reduced to a statement about purely pragmatic features of knowledge ascriptions. The apparent uniformity of mathematical practice suggests that the claimed context sensitivity should be systematic, and (therefore) accessible by empirical means.
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45 (H2) The actual possession of a formal proof by the epistemic subject X is sufficient for ascribing knowledge to X. In Section 3.2, it is argued that truth criteria for “X knows that p is true” which demand the actual possession of a formal proof of p are too restrictive, because as a consequence nearly no mathematician would have much more than non-trivial mathematical knowledge. Yet, for any current philosophical conception of knowledge, the actual possession of a formal proof of p is sufficient to know that p is true. Therefore, we also expect to find in mathematical practice that the possession of a formal proof of p is sufficient to count as knowing that p is true. Otherwise, it would appear questionable whether the endeavour of combining the analytical, philosophical conception of mathematical knowledge with the concept of knowledge used in mathematical practice might be successful.
46 (H3) Mathematicians do not necessarily demand a formal proof in order to ascribe knowledge. Formal proofs are rarely used in mathematical practice. Based on this, the authors of [Löwe & Müller 2008] argue that the actual possession of a formal proof of p is not a reasonable criterion for knowing that p is true. Accordingly, the standards for knowledge ascriptions actually used in mathematical practice should not include the constraint that X must actually possess a formal proof of p in order to know that p is true.
4.4 Selected results
47 In the following, we will present selected results of Part I and II by giving an excerpt of the questions together with the corresponding data. As the questionnaire had 74 questions in total, we will only present those questions and results that have been significant for the interpretation regarding our hypotheses (H1), (H2) and (H3).
48 Part I These are two examples of questions participants were asked in the first part of the questionnaire: “Is mathematical knowledge objective?”
response frequency count (∑ 74)
yes 82.4% 61
no 17.6% 13
“ Please select to which degree you accept the following statement: ‘One can precisely define what a mathematical proof is.”’
response frequency count (∑ 74)
strongly agree 28.4% 21
agree 60.8% 45
disagree 9.5% 7
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strongly disagree 1.4% 1
49 Participants who gave a positive answer to the latter question, either “strongly agree” or “agree”, were then asked in a free-text question to give a definition of “mathematical proof”. We’ll give selected quotes from the answers. The majority of answers gave a formal definition of mathematical proof. The quotes below show some less frequently given answers. Note that the second and fourth answer still state that the informally proven theorem has to be formally derivable:11 - “Formally: from a given set of deductive rules, and a set of axioms, a sequence of statements (machine-verifiable in the correctness of application of the rules), starting with hypothesis and ending with conclusion. Ideally, anyway.” - “I would defined a proof fundamentally as argument that convinces mathematicians, less fundamentally as an argument that can be formalised and proven mechanically.” - “A convincing argument that instills belief that it is possible to construct a sequence of formal logical steps leading from generally accepted axioms to the given assertion.” - “A finite sequence of statements following logically from each other, that begin with a given set of axioms and have the statement to be proved as a conclusion. In actual mathematical practice, this sequence tends to be shortened and written in some human language rather than pure symbolic logic, so the only difficulty that may arise in defining what a ‘real-life’ mathematical proof is, is to decide what constitutes an acceptable abbreviation of the hypothetical, full-length logical proof.”
50 Part II In the second part, people were led through four scenarios. Each screen of the online questionnaire contained a piece of the story, and at the end of each screen the participants were repeatedly asked whether they would ascribe knowledge to the protagonist of the scenario or not. In the following, we will give excerpts from Scenario 1 and Scenario 3. The ongoing story of each scenario will be shortly summarized when parts are left out.
51 In Scenario 1, the protagonist was a PhD student named John: “Scenario 1 John is a graduate student, and Jane Jones, a world famous expert on holomorphic functions, is his supervisor. One evening, John is working on the Jones conjecture and seems to have made a break-through. He produces scribbled notes on yellow sheets of paper and convinces himself that these notes constitute a proof of his theorem.”
52 Then, participants were asked to answer the following question: “Does John know that the Jones conjecture is true?”
53 On the following screens, the story continued, and the question “Does John know that the Jones conjecture is true?” was repeated several times. In the final part of the story, John and his supervisor jointly prove the Jones Conjecture, and publish their proof in a mathematical journal of high reputation: “Eighteen months later, the editor accepts the paper for publication, based on a positive referee report.”
54 At this point of the story, the majority (84.9%) of the participants gave a positive answer to the question: “Does John know that the Jones conjecture is true? ”
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response frequency count (∑ 66)
yes 28.8% 19
almost surely yes 56.1% 37
almost surely no 3.0% 2
no 4.5% 3
can’t tell 7.6% 5
55 On the next screen, which was the last of Scenario 1, the story ended with: “After his Ph.D., John continues his mathematical career. Five years after the paper was published, he listens to a talk on anti-Jones functions. That evening, he discovers that based on these functions, one can construct a counterexample to the Jones conjecture. He is shocked, and so is professor Jones.”
56 Now, 61.3% of the participants gave a positive answer on the question whether John knows that the Jones conjecture is false : “Does John know that the Jones conjecture is false? ”
response frequency count (∑ 62)
yes 14.5% 9
almost surely yes 46.8% 29
almost surely no 6.5% 4
no 8.1% 5
can’t tell 24.2% 15
57 On the same screen, the participants were also queried:12 “ Did John know that the Jones conjecture was true on the morning before the talk? ”
58 The result is somehow astonishing, because still, 71% gave a positive answer to this question:
response frequency count (∑ 62)
yes 24.2% 15
almost surely yes 46.8% 29
almost surely no 4.8% 3
no 14.5% 9
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can’t tell 9.7% 6
59 Note that when answering these last two questions, participants already had the information that John has discovered a counterexample to the Jones conjecture.
60 After finishing Scenario 1, we asked the participants to give comments on the scenario in a free-text field. These are some selected quotes, emphasizing different factors that influenced the answers in Scenario 1: - “How important is the Jones conjecture? How large is the community?” - “My answers would have been very different with different time frames mentioned.” - “I don’t know John or Bob, so I don’t have a good feel for how rigorously they work.”
61 In Scenario 3, the protagonist was a student of mathematics named Tom, and the setting was an oral examination at the end of the semester: “Scenario 3: Tom Jenkins is a student of mathematics and has to pass an oral exam at the end of the algebra lecture held by his professor Robin Smith. Tom did some oral exams before, so he is not too nervous, and is able to pay concentrated attention to the professor’s questions during the whole exam. At some point of the exam, Smith asks Tom for the proof of a certain algebraic theorem T1. The proof consists mainly of a tricky application ofthe fundamental theorem on homomorphisms and was conducted in one lecture on the blackboard. Tom is able to give a rather technical, but absolutely correct step-by-step proof in full detail.”
62 83.3% of the participants gave a positive answer to the question: “Does Tom know that T1 is true? ”
response frequency count (∑ 54)
yes 46.3% 25
almost surely yes 37% 20
almost surely no 1.9% 1
no 1.9% 1
can’t tell 13% 7
63 In the proceeding story, professor Smith challenges Tom with questions concerning the ideas behind the proof of T1, and further applications of these ideas. Tom fails at all these questions: “The exam continues with some questions about definitions, and after some minutes Smith asks Tom to explain why the general idea of how to apply the fundamental theorem on homomorphisms in the proof of the former theorem is also fruitful to prove a second algebraic theorem T2. Tom completely fails in his answer”
64 Still, 83.3% gave a positive answer to the question whether Tom knows that T1 is true. There is only a slight adjustment from “yes” towards “almost surely yes”: “Does Tom know that T1 is true?”
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response frequency count (∑ 54)
yes 40.7% 22
almost surely yes 42.6% 23
almost surely no 3.7% 2
no 1.9% 1
can’t tell 11.1% 6
65 On the same screen, participants were asked: “Did Tom know that the first theorem T1 was true before he failed in answering Professor Smith’s last question correctly?”
66 Again, there is nearly no quantitative effect on the positive knowledge ascriptions, 83% gave a positive answer. Note the slight adjustment from “almost surely yes” towards “yes” compared with the answers corresponding to the preceding screen:
response frequency count (∑ 53)
yes 47.2% 25
almost surely yes 35.8% 19
almost surely no 1.9% 1
no 1.9% 1
can’t tell 13.2% 7
67 The story continued: “Smith asks Tom to formulate the general idea behind the proofofthe first theorem T1. Tom fails in his answer.”
68 Note that although there is a clear shift from positive to negative answers (13.2%) to the next question, there is no effect on the results concerning the qualitative behavior of positive and negative knowledge ascriptions: “Does Tom know that the first theorem T1 is true? ”
response frequency count (∑ 53)
yes 32.1% 17
almost surely yes 37.7% 20
almost surely no 11.3% 6
no 5.7% 3
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can’t tell 13.2% 7
69 After finishing Scenario 3, we asked the participants again to give comments on the scenario in a free-text field. These are some selected quotes. The quotes support the quantitative result that a correct step-by-step proof, at least when it is given by the epistemic subject, is sufficient for knowledge ascriptions or knowledge claims:13 - “He knows the truth of the theorems, because these are well known and proved theorems.” - “Once you are sure that you gave a correct proof of theorem T1, you need not revise your opinion on the truth of T1.” - “Tom knew all the time that T1 was true. He could give a complete and correct proof, after all!” - “Well you don’t need to understand the idea behind the proof of some theorem [...] and to some point the idea is not important at all—just as I said before: symbol processing.”
4.5 Summary and interpretation of the selected results regarding (H1), (H2) and (H3)
70 In the following, we will propose some interpretation of the above presented results.
71 First of all, there appears to be a certain tension between the results of Part I and II: - In Part I, the majority of the free-text answers to the question on the definition of “mathematical proof” refer to the definition of formal proof. In contrast, the quantitative results from Scenario 1 in Part II affirm (H3): Formal proof is not necessary for knowledge ascriptions in mathematical practice (cf. below). - In Part I, 82.4% of the participants answered that mathematical knowledge is objective. In contrast, the free-text comments on the scenarios in Part II show that the standards for knowledge ascriptions that are taken to be appropriate seem to depend on less objective factors like the size of the community, the importance of the proven theorems, or on time frames.
72 Regarding the three hypotheses (H1), (H2) and (H3) formulated in Section 4.3 the following can be observed:
73 (H1) Whether mathematicians ascribe knowledge based on claimed proof depends essentially and systematically on contextual factors. The free-text comments in Part II show that whether subjects ascribe knowledge or not may depend on: - the size of the community of the corresponding branch of mathematics, - the number of referees, - what is at stake, - how important the proven theorem is, - the epistemic subject (e.g., working habits), - time frames.
74 This suggests a systematic context sensitivity of standards for knowledge ascriptions, as it seems to be possible to categorize the relevant contextual factors beyond the actual, concrete context.
75 The quantitative results from Scenario 1 point to the conclusion that this sensitivity is not contingent, but rather essential. After the participants had received the information
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that John has discovered a counterexample to the Jones conjecture, they were asked both if he does now know that the Jones conjecture is false, and if he did know before his discovery that the Jones conjecture was true. Assuming that the appropriateness of the ascription “John knows that the Jones conjecture is false” implies the appropriateness of the ascription “John does not know that the Jones conjecture is true”, the results suggest that the standards for knowledge ascriptions that are taken to be appropriate in mathematical practice are not strictly invariant, but may for example depend on the epistemic standards in play at the context of the epistemic subject (John in this case): - Does John know that the Jones conjecture is false? 61.3% ‘yes’ or ‘almost surely yes’ - Did John know that the Jones conjecture was true on the morning before the talk? 71% ‘yes’ or ‘almost surely yes’
76 This answering profile was the most frequently represented one, but is incompatible with strictly invariantist standards for knowledge ascriptions - according to the latter, the results for “Did John know that the Jones conjecture was true on the morning before the talk?” should have been a majority of negative answers. Our interpretation is backed up by the free-text comments on Scenario 1 given by participants who answered “can’t tell” on the last two questions of the scenario. The following quotes may serve as paradigmatic examples here: - “[...] There is no objective truth, even in mathematics. There is only mathematicians who may be convinced one way or another, and a ‘proof’ (or a ‘counterexample’) may be a means to convince some/many/most of them.” - “I have consistently interpreted the word ‘know’ in a rather narrow sense. Had I been John, I would have claimed for myself knowledge once I was reasonably confident that I had inspected every detail of a proof—but I might have been wrong in believing that I knew. [...]”
77 Though knowledge ascriptions seem to work quite well in mathematical practice, participants reflecting on the requirements of a theoretical, rather invariantist notion of “to know” weren’t able to decide whether to ascribe knowledge or not in the given scenario anymore.
78 (H2) The actual possession of a formal proof by the epistemic subject X is sufficient for ascribing knowledge to X. The quantitative results as well as the free-text comments from Scenario 3 affirm this hypothesis.
79 (H3) Mathematicians do not necessarily demand a formal proof in order to ascribe knowledge.
80 This hypothesis appears to be affirmed, again by the results from Scenario 1 in Part II: 61.3% of the participants gave a positive answer on the question whether John knows that the Jones conjecture is false, after they had received the information that he has discovered the counterexample. 71% also gave a positive answer to the question whether he still knew that the Jones conjecture was true before he discovered the counterexample. If a formal proof had been demanded by these participants, they would have committed themselves rather consciously to the claim that a formal proof of both the Jones conjecture and its negation would be possible, and thus to an apparent contradiction: Formal derivability of both the Jones conjecture and its negation yields the inconsistency of the axioms of the formal system that is used, but the notion of formal proof includes the consistency of the axioms.
81 The results suggest an even stronger conclusion: For readings of formaliz-ability criteria that necessarily entail the logical possibility to prove p formally (with respect
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to a fixed formal system), the same contradiction holds. So, it seems that at least these formalizability criteria for knowledge ascriptions are not necessarily employed by mathematicians.
5 Conclusions
82 We saw that the effectively employed conditions for knowledge ascriptions in mathematical practice seem to depend on contextual factors, and even that formal provability of a mathematical statement p appears to be not necessary for ascribing knowledge of p. On the other hand, actually available formal proof seems to be sufficient for ascribing mathematical knowledge, and in spite of the relative use of knowledge ascriptions, there is still an emphasis on the objectivity of mathematical knowledge and proof in the back of working mathematicians’ minds. What conclusions can already be drawn from these observations towards Step 3 of our agenda, and what lessons can be learned regarding a possible iteration of Step 1 and 2?
5.1 Towards Step 3
83 The considerations we have undertaken in Section 4.5 point to some general incompatibility of the empirical results and purely internalistic, strictly invariantist readings of formalizability criteria (recall that in Section 3.2, we have only discussed particular variants of such readings of (*)). The use of knowledge ascriptions in mathematical practice seems to depend on both internal and external contextual factors. Factors related to the epistemic subject, such as working habits, are internal factors, whereas time frames, size of the community, number of referees checking a claimed proof, or importance of the proven theorem are external factors. Therefore, a specification of (*) that can still account for our empirical results should allow for some weaker standards for knowledge ascriptions than strict invariantist standards, and it should also be capable of incorporating external factors.
84 We would like to remind the reader of our conjecture about formalizability criteria fulfilling (T), (Inv), and (Int), stated in Section 3.2. Employing the terminology used in that section, an incorporation of the external factors may yield a relative notion of formalizability (i.e., a weakening of (T)), and a quite externalist understanding of “having available a formalizable proof” (i.e., a weakening of (Int)) in turn: Whether a given proof is formalizable could be specified relative to time frames involved, size of the community, number of referees checking the proof, or importance of the proven theorem. Hence, an epistemic subject X could have available a formalizable proof without having cognitive access to that fact. Factors related to the epistemic subject X, like working habits, or maybe certain mathematical skills, can be modeled by an appropriate specification of “available”. Depending on the particular role and significance of these contextual factors (which has to be investigated further), their incorporation may also have a weakening of (Inv) as a result.14 As mentioned in the interpretation of the empirical results regarding (H1) (cf. Section 4.5), shifts of X’s context seem to be more relevant than shifts in the ascriber’s context concerning the epistemic standards for knowledge ascriptions in this regard.15
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85 Therefore, what we end up with in some kind of first order approximation is that an externalistic, subject sensitive reading of (*) might indeed be better suited than the traditional, internalistic strict invariantist readings discussed in Section 3.2.
5.2 Outlook—starting points for an iteration of Step 1 and 2
86 Regarding Step 1, the next iteration of our three-step research programme will thus be devoted to a theoretical discussion of possible candidates for an exter-nalistic, subject sensitive reading of (*). As a starting point, we will investigate Azzouni’s derivation indicator view of proof developed in [Azzouni 2004], and Manin’s notion of degrees of proofness [Manin 1977, 1981] as candidates for the respective specification of “formalizable proof”.
87 Regarding Step 2, it should then be investigated whether and how these candidates for readings of (*) fit our empirical results. To this end, some of these results have to be refined in additional empirical studies. For example, the tension between the empirical results concerning mathematicians’ abstract concepts of mathematical knowledge and proof, and the empirical results concerning knowledge ascriptions in mathematical practice, has to be investigated further. Are the conceptions of knowledge and proof used in mathematical practice inconsistent, or is the tension between the results due to different, but compatible, aspects? Two other issues for further empirical investigation are to clarify the role of contextual factors for the epistemic standards employed in mathematical practice, and the role of different factors determining a possible relative notion of formalizability. Are there other or additional factors than those mentioned by the participants of our first study? Part of these questions are being pursued empirically in our ongoing qualitative interview study mentioned in Section 4.2.1.
BIBLIOGRAPHY
AZZOUNI, JODY — 2004 The Derivation Indicator View of Mathematical Practice, Philosophic, Mathematical, 12 (2), 81-106.
BLOOR, DAVID — 1996 Scientific Knowledge: A Sociological Analysis, Chicago: Chicago University Press. — 2004 Sociology of Scientific Knowledge, Handbook of Epistemology, I. Ni-iniluoto et al. (eds.), Dordrecht: Kluwer, 919-962.
BRENDEL, ELKE — 1999 Wahrheit und Wissen, Paderborn: Mentis.
CARSTON, ROBYN — 1991 Implicature, Explicature, and Truth-Theoretic Semantics, Pragmatics. A Reader, Steven Davis (ed.), New York: Oxford University Press, 33-51.
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DETLEFSEN, MICHAEL (ED.) — 1992 Proof, Logic and Formalization, London: Routledge.
ERNEST, PAUL — 1998 Social Constructivism as a Philosophy of Mathematics, Albany, New York: SUNY Press.
GETTIER, EDMUND — 1963 Is Justified True Belief Knowledge?, Analysis, 23, 121-123.
GRICE, HERBERT P. — 1975 Logic and Conversation, Syntax and Semantics. Vol 3: Speech Acts, P. Cole & J. Morgan (eds.), New York: Academic Press.
HEINTZ, BETTINA — 2000 Die Innenwelt der Mathematik. Zur Kultur und Praxis einer beweisenden Disziplin, Wien: Springer.
KITCHER, PHILIP — 1984 The Nature of Mathematical Knowledge, New York: Oxford University Press.
KLAMMER, BERND — 2005 Empirische Sozialforschung, Konstanz: UVK.
KOMPA, NIKOLA — 2001 Wissen und Kontext, Paderborn: Mentis.
LÖWE, BENEDIKT & MÜLLER, THOMAS — 2008 Mathematical Knowledge is Context-Dependent, Grazer Philosophische Studien, 76, 91-107.
MACFARLANE, JOHN — 2005 The Assessment Sensitivity of Knowledge Attributions, Oxford Studies in Epistemology, 1, 197-233.
MANIN, YURI I. — 1977 A Course in Mathematical Logic, New York, Heidelberg, Berlin: Springer. — 1981 A Digression on Proof, The Two-Year College Mathematics Journal, Vol. 12(2), 104-107.
MARKOWITSCH, JÖRG — 1997 Metaphysik und Mathematik. Über implizites Wissen, Verstehen und die Praxis in der Mathematik, PhD Thesis, Universität Wien.
PUTNAM, HILARY — 1992 Brains in a Vat, Skepticism: A Contemporary Reader, K. DeRose & T.A. Warfield (eds.), Oxford: Oxford University Press.
RAV, YEHUDA — 1999 Why Do We Prove Theorems?, Philosophia Mathematica, Vol. 7(3), 5-41.
ZACH, RICHARD — 2003 Hilbert’s Program, Stanford Encyclopedia of Philosophy (Fall 2003 Edition), Edward N. Zalta (ed.). http://plato.stanford.edu/entries/hilbert-program/
NOTES
1. Note that the transformation relations include the case that a formalizable proof is already a formal proof. Therefore, all formal proofs are formalizable.
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2. In its final version, Hilbert’s programme towards a new foundation of mathematics was proposed by David Hilbert in 1921. It aims at a formalization of mathematics and at a proof that the axioms of the formal system that is used are consistent [Zach 2003]. Hilbert’s thesis is actually weaker than Hilbert’s programme itself, which, as is known, failed as a result of Godel’s theorems about the completeness and incompleteness of formal systems. Whether Hilbert’s thesis is true or not may strongly depend on which reading of “can be transformed” is chosen, and therefore, on which meaning of “formalizable proof” is used. 3. The Gettier examples show that in certain cases, justified true belief is not sufficient for knowledge. Gettier published his examples in 1963 [Gettier 1963]. The so-called sceptical challenge derives from the possibility of sceptical scenarios like Descartes’ Evil Demon or Putnam’s brain in a vat scenario [Putnam 1992]. 4. Note that when using the term ”context sensitivism“, we do not distinguish between contextualism and a sensitive form of invariantism, namely subject sensitive invariantism [MacFarlane 2005]. Our main argument in this section hits strict invariantist readings of formalizability criteria, and will not be in favor of either sensitive invariantism or contextualism. We will touch this issue again in Section 5.1 as part of the outlook on Step 3 of our agenda. 5. The “only if” part of this assumption may be seen as a version of one of the Gricean conversational maximes: “Do not say what you believe to be false”, falling under the category of Quality of his cooperative principle of conversation [Grice 1975]. The “if” part may not be valid in general, but reduces to a plausible methodological premise in the setting of our empirical research project, a questionnaire study. 6. Referring to the well-known semantics-pragmatics distinction, our first assumption emphasizes that this shift does not imply a restriction to “pure pragmatics” of knowledge ascriptions, leaving the semantics as an autonomous constituent of their meaning aside. At least truth-conditional semantics is not taken to be autonomous with respect to pragmatics in general, e.g. [Carston 1991, 47—48]. 7. It is important to note that the restriction to knowledge ascriptions based on claimed proof does not coincide with a restriction to an internalist epistemology, neither regarding knowledge nor regarding justification. One might be tempted to bring this up as an objection, as knowledge ascriptions based on claimed proof seem to focus too much on the epistemic subject’s argumentative practice [Brendel 1999, 236], entailing that all justifying factors are part of what is actually accessible to the epistemic subject. However, this depends essentially on the particular specification of “available” and “formalizable”. As we have already seen, there are a lot more notions of availability than just the narrow notion of immediate, conscious cognitive access. Another obvious way to incorporate external factors in formalizability criteria is to exploit the possibility that an epistemic subject may have available a formalizable proof without having (conscious) cognitive access to the fact that the proof is formalizable (cf. Section 5.1). 8. In his PhD Thesis, Jorg Markowitsch also refers to results from interviews with mathematicians [Markowitsch 1997]. 9. The follow-up interview study mentioned above shall also serve as a control mechanism in this regard. 10. In the presentation of the results in Section 4.4, we will give the total count of valid answers from the target group for each reported multiple choice question. 11. The orthography of all quotations has been corrected, but the grammar has been left as it was in the original responses. 12. Participants were able, though not requested, to switch back and forth between the different screens. 13. Concerning orthography and grammar of questionnaire quotations, cf. Footnote 11.
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14. Note that even if the context dependency of epistemic standards might turn out to be an empirical phenomenon of knowledge ascriptions in general, the crucial point would be that mathematics is no exception. 15. A general epistemology that could account for these empirical results might thus be subject sensitive invariantism rather than contextualism in the narrow sense [MacFarlane 2005, 198-199]. Cf. also Footnote 4.
ABSTRACTS
We investigate the truth conditions of knowledge ascriptions for the case of mathematical knowledge. The availability of a formalizable mathematical proof appears to be a natural criterion: (*) X knows that p is true iff X has available a formalizable proof of p. Yet, formalizability plays no major role in actual mathematical practice. We present results of an empirical study, which suggest that certain readings of (*) are not necessarily employed by mathematicians when ascribing knowledge. Further, we argue that the concept of mathematical knowledge underlying the actual use of “to know” in mathematical practice is compatible with certain philosophical intuitions, but seems to differ from philosophical knowledge conceptions underlying (*).
Nous examinons les conditions de vérité pour des attributions de savoir dans le cas des connaissances mathématiques. La disposition d’une démonstration formalisable semble être un critère naturel : (*) X sait que p est vrai si et seulement si X en principe dispose d’une démonstration formalisable pour p. La formalisabilité pourtant ne joue pas un grand rôle dans la pratique mathématique effective. Nous présentons des résultats d’une recherche empirique qui indiquent que les mathématiciens n’employent pas certaines spécifications de (*) quand ils attribuent du savoir. De plus, nous montrons que le concept de savoir mathématique qui est à la base de l’emploi effectif du mot « savoir » de la pratique mathématique est tout à fait compatible avec certaines intuitions philosophiques mais apparaît comme différent des concepts philosophiques formant la base de (*).
AUTHOR
EVA MÜLLER-HILL Rheinische Friedrich-Wilhelms-Universität Bonn, Germany
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Conceptions of Continuity: William Kingdon Clifford’s Empirical Conception of Continuity in Mathematics (1868-1879)
Josipa Gordana Petrunić
Introduction
1 In the edited collection Conceptions of Ether: Studies in the history of ether theories [1981], G. N. Cantor and M. J. S Hodge write that, in presenting a volume composed of various case studies in the history of ether concepts, their book introduces readers “to the broadest themes in the scientific thought of the eighteenth and nineteenth centuries” [Cantor & Hodge 1981, ix].1 Likewise, histories of “continuity” can introduce historians to the broadest themes in eighteenth- and nineteenth-century mathematics. “Continuity” is one of the fundamental concepts in calculus, and its history is by no means simple or straightforward. The aim of this paper is to show how one mathematician, namely William Kingdon Clifford (1845-1879), conceived of mathematical “continuity”, how he used it, and how he subtly redefined it as part of his grander philosophical project—to prove that scientific theories based on action-at-a- distance principles (i.e., instantaneous action across excessively large or infinitesimally small expanses of space) constitute poor means of explaining physical phenomena.
2 As a mid-19th century empiricist—influenced as much by the ideas of Charles Darwin and Herbert Spencer as he was by the ideas of Bernhard Riemann, Sir William Rowan Hamilton and Hermann Grassmann—Clifford’s approach to continuity can be summarized as follows: continuity is a conventional mathematical tool based on empirical evidence. Its mathematical definition is an abstraction of the assumed existence of continuity in space and time. Continuous space implies that there are no gaps or moments of non-existence in the fabric of the universe; continuous time implies that there are no gaps in the fabric of forward-moving time. Specifically with regards
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to the physical aspect of continuity, Clifford invoked the image of a continuous medium that pervades the entire universe (i.e., an ether).
3 Thus, for Clifford, the techniques of calculus are conventional tools that describe phenomena in space. The accuracy of the descriptions gained from the use of such tools (for example, descriptions depicting the rate of acceleration of an object falling towards Earth) is fundamentally dependent upon the truth of the assumption that continuity exists in the ontological structure of the universe. Following in the footsteps of a quintessentially British-empiricist tradition, Clifford accepted that there is no way to know certainly whether the universe really is continuous; he acknowledged that many physical phenomena might in fact be the product of discontinuities in space, rendering analytical techniques useless and inappropriate. However, Clifford contended that if scientists correctly adopted the assumption that continuity is true of the structure of the universe (as Clifford himself believed it to be), then they must avoid the notion of “force” as a causal explanation of phenomena. Forces, by their very nature, are a- physical; they exist independently of the material bodies they act upon. Newtonian gravity is a case in point—the belief that two bodies simultaneously express an attraction towards one another implies that the “force” of attraction operates across immense tracks of space instantaneously and thereby causes bodily motion. Such action amounts to a “discontinuity” in space, meaning that physical theories often devolve into metaphysical speculation. Clifford argued that scientists and mathematicians who accurately accept the implications of analytical techniques must accept certain limitations to their scientific theorizing. They must accept that material interactions alone can result in changes of motion. For example, continuity implies that “energy” (then a newly emergent concept in physics) can only pass from one piece of matter to another; it cannot be transferred through a void.
4 Clifford’s contentions with regards to continuity would not have been out of place among many of his contemporaries in the 1870s, given that many of his colleagues sympathized with the empiricist undertones in his foundational arguments. However, a mathematical philosophy of the type that he was advocating was not easily accepted by the whole of the mathematical community either. Indeed, other mathematicians were interpreting “continuity” in radically different ways at the same time that Clifford was pushing his empiricist agenda. This is demonstrated in Richard Dedekind’s (1831-1916) account of continuity. Dedekind and Clifford make useful comparative case studies because both mathematicians clearly defined their guiding conceptions of “continuity” by the mid-1860s and both mathematicians published their views on continuity primarily in the 1870s. Both practitioners were also deeply interested in exploring the degree to which mathematical “continuity” was necessarily reliant upon physical structures for its ontological content. In addition, both Dedekind and Clifford were influenced by Bernhard Riemann (1826-1866), who was a colleague of Dedekind’s, and whose work in non-Euclidean geometry was of fundamental importance to Clifford’s mathematical research. As a result, the mathematicians wholeheartedly adopted many of Riemann’s conventionalist musings on the foundations of mathematics.
5 Yet, for all of their similarities, Dedekind and Clifford also diverged significantly in terms of their eventual definitions of “continuity”. For Dedekind, mathematical “continuity” is made precise by appealing to infinitesimally small intervals on a number line. Thus, “continuity” requires a formal definition in mathematics that is independent of physical truths in the universe. Dedekind defined “continuity” through
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the use of the mathematical concept known as an “infinitesimal”. He argued the “infinitesimal” is not based upon spatial or geometrical intuition. On Dedekind’s view, the infinitesimal measuring stick that mathematicians embody in the notion of a “limit” can be rendered meaningful through purely mathematical (i.e., arithmetical) construction. Conversely, Clifford argued the “infinitesimal” is fundamentally and irrevocably empirical; it cannot be understood without appealing to descriptive properties of material space.
6 In this paper, I will explore the formal mathematical definition of continuity that Dedekind provided in his account of real numbers (initially written in the 1850s, but not translated into English until the 1870s) and compare it to Clifford’s own materialist account (published throughout the 1870s). The aim is to highlight the sociological claim that mathematical and scientific concepts, such as “continuity”, are best understood as bearing a family-like resemblance to a plethora of differing notions. “Continuity” has been defined in a plurality of ways by different practitioners throughout history, and it has been used to serve their differing interests.
1. On Continuity and Limits
7 From a contemporary standpoint, any first year course in calculus introduces students to the intuitive, yet formally elusive, concept of“continuity.” Consider for example the introductory remarks made in one such textbook. Here the student is told: The study of two concepts essential for calculus [is] limits and continuity. Although the ideas behind them appear simple, they are in fact difficult to understand. (It is often the case in mathematics that the simplest-seeming ideas are the hardest.) It is easy to grasp the idea of a limit, and the idea of continuity. To say that an infinite
sequence of numbers x1, x2, x3,..., approaches a number x as a limit simply means that the further we go in the sequence, the closer we get to x, and we can get as close as we want simply by going far enough out in the sequence. Continuity is even easier— a function f is continuous on an interval [a, b] if it is possible to sketch its graph over the interval without lifting the pencil from the paper... Despite this intuitive simplicity, however, both ideas have proved extremely difficult to formalize in mathematical language. [Voorhees & Chun 2004, 79]
8 The student who follows such a lesson is usually then introduced to Zeno’s paradox in which Achilles and the tortoise race; the tortoise is given a head start and Achilles is never able to catch up because he must always travel half the distance between himself and the tortoise ad infinitum. This is presented as the historical antecedent to the modern concept of “continuity”; the notion of a “limit” is introduced as the accepted resolution to Zeno’s paradox: The mathematical resolution of the paradox is to note that the sequence of values
x1, x2, x3,..., converges to a limit x, and that this is the point at which Achilles passes the tortoise. Even today, however, philosophers (but not mathematicians) argue about whether this paradox has actually been resolved. [Voorhees & Chun 2004, 81]
9 A full account of the various contemporary accounts of continuity is beyond the remit of this paper. However, it is useful to note that within the world of calculus education, the now accepted “precise” account of continuity is as follows: The limit of a function as x approaches a can often be found simply by calculating the value of the function at a. Functions with this property are called continuous at a. We will see that the mathematical definition of continuity corresponds closely
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with the meaning of the word continuity in everyday language. (A continuous process is one that takes place gradually, without interruption or abrupt change.) Definition: A function f is continuous at a number a if
Notice that Definition 1 requires three things if f is continuous at a: 1. f (a) is defined (that is, a is in the domain of f),
2. limx→a f(x) exists,
3. limx→a f(x) = f(a). The definition says that f is continuous at a if f(x) approaches f(a) as x approaches a. Thus, a continuous function f has the property that a small change in x produces only a small change in f(x). In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a, or f has a discontinuity at a, iff is not continuous at a. Physical phenomena are usually continuous. For instance, the displacement or velocity of a vehicle varies continuously with time, as does a person’s height. But discontinuities do occur in such situations as electric currents. Geometrically, you can think of a function that is continuous at every number in an interval as a function whose graph has no break in it. The graph can be drawn without removing your pen from the paper. [Stewart 2003, 102]
10 We see that the precise account of continuity (propositions 1,2, and 3) relies, ultimately, upon physical or geometrical intuitions of uninterrupted space and time and the associated mental image of an infinitesimal interval. However, the language used to define it invokes formal symbols, suggesting an independence from physical concerns. Hence, the modern conception of “continuity” is, in many ways, a mélange of previously distinct approaches.
11 It is in rendering problematic the now accepted usage of “continuity” that historians and sociologists can begin to see heterogeneity in the historical definitions offered for this seemingly settled and homogenous concept. Indeed, mathematicians have used the word “continuity” to refer to very different things at different times. A quick scan of literature spanning the past 10 decades indicates that rarely has there existed a consensus over the meaning of “continuity” in mathematics, physics, or philosophy [Pitkin 1906]. Early 20th century discussions of “continuity” include, for example, Philip E.B. Jourdain’s 1908 account of “irrational numbers” in which he argued that “irrational numbers” are often taught to students by appealing to ambiguous and poorly defined notions of spatial extension [Jourdain 1908], while in 1933 G. J. Whitrow argued that the problem with relying upon the formal definition of limits in calculus is that the “continuum”, upon which it is based, is nothing but “an ingenious theoretical substitution for the vague empirical notion of the continuous” [Whitrow 1933, 157]. Jose A. Benardete later wrote that rational and real numbers are arithmetical approximations to the “continuum”—they are based on an intuitive, spatial sense of continuity as represented in geometrical magnitude [Benardete 1968]. Conversely, Ian Meuller argued in the mid-20th century that the “continuum” could, and should, be rendered intelligible in terms of discrete units only—not geometrical magnitudes. Meuller believed practitioners could develop new and independent notions of “continuity” without having to appeal to any geometrical or physical phenomenon as referents [Meuller 1969].
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12 More recently, the question of “continuity” and its place in the history of mathematics has constituted the unifying theme for a colloquium held in France in 1990. That gathering resulted in the compendium entitled Le Labyrinthe du Continu [1992]. The editors recount: La question du continu est à tout égard privilégiée : par son ancienneté, puisque, on le sait, la philosophie et les mathématiques venaient juste de naître en Grèce lorsqu’elle fut pour la première fois formulée ; par sa permanence, puisqu’elle n’a pour ainsi dire jamais cessé d’occuper le devant de la scène, suscitant des débats profonds, passionnés, incertains ; par sa centralité dans le champ scientifique, puisqu’elle est aujourd’hui clairement un enjeu non seulement pour la logique et les mathématiques, mais aussi pour la physique et les sciences cognitives. [Salanksi & Sinaceur 1992, III]
13 In other words, continuity’s definition, use, and legitimate place within mathematical, physical and philosophical speculation are still open to considerable debate. The aim thus far has been to highlight the fact that debates over “continuity” have continued unabated despite a seemingly standardized view of the formal notion in mathematics.
2. Continuity and Dedekind’s Approach
14 Dedekind was writing on “continuity” at a time when Clifford was finishing his undergraduate studies at Cambridge.2 The two mathematicians were both deeply influenced by the works of Bernhard Riemann. Dedekind had been Riemann’s contemporary as a student at Göttingen in the 1850s; Clifford would later become the translator and advocate of Riemann’s work in Great Britain in the early 1870s. As a result of their common Riemannian background, both Dedekind and Clifford shared many fundamental beliefs regarding the nature of mathematical knowledge. Both, for example, thought that mathematical knowledge was constructed and conventional at root. Yet, they each defined “continuity” in notably different ways. It is worthwhile to briefly explore Dedekind’s approach to the concept in order to observe the sociological claim that mathematicians draw on social resources (including philosophical belief systems) to interpret and reconstruct mathematical notions idiosyncrat-ically, recreating the concepts they engage with along the way.
15 Largely due to his explicit essay on the topic, “Continuity and Irrational Numbers”, which was conceived in the 1850s but not published until the 1870s, Dedekind usually comes to mind today when mathematicians think of the formalisation of continuity. Consider, for example, the opinion of Walter Pitkin, who states: The problem of continuity enters into mathematics, physics and philosophy in one guise or another. It seems to have reached a satisfactory solution only in the first- named field, where the theory of continua of higher order, and the Dedekind theory of the nature of irrationals, appear to have brought about that much-desired freedom from paradoxes which the physicist and philosophical geometer cannot attain. [Pitkin 1906, 597]
16 Dedekind was among those who rejected the use of physical intuition and empirical observation in the behaviour of curves as the foundation for a mathematical conception of continuity.
17 As a former student of Gauss‘s at the University of Göttingen from 1850 to 1852, Dedekind completed his Habilitation at the same time as Bernhard Riemann in 1854. Following Gauss’s death, he was mentored both by Riemann and J.P.G. Lejeune-
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Dirichlet, and later began to teach at Göttingen. By 1874, having moved to Zürich, Dedekind had become influenced by Cantor’s attempts to introduce the “freedom of definition” into mathematics, after which he devoted most of his efforts to the development of set theory. The question of continuity and its links to infinitesimal analysis (as advanced by A.L. Cauchy in particular) was thus an early concern for Dedekind. Yet, from his introductory remarks to the essay indicated above, it is also clear that “continuity” remained a concern for him well into the 1870s.
18 In the preface to his work on irrational numbers, Dedekind wrote that his interest in the concept of continuity peaked when he was hired to teach calculus at the Polytechnic in Zürich in 1858. In appealing to “the notion of the approach of a variable magnitude to a fixed limiting value,” Dedekind said he had recourse to “geometric evidences” which were highly useful from a didactic point of view. Yet, that “this form of introduction into the differential calculus can make no claim to being scientific, no one will deny,” he argued [Dedekind 1963, 1]. Dedekind lamented that, Even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. [Dedekind 1963, 2]
19 Dedekind’s sole objective, therefore, was to find a “purely arithmetical and perfectly rigorous foundation for the principles of infinitesimal analysis” such that “continuity” would finally get a non-physical definition. Dedekind’s belief was that a truly “scientific basis” for differential calculus and infinitesimal analysis is to be found not in the observation of geometric behaviour (of curves) or in any ambiguous spatial intuitions, but in arithmetic, where “arithmetic” is to be understood as the theory of discrete numbers.
20 For Dedekind, the “whole of arithmetic” is an extension of the simple act of counting and “counting itself [is] nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding” [Dedekind 1963, 4]. When combined with the four basic arithmetic operations (addition, multiplication, subtraction, and division), the chain of these numbers proves to be an “exceedingly useful instrument.” The series of positive counting numbers always allows for addition and multiplication; it does not, however, allow for subtraction and division in all instances. It was this limitation that had propelled humans to “creatively” construct negative numbers and fractions in order to produce a more “complete” set of numbers (the set of rational numbers) which possesses two key characteristics: 1) Any of the four operations can be performed on any two elements in the system, except in the case of division by 0; and, 2) The system “forms a well-arranged domain of one dimension extending to infinity on two opposite sides” [Dedekind 1963, 5].
21 Dedekind admitted that the latter characteristic seemed to have a geometric underpinning. To avoid the “appearance” that “arithmetic was in need of ideas foreign to it,” Dedekind chose to focus on the “purely arithmetic” properties of the rational number system, which he claimed were dependent only upon the notion of a “cut” (Schnitt) [Dedekind 1963, 5]. The “cut” in question referred to the fact that for any
rational number, a, the system of rational numbers divides into two sets: A1, and A2. “Every number of the first class A1 is less than every number of the second class A2,”
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he wrote [Dedekind 1963, 6]. An analogy could be made between a straight line with an arbitrarily chosen origin (o) and the rational number series. In such a model, every rational number is represented by one point on this line. On the straight line, L, “there [would be] infinitely many points which correspond to no rational number,” he noted [Dedekind 1963, 8]. Therefore, the straight line is infinitely “richer in point-individuals than the domain R of rational numbers in number-individuals” [Dedekind 1963, 9]. To account for all “phenomena in the straight line” (that is, to account for all “cuts” on the line) the mathematician must construct new numbers “such that the domain of numbers shall gain the same completeness, or as we may say at once, the continuity, as the straight line”3 [Dedekind 1963, 9].
22 Dedekind further noted that irrational numbers are often introduced with reference to the conception of “extensive magnitudes.” Such “magnitudes” have not, however, been “carefully defined” and pedagogical appeals to extensive magnitude mean that mathematicians have been explaining “number” as the product of a process of measuring [Dedekind 1963, 9]. Dedekind’s objective was to separate himself from that tradition in order to demonstrate that the basis for mathematical continuity is arithmetical; thus, it is not reliant upon measurement or upon the assumed existence of continuity in space, which underlies any process of physical measurement.
23 Historically, the comparison of the domain R (rational numbers) with a straight line has “led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, which we ascribe to the straight line completeness, absence of gaps, or continuity,” Dedekind wrote [Dedekind 1963, 10]. Naturally, mathematicians have been left wondering what continuity actually refers to. “Everything must depend on the answer to this question” and “only through it shall we obtain a scientific basis for the investigation of all continuous domains” [Dedekind 1963, 10]. For Dedekind, the answer is not to be found in “vague remarks about the unbroken connection in the smallest parts” [Dedekind 1963, 10]. Rather, a precise account of continuity requires a definition of the following sort: If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one [point] which produces this division of all points into two classes, this severing of the straight line into two portions. [Dedekind 1963, 11]
24 This tool can be understood to be a cut upon a line which is not represented by a
rational number. Whenever we have a cut (A1, A2), in which A1 is the lower limit of the
number and A2 is the upper limit (neither of which is produced by a rational number) then mathematicians must create a new irrational number, which should be regarded as
completely defined by the cut (A1, A2). In effect, every irrational number becomes an interval on a line in which the upper and lower limits of the interval are the same.
25 To obtain a full definition of continuity, mathematicians also need to consider the
relationship between two cuts (A1, A2) and (B1, B2). If two cuts produce two numbers α and γ, then the principle of continuity indicates that there is an infinite number of different numbers β (that is, an infinite number of cuts or intervals) lying between those two numbers. The upshot of this definition is that in considering real number operations such as √2 x √3 = √6,4 mathematicians must appeal to the idea of continuous functions and limiting values. For Dedekind, arithmetic “continuity” is one based on limiting values defined by infinitesimal intervals. In particular, if we have two numbers α and β, which are characterized by the properties that if ∊ is an arbitrarily small,
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positive magnitude then x < α + ∊ and x > β — ∊, but never x < α — ∊ or x < β + . If α and β are different from one another, “However small be the positive magnitude ∊, we always have finally x < α + ∊ and x > β — ∊ [where] x approaches the limiting value α” [Dedekind 1963, 27]. These examples demonstrate the fundamental connection between the principle of continuity and infinitesimal analysis, Dedekind claimed; they also indicate the manner in which continuity can be arithmetically presented and thus severed from any geometrical roots.
26 Dedekind noted that although most of his readers would find this definition of continuity to be common sense, and although they might even be disappointed to learn by such “commonplace remark[s] the secret of continuity”, he could not prove the truth of his own claims: I may say that I am glad if every one finds the above principle so obvious and so in harmony with his own idea of a line; for I am utterly unable to adduce any proof of its correctness, nor has any one the power. [Dedekind 1963, 11]
27 In other words, the assumption of continuity is nothing other than the axiom by which mathematicians attribute continuity to the line. The notion of “continuity” is an axiomatic one. It is not empirical or contingent upon the observation of motion in space. It is an assertion—it is an exercise in logic. Because it is a created concept, Dedekind concluded: If space has at all a real existence it is not necessary for it to be continuous; many of its properties would remain the same even were it discontinuous. And if we knew for certain that space was discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps, in thought, and thus making it continuous; this filling up would consist in a creation of new point-individuals and would have to be effected in accordance with the above principle“. [Dedekind 1963, 12]
28 In a recent analysis of Dedekind’s account of continuity, Roger Cooke tells us that Dedekind’s aim in linking the definition of continuity to infinitesimal analysis was to address a ”foundation problem that had lain beneath the surface of analysis since the time of Descartes“ [Cooke 2005, 553]. It was a foundational problem that the ancients had dealt with under the rubric of ”incommensurables“—namely, the problem of how to express diagonals and sides of squares that could not be represented as ratios or integers. The problem facing mathematicians right up until Descartes’ time, says Cooke, is that ”geometric magnitudes had never been systematized in terms of arithmetic rules, since they had never been thought of as numbers“ [Cooke 2005, 554]. As Cooke explains, Descartes was among the first to interpret the product of two lengths as a line as opposed to an area. His analytic geometry was not, however, without its problems. The mean-value theorem, for example, relied on some notion of ”continuity“ to, Guarantee that a curve containing points on both sides of a line must intersect the line. But, as the Pythagoreans had shown, the numerical version of this theorem was false: the point of intersection might very well not correspond to any number. It was incorrect to call the intersection an irrational number, since there was no articulated theory of irrational magnitudes that allowed them to be added or multiplied like numbers. Algebraic rules such as √ab = √a x √b applied to magnitudes by fiat, but were difficult to prove, even geometrically; and no one had produced any corresponding arithmetic rules, or even a non-tautological definition of the square root of a non-integer. Any such definition first of all begged the question of the existence of the object defined; and if existence is granted by appeal to geometry, the rules for treating lengths as numbers still needed to be formulated
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and proved correct. Such was the situation that confronted Dedekind. [Cooke 2005, 555]
29 Working within this tradition, Dedekind had become concerned over the possibility that continuity was invoked whenever mathematicians were implicitly (or explicitly) referring to ambiguous geometric conceptions (i.e., in their descriptions of the real number line as a geometric magnitude). In line with Riemann and Gauss before him, Dedekind’s aim was to prove that new number systems could be constructed, as he did not think they were dependent upon a priori physical truths [Cooke 2005, 559]. On Dedekind’s view, mathematicians construct their tools, including their accounts of continuity, in order to carry on with their research. They adopt such constructed rules as axiomatically true statements about the systems within which they work. In arguing for such an account of mathematics, Dedekind was engaged in the process of separating mathematical constructs from both scientific theorizing and physical existence. ”Continuity“, he argued, is based on arithmetical rules which involve the notion of a ”cut“. It exists as a constructed mathematical tool rendered axiomatically true by fiat, and used in mathematical puzzle-solving, though not necessarily in scientific theorizing.
3. Clifford’s Continuity
30 Clifford’s critical view of continuity underpins the almost virulent empiricism that pervades his entire mathematical project. For Clifford, ”continuity“ is to be understood as the assumption that there are no physical, non-material gaps in the fabric of space. It is empirically justifiable; it makes sense to believe that space is a continuous fabric of some sort of material ether. It makes sense to believe that space exists everywhere— that there are no gaps of non-existence from this end of the room to the other. It also makes sense to believe that there are no gaps in the continuity of time; there are likely no moments of non-existence that follow moments of existence. In so far as we trust our sensory organs, we are justified, therefore, in believing in the continuity of space and time. Hence, we are justified in basing our mathematical description of physical phenomena upon the assumption that continuity exists as a physical fact.
31 Clifford’s mathematical concerns with continuity cannot be distinguished from his other conceptual commitments, in particular his view on the knowledge of causes. Calculus, he argued, involves the art of describing the rate of change in motion of a point or an object. ”Continuity“, upon which this description is based, is fundamentally based upon a justified belief in spatial and temporal continuity in the universe. Note that, for Clifford, mathematics is fundamentally empirical, as well as conventional—the former being due to the fact that we can know nothing beyond the provisions of our sensory organs, and the latter being due to the fact that all of our knowledge, no matter how well justified, is ultimately limited by the finite perceptual capacities of our sensory organs. All knowledge is therefore empirically derived and open to future revision. Thus, in considering Clifford’s stand on ”continuity“ we need to consider two inter-related aspects of his philosophy: 1. Empirical experience (observation) as the basis of all mathematical knowledge; 2. ”Continuity“ as a hypothetical property of space.
32 Clifford’s acceptance of ”continuity“ as a spatial property highlights his general philosophy of science, which advocated for an ardent materialism in which cause-and-
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effect relationships could only be accounted for via a series of contiguous particle motions in space. Throughout the 1870s, Clifford felt the need to discuss and explain these beliefs in great detail. His popular lectures and his Common Sense of the Exact Sciences (published posthumously in 1885) were aimed at non-specialist audiences for whom the notion of continuity would have been new, or at least unfamiliar, terrain. For instance, when he died, Clifford was engaged in writing a textbook for his first year students at University College London (UCL). In part this was because Clifford was working within a society that favoured and lauded popularizations of science and mathematics. Scientists and popular lecturers such as T.H. Huxley, John Tyndall, and even P.G. Tait (who popularized links between science and religion) had made it the duty of experts to explain their expertise in palatable and popular forms, even when writing for expert audiences. Clifford was hypersensitive to this fact given his employment at UCL, an institution that had been established with the aim of extending the scope of higher education to include students who had emerged from families not traditionally associated with Cambridge or Oxford (i.e., sons of merchants in London, and sons of secular families).5
33 Clifford, however, also devoted a significant amount of space in his expert mathematical papers to discussing the concept. He used those publications to advance controversial claims regarding the foundations of ”number“ and ”quantity“. One such claim was that ”number“ ought to be understood as motion—as a ”step in space“— where a ”step“ could be understood as a simple geometrical displacement. Thus, all number would be geometrical at root, unless specified as ”discrete“ in nature. ”Continuity“ would therefore constitute a fundamental property of ”number“ understood as ”continuous magnitude“. The implication here is that a certain kind of physical space is necessary in order for numbers and their related operations to exist. These specific claims set Clifford apart from the conventionalist philosophy that defined the works of people such as Dedekind, because they indicate Clifford’s deeply empirical approach to mathematical knowledge. His was an approach that accepted the conventional nature of mathematical concepts, but only in so far as those conventions were the product of empirical observations and inferences about the space within which motion is possible.
34 For Clifford, all we can ever know is that which is empirically inferred. It is important, therefore, to qualify Clifford’s view of empirical inference. While it is the case that empirical evidence is the best judge of what does and does not exist in this world, direct empirical knowledge is limited. It is circumscribed by the finite reach of our sensory capacities. Because of this, Clifford accepted that our mathematical descriptions can only ever be conventional expressions of what we believe to be true about the world. In order to advance mathematical and scientific knowledge, experimentalists and mathematicians had to be willing to speculate upon, and use, concepts related to the nature of the world as they knew it indirectly. An example of this approach is found in Sir William Rowan Hamilton’s work on quaternions which, Clifford claimed, demonstrated that complex numbers were not special, mystical, or imaginary ”quantities“.6 Rather, they could be meaningfully understood when thought of as geometrical operators that caused bodies to rotate in space. Accepting such an interpretation required that mathematicians revise their beliefs regarding the universality of certain mathematical principles, in particular the principle of commutativity. It also required that mathematicians quench any desire to ideologically
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preserve commutativity in light of the useful tools that could emerge when adherence to such traditional principles were abandoned.
35 An early indication of what Clifford had in mind by advancing such a philosophy of mathematics is found in his paper, ”On the Space Theory of Matter“ (delivered to the Cambridge Philosophical Society in 1870 and published in its proceedings in 1876)7 [Clifford 1876b]. Although it is unclear when Clifford first read the German version of Riemann’s paper on the ”Hypotheses that lie at the base of geometry“ (1854), it is likely that upon reading Riemann in the mid-1860s (near the end of his university career), he began to further develop his view of mathematical knowledge as an empirical craft. It is also at that time that he likely began to develop his specific views regarding the physical nature of mathematical ”continuity“. In his 1870 presentation, Clifford posited that continuity was a workable hypothesis, which enjoined scientists and mathematicians to adhere to a fundamentally material view of the universe such that the interaction between physical entities (say, of atoms at the most basic level of material reality) would dominate descriptions of physical phenomena. Clifford wrote that if scientists were to adopt the Riemannian hypothesis that space might not be flat (i.e., that it might possess an inherent curvature) then the continuity hypothesis would lead to the view that non-flat geometrical structures could cause physical phenomena. In other words, continuous geometrical displacements in non-flat space might manifest themselves in the form of seemingly discontinuous phenomena perceived by humans in Euclidean space (i.e., as gravity, electromagnetism, or colour change) [Clifford 1876b, 21].
36 Clifford’s justification for advancing such a hypothesis was in part physiological. A scientist’s sensory perceptions are limited, he wrote. Consider the fact that the axioms of plane geometry work on a piece of paper, while in reality, we know the piece of paper is covered with “small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true” [Clifford 1876b, 21]. The scientist’s limited experience of space may suggest that the axioms of solid geometry are more or less true for finite portions of space, but we “have no reason to conclude that they are true for very small portions, and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space” [Clifford 1876b, 21]. Clifford combined his views of geometry with his overarching belief in the existence of spatial and temporal continuity in order to make the following claims: (1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them; (2) That this property of being curved, or distorted, is continually being passed on from one portion of space to another after the manner of a wave; (3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial; (4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity [Clifford 1876b, 21-22].
37 This paper stands as one of many in which Clifford built up a new philosophical approach to mathematical and scientific theorizing. In Clifford’s worldview, if mathematicians and theorists were to consistently abide by the physical principle of “continuity”, explanations of seemingly discontinuous events (such as gravitational
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attraction between bodies) could lead to new conceptualizations of spatial structure (i.e., non-flat spaces).
4. Clifford the Riemannian
38 Based on his view of continuity, Clifford developed operational tools which he called bi- quaternions. In so doing, he aimed to expand upon both Hamiltonian quaternions as well as Grassmannian algebra.8 Clifford’s bi-quaternions constituted another form of non-commutative algebra involving operators that could be used to map the motion of rigid bodies in Euclidean and non-Euclidean spaces. Throughout the 1870s, Clifford developed his bi-quaternions in the hopes of linking the seemingly empirical reality of Euclidean space to a geometric model that would be able to describe rigid body motion in non-flat spaces. Lauded in a posthumous review (by his biographer H. J. Stephen Smith), Clifford’s contributions on bi-quaternions included: “Preliminary Sketch of Bi- quaternions” [1873], “A Further Note on Bi-quaternions”9 [1876], and his book-length account Elements of Dynamic: an introduction to the study of motion and rest in solid and fluid bodies [1878]. In those works, Clifford outlined his development of various new mathematical operators called “rotors”, “motors” and “bi-quaternions”, which resulted from his continuing interest in the description of physical motion in Riemannian manifold space. As Smith wrote, Riemann’s ideas of a constant curvature in space lies at the bottom of Clifford’s theory of bi-quaternions, to which he devoted much continuous thought, and which was the origin of his researches into the classification of geometric algebras.10 [Tucker 1882, xlv]
39 Indeed, Clifford’s work constituted a major break with traditional geometrical thinking in that: Some men who have an ardent love for new knowledge find it difficult to maintain an unflagging interest in geometry, because they regard it as a purely deductive science of which the first principles (axioms, postulates, and definitions), whether derived from experience or not, are unquestionable and contain implicitly in themselves all possible propositions concerning space. Thus, the unknown, or at least the unforeseen, seems to be excluded from geometry, because whatever may be found out hereafter must be latent in what is already known“. [Tucker 1882, xxxl]
40 Such staid attitudes had begun to change in Great Britain in the 1860s, and they became more prominent throughout the 1870s, largely due to the popularizing efforts by Hermann von Helmholtz and Clifford. Those two mathematicians defended the usefulness and justifiability of non-Euclidean geometrical models in mathematical physics. As Tucker recounted: Upon the view put forward by Riemann and adopted by Clifford, the essential properties of space have to be regarded as things still unknown, which we may one day hope to find out by closer observation and more patient reflection, and not as axioms to be accepted on the authority of universal experience, or of the inner consciousness. [Tucker 1882, xl]
41 Both Clifford and Helmholtz emphasized the conventional and hypothetical nature of the Euclidean axioms and the consequent need for an empirically-based revision of Euclidean-dominated mathematics and research programmes found within the university settings of Cambridge, UCL, and other British educational institutions. Clifford’s concerns regarding ”continuity“ formed part of this wider empiricist agenda,
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which included the introduction of non-Euclidean geometries to mathematical physics and the reform of mathematics education in general. Clifford argued that bi- quaternions were the result of observing ”continuity“ in physical processes, which he felt required more powerful conventional description than that provided by the Cartesian coordinate system.
42 Tucker recounts four assumptions Clifford made regarding the underlying ”ordinary conceptions about space“. Those ”conceptions“ included the following: - Spatial continuity; - Spatial flatness at microscopic levels; - Spatial similarity at every point; - The possibility of the existence of figures similar to one another, but on different scales of magnitude [Tucker 1882, xl].
43 As mentioned earlier with regards to ”continuity“, Clifford was philosophically ready, To adopt either of the two opposite hypotheses that space is continuous or that it is discontinuous, while admitting fully that no phenomena have yet been observed which point to its discontinuity. [Tucker 1882, xli]
44 In his research, Clifford found it productive to assume that continuity did exist, while the other axiomatic claims in mathematics could be revised around it.
45 With regards to the second assumption, Clifford followed Riemann in arguing that mathematicians and scientists had to be willing to reject the notion that between every three points very near to one another there always exists a triangle with an internal angle value of 180 degrees. The idea that objects may not stay the same shape as they move through space suggests that space might not be homogeneous. Clifford thought the topological structure of space might vary from one section of space to another. Thus, our perceptions of an object in one section of space compared to our perceptions of that same object in another section of space might yield very different empirical results. These views underpinned Clifford’s belief that action-at-a-distance ”forces“ (such as Newtonian gravity) were describable in terms of simple movements (or displacements) of bodies in non-flat or elliptic space. Indeed, for Clifford, any empirical perceptions of inexplicable, mysterious, or mystical ”forces“ were an indication of a non-Euclidean ontological curvature requiring a new form of geometrical description.
46 One of the seminal influences that helped to shape Clifford’s views on these matters can be found in Riemann’s paper, ”On the Hypotheses which Lie at the Bases of Geometry“ (originally delivered in 1854). In that paper, Riemann argued that ”geometry assumes, as things given, both the notion of space and the first principles of constructions in space“ [Riemann 1867, 55]. According to Riemann, geometry contains definitions of these assumptions; yet, those definitions remain merely ”nominal“: The true definitions appear in the forms of axioms. The relation of these assumptions remains consequently in darkness. We neither perceive whether and how far their connection is necessary, nor, a priori, whether it is possible. [Riemann 1867, 55]
47 Riemann set as his own task that of clarifying those nominal relationships by reconsidering the idea of magnitudes, in particular ”extended magnitudes, including space“ [Riemann 1867, 55]. He aimed to construct ”the notion of a multiply extended magnitude out of general notions of magnitude“, the upshot of which is the fact that multiply extended magnitudes require differing measure-relations [Riemann 1867, 55]. Different spatial structures require different metric systems.
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48 One consequence of Riemann’s reconsideration of measure relations in variously extended magnitudes is that ”the propositions of geometry cannot be derived from general notions of magnitude, but (...) the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience“ [Riemann 1867, 56]. Triply extended magnitudes (space of three dimensions) that can be measured using Euclidean lines and line segments on a Cartesian graph, constitute only one particular case [Riemann 1867, 56]. That is to say the system Euclid had established constitutes only one system of measure-relations in space. Many other systems could, in theory, be created.
49 In Riemann’s opinion, the Euclidean construction should therefore be considered to be an hypothesis: ”These matters of fact are, like all matters of fact, not necessary, but only of empirical certainty; they are hypotheses“ [Riemann 1867, 56]. Scientists could investigate the ”probability’ that humans live within a spatial fabric that abides by Euclidean measure relations, but the establishment of such probabilities would be dependent upon the powers of empirical observation open to us. Therefore, it remains incumbent upon mathematicians to always question the validity or “justice” of “their extension beyond the limits of observation” [Riemann 1867, 56]. Riemann, therefore, suggested that mathematicians extend their systems to include models that could describe and grasp that which is not directly observable.
50 In clarifying his notion of an n-ply extended magnitude, or “manifold”, Riemann considered two types of manifolds: continuous and discrete. This distinction is one that Clifford would later reproduce in his own mathematical papers and in the Common Sense ofthe Exact Sciences. Examples of discrete manifolds, Riemann wrote, are common in everyday language. Any group of distinct things, such as the letters in the alphabet, can be considered to be a discrete manifold. Continuous manifolds, on the other hand, are more difficult to imagine. One common example is the position of perceived objects, when definite portions of manifoldness are distinguished by boundaries, which themselves take up no space11 [Riemann 1867, 57].
51 In addition, the difference between discrete and continuous “quantity” can be made clear by discussing discrete number in terms of “counting” and continuous magnitude in terms of “measuring”. Riemann wrote that “measure consists in the superposition of the magnitudes to be compared. It therefore requires a means of using one magnitude as the standard for another” [Riemann 1867, 57]. In the absence of a “standard”, two magnitudes can only be compared “when one is a part of the other, in which case we can only determine the more or less and not the how much” [Riemann 1867, 57]. The question of “How much?” is, therefore, only applicable in a continuous manifold when discussing “ How many units of measure?’ and not ” How much of the thing itself?’ since a continuous system is not open to discrete divisions12 [Riemann 1867, 57].
52 “Continuity” serves as the basis of continuous manifolds, as it implies a certain kind of spatial structure-one in which a point or object can move about in a variety of directions without interruption. The number of directions of movement determines the type of space being defined. “If one regards the variable object instead of the determinable notion of it,” Riemann wrote, the construction may be described as a “composition of a variability of n dimensions and a variability of one dimension” [Riemann 1867, 58]. Riemann argued that in any given manifoldness (any n-ply extended space) there are n determinations of quantity which, together, define the position of the object (for example, in Cartesian space, x—, y—, and z—coordinates
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provide three “quantities” that together give us the precise location of an object). In any manifold space, “The determination of position in it may be reduced to n determinations of magnitude” [Riemann 1867, 59]. Measure-relations can be studied in the abstract, because their dependence upon one another can be represented by formulae. However, geometric representations always lay at the bases of these formulae.
53 In sum, Riemann’s working hypothesis was that position-fixing can be reduced to quantity-fixing in an n-dimensioned manifoldness. If the position of an object changes, its displacement can be represented by a relative change in quantities (measure- relations) in a given number of directional dimensions. In cases in which the manifold contains no curvature, the equivalent mathematical model at play is a Euclidean one. However, Riemann contended that such a model should be considered to be the exception and not the norm, because in such spaces figures could be moved about with no change in their size or shape.13
54 In the final section of his 1854 paper, Riemann elaborated upon the physical implications of his research by arguing that what remained to be done was to determine “the degree to which these assumptions are borne out by experience” [Riemann 1867, 67]. Indeed, “It is upon the exactness with which we follow phenomena into the infinitely small that our knowledge of their causal relations essentially depends,” he wrote, adding that it is possible, and even likely, that conventional Euclidean measurements based upon “the notion of a solid body and of a ray of light, cease to be valid for the infinitely small.” Instead, the only believable assumption that might remain is one of spatial continuity—all else needs to flow from this conception of the universe, thus leading to the rejection of action-at-a-distance principles, and other “discontinuities” prominent in scientific theorizing [Riemann 1867, 68]. The entire structure of good science, therefore, depends upon recognition of this fact. Riemann concluded: Researches starting from general notions, like the investigation we have just made, can only be useful in preventing this work from being hampered by too narrow views, and progress in knowledge of the interdependence of things from being checked by traditional prejudices. [Riemann 1867, 69]
55 Thus, the mathematician must be willing to engage in another domain of knowledge, “another science” altogether, in order to determine and devise good mathematics— namely, physics.
56 In a recent account of Riemann’s seminal paper, Jeremy Gray argues that Riemann’s lecture (published posthumously in German in 1867) was to prove “central to the overthrow of Euclidean geometry as the source of geometrical ideas” [Gray 2005, 506]. The idea of a multiply extended magnitude was, roughly, something that could be measured by a given number of coordinates. Riemann concluded that geometry could be thought of as dealing with various spaces (n-fold extended magnitudes) in which distances were to be measured using an infinitesimal ruler [Gray 2005, 510]. If the infinitesimal measuring rod could be placed anywhere and still offer the same scale of measurement, it would mean that the space being measured is constant in curvature. Thus, the sum of the angles in any one triangle in such a spatial model would be equivalent to the sum of triangles at all places. However, Riemann ultimately wanted to discuss the measurement of distances between points without having to rely on Euclidean space.14 In addition, at the time of Riemann’s Habilitation, Gray writes, he was
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working in Wilhelm Weber’s physical laboratory. Riemann had therefore been occupied with issues of gravitation, electricity, and magnetism—all of which had the property of acting across vast distances at “enormous speeds” [Gray 2005, 513]. Gray thus accounts for Riemann’s approach to mathematics in the following way: [Riemann] apparently sought to explain this by imagining that the fabric of space was in some way subtly altered, and distortions in it spread like ripples. He was not able to work this up into a coherent theory, but it seems clear that this idea of conceptually rethinking the nature of space in a direct physical context accounts for many features of his lecture. It underlines the avowedly empirical nature of the lecture, it is in line with Riemann’s attempt to make proposals that allow one to explain phenomena, and it explains the somewhat anti-Newtonian rhetoric. [Gray 2005, 513]
57 Clifford adopted all of these Riemannian themes. He assumed that space is not discontinuous. Rather, he argued, the principle of “continuity” prevails and mathematical-physical representations of dynamics had to adapt to that fact. Spatial “continuity” allowed for the transmission of action between cause and effect. Clifford’s repeated attacks on gravity—a force that he underlined as problematical—drew heavily from Riemann’s own attacks on such action-at-a-distance concepts. Clifford’s idiosyncratic approach to these topics, however, comes from his engagement with both Hamiltonian quaternion analysis and Grassmannian algebra, both of which aided in his development of bi-quaternions. Clifford’s bi-quaternions kinematically describe the motion of rotating rigid bodies. They offer the possibility of describing physical phenomena in continuous, non-flat space—this being an extension, in Clifford’s opinion, of Riemann’s own work.
5. A Further Note on Bi-Quaternions
58 Clifford built upon Riemann’s distinction between discrete and continuous manifolds in his “Further Note on Bi-quaternions” by highlighting the difference between discrete numbers and continuous “steps in space”. The notion of a “step in space” constituted a unifying theme throughout Clifford’s works, in which he described arithmetic, algebra, and calculus in terms of simple “steps in space”. For the purposes of this paper, I will focus on Clifford’s brief discussion of the theme in the “Further Note on Bi- quaternions”. This paper demonstrates Clifford’s view that mathematical concepts do reflect the “continuity” of physical space and, thus, they can be described as “steps in space”.
59 Recall that, for Clifford, “continuity” is ultimately an assumption. Mathematical knowledge remains a conventional language open to indefinite revision limited only by that which is empirically possible. Consider the following equation: 2 x 3 = 6
60 Following in the tradition of the Newtonian polar coordinates, which posited that a set of polar coordinates (r, θ) does not define a point uniquely, Clifford suggested that an arithmetic operation is alone open to multiple representations. In the equation above, two interpretations exist. In the first instance, “3” can be considered a “concrete number of things” in which “2” is the operation of doubling such that we would read the entire equation as “doubling three marbles makes six marbles” [Clifford 1876b, 385]. In the second instance, both “2” and “3” can be abstract numbers which affirm the existence of a third number “6”. That third number has a definite relation to the
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other two numbers only as their product (i.e., as the result of an operation, rather than as a quantity on its own) [Clifford 1876a, 385]. Clifford contended that “various meanings” could be given to the numbers “2” and “3”: We have regarded 2 as a symbol of operation, 3 as a concrete number, and 6 as a concrete number. But we may also regard all three symbols as symbols of operation, and so read the formula 'doubling the triple of anything makes the sextuple of it’. [Clifford 1876a, 385]
61 Similarly, the equation abc = d possesses two interpretations, Clifford wrote: 1) a times b times c things makes d things; 2) a times b times c times anything makes d times that thing [Clifford 1876a, 386].
62 The last symbol can be regarded either as a concrete number or as a symbol of operation. Such varying interpretations allow one to easily extend the concepts of addition, subtraction, and other arithmetical processes to “steps”, Clifford contended. For example, he provided a double meaning to the symbols “+” and “—”. In the first instance, the symbols would indicate the direction of a “step”. The symbol (+3) would therefore indicate a step of 3 units forward, while the symbol (-3) would indicate a step of three units backwards. “When these symbols are attached to an operation performed upon steps, they mean retaining and reversing respectively,” Clifford explained [Clifford 1876a, 386]. Thus, the equation (-2)(+3) = (-6) would be open to two interpretations: 1) Doubling a step of three forward and reversing it makes a step of 6 backward; 2) [Or] To triple a step and retain its direction then to double and reverse it is the same as to sextuple and reverse it [Clifford 1876a, 387].
63 Given the controversy surrounding “quaternions” as initially proposed by Sir William Rowan Hamilton in the first half of the 19th century (Pycior 1976), and given that the debate regarding the foundations of mathematical “number” (in particular, negative numbers and the square roots of negative numbers) had not entirely died down by the time Clifford chose to develop and extend Hamiltonian quaternions and Grassmannian algebra, it is clear that this seemingly basic introduction to the interpretability of mathematical concepts in the “Further Note” in fact constituted an important philosophical statement on Clifford’s part. Clifford’s aim was to demonstrate that since simple mathematical concepts such as addition and subtraction could be treated in an equivocal and malleable way, so too could more complex concepts such as Hamiltonian vectors. The “rules” of arithmetic were by no means an ideological barrier to the establishment of new mathematical system, especially if the system in question upheld the principle of geometrical and physical continuity (as we will see). In Clifford’s opinion—and against many of Hamilton’s critics— quaternion algebra did not necessitate any philosophical conundrums or lead to any troubling debates over the a priori status of mathematical knowledge. It was as simple a step in logic for an empiricist such as Clifford to afford a dual interpretation to the new system of non- commutative algebra.
64 In order to see how Clifford further extended Hamiltonian quaternions on the basis of “continuity” as a physical fact, it is useful to provide a brief account of what it is that Hamilton was actually claiming. It was during his time as a student at Trinity College, Dublin, from 1823 to 1827, that Hamilton had been exposed to the French works of P.S. Laplace, J.L. Lagrange, S.D. Poisson and S.F. Lacroix [Lewis 2005, 460]. Some of those influences led him to consider a problem that many mathematicians in the 1830s had considered to be one of the most important of the day: how to extend the system of number pairs (which represented complex numbers), to triples of numbers such that
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the same operational properties were preserved [Lewis 2005, 461]. In his account of Hamilton’s efforts to do so, Lewis summarizes the problem in the following way: A complex number z = a + bi can be represented in the Euclidean plane by the directed line segment from the origin to the oint with real number coordinates (a, b). On multiplying z by i(= √–1), the result would be the segment from the origin to (—b, a) which can be regarded as the result of rotating the original segment 90° counter-clockwise about the origin. The problem could thus be put in a geometrical and somewhat more general fashion: How can this mathematical operation, represented by rotation about a point in the plane, be extended to rotation about a line in three dimensions? Expressed this way the answer turns out to be that four numbers, not three, are required. [Lewis 2005, 461]
65 Hamilton’s construction was to introduce the now well-known imaginary entities, i, j, k, which abided by the following rules: i2 = j2 = k2 = –1 ij = k jk = i and, ijk = –1
66 The resulting “hypercomplex number”—or Hamilton’s quaternion (q)—was formed as: a + bi + cj + dk, where a, b, and c are the real number components of the expression. Hamilton made the following claims in regards to the expressions β ÷ α = q and q x α = β: The quotient q can be regarded as an operator that produces one directed line segment from another. If q is a ‘tensor’, or signless number, then it affects only the length of α. If q is a sign (+ or — ) it changes the direction of α. If it is a real number then q may have the effect of changing both the direction and length. If it is a ‘vector-unit’ (or ‘quadrantal vector’), i, j, k, then the effect is to turn α right- handedly through 90 degrees in a plane perpendicular to the vector-unit. Hamilton points out that a multiplication of a vector-unit, say i, by itself results in a rotation of 180°, i.e., the same as multiplying by —1 (reversing its direction) or i2 = —1. [Lewis 2005, 464]
67 Hamilton had called the addition of a scalar quantity and a vector a “quaternion”. It was an entity that contained both a scalar part and a vector part and caused rotation about an axis with a magnitude change in a given vector.
68 Clifford repeated much of Hamilton’s approach in introducing his own bi-quaternions to readers in the 1870s. If mathematicians assumed that the law of addition of vectors held in a plane, such that AB + BC = AC, then they could interpret the “so-called imaginary or impossible quantities [as] the operators which convert one vector into another,” Clifford wrote [Clifford 1876a, 386] (See Figure 1).
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FIGURE 1: [Clifford 1882, 414]
69 On this interpretation, i operates on a vector causing it to turn counterclockwise through a right angle so that: i x OE = OA.
70 If a = OM/OA and b = MB/OA, where a and b are ratios of vectors in a line (as defined above), then Clifford argued that every expression of the form a + bi is a ratio of two vectors [Clifford 1876a, 387]. By giving “proper values” to a and b every vector in the plane can be represented by (a x OA) + (b x OB’) [Clifford 1876a, 387] such that: OA = j; OA’ = k; then ij = k; ik = — j.
71 Turning to the problem of vectors in space, Clifford noted that “the operation which makes one vector into another is of the form a + bq”, where Q turns through a right angle in the plane of the two vectors.
72 Note that Clifford’s idiosyncratic approach to this notion was to search for a mechanism by which vectors were not limited to turning through right angles in their plane of existence. Rather, he wanted to develop a mechanism by which one could transform a vector in a given plane into a vector in an alternative plane. To do so, he had to extend the notion of vectors as directed lines (or steps) in space to directed, positioned, and rotating lines (or steps) in space. The emphasis on position in Clifford’s bi-quaternions is largely a product of his engagement with Riemannian geometry, as explained earlier. Riemann had emphasized the crucial aspect of position in space, given that the same body in two different locations of space might behave differently depending on the geometrical curvature of that particular segment of space (thus, what we perceive of as being discontinuous “forces” acting on bodies is explicable by reference to continuous motion in non-flat spaces). Thus, in Clifford’s extension of Hamiltonian quaternions, the principle of continuity remains intact—bi-quaternions operate on entities that are composed of both rotors and vectors (namely, “motors”) in continuous physical space. If space were discontinuous, the mathematical use of these operators to describe physical phenomena would not be possible, because of spatial discontinuity, would render biquaternion operations meaningless.
73 Thus, Clifford clarified the construction of a bi-quaternion in the following way. First, “the velocity of a rigid body may be represented in one way only as a rotation-velocity (ω) about a certain axis combined with a translation-velocity (v) along that axis”
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[Clifford 1873, 182]. This combined velocity is what he called a twist-velocity about a particular screw, where the axis of the screw was the axis of rotation, and its pitch was the ratio v/ω (the ratio of the translation to the rotation). Clifford then suggested the following terms: 1. A “screw” is the geometrical form resulting from the combination of an axis, or straight line, with a pitch, which is a linear magnitude; 2. A “wrench” is the association of a screw with a magnitude “whose dimensions are those of a force”; 3. Finally, a “twist-velocity” is the association of a magnitude whose dimensions are those of an angular velocity [Clifford 1873, 183].
74 Along with these new basic spatial-geometrical entities, Clifford defined his more complex creations such as “motors” in the following way: Just as a vector (translation-velocity, or couple) is magnitude associated with direction, and as a rotor (rotation-velocity, or force) is magnitude associated with an axis, so this new quantity, which is the sum of two or more rotors (twist-velocity or wrench) is magnitude associated with a screw. [Clifford 1873, 183]
75 The simplest example of a “motor” involved the general motion of a rigid body.
76 With these new tools in hand, and assuming continuity to hold as a spatial fact, Clifford explained the operation of quaternions and bi-quaternions as follows. If we have the vectors AB and AC, both of which start from any arbitrary point A, we can make AB identical to AC by “turning it round an axis through A perpendicular to the plane BAC until its direction coincides with that of AC, and then magnifying or diminishing it until it is of the same length as AC” [Clifford 1873, 183] (see Figure 2).
FIGURE 2: [Clifford 1882, 228]
77 The ratio of the two vectors is the combination of an ordinary numerical ratio (magnitude change) with a rotation (direction change). In Hamilton’s terms, this means that a quaternion is the product of a tensor and a ver-sor. Since point A is perfectly arbitrary, “this rotation is not about a definite axis, but is completely specified when its angular magnitude and the direction of its axis are given” [Clifford 1873, 184]. The mathematician can therefore write that: AC/AB = q [ quaternion].
78 For Clifford, this is an operation which, Being performed on AB, converts it into AC, so that q . AB = AC. The axis of the quaternion is perpendicular to the plane BAC; and it is clear that the quaternion operating upon any other vector AD in this plane will convert it into a fourth vector AE in the same plane, the angle DAE being equal to BAC and the lengths of the four lines proportionals. [Clifford 1873, 184]
79 The problem, as mentioned earlier, was that a quaternion could only operate upon a vector that was perpendicular to its axis: “If AF be any vector not in the plane BAC, the expression q . AF is absolutely unmeaning,” Clifford wrote [Clifford 1873, 184]. He,
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therefore, set as his task the construction of an operator by which one could convert a rotor into another rotor. In so doing he found that he had to (see Figure 3): Turn A about the axis AC into the position AB’, parallel to CD. Then slide it along this axis into the position CD’. Lastly, magnify or diminish it in the ratio of CD’ to CD. The first two operations may be regarded as together forming a twist about a screw whose axis is AC and whose pitch is
[Clifford 1873, 184]
FIGURE 3: [Clifford 1882, 228]
80 Clifford explained that the ratio of two rotors constitutes the combination of an ordinary numerical ratio with a twist, where a twist indicates a definite “screw” and is “only specified when its angular magnitude and the screw (involving direction, position, and pitch) [are] given” [Clifford 1873, 185]. He then made the following analogy: “Just as the rotation (versor) involved in a quaternion is the ratio of two directions, so the twist involved in the ratio of two rotors is really the ratio of their axes”. Thus, by Using the expression tensor-twist to mean the ratio of two rotors (which is in fact a twist multiplied by a tensor), we may say that a tensor-twist can operate upon any rotor which meets its axis at right angles. [Clifford 1873, 185]
81 Like the quaternion, Clifford’s ratio of two rotors also had a “restricted range of operation” [Clifford 1873, 185]. The question Clifford needed to answer was: “What is the operation that converts one motor [which performs a screw operation] into another?” [Clifford 1873, 185]. When two motors have the same pitch, he said, the answer is easy: their ratio is a tensor-twist. The tensor is the ratio of their magnitudes, and the twist is the ratio of their axes.15 The more general case, however, in which the pitches of two motors are different, is not so simple to solve. If a motor consists of a rotor part and a vector part, then its pitch is determined by the ratio of these two parts. Clifford determined this ratio by combining a “suitable vector with a motor” such that
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the pitch could be made to be “anything we like, without altering the rotor part” [Clifford 1873, 186].
Geometrical Quantity Example Ratio Form
Sense on a straight Vector on a Signed Ratio Addition or Substraction line straight line ( + , or —)
Complex ratio Direction in Plane Vector in Plane Complex quantity [where √–1 is an operator that turns vectors]
Translation, Direction in Space Vector in Space Quaternion Couple
Axis Rotor Rotation-Velocity, Force Twist
Twist-Velocity, System Screw Motor Biquaternion of Forces
82 Therefore, to determine the operation that could convert motor A into motor B, Clifford first let B’ be a motor that had the same rotor part as B and the same pitch as A. To convert motor A into vector β (where β is a vector parallel to the axis of B), he introduced the symbol (ω) “whose nature and operation will at first sight appear completely arbitrary, but will be justified in the sequel.” Analogously to quaternions, “the symbol ω, applied to any motor, changes it into a vector parallel to its axis and proportional to the rotor part of it” [Clifford 1873, 186]. This operation (ω) changes a rotation about an axis into a translation parallel to that axis. The ratio of two motors can then be expressed as the sum of two parts—one of which is a tensor-twist and the other of which is w multiplied by a quaternion [Clifford 1873, 188]. This ratio is Clifford’s biquaternion.
83 Clifford concludes by offering the following table of concepts to his readers to clarify the relationships that he has created between his various new entities in the extension of quaternion mathematics—entities that operate upon physical bodies in continuous space.
6. Specialist and Non-Specialist Statements
84 It is possible now to paint the picture of a mathematician who could have chosen to pursue research based on discontinuous spatial models (such as action-at-a-distance theories), but who chose, not to do so. Rather, Clifford pursued research based on models of elliptic and non-flat spaces, founding his pursuits on the assumption that continuity existed as a structural aspect of space. His aim was to develop tools that would help to describe rigid body motion by extending what he thought was a useful starting concept (namely, Hamilto-nian quaternions). The importance of “continuity” as the underlying basis for Clifford’s advanced mathematical constructions is
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highlighted clearly in two of Clifford’s early lectures on related topics—the first being “On the theories of the physical forces” (delivered in 1870 to the Royal Institution), and the second being “Atoms” (delivered in 1872 to a popular audience at the Sunday Lecture Society).
85 “On the theories of the physical forces” describes what Clifford deemed to be the three historical stages of scientific belief, as well as his corollary belief in the need to encourage a new “fourth” stage of scientific thought. The first stage, Clifford argued, had involved the basic observation of phenomena such as planetary motions; the second stage had involved theological reasoning as to the causes of those phenomena. In the third stage—the “era of modern science”—partially materialist conceptions of cause (such as “force”) had come into play. However, those conceptions often relied upon metaphysical assumptions. In the new fourth stage of science—the stage that Clifford felt himself to be engaged in as a reformist Victorian mathematician—the conception of “force” would disappear altogether. Clifford’s aim in developing bi- quaternions was to encourage a movement towards that “fourth” stage, by designing mathematical tools that would precipitate the transition away from action-at-a- distance principles (and their inherent discontinuities) and towards new theories of continuous, non-flat spatial ontology and geometry.
86 In order to achieve these goals scientists had to confront the underlying question: “What is it that lies at the bottom of thing?” In Clifford’s view, this question is composed of two distinct queries: 1) Why do things happen? 2) What is it precisely that does happen? [Clifford 1870, 110]
87 The first question, he stated, is external to the province of science, Clifford argued. It is a question whose answer lay outside the domain of human knowledge.16 For the scientific inquirer, there is no hope of answering the question “Why?”, unless an appeal to speculative metaphysics comes into play. The second question is one that Science can aim to answer. But in order for science to supersede the enquiry into why things happen, and to advance into the domain of what is happening, Clifford argued that scientists must rely upon the “hypothesis of continuity”. This is because the “hypothesis of continuity involves such an interdependence of the facts of the universe as forbids us to speak of one fact or set of facts as the cause of another set of facts” [Clifford 1870, 111].
88 In his popular account of continuity, Clifford wrote: Things frequently move. Some things move faster than others. Even the same thing moves faster at one time than it does at another time. When you say that you are walking four miles an hour, you do not mean that you actually walk exactly four miles in any particular hour; you mean that if anybody did walk for an hour, keeping all the time exactly at the rate at which you were walking, he would in that hour walk four miles. But now suppose that you start walking four miles an hour, and gradually quicken your pace, until you are walking six miles an hour. Then this question may be asked: Suppose that anybody chose a particular number between four and six, say four and five-eighths, is it perfectly certain that at some instant or other during that interval you were walking at the rate of four miles and five- eighths in the hour? Or, to put it more accurately, suppose that we have a vessel containing four pints of water exactly, and that somebody adds to it a casual quantity of water less than two pints. Then is it perfectly certain that between these two times, when you were walking at four miles an hour, and when you were walking six miles an hour, there was some particular instant at which you were
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walking exactly as many miles and fractions of a mile an hour as there are pints and fractions of a pint of water in the vessel? The hypothesis of continuity says that the answer to this question is yes: and this is the answer which everybody gives nowadays; which everybody has given mostly since the invention of the differential calculus. [Clifford 1870, 112]
89 This conclusion is a matter of “fact”, and not “calculation,” Clifford contended. In other words, the hypothesis of continuity is a hypothesis about the nature of matter and the nature of motion; it is a statement about the ontology of the universe. In Clifford’s view, the dependence of calculus upon “continuity” indicates that calculus describes motion in space as it exists. Calculus, therefore, is not an abstract mathematical language disjointed from empirical observation, as Dedekind and his supporters would have argued. The techniques of calculus involve measurements of space over time. The mathematical abstraction of these techniques into symbolic language does not sever their meaning from those empirical and physical roots.
90 While it might seem obvious to the modern reader that calculus involves an empirical measure of objects in motion, Clifford’s audiences would not necessarily have considered mathematics, in general, to be a fundamentally empirical field of knowledge. The vehemence with which Clifford emphasized the empirical basis of continuity indicates the ideological nature of his statements. In taking such a stand, Clifford was replying to members of the Analytic Society of the early 1820s, and their later followers at Cambridge, as well as formalized approaches to continuity such as Dedekind’s, which sought to establish meanings for mathematical symbols that were internally defined (i.e., not linked to external phenomena). For instance, in a subtle criticism of such thinkers, Clifford invoked the physical image of “continuity” by describing two rows of points that represent a series of positions in space. The lower row of points represents a series of instants in time in which it is conceivable that some thing might exist. At the instant of time represented by the first point in the lower row, the object in question holds a position in space represented by the first point in the upper row. That object only exists there for an instant, and then it disappears, such that at the “succeeding instants where the lower points have no points directly above them, the thing is nowhere at all” [Clifford 1870, 116]. At the instant of time when there is a space-dot above the object, the thing exists in that space-position. A motion is discontinuous, therefore, when the thing is in “different places at different times, though it is not at all times that it exists at all.” In a discontinuous world, in other words, an object appears and disappears between points in space. When it disappears, it has no existence, which means that the object goes through a series of creations, destructions, and re-creations as time moves forward. Mathematically, such discontinuities arise when “the thing passes from one position to another distant from it without going through any intermediate position” [Clifford 1870, 116]. If the dots are very close together, they would appear to be continuous to the general observer and “this is the sort of representation of what we might have to suppose if we did not assume the truth of the law of continuity,” Clifford stated [Clifford 1870, 117]. In that discontinuous world, however, the path from one of the end-dots to the other would be composed of a series of “discrete positions”.
91 In more technical terminology, Clifford described the mathematical “hypothesis of continuity” as the claim that (see Figure 4), The motion of N asserts that not merely N itself moves without any jumps, but that the rate at which N is going changes gradually without any jumps, and
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consequently that the direction of P’s motion changes gradually; or that the curve described by P cannot have a sharp point. [Clifford 1870, 119]
FIGURE 4: [Clifford 1870, 118]
92 Thus, one can calculate the rate of change of the curve representing the rate of change of another curve (i.e., a 2nd order derivative). Indeed, the mathematical hypothesis of continuity asserts that this process can be repeated until there is no longer a rate of change to measure. Ultimately, however, these more advanced mathematical aspects of “continuity” are still based upon the physical truth of continuous space: The hypothesis of the perfect continuity of N’s motion asserts that all these points move continuously without any jumps. [Whereas] a jump made by any one of these points, being a finite change made in no time, would be a change made at an infinite rate; the next point, therefore, and all after it, would go right away from 0, and disappear altogether. We may express the law of continuity also in this form; that there is no infinite change of any order. [Clifford 1870, 121]
93 Ultimately, the hypothesis of continuity states that existent bodies never lose their existence in space.
94 Thus, for Clifford, mathematical “continuity” is dependent upon what happens to an object in space as time progresses. If space were composed of discrete units, then the movement of the object must involve leaps from one segment of space to another with no travel in-between. It would involve what Clifford would consider to be an instantaneous action-at-a-distance. And if one were to try to measure a “rate of change” based on the size of a jump from one dot to another, one would be led to the consideration that, This rate might obviously change by jumps as violent and sudden as those of the thing itself; at any instant, when the thing was non-existent, its rate would be non- existent, and whenever the thing came into existence its rate would suddenly have a value depending on how far off its last position was. [Clifford 1870, 118]
95 In a discontinuous world there could be no such thing as a meaningful “rate of motion.” There would be no meaningful use of the calculus, and certainly no meaningful use of quaternions or vector transformations. As a spatial model, discontinuity would mean that calculus could not accurately represent the world as it is. The measurement of distances between points or bodies in space would become a useless task, as it would tell us nothing about the intervening segments of space. Mathematics as a descriptive science would fall apart, as it would have no role to play in describing the causal
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picture of the universe. Clifford thus acknowledged that science post-Newton and post- Leibniz could only move forward on the justifiable assumption that continuity holds true in space and time. Otherwise, the entire structure of calculus would have to be done away with and new tools for measuring motion in discontinuous space would have to be developed.17
7. On Ether as the Necessary Result of the Continuity Hypothesis
96 Perhaps not surprisingly, Clifford’s reliance upon the necessity of “continuity” in space, and upon the physiological predisposition of human beings to believe in it, was deeply intertwined with his belief in the existence of the ether-a medium that guaranteed a material world in which cause and effect could be explained via the interactions of particles. From Clifford’s perspective, an all-pervading ether precluded the need for metaphysical explanations of physical phenomena. Delivered to the Sunday Lecture Society in 1872 (a popular forum), Clifford’s essay, entitled “Atoms”, expounded upon the link between “continuity” and the ether and also the link between mystical beliefs in “God” and action-at-a-distance “forces.”
97 Taking the example of a person rubbing a wet finger along the rim of a glass, Clifford argued that in order to explain the phenomena of “sound” (i.e., the perception of musical waves) scientists had to acknowledge the existence of a “surrounding” framework through which waves could travel such that a person distant from the origin of the sound could perceive it. At its most basic level, that surrounding framework (the ether) is composed of “atoms”, which pervade the entire universe such that they are always in contact with one another. According to Clifford, this understanding of the universe’s continuous physical structure was no longer deniable: The ‘atomic theory’ (...) is no longer in the position of a theory, but that such of the facts as I have just explained to you are really things which are definitely known and which are no longer suppositions; that the arguments by which scientific men have been led to adopt these views are such as, to anybody who fairly considers them, justify that person in believing that the statements are true. [Clifford 1872, 163]
98 The “undulatory theory” of light was a scientific theory that Clifford also considered to be consistent with his view of the universe, as it allowed for the transference of “energy” between contiguous entities with no discontinuous jumps in space. In other words, light consists of waves transmitted through a physically continuous medium. Clifford’s aim in drawing out this example was to convince mathematicians that if they were to rely upon the fundamentally physical notion of continuity, then they would be well advised to also recognize the useful corollary outcomes of that choice. One such outcome was the fact that, in order for light waves to travel from one location in the universe to another, there had to exist a mechanical series of entities between the two positions such that light waves never disappeared and reappeared in the process of transmission. From a Cliffordian perspective, therefore, if scientists and mathematicians hoped to make meaningful claims about the universe, they were compelled to adopt a thorough belief in the continuity of space and time. In so doing, they had to resist conventionalising mathematical knowledge such that “continuity”
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became a mere logical tool, or symbolical axiom, defined intrinsically and thus severed from the very empirical inferences that had given rise to the notion in the first place.
Conclusion
99 Clifford’s final statement on continuity comes in the form of his monograph, The Common Sense of the Exact Sciences (published posthumously in 1885 and edited by Karl Pearson). In his Common Sense text Clifford brought together the various strands of his continuity theorizing, linking them to his views on empiricism, physiology, non- Euclidean geometry, and bi-quaternions. In a chapter entitled “Of Mass and Force”, he wrote that: The custom of basing our ideas of motion on these terms “matter” and “force” has too often led to obscurity, not only in mathematical, but in philosophical reasoning. We do not know why the presence of one body tends to change the velocity of another; to say that it arises from the force resident in the first body acting upon the matter of the moving body is only to slur over our ignorance. All that we do know is that the presence of one body may tend to change the velocity of another, and that, if it does, the change can be ascertained from experiment.18 [Clifford 1885, 243]
100 For Clifford, the question of whether mathematical “continuity” ought to be defined independently of physical continuity is a question about what constitutes good scientific and mathematical theorizing. Given that empirical sensory observation can only tell us so much about the world “in itself”, scientists are heavily reliant upon their assumptions about how it ought to be.
101 For Clifford, therefore, it is sensible to assume continuity exists in the physical structure of the universe and to use this principle in order to define and develop new geometrical models that account for what often appear to be discontinuous events (i.e., gravity and other action-at-a-distance phenomena).
102 The Cliffordian case study demonstrates that “continuity” has served various scientific, mathematical, philosophical and even physiological roles throughout history. In Clifford’s case, it formed an intricate part of his philosophy of science, and it underpinned his deeply materialist philosophy of space. In comparing his account to that of Dedekind’s arithmetization of the concept, it is possible to highlight the fact that in these divergent interpretations of a fundamental concept in mathematics, a finitist picture emerges—one in which the concept of “continuity” does not come pre- packaged with a series of understood and consensual meanings bound together. Rather, “continuity”, as with all mathematical concepts, is underdetermined. Practitioners are free to idiosyncratically draw upon the wide-spread cultural resources available to them in adopting, redefining and using such concepts in previously unknown ways. In this conceptually underdetermined world, it becomes clear that, for someone like Dedekind, the motivation to define the concept of continuity stemmed from a belief in the abstract and conventional (if not language-like) nature of mathematics as distinct from any physical concerns. For Clifford, on the other hand, the definition and use of continuity stemmed from his thoroughly physics-minded agenda, which ultimately placed limits on the bounds of what could constitute good scientific theorizing (i.e., appeals to materialist descriptions of cause and effect) and good mathematical practice (i.e., the development of descriptive geometrical tools that would preserve continuity and account for physical phenomena). Each practitioner viewed his own account as the
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correct one; each practitioner thus managed to redefine the bounds of proper use for this seemingly simple and intuitively clear notion.
BIBLIOGRAPHY
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BENARDETE, J. A. — 1968 Continuity and the Theory of Measurement, The Journal of Philosophy, 65(14), 411-430.
BOI, L. — 1997 Géométrie elliptique non-euclidienne et théorie des biquaternions chez Clifford : l’élaboration d’une algèbre géométrique. Le Nombre, une hydre à n visages : Entre nombres complexes et vecteurs, Paris : Fondation Maison des Sciences de l’Homme, 209-238.
CANTOR, G. N. & HODGE, M. J. S. — 1981 Conceptions of Ether: Studies in the History of Ether Theories 1740-1900, Cambridge: The Cambridge University Press.
CLIFFORD, W. K. — 1870 On Theories of Physical Forces, Lectures and Essays, London: MacMillan and Co., 1879, 109-123. — 1872 Atoms, Lectures and Essays, London: MacMillan and Co., 1879, 158-189. — 1873 Preliminary Sketches of Bi-quaternions, Mathematical Papers, London: MacMillan and Co., 1882, 181-196. — 1876a Further Note on Bi-quaternions, Mathematical Papers, London: MacMillan and Co., 1882, 385-394. — 1876b On the Space-Theory of Matter (Abstract), Mathematical Papers, London: Macmillan and Co., 1882, 21-22. — 1878 Elements of Dynamic, An Introduction to the Study of Motion and Rest in Solid and Fluid Bodies, Part I. Kinematic, London: Macmillan and Co. — 1879 The Philosophy of the Pure Sciences, Lectures and Essays, London: MacMillan and Co., 301-409. — 1882 Mathematical Papers, London: MacMillan and Co. — 1885 Common Sense of the Exact Sciences, London: Sigma Books, 1947.
COOKE, R. — 2005 Richard Dedekind, Stetigkeit und Irrationale Zahlen (1872), Landmarks Writings in Western Mathematics, 1640-1940, Amsterdam: Elsevier, B.V., Chapter 43, 553-563.
CROWE, M. J. — 1967 A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, Notre Dame and London: University of Notre Dame Press.
DEDEKIND, R. — 1963 Continuity and Irrational Numbers. Essays on the Theory of Numbers, New York: Dover, 1-27.
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EVANS, M. G. — 1955 Aristotle, Newton and the Theory of Continuous Magnitudes, Journal of the History of Ideas, 16(4), 548-557.
FOWLER, D. — 1992 Dedekind’s Theorem: √2 x √3 = √6, The American Mathematical Monthly, 99(8), 725-733.
GRAY, J. — 2005 Bernhard Riemann, Posthumous Thesis ‘On the hypotheses which lie at the foundation of geometry’ (1867). Landmark Writings in Western Mathematics, 1640-1940, Amsterdam: Elsevier B.V., Chapter 39, 506-520.
JOURDAIN, P. E. B. — 1908 The Introduction of Irrational Numbers, The Mathematical Gazette, 4(69), 201-209. — 1867 Law and Bye Laws of the Cambridge Philosophical Society, The Cambridge University Calendar For the Year 1866, Cambridge: Deighton, Bell, and Co., 591-605.
LEWIS, A. C. — 2005 William Rowan Hamilton, Lectures on Quaternions (1853). Landmark Writings in Western Mathematics, 1640-1940, Amsterdam: Elsevier B.V., Chapter 35, 460-469.
LIGHTMAN, B. — 1997 “The Voices of Nature”: Popularizing Victorian Science, Victorian Science in Context, Chicago: University of Chicago Press, Chapter 9, 187-211.
MEULLER, I. — 1969 Zeno’s Paradoxes and Continuity, Mind, New Series, 78(309), 129-131.
PITKIN, W. B. — 1906 Continuity and Number, The Philosophical Review, 15(6), 597-605.
PYCIOR, H. M. — 1976 Ph.D. Thesis, The Role of Sir William Rowan Hamilton in the Development of British Modern Algebra, Cornell University. — 1997 Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra Through the Commentaries on Newton’s Universal Arithmetick, Cambridge: Cambridge University Press.
RIEMANN, B. — 1868 Ueber die Hypothesen, welche der Geometrie zu Grunde liegen, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13, 1867. Cited according to the English translation by William Kingdon Clifford, On the Hypotheses which Lie at the Bases of Geometry, Mathematical Papers, London: MacMillan and Co., 1882, 55-71.
SALANSKIS, J.-M. & SINACEUR, H. — 1992 Le Labyrinthe du Continu : Colloque de Cerisy, Paris : Springer-Verlag.
STEWART, J. 2003 2.5 Continuity, Calculus, 5th Edition, Belmont: Brooks/Cole - Thomson Learning, 102-110.
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WARWICK, A. — 2003 Masters of Theory: Cambridge and the Rise of Mathematical Physics, Chicago: University of Chicago Press.
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WHITROW, G. J. — 1933 Continuity and Irrational Number, The Mathematical Gazette, 17 (224), 151-157.
NOTES
1. In their account of ether histories, Cantor and Hodge identify five categories of “ether' concepts that have been discussed and used by natural philosophers, mathematicians, and physicists over the course of the past century. The authors write: ”We might introduce the historical study of ether theories by stating simply what ether is. But two reflections should make us pause before attempting this. First, although essentialism is now favoured again in reputable quarters, the lessons of Wittgenstein and others still stand. For many kinds of things it is futile to seek a common and distinctive essence. For many terms it is misleading to demand a definition specifying the conditions necessary and sufficient for their application. So it is with ether. We cannot usefully indicate properties that all ethers must and only ethers can have. Second, even if we could, we should not wish to make the attempt independently of historical inquiry. For definition demarcation of one kind of theory from others may serve merely to separate lines of theorising that in fact developed together, or to conflate those that developed apart“ [Cantor & Hodge 1981, 1]. I could repeat Cantor’s and Hodge’s reflections here as an introduction to my own account of ”continuity' as it appeared throughout the 19th century and earlier. “Continuity” was not a clearly defined mathematical notion. It was not a perfectly understood physical notion. It was used by various authors in various ways serving their varying interests. In short, it is a concept with a history, and the historical construction of its meaning is still ongoing. 2. In the Preface to his essay “Continuity and Irrational Number”, Dedekind wrote that in 1858 he found himself “obliged to lecture upon the elements of the differential calculus,” and he “felt more keenly than ever before the lack of a really scientific foundation for arithmetic” [Dedekind 1963]. 3. The emphases are Dedekind’s own. 4. In a celebratory account of Dedekind’s contribution to the field of number theory, David Fowler writes that mathematicians often have a “naive belief that arithmetical operations on decimals pose no problems.” Fowler contends that Dedekind’s definition of continuity allows us to recognize that stating 0.999... = 1 000 is problematical unless we first recognize and adopt a synthetic approach to the creation of new types of numbers. Such an approach is necessary, in Fowler’s view, for the resolution of the equation √2 x √3 = √6 which presupposes that 0.999.. = 1 000 is a legitimate identity to make [Fowler 1992, 725—733]. 5. Lightman notes that efforts to popularize science peaked in the late Victorian period, by which point journals and magazines were hiring non-specialists to journalistically present mathematical and scientific discoveries in accessible formats [Lightman 1997]. 6. See [Pycior 1997] and [Pycior 1976] for discussions of the mysterious nature accredited to imaginary numbers well into the 19th-century. 7. The Cambridge Philosophical Society was set up in 1819 to promote ”scientific inquiry“ and facilitate ”the communication of facts connected with the advancement of Philosophy and
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Natural History’. We can consider this paper as one delivered to an audience composed of experts, and students on their way to becoming mathematical experts. 8. Only some of the Hamiltonian aspects of Clifford’s work will be highlighted here; I elaborate upon the Grassmannian aspects of his work in my PhD thesis (to be submitted June 2009). 9. These two papers were left untitled by Clifford; their respective names are to be credited to Robert Tucker. 10. According to Tucker, nine-tenths of the papers in his Mathematical Papers are ”geometrical“ in nature. 11. Riemann called these portions “quanta”. 12. Riemann noted that it was this distinction that formed “a general division of the science of magnitude in which magnitudes are regarded not as existing independently of position, and not as expressible in terms of a unit, but as regions in a manifoldness”. By the time Riemann’s paper was published in 1854, such research had become a necessity. The previous lack of such considerations was “no doubt a chief cause why the celebrated theorem of Abel, and the achievements of Lagrange, Pfaff, Jacobi for the general theory of differential equations, have so long remained unfruitful” [Riemann 1867, 57]. 13. This claim was fundamental for both Helmholtz and Clifford’s respective philosophies of science, in which they each sought to defend the view that non-Euclidean geometry is physiologically justifiable because, even in uniformly curved space, we do not perceive any change in shape of a body as it moves (even though the underlying spatial structure might be non-flat). 14. Gray writes that Riemann’s “inspiration here was Gauss’s discovery of curvature, which Gauss showed was something that could be determined from quantities measured in the surface alone” [Gray 2005, 509]. 15. We are led to this conclusion, he wrote, “by considering each motor as the sum of two rotors which do not intersect” [Clifford 1873, 186]. 16. For an account of Clifford’s refusal to consider the “Why?” in scientific theorizing, consider the following polemical statement: “Reflect on the fact that for a single particle, quite irrespective of everything else—the history of eternity is contained in every second of time, and then try if you can find room in this one stifling eternal fact for any secondary causes and the question why? (...) Why does the moon go around the Earth? When the solar system was nebulous, anybody who knew all about some one particle of nebulous vapour might have predicted that it would at this moment form part of the moon's mass, and be rotating about the earth exactly as it does. But why with an acceleration inversely as the square of the distance? There is no why (...) The cause is only the fact that at some moment the thing is so—or rather, the facts of one time are not the cause of the facts of another, but the facts of all time are included in one statement, and rigorously bound up together” [Clifford 1870, 123]. 17. Clifford wrote often about the “fluxion” method that Newton had developed, although it will be clear to any observer that Clifford—as with many mathematicians of the late-19th century— interpreted the Newtonian fluxion and the Leibnizian derivative as two sides of the same coin. Clifford’s goal was to link the assumption of “continuity” to the scientific usefulness of the Newtonian fluxional method. 18. Although Pearson was responsible for heavily editing this section of the book, Clifford makes similar claims in various lectures delivered throughout the 1870s, many of which were later published in his Lectures and Essays. Thus, the statement is a fair representation of Clifford’s own views on the subject.
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ABSTRACTS
The concept of continuity is fundamental to contemporary mathematical analysis. However, the seemingly well-founded definition currently used for this concept is just one of many versions that have been historically proffered, used and reshaped by mathematical practitioners over the centuries. In looking at the particular manner in which William Kingdon Clifford (1845-1879) defined and negotiated the physical underpinnings to “continuity” in his works, and by comparing that account to Richard Dedekind’s (1831-1916) conventionalist definition of the concept, it becomes clear that “continuity” has served differing interests over the centuries and has been shaped by the process of applying it to various philosophical and mathematical projects.
Le concept de continuité est fondamental pour l’analyse mathématique contemporaine. Cependant, la définition actuellement employée, apparemment bien fondée de ce concept, n’est que l’une des nombreuses versions historiquement énoncées, utilisées et affinées par les mathématiciens au travers des siècles. Cet article présente la façon dont William Kingdon Clifford (1845-1879) a façonné ce concept en lui donnant des bases physiques. La présentation de l’effort de Richard Dedekind (1831-1916) pour établir mathématiquement cette notion dans une perspective conventionnaliste permettra de mieux apprécier la spécificité de chacune des deux démarches. Cette étude montrera quel rôle historique la continuité a joué dans plusieurs projets mathématiques et philosophiques, et comment elle a été façonnée à partir des différents centres d’intérêt de ses utilisateurs.
AUTHOR
JOSIPA GORDANA PETRUNIĆ Science Studies Unit, University of Edinburgh
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Husserlian and Fichtean Leanings: Weyl on Logicism, Intuitionism, and Formalism
Norman Sieroka
1 Introduction
1 Hermann Weyl was one of the greatest mathematicians of the last century. Within the last few years, his philosophical writings have also gained attention, in particular the way they were influenced by phenomenology as presented by its founder, Edmund Husserl. (See, for instance, [Ryckman 2005].) When he was a student of mathematics in Gottingen under David Hilbert, Weyl attended lectures by Husserl, who was then a professor of philosophy there.
2 However, after gaining his first chair in mathematics at the ETH in Zurich, Weyl, as he himself put it, got “deeply involved in Fichte” [Weyl 1946, 1]. (Translations throughout the paper are mine.) As further evidence of this involvement, Weyl attended a reading group on Fichte [Weyl 1948, 381] and also took extensive notes on Fichte’s major works [Weyl (undated I)]. Weyl’s strong involvement in the work of a German Idealist seems at first glance astonishing and, at least from the received view in contemporary philosophy, needs some explanation. Historically, Weyl’s interest in Fichte was initiated by his Zurich colleague, the philosopher and Fichte expert Fritz Medicus. As Weyl acknowledges, it was also through Medicus that “the theory of relativity, the problem of the infinite in mathematics, and finally quantum mechanics became the motivations for my attempts to help clarifying the methods of scientific understanding and the theoretical picture of reality as a whole” [Weyl 1946, 1].
3 In this paper I will present Weyl’s readings of phenomenology and Fichte’s “constructivism” in relation to the philosophy of mathematics. In Section 2 Weyl’s 1918 work on the continuum will be discussed against the background of the logicist programme, which attempted to reduce mathematics to logic and set theory, and how Weyl opposed it. Whereas in this work he refers to both, Husserl and Fichte, as his
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allies, Weyl made a clear separation between the two philosophers during the debate between intuitionism and formalism in the early 1920s. As discussed in Section 3, Weyl at that point linked intuitionism, which was favoured by him for only a very brief period, to Husserlian phenomenology and linked formalism to Fichtean constructivism. I will show that a similar distinction can also be found in the work of Husserl’s model student in the philosophy of mathematics, Oskar Becker.
4 As the earlier quote from Weyl suggests, the debate about the problem of the infinite as it occurs in the analysis of the continuum, and in the intuitionism-formalism debate cannot be strictly separated from issues in physics. (See, for instance, [Sigurdsson 1991] and [Scholz 2001].) Indeed in the middle of the 1920s Weyl developed what he called an “agens theory” of matter, which attempted to accommodate recent results in relativity theory and atomic physics and which was also strongly influenced by Weyl’s reading of Fichte. However, since this has been dealt with in detail elsewhere [Sieroka 2007], it will not be addressed in this paper.
5 The overall aim of this paper is to reconstruct how Weyl philosophically locates his own position during the debates between logicism, intuitionism and formalism. Although one obviously does not need to agree with the strong association of intuitionism and phenomenology, for instance, the following investigation shows that Weyl’s claims and distinctions are understandable through considering the historical constellation in which his reading of Husserl and Brouwer arose.
6 This paper also contributes to two more detailed issues. Over the last few years, studies of Weyl have acknowledged his transcendental idealism but have mainly focused on its Husserlian leanings. The present study may help to get a more detailed picture of Weyl’s idealism by locating his Fichtean leanings more properly (in particular since the few works that exist in this context are to some extent misleading, as will be considered below). The other issue concerns contemporary research that takes Fichte mainly to be a philosophical foundationalist who is disclosing some mystical self-relation of the ego. (See, for instance, [Henrich 1967].) In contrast to this well-known view, the present paper (following Weyl) indicates a constructivist reading of Fichte which seems not only possible but also very fruitful for philosophical debates about mathematics and physics.
2 Against logicism
7 In 1918 Weyl wrote a book entitled Das Kontinuum which attempted to give a foundation to analysis (to the calculus). At the very beginning of this book, Weyl engaged in a discussion of the pertinent antinomies, the most famous of which is the one by Russell about the set of all sets that does not contain itself as a member. The main point I want to make is that here Weyl refers to Fichte for the first time. He reveals one of those pertinent antinomies, namely the Grelling paradox (discussed below), as “scholasticism of the worst sort” and that asserts that, in this case, “one must seek advice from men, the names of whom cannot be mentioned among mathematicians without eliciting a sympathetic smile—Fichte, for instance” [Weyl 1918, 2]. The particular passage Weyl refers to in this quote is likely to be the lectures XXVIII-XXX of Fichte’s Transzendentale Logik with their prominent distinction between finite and infinite judgements [Fichte 1834, 265-289].
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8 Fichte was not a particularly gifted mathematician, as confirmed by the comments of several of his contemporaries. It was said, for instance, that “there was never a mathematical construction understood by him [Fichte]” and “when it comes to mathematics, I absolutely refuse to tolerate any instruction by Mr. Fichte” (quoted in [Radbruch 2003, 254]). Weyl, in his notes on Fichte, at one point writes: “Fichte is particularly funny, when he tries to express his philosophical constructs in mathematical form” [Weyl (undated I), 4]. So, rather than the pseudo-mathematical formalisations that one can find in Fichte at times, it is the overall philosophical project of constructivism that Weyl appreciates in Fichte. (Since the last quote refers to the section on the constitution of space in the Grundriß des Eigentümlichen der Wissenschaftslehre [Fichte 1795, 71-76], the interpretation in [Scholz 2004] might be confronted with a problem, for it draws heavily on exactly this passage to show parallels with Weyl’s analysis of the problem of space.)
9 More broadly speaking, what Weyl is opposed to when writing about a “scholasticism of the worst sort” is logicism. For him, all attempts to reduce mathematics to logic and set theory are mainly a symptom of a typical “mathematician’s disease” [Weyl 1919, 44]. To use Weyl’s own example, it is mainly by an uncomprehending application of logic that the so-called antinomies, like Grelling’s paradox about the word “heterological”, appear. An adjective is said to be heterological if it is not the case that “the word itself possesses the property which it denotes” [Weyl 1918, 2]. Therefore, “long” is heterological since it is not a long word, whereas “short” is indeed short and hence not hetero- but autological. The alleged antinomy now appears if one asks whether the adjective “heterological” is itself heterological or not. Weyl claims that all alleged antinomies can be removed by using an intensional logic, which accepts that “the meaning of a concept is the logical prius as opposed to its extension” [Weyl 1919, 44]. As a philosophical point of reference for his own view, Weyl again mentions Fichte’s Transzendentale Logik. Alongside this work, however, Weyl now also refers to Husserl’s Logische Untersuchungen [Husserl 1900/01] as an important philosophical work opposed to an unlimited extensional view of logic.
10 In Das Kontinuum Weyl discusses the concept of the continuum both in relation to our experience of the continuous flow of time and in its mathematical use, wherein the set of real numbers is assumed to form a continuum. Weyl claims that his investigation contributes to “the epistemological question concerning the relation between the immediately (intuitively) given and the formal concepts (in the sphere of mathematics) by which, in geometry and physics, we seek to construct this given” [Weyl 1918, iv]. According to Weyl, it is of central importance for the philosophy of both mathematics and physics to distinguish between a sphere of construction and a sphere of something immediately given (a sphere of intuition). Together with distinguishing these two, according to Weyl it is also important to investigate their mutual relation.
11 In the case of mathematics, this distinction of spheres remains the main contrast in Weyl’s writings about the debate between intuitionism and formalism, to which I will turn in the next section. (For a more detailed discussion of the issues in the philosophy of physics see [Sigurdsson 1991], [Ryckman 2005] and also compare Weyl’s harsh critique of Rudolf Carnap’s Der Raum [Weyl 1922].)
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3 Passive intuitionism and active formalism
12 After about 1912, intuitionism was mainly developed by the Dutch mathematician L.E.J. Brouwer. (See, for instance, [Brouwer 1912].) In this approach, mathematics is understood in terms of the intuitive and exact thinking of an individual mathematician. So, instead of seeking out an inner-mathematical foundation via logic and set theory, as in logicism, intuitionism aims at a foundation “from outside,” that is, via some non- mathematical given. In contrast, formalism as elaborated especially by David Hilbert is mainly concerned with the completeness and consistency of a formal system and not with its “truth” in any narrow sense. (See, for instance, [Hilbert 1928].)
13 For a brief period, Weyl strongly defended intuitionism. In his 1921 paper Über die neue Grundlagenkrise der Mathematik, he even talks about Brouwer as bringing “the revolution” that promises mathematicians’ “salvation from an evil incubus” [Weyl 1921, 158, 179]. (The former quote afterwards developed a philosophical history of its own. Famously, a few years later Frank Ramsey wrote about the “Bolshevik menace of Brouwer and Weyl” [Ramsey 1925, 219], which in turn led Ludwig Wittgenstein to call Ramsey a “bourgeois thinker” [Wittgenstein 1977, 17].) However, Weyl’s alliance with intuitionism quickly and considerably weakened, when it became apparent that intuitionism was a very ambitious project employing concepts which, at least from the point of view of a practising mathematician, were very restrictive.
14 Since the use of the law of the excluded middle is limited in intuitionism, proofs by contradiction, which are quite common in mathematics, often failed to be reproducible within this approach. Thus, as Weyl once put it, with intuitionism a huge part of mathematics “goes up in smoke” [Weyl 1925, 534]. This is not to say that Weyl in his later work did not adhere to some details of the intuitionist treatment of the infinite, but as a practising mathematician and theoretical physicist he retreated from the general foundational ambitions of intuitionism. In this respect, Weyl not only claimed a victory for formalism but also claimed a strong philosophical implication that along with intuitionism comes a “decisive defeat of pure phenomenology” [Weyl 1928, 149].
15 (Although, as a whole, this sentence from [Weyl 1928] is a conditional— saying that if intuitionism fails, then phenomenology would be defeated—it is evident from the context that Weyl holds the antecedent to be true; see also the quotation from Weyl in [Becker 1926-33, 249-250]. Moreover, even after Godel had proven his incompleteness theorems in 1931, which gave a considerable blow to Hilbert’s original programme, Weyl was not much concerned because of the wider notion of formalism and construction he had gained by then; see below, and for a more detailed discussion of Godel’s theorems [Weyl 1949a, AppendixA] and [Weyl 1951, 465, 494].)
16 In a letter to Husserl’s pupil Oskar Becker written in 1926, Weyl maintained that with the victory of formalism “ theoretical construction begins to look like the main epistemological problem that in the end is not bound to some intuitive foundation as an absolute and unexceedable frame” [Becker 1926-33, 249-250] (cf. also [Mancosu and Ryckman 2002]). Here Weyl connects intuitionism and phenomenology more closely via the concept of “intuition” (see below) and, speaking in modern terms, claims that both run afoul of the so called “myth of the given”, the idea that there is something immediately given which at the same time is a piece of non-refutable and proper knowledge on which all other knowledge is firmly based [Sellars 1956]. Because many wrote about this during the second half of the twentieth century, it should be enough
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to note that already Fichte opposed this “myth”. For instance, in his Transzendentale Logik, Fichte writes that “in knowledge there are no separated and single entities [...]. That someone might know about entities as being different from each other entails that he already understands this very difference, that he contrasts (entgegensetzt) and compares these entities; therefore, that he already has a general concept of these entities” [Fichte 1834, 12].
17 According to Weyl, intuitionism cannot adequately account for theoretical construction and also for the creativity (das Schöpferische) of science. For him, this goes along with a distinction between two approaches in philosophy, one that starts from an immediately given and from a passive viewing (passive Schau) and one that defends an active constructivism. As Weyl puts his own position in a pointed remark, “we do not have the truth; it does not suffice merely to open your eyes wide and then we shall see it; but we must act and parts of the truth will be revealed to us by our very action” [Weyl (undated II), 6-7]. (This quote is taken from an English manuscript, but nearly the same phrasing also occurs in one of Weyl’s German papers [Weyl 1949b, 334].) It seems possible and perhaps even plausible to distinguish those two approaches, as Weyl does, along the lines of the philosophies of Husserl and Fichte. Indeed, one of the most important and even peculiar concepts of Husserlian phenomenology, which distinguishes it from nearly all other philosophies of the twentieth century, is the Wesensschau: the assumption that we can “intuit” or rather “view essences” if we reflect upon our mental acts [Tieszen 2005, 69].
18 In contrast, Fichte’s philosophy, the “Doctrine of Knowledge” Wissenschaftslehre, famously starts from the ego as acting. At least in the early Fichte, this involves a constructivism with a strong kind of pragmatist dimension. Fichte maintains that “the Doctrine of Knowledge is to be a pragmatic history of the human mind” [Fichte 1794b, 141]: If our Doctrine of Knowledge gives an accurate account […], then it is absolutely certain and infallible […]; however, the question is exactly this, whether and to what extent our account might be appropriate; and thereupon we can never give a rigorous proof, but only one in terms of probabilities. […] We are not the legislators of the human mind, but its historiographers; we are not, of course, journalists, but rather writers of pragmatic history. [Fichte 1794a, 69]
19 Thus, Fichte is certainly a constructivist, but need not necessarily be read as the stubborn aprioristic philosopher he is often taken to be. There is a possible and consistent pragmatist, anti-foundationalist reading of Fichte which can also be found in the recent literature [Rockmore 1995]. Weyl also was aware of this dimension and claimed that Fichte entered but did not fully develop a “third realm” over and above realism and idealism [Weyl 1925, 540]. To use Weyl’s own phrase, there is a potential in Fichte which could be further developed into a modern “symbolic constructivism” [Weyl 1949b]. So, with respect to the distinction between the aforementioned philosophical approaches, it is clear that for Weyl, Fichte, rather than Husserl, was on the right track: [As compared to Husserl, Fichte] is anything but a phenomenologist, he is a constructivist of the purest sort, who—without looking left or right—goes his own headstrong way of construction. [...] As for the antagonism between constructivism and phenomenology, my sympathy is wholly on his side. However, the way a constructive procedure that leads towards symbolic representation, not a priori, but with continuous reference to experience, the way such a procedure can actually be carried out is shown by physics [...]. [Weyl 1954, 641, 643]
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20 Again, the above quote shows Weyl’s rather broad leanings to Fichtean constructivism, leanings that indeed allow him to even include Hilbert’s axiomatic method in a wider family of constructivist approaches to the mathematical sciences. (That this quote also shows the blurry boundary between mathematics and theoretical physics is, again, not my topic here.) Given that this quote is from 1954, it also shows how Weyl continued to keep his earlier Fichtean motifs, for already in 1925 Weyl was writing about “theoretical construction” and “symbolic representation” in relation to Fichte [Weyl 1925, 540-542]. And so Weyl’s “symbolic constructivism” can indeed be seen as an interpretation or elaboration of Fichtean philosophy, one that follows the frame of an anti-foundationalism rather than following the peculiar details of some a priori transcendental deductions that allegedly once and for all ground the exact sciences or their central concepts. Thus, according to Weyl, adopting a broad Fichtean framework is compatible with acknowledging the particular methods and empirical findings in the exact sciences. (Besides, this also explains Weyl’s reference to Fichte rather than to one of the other most eminent German Idealists. Schelling and Hegel tend to be much more patronising and prescriptive when it comes to concrete practices and findings in the natural sciences, so that one would have many more problems reading them in this anti-foundationalist fashion.)
21 To avoid misunderstanding Weyl’s distinction between an intuitionist- phenomenological and a formalist-constructivist approach, it should be added that there are, of course, differences between Brouwer and Husserl. In particular, given the importance of constructibility in intuitionism one might even argue that intuitionism would belong on the side of active construction rather than passive viewing. And so, as already mentioned, Weyl’s distinction here must be viewed against his historical background. For it is rather the notion of “intuition” (Anschauung) that is important for him and from which the association of intuitionism and phenomenology appears plausible.
22 Indeed, Husserl himself never engaged in a detailed discussion of intuition-ism, though it is documented that he met Brouwer in Amsterdam in 1928 and was very much impressed by him [van Atten 2004, 205]. Interestingly enough, with respect to his Philosophie der Arithmetik and his notion of “epitome” (Inbegriff) therein [Husserl 1891, 74-79], Husserl can be seen as being near to classical mathematics and to logicism [van Atten 2005, 113]. On the other hand, within recent philosophy of mathematics also a strong constructivist reading of Husserl has been proposed, particularly by Tieszen [Tieszen 2005, passim]. However, even here Weyl’s distinction between different philosophical leanings remains significant. For Tieszen’s phenomenological approach moves towards intuitionism at times, for instance, when he (like Brouwer) argues for a primacy of ordinal numbers over cardinal numbers [Tieszen 1989, 105]. This is opposed to what Husserl himself does in his Philosophie der Arithmetik, but seems in line with Husserl’s later investigations on time-consciousness, where a sequential (ordered) structure lies at the foundation of all mental acts.
23 Thus, despite minor discrepancies, Weyl’s distinction certainly still has a point to it. By hinting at the problem of passive viewing, Weyl very early on put a finger on a sore spot in Husserl’s system. Weyl’s critique, according to which it is not enough merely to “open our eyes wide” and to “see”, dates from a time when none of Husserl’s writings on passivity has yet been published. That later on Husserl deals with this topic so intensively in his Analysen zur passiven Synthesis [Husserl 1966], his Formale und
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transzendentale Logik [Husserl 1929] and also in Erfahrung und Urteil [Husserl 1939] shows how delicate it became for his phenomenology.
24 Oskar Becker also argued for a similar distinction of philosophical approaches in his 1927 work, Mathematische Existenz. Unlike his teacher Husserl, Becker was engaged in the mathematical details of intuitionism, such as choice sequences. He claimed that intuitionism and Husserlian phenomenology have common roots in the Kantian notion of intuition, whereas formalism is said to have Leibniz as its philosophical ancestor [Becker 1927, 714-747; Becker 192633, passim]. Not only does Becker explicitly mention Weyl as a defender of formalism, but also emphasises that Weyl’s own interest in Leibniz originated from reading Fichte [Becker 1927, 726-727, 768]. Unfortunately, Becker does not develop this any further. But from the above statements, Weyl’s reliance upon Fichte should now be obvious. Indeed, Weyl himself also explicitly states his high regard for Leibniz and, like Becker, closely relates him to Hilbertian formalism [Weyl 1925, 540]. For completeness it should be added that Fichte too thought highly of Leibniz, referred to Leibniz’ concept of intuition [Fichte 1834, 10] and even called him “the only ever convinced person (der einige Überzeugte) in the history of philosophy” [Fichte 1797/98, 96].
25 Though Becker’s distinction is similar to Weyl’s (one might even speculate whether Becker might have taken it over from Weyl), their alliances are completely different. In opposition to Weyl, the phenomenologist Becker expected that formalism would be defeated soon. Becker’s own critique on Hilbert, however, is not the issue here and has already been critically examined elsewhere [Peckhaus 2005].
4 Conclusion
26 Let me briefly summarise my sketch of Weyl’s philosophy of mathematics from about 1918 to 1928. At the beginning of this period, by relying on both Fichte and Husserl, Weyl refrained from the logicist attempt to reduce mathematics to logic and set theory. Later on, after his brief alliance with Brouwer in 1921, Weyl distinguished between an intuitionistic-phenomenological and a formalistic-constructivist approach. At that time, Weyl came to believe in the superiority of Fichtean constructivism over the passive Husserlian viewing of essences, which seems not to do justice to the creativity of work done in mathematics and theoretical physics.
27 Thus, this paper has shown how Weyl’s intellectual development and his claims about phenomenology, intuitionism, constructivism and formalism followed from certain historical constellations. Arguably, some of these claims are plausible more generally or at least are of particular interest insofar as they are not results of some detached “arm-chair philosophy” but rather originated from Weyl’s actual scientific work in mathematics and physics. Having said that, whether Weyl’s interpretation of Fichte and Husserl is tenable in detail from the perspective of contemporary philosophical scholarship has not been my main interest here (though I have mentioned some recent literature supporting this reading). To be sure, it never seemed to be Weyl’s aim to become an expert Fichte or Husserl scholar. Yet, especially in the case of Fichte, Weyl’s work may help to establish a possible and consistent reading which is of interest to the philosophy of science, an area in which so far Fichte has been neglected nearly completely (except perhaps for such works as [Heidelberger 1998]).
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Acknowledgements I am grateful to the audience at the 17th Novembertagung (Edinburgh, 2006), to two anonymous referees, and in particular to Peter Pesic.
BIBLIOGRAPHY
BECKER, OSKAR — 1927 Mathematische Existenz. Untersuchungen zur Logik und Ontologie mathematischer Phänomene, Jahrbuch für Philosophie und phänomenologische Forschung, 8, 440-809. — 1926-33 Briefwechsel mit Dietrich Mahnke, Oskar Becker und die Philosophie der Mathematik, V. Peckhaus (ed.), 245-278, Munich: Wilhelm Fink, 2005.
BROUWER, LÜITZEN EGBERTÜS JAN — 1912 Intuitionism and Formalism, L. E. J. Brouwer: Collected Works I, A. Heyting (ed.), 123-138, Amsterdam/Oxford: North-Holland Publishing Company, 1975.
FICHTE, JOHANN GOTTLIEB — 1794a Über den Begriff der Wissenschaftslehre oder der sogenannten Philosophie, Stuttgart: Reclam, 1991. — 1794b Grundlage der gesamten Wissenschaftslehre, Hamburg: Meiner, 1997. — 1795 Grundriß des Eigentümlichen der Wissenschaftslehre, Hamburg: Meiner, 1995. — 1797/98 Versuch einer neuen Darstellung der Wissenschaftslehre, Hamburg: Meiner, 1984. — 1834 Transzendentale Logik, Leipzig: Meiner, 1922.
HEIDELBERGER, MICHAEL — 1998 Die Erweiterung der Wirklichkeit im Experiment, Experimental Essays - Versuche zum Experiment, M. Heidelberger & F. Steinle (eds.), 71-92, Baden-Baden: Nomos.
HENRICH, DIETER — 1967 Fichtes „Ich“, Selbstverhältnisse, D. Henrich (ed.), 57-82, Ditzingen: Reclam, 1993.
HILBERT, DAVID — 1928 Die Grundlagen der Mathematik, Leipzig: B. G. Teubner.
HUSSERL, EDMUND — 1891 Philosophie der Arithmetik, Den Haag: Martinus Nijhoff, 1970. — 1900/01 Logische Untersuchungen, 3 volumes, Tübingen: Niemeyer, 1993. — 1929 Formale und transzendentale Logik, Hamburg: Meiner, 1992. — 1939 Erfahrung und Urteil, Prag: Academia. — 1966 Analysen zur passiven Synthesis, Den Haag: Martinus Nijhoff.
MANCOSU, PAOLO & RYCKMAN, THOMAS — 2002 Mathematics and Phenomenology: The Correspondence between O. Becker and H. Weyl, Philosophia Mathematica, 10, 130-202.
PECKHAUS, VOLKER — 2005 Impliziert Widerspruchsfreiheit Existenz? Oskar Beckers Kritik am formalistischen
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Existenzbegriff, Oskar Becker und die Philosophie der Mathematik, V. Peckhaus (ed.), 79-99, Munich: Wilhelm Fink.
RADBRUCH, KNUT — 2003 Die Bedeutung der Mathematik für die Philosophie bei Fichte, Fichte Studien, 22, 251-263.
RAMSEY, FRANK P. — 1990 Philosophical Papers: F. P. Ramsey, D. H. Mellor (ed.), Cambridge: Cambridge University Press. — 1925 The Foundations of Mathematics, Proceedings of the London Mathematical Society (ser.2), 25, 338-84. Cited according to [Ramsey 1990, 164-224].
ROCKMORE, TOM — 1995 Fichtean Circularity, Antifoundationalism, and Groundless System, Idealistic Studies, 25, 107-124.
RYCKMAN, THOMAS A. — 2005 The Reign of Relativity: Philosophy in Physics 1915-1925, Oxford: Oxford University Press.
SCHOLZ, ERHARD — 2001 Weyls Infinitesimalgeometrie 1917-1925, Hermann Weyl’s ”Raum-Zeit-Materie“ and a General Introduction to His Scientific Work, E. Scholz (ed.), 48-104, Basel: Birkhäuser. — 2004 Hermann Weyl’s Analysis of the ’Problem of Space’ and the Origin of Gauge Structures, Science in Context, 17, 165-197.
SELLARS, WILFRID — 1956 Empiricism and the Philosophy of Mind, Cambridge (Mass.): Harvard University Press, 1997.
SIEROKA, NORMAN — 2007 Weyl’s “Agens Theory” of Matter and the Zurich Fichte, Studies in History and Philosophy of Science, 38, 84-107.
SIGURDSSON, SKÜLI — 1991 Hermann Weyl, Mathematics and Physics, 1900-1927, PhD Thesis, Harvard University, Department of the History of Science.
TIESZEN, RICHARD — 1989 Mathematical Intuition, Dordrecht: Kluwer. — 2005 Phenomenology, Logic, and the Philosophy of Mathematics, New York: Cambridge University Press.
VAN ATTEN, MARK — 2004 Intuitionistic Remarks on Husserl’s Analysis of Finite Number in the “Philosophy of Arithmetic”, Graduate Faculty Philosophy Journal, 25, 205-225. — 2005 Phenomenology’s Reception of Brouwer’s Choice Sequences, Oskar Becker und die Philosophie der Mathematik, V. Peckhaus (ed.), 101-117, Munich: Wilhelm Fink.
WEYL, HERMANN — 1968 Gesammelte Abhandlungen, 4 volumes, Berlin: Springer. — 1918 Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis, Leipzig: Veit & Comp. — 1919 Der circulus vitiosus in der heutigen Begründung der Analysis, Jahresbericht der Deutschen Mathematikervereinigung, 28, 85-92. Cited according to [Weyl 1968, II, 43-50]. — 1921 Über die neue Grundlagenkrise der Mathematik, Mathematische Zeitschrift, 10, 39-79. Cited according to [Weyl 1968, II, 143-180]. — 1922 R. Carnap: Der Raum. Ein Beitrag zur Wissenschaftslehre. Buchrezension, Jahrbuch über die
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Fortschritte der Mathematik, 48, 631-632. — 1925 Die heutige Erkenntnislage in der Mathematik, Symposion, 1, 1-23. Cited according to [Weyl 1968, II, 511-542]. — 1928 Diskussionsbemerkungen zu dem zweiten Hilbertschen Vortrag über die Grundlagen der Mathematik, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, 6, 86-88. Cited according to [Weyl 1968, III, 147-149]. — 1946 Eine Ansprache [zum Geburtstag von Fritz Medicus], Zürich: ETH-Bibliothek, Archive, Hs 91a: 16. — 1948 In memoriam Helene Weyl, Arnold Zweig, Beatrice Zweig, Helene Weyl: Komm her, wir lieben Dich. Briefe einer ungewöhnlichen Freundschaft zu dritt, I. Lange (ed.), 379-390, Berlin: Aufbau-Verlag, 1996. — 1949a Philosophy of Mathematics and Natural Science, Princeton, Princeton University Press. — 1949b Wissenschaft als symbolische Konstruktion des Menschen, Eranos-Jahrbuch, 1948, 375-431. Cited according to [Weyl 1968, IV, 289-345]. — 1951 A Half-Century of Mathematics, The American Mathematical Monthly, 523-553. Cited according to [Weyl 1968, IV, 464-494]. — 1954 Erkenntnis und Besinnung (Ein Lebensrückblick), Studia Philosophica, Jahrbuch der Schweizerischen Philosophischen Gesellschaft / Annuaire de la Société Suisse de Philosophie. Cited according to [Weyl 1968, IV, 631-649]. — (undated I) Notizen zu Fichte, Zürich: ETH-Bibliothek, Archive, Hs 91a: 75. — (undated II) Notizen über Symbolismus: Praxis, Theorie und Magie der Zahlen, Zürich: ETH- Bibliothek, Archive, Hs 91a: 22.
WITTGENSTEIN, LUDWIG — 1977 Culture and Value, G. H. von Wright (ed.), Chicago: University of Chicago Press.
ABSTRACTS
Around 1918 Hermann Weyl resisted the logicists’ attempt to reduce mathematics to logic and set theory. His philosophical points of reference were Husserl and Fichte. In the 1920s, Weyl distinguished between the position of these two philosophers and separated the conceptual affinity between intuitionism and phenomenology from the affinity between formalism and constructivism. Not long after Weyl had done so, Oskar Becker adopted a similar distinction. In contrast to the phenomenologist Becker, however, Weyl assumed the superiority of active Fichtean constructivism over the passive Husserlian view of essences. The present paper discusses this development in Weyl’s thought. Though not all of Weyl’s claims about Husserl and Fichte can be maintained in detail, I will argue for the general plausibility of Weyl’s distinction.
Vers 1918 Hermann Weyl abandonnait le logicisme et donc la tentative de réduire les mathématiques à la logique et la théorie des ensembles. Au niveau philosophique, ses points de référence furent ensuite Husserl et Fichte. Dans les années 1920 il distingua leurs positions, entre une direction intuitionniste-phénoménologique d’un côté, et formaliste-constructiviste de l’autre. Peu après Weyl, Oskar Becker adopta une distinction similaire. Mais à la différence du phénoménologue Becker, Weyl considérait l’approche active du constructivisme de Fichte comme supérieure à la vision d’essences passive de Husserl. Dans cet essai, je montre ce développement dans la pensée de Weyl. Bien que certains points dans sa réception de Husserl et Fichte ne soient pas soutenables, je vais argumenter en faveur de la distinction fondamentale mentionnée ci-dessus.
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AUTHOR
NORMAN SIEROKA ETH Zürich
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Les journaux de mathématiques dans la première moitié du XIXe siècle en Europe1
Norbert Verdier
1 Nous ne revenons pas ici sur l’intérêt que représente l’étude des journaux pour faire de l’histoire, de l’histoire des mathématiques plus particulièrement. Un journal, par ses contenus, ses réseaux d’auteurs et sa périodicité, est un marqueur du temps qui passe sur une période donnée. Jean Dhombres a pointé et synthétisé les éléments essentiels qui montrent la pertinence qu’il y a à se focaliser sur l’étude d’un journal [Dhombres 1994]. Son article introduit une série d’études consacrées à certains journaux pris chacun comme une entité.
2 Notre étude ici n’est pas focalisée sur un journal mais sur les premiers journaux de mathématiques, à l’échelle européenne, car l’étude de journaux est un objet approprié pour cerner au plus près les dynamiques qui se façonnent, les champs d’études qui finissent par s’instaurer et les réseaux d’auteurs qui ne cessent de se croiser et s’entrecroiser.
3 Plusieurs études ont déjà été consacrées à ce sujet [Ausejo & Hormigon 1993] et [Goldstein, Gray & Ritter 1996]. Ces études sont essentiellement constituées d’une juxtaposition d’études isolées associées à tel ou tel journal. La plupart du temps, les liens avec les autres journaux ne sont pas mentionnés. Notre objectif ici est autre. Il consiste, en premier lieu, à étudier sinon l’interdépendance au moins les relations qui unissent ou désunissent les premiers journaux de mathématiques. Parallèlement à cette étude, nous nous livrons à une étude bibliographique pour compléter ou réactualiser les bibliographies accompagnant les travaux cités. Notre étude de l’offre éditoriale sur le plan des mathématiques suit un ordre chronologique. En nous appuyant sur des éléments d’archives qui n’ont pas encore été exploités, nous commençons par étudier le regard que portait Gergonne – le fondateur principal du premier journal (français) de mathématiques – sur la presse mathématique de son temps. Dans un deuxième temps, nous confrontons les propos de Gergonne au paysage éditorial qu’il dépeint, un paysage éditorial dont il se voit l’inspirateur. Dans un dernier temps, nous étudions la
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structuration progressive de l’espace éditorial au cours des années trente et quarante en France, en Allemagne et en Angleterre.
1825 : le regard de Gergonne sur la presse de son temps
4 Gergonne est le (co)-fondateur, avec Joseph-Esprit Thomas Lavernède [Gérini 2002, 116], du premier journal de mathématiques français2 exclusivement dédié aux mathématiques : les Annales de mathématiques pures et appliquées. Il a été fondé en 1810, non pas à Paris, mais dans une ville moyenne du sud de la France, Nîmes. Dans le prospectus de lancement, il est déclaré : C’est une singularité assez digne de remarquer que, tandis qu’il existe une multitude de journaux relatifs à la Politique, à la Jurisprudence, à l’Agriculture, au Commerce, aux Sciences Physiques et naturelles, aux Lettres et aux Arts, les sciences exactes, cultivées aujourd’hui si universellement et avec tant de succès, ne comptent pas encore un seul recueil périodique qui leur soit spécialement consacré. [Gergonne & Lavernède 1810-1811]
5 Dans une note de bas de page, Gergonne explique comment il démarque sa publication des deux publications de l’École polytechnique que sont la Correspondance de Hachette et le Journal de l’Ecole polytechnique : On ne saurait, en effet, considérer comme tels, le Journal de l’école Polytechnique, non plus que la Correspondance que rédige M. Hachette : recueils très précieux sans doute, mais qui, outre qu’ils ne paraissent qu’à des époques peu rapprochées, sont consacrés presque exclusivement aux travaux d’un seul établissement. [Gergonne & Lavernède 1810-1811]
6 Les affirmations de Gergonne sont confirmées par les études de ces journaux effectuées par Loic Lamy [Lamy 1995], pour le Journal de l’Ecole polytechnique, et par Brigitte Dody, pour la Correspondance de Hachette [Dody 1994]. Ce sont deux journaux institutionnels au service d’une école : l’École polytechnique.
7 Les Annales de mathématiques pures et appliquées ont été analysées selon des voies différentes et complémentaires par Jean Dhombres, Mario H. Otero et Christian Gérini : les deux premiers se sont attachés à démontrer que les Annales sont le journal d’un homme seul au profit d’une communauté enseignante [Dhombres & Otero 1993], alors que Christian Gérini s’est focalisé sur les apports scientifiques et épistémologiques des Annales [Gérini 2002]. Nous allons nous focaliser sur un document inédit (la correspondance entre Gergonne et un de ses auteurs, Talbot) pour éclairer le contexte éditorial des années vingt.
8 Cette correspondance est constituée de cinq lettres adressées de Gergonne à Talbot ; elles sont archivées au Fox Talbot Museum (FTM) en Angleterre dans la « Lacock Abbey Collection ». Ces lettres nous intéressent ici pour deux raisons : elles éclairent la façon dont Gergonne gérait sa publication et elles offrent un point d’entrée sur la presse mathématique du temps de Gergonne. La première lettre n’est pas datée (mais est antérieure à 1822) ; elle indique quel rôle jouait Talbot à l’égard des Annales de Gergonne : J’ai l’honneur de faire agréer à Monsieur W.H. Talbot mes sincères remerciements pour ses offres gracieuses. Le seul service qu’il puisse me rendre en Angleterre est de me tenir au courant, lors qu’il en aura le loisir, des découvertes mathématiques qu’on pourra y faire, et de vouloir bien continuer à m’adresser les résultats de ses propres recherches. [Gergonne, FTM]
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9 Talbot joue le rôle d’un « passeur de sciences » entre l’Angleterre et la France. Il publie directement des articles dans les Annales, en répondant le plus souvent à des questions posées [Talbot 1822-1823a, 1822-1823b, 1823-1824a, 1823-1824b, 1823-1824c & 1823-1824d] ou propose à Gergonne des contributions d’autres auteurs. Une lettre, datée du 31 mai 1824, indique ainsi un refus de la part de Gergonne : J’aurais bien désiré pouvoir dire un peu de bien du petit ouvrage que Monsieur W. H. Talbot a eu la bonté de m’adresser. J’aurais même fait un article de mon recueil à son sujet, pour peu qu’il en eût valu la peine. Il faut que d’abord l’auteur, fort jeune sans doute, ne sache pas bien positivement en quoi consiste le problème de la résolution générale des équations algébriques ; et la faute en serait à ceux qui l’ont enseigné. Il faut ensuite qu’il n’ait rien lu de ce qui a été écrit sur la résolution des équations numériques. La méthode de M. Budan3 assez peu estimée en France, me semble, en particulier, infiniment supérieure à la sienne. [Gergonne FTM, 1824]
10 La fin de la lettre est un peu plus encourageante : J’éprouve beaucoup de plaisir à faire connaître en France ce qui se fait de bon au dehors ; je désire, en conséquence que Monsieur Talbot puisse bientôt m’offrire [sic] l’occasion de me dédommager de l’espèce de désappointement que j’ai éprouvé en parcourant ce petit écrit ; et je le prie de vouloir bien agréer l’assurance de toute mon estime et de tout mon dévouement. [Gergonne FTM, 1824]
11 Après cette date, Talbot ne publie plus dans les Annales de Gergonne. Pourtant les relations entre les deux hommes restent, semble-t-il, franches et courtoises. Une autre lettre de Gergonne à Talbot, datée du 16 décembre 1826, va constituer le fil directeur de cet article. Gergonne écrit de Montpellier à Talbot qui réside alors à Paris à l’hôtel Maurice (rue Saint Honoré) : Depuis l’interruption de nos relations, vous aurez sans doute remarqué, Monsieur, la naissance de deux recueils à l’imitation du mien : l’un est la Correspondance publiée à Bruxelles par MM. Quetelet et Garnier, et dans laquelle ce dernier m’a souvent copié textuellement sans me citer. Ils ont là parmi leurs rédact [ sic] collaborateurs un M. Dandelin qui a du mérite. [Gergonne FTM, 1826]
12 Gergonne poursuit sa lettre en évoquant ce qui se passe outre Rhin : L’autre recueil est celui que M. Crelle publie en allemand à Berlin. Je viens d’en recevoir les trois premiers cahiers dont je n’ai encore lu que la table des matières. M. Schmidten y a reproduit ce me semble un mémoire qu’il avait déjà publié dans mes annales; j’y ai reconnu aussi un mémoire de M. Louis Olivier sur la résolution générale des équations dont j’ai en main le manuscrit en français depuis plus de deux ans, et que j’ai refusé dans le tems de publier, parce que cela ne mène à rien et ne saurait franchir la barrière qu’on rencontre au delà du 4em degré. [Gergonne FTM, 1826]
13 Gergonne, dans sa lettre, ne fait pas que pointer l’existant. Il pointe également une carence éditoriale au niveau de l’Angleterre. Il questionne ainsi Talbot sur la situation outre-Manche : Il est surprenant que, dans un pays aussi éclairé que votre Angleterre l’introduction des écrits relatifs aux sciences éprouve tant de gêne et d’entraves. Que l’on en use ainsi pour les objets de mode, à la bonne heure ; la difficulté de se les procurer ne fera même qu’en rehausser le prix ; mais il devrait en être tout autrement de tout ce qui tend à perfectionner l’intelligence : Les Ministres viennent d’obtenir un bill d’indemnité pour l’introduction des céréales ; mais non in solo pane vivit homo ; et ce qui nourrit l’esprit ne devrait pas être moins favorablement traité que ce qui nourrit le corps. [Gergonne FTM, 1826].
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14 Gergonne a beau soutenir que l’homme ne vit pas que de pain (non in solo vivit homo) mais aussi de mathématiques, il n’est pas entendu. La première publication exclusivement réservée aux mathématiques tarde à venir en Angleterre.
La poussée éditoriale de 1825-1826
15 Nous allons dans ce qui suit étudier plus précisément le paysage éditorial dépeint par Gergonne. Il évoque deux journaux qu’il considère comme étant « à l’imitation du sien ». Qu’en est-il? Dans quelle mesure les autres journaux mathématiques se sont-ils inspirés des Annales de Gergonne ? Les Annales de Gergonne se sont-elles inspirées de ces nouvelles publications? Est-ce que le paysage éditorial de Gergonne est à compléter ? Autant de questions qui pilotent et structurent cette partie.
À Gand et à Bruxelles en 1825 avec la Correspondance mathématique et physique
16 En 1825, MM. Garnier « professeur de Mathématiques et d’astronomie à l’université de Gand » et Quetelet « Professeur de Mathématiques, de Physique et d’Astronomie à l’Athénée de Bruxelles », tous les deux « Membres de l’Académie royale des Sciences et Belles-Lettres de Bruxelles », lancent la Correspondance mathématique et physique. Les circonstances de la création du journal et les relations entre Quetelet et Garnier sont bien connues grâce aux travaux de Droesbeke et d’Elkhadem [Droesbeke 2005], [Elkhadem 1978].
17 Dans le prospectus de lancement [Garnier & Quételet 1825], les auteurs justifient leur démarche éditoriale. Ils partent du constat que dans le royaume4 n’existe aucun « recueil consacré aux sciences Mathématiques et Physiques », un recueil « qui permette à ceux qui les cultivent, d’établir entre eux un commerce scientifique, et qui, entre autres avantages, offre celui de garantir à chacun la propriété et la prompte publicité des résultats de ses recherches ». Ensuite, avec précaution, ils expriment leurs intentions : « Le Journal dont nous hazardons le 1er numéro 5 offrira, quant aux Mathématiques, deux grandes divisions : l’une consacrée aux Mathématiques élémentaires et transcendantes pures et appliquées ; l’autre à la revue des travaux scientifiques » [Garnier & Quételet 1825].
18 Il s’agit d’un journal à visée nationale. Le mot « Royaume » revient trois fois dans le prospectus de lancement et une des rubriques du Journal est constituée d’« éloges des Géomètres tant anciens que modernes qui ont honoré notre pays » [Garnier & Quételet 1825]. La liste des souscripteurs insérée dans le premier tome est aussi assez significative : sur une centaine de souscripteurs (99), tous sont du Royaume (La Haye, Bruxelles, Utrecht, Leyden, Groningen, Liège, Louvain, Gand, Delft, Tournay, Anvers, Bruges, Luxembourg, Na-mur, Francker, Maestricht, Gand, Charleroy, Nevele, Amsterdam, St Josse ten Noode, Bergen-op-Zoom, Deventer, Vilvorde), sauf quatre souscripteurs français : Delezenne, Gergonne, le baron de Férussac et Hachette. La liste de MM. Les Abonnés, une liste qui précise-t-on doit être « continuée dans les nos sui- vans » (mais qui ne l’est pas) est aussi significative. Sur 64 abonnés, tous sont du Royaume.
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19 Gergonne reproche à Garnier de le copier sans le citer. En fait, Garnier écrit de nombreuses notes qu’il extrait d’autres publications. Dans le corps du texte, Garnier ne précise pas toujours la source (comme cela se fait très couramment à l’époque) mais dans la table des matières, il le fait, sans être précis sur la référence de l’article. Il se contente de citer la publication d’origine, les « Annales de Nîmes » pour les Annales de Gergonne, par exemple. Pour la seule année 1825, Garnier extrait deux articles de géométrie, une « note sur les polyèdres » et une note « sur la division d’une ligne droite en un nombre quelconque de parties égales » des « Annales de Nîmes » [Correspondance mathématique et physique I, 1825, 64-66; 113-114]6. Ce sont des notes extraites d’articles de Gergonne lui-même [Gergonne 1824-1825a], [Gergonne 1824-1825b et c]. Sont également publiés deux extraits de lettres, à contenu mathématique, de Gergonne à Quetelet [Correspondance mathématique et physique, I, 1825, 149; 268-270].
20 De son côté, Gergonne publie dans ses Annales ce « M. Dandelin qui a du mérite ». En 1824-1825, il publie les « Recherches nouvelles sur les sections du cône et sur les hexagones inscrits et circonscrits à ces sections » [Dandelin 18241825]. En 1825-1826, il publie ses « Usages de la projection stéréographique en géométrie » [Dandelin 1825-1826]. Gergonne précise en fin d’article : Quant à nous, quelque plaisir que nous ayons à faire connaître à nos lecteurs les élégantes recherches de M. Dandelin, nous désirons sincèrement que le recueil périodique dans lequel il en consigne les résultats soit assez répandu en France pour que nous puissions nous dispenser à l’avenir de les entretenir des choses intéressantes qui s’y trouvent contenues, et nous borner à en faire à nous-même notre profit. [Dandelin 1825-1826, 327]
21 En effet, la même année dans la Correspondance, Quetelet extrait également deux articles de Dandelin à partir d’une de ses publications académiques [Correspondance mathématique et physique, I, 1825, 256-264; 316-322]. Les deux articles de Dandelin constituent à eux seuls toute la section de « Géométrie » du journal. Les propos de Gergonne interpellent sur la diffusion de la Correspondance en France. L’examen des catalogues des bibliothèques universitaires va dans le sens souligné par Gergonne : la Correspondance était sans doute très peu diffusée en France7.
À Berlin en 1826 avec le Journal für die reine und angewandte Mathematik
22 La Correspondance provoque un réflexe identitaire à Berlin chez Crelle, dont le parcours a été étudié par Jean-Pierre Friedelmeyer [Friedelmeyer 1995, 1998]. Crelle, qui a déjà participé en tant qu’auteur aux Annales de Gergonne, lance, en janvier 1826, son Journal für die reine und angewandte Mathematik. Il se positionne par rapport aux deux journaux de langue française : « Depuis 16 ans existe sans interruption en français un journal mathématique : « Les Annales de mathématiques pures et appliquées », ouvrage périodique rédigé par M. Gergonne à Montpellier, et un autre à Bruxelles est en voie de naître » [Crelle 1826]. Les intentions de Crelle sont clairement affichées : Comme donc un Journal est dans les faits un moyen très efficace pour développer une science et la diffuser, pour la fermer aux influences étrangères, et la protéger des sujétions, des modes, des autorités, des écoles, des respects et la conserver dans le domaine libre de la pensée, il vaut la peine d’essayer s’il est possible de donner vie et croissance à une telle publication en langue allemande pour les mathématiques. [Crelle 1826]
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23 Crelle donne effectivement « vie et croissance » à un journal de mathématiques mais les intentions éditoriales affichées, mettant en avant une idée de repli sur la Prusse, sont allègrement bousculées. Beaucoup de mémoires sont publiés en français ou en latin (d’autres encore, dans une moindre mesure, en anglais). Les décomptes effectués par Crelle dans son bilan de 1855 [Crelle 1855] sont sans appel : sur 29 ans et 8 mois de publications environ deux tiers des feuilles sont en allemand. Sur le tiers publié dans une autre langue, les deux tiers sont en français, presque un tiers en latin, et le reste (quantitativement négligeable), en anglais ou en italien.
24 Ainsi, plusieurs Français publient des mémoires dans le Journal de Crelle. Sur la première décennie du Journal, entre 1826 et 1835, nous dénombrons une quarantaine de contributions dues à Clapeyron (1), Cournot (2), Lamé (1), Libri (10), Liouville (7), Navier (1), Poisson (9) et Poncelet (7). Elles sont essentiellement de Libri, Liouville, Poisson et Poncelet. D’une certaine façon, et dans une mesure qu’il conviendrait de nuancer, le Journal de Crelle est une sorte de caisse de résonance de ce qui s’écrit à Paris ou plus exactement de ce qui s’est écrit à Paris (car parmi ces contributions beaucoup sont des reprises de textes académiques anciens déjà publiés à Paris). Nous reviendrons dans la prochaine partie sur les contributions de Liouville et en fin de ce paragraphe sur les contributions de Poncelet.
25 Gergonne, dans sa lettre, faisait référence à un mémoire de Schmidten publié dans le premier tome du Journal de Crelle. Schmidten y est présenté comme « weiland Professor der Mathematik zu Copenhagen »8 [Schmidten 1826]. Gergonne a effectivement déjà publié cet auteur, à trois reprises [Schmidten 1820-1821, 1821-1822, 1822-1823]. L’article publié dans le Journal de Crelle est une reprise partielle du premier article publié dans les Annales de Gergonne. Cet article a eu une certaine réception puisqu’il est cité dans un autre mémoire anonyme dû à un certain RMS, quelques années plus tard, dans le Journal de Gergonne [RMS 1829-1830]. Schmidten résidait probablement en France en cette période, puisque son dernier article est signé « Plombières, 24 juillet 1822 » [Schmidten 1822-1823, 131]. Après cet intermède dans les Annales de Gergonne, Schmidten, dont certains des apports ont été étudiés par Dominique Tournès dans sa thèse [Tournès 1996, 245-304], publie deux articles dans le Journal de Crelle. Gergonne évoque également le mémoire de Louis Olivier qu’il a refusé. Il semblerait donc que Louis Olivier ait essayé d’être publié dans les Annales de Gergonne avant, lui aussi, de publier (assez massivement) dans le Journal de Crelle. Il y publie douze mémoires au total. Ces contributions d’Olivier ont été étudiées par Henrik Kragh Sorensen [Sorensen 2006]. Nous ne disposons, malgré de nombreuses recherches dont certaines sont encore en cours, d’aucune information biographique concernant Louis Olivier.
26 Dans le Journal de Crelle, quelques articles mentionnent explicitement les Annales de Gergonne. Dans le tome XVII, en 1826-1827, des Annales de Ger-gonne est posée une question de géométrie descriptive : « Construire rigoureusement la droite qui coupe à la fois quatre droites données dans l’espace, non comprises deux à deux dans un même plan » [Annales de mathématiques pures et appliquées, 17, 1826-1827, 83]9. L’année suivante une « solution du problème de géométrie descriptive énoncé à la page 83 du présent volume » est publiée [Annales de mathématiques pures et appliquées, 18, 1827-1828, 182184]. En fait, il s’agit de la solution de Bobillier alors « professeur à l’École des arts et métiers de Châlons-sur-Marne » et de Garbinski « professeur à Varsovie ». Steiner, en 1827, fait paraître dans le Journal de Crelle son article intitulé « Auflösung einer Aufgabe aus den Annalen der Mathematik von Herrn Gergonne » [Steiner 1827].
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27 Un second exemple de va-et-vient entre les Annales de Gergonne et le Journal de Crelle mérite d’être cité. Dans le tome XIX des Annales de Gergonne (1828-1829) [Annales de mathématiques pures et appliquées, 19, 1828-1829, 256] sont proposés des théorèmes d’arithmétique dont : « Tout nombre entier est diviseur d’un nombre exprimé par une suite de 9 suivis de plusieurs zéros » [Annales de mathématiques pures et appliquées, 19, 1828-1829, 256]. Dans le tome suivant, est apportée une réponse « par un abonné » sous forme d’une lettre datée de « Lyon, le 17 novembre 1829 » [Annales de mathématiques pures et appliquées, 20, 1829-1830, 304-305]. Une note de bas de page de Gergonne est rédigée : À la page 185 du Vme volume du Journal de M. Crelle, on trouve une démonstration de ce théorème qui, bien qu’assez brève, est beaucoup moins élémentaire que celle qu’on vient de lire ; mais à la page 296 du même volume, on en trouve une autre démonstration directe fort simple, ainsi que celle d’un théorème plus général. [Annales de mathématiques pures et appliquées, 20, 18291830, 304-305]
28 En effet, J.A. Grünert, présenté comme « prof. des math. à Brandenbourg » présente dans son article de 1830 une « démonstration d’un théorème d’arithmétique proposé dans les annales de mathématiques de Mr. Gergonne, tom. XIX. p. 256 » [Grünert 1830]. Dans le même tome, une synthèse intitulée « Deux théorèmes sur les nombres » [Journal für die reine und angewandte Mathematik, 5, 1830, 296]10 est présentée. Une démonstration « par un abonné » élémentaire et concise est proposée ; l’éditeur (Crelle) poursuit en proposant un théorème plus général : « Tout nombre entier est diviseur d’un nombre exprimé par une suite de périodes de chiffres donnés, suivis de plusieurs zéros » [Journal für die reine und angewandte Mathematik, 5, 1830, 296]. Gergonne reprend la réponse de Crelle pour en faire un article qu’il attribue à Crelle [Crelle 1829-1830].
29 Nous terminons par la publication des textes de Poncelet car elle éclaire sur la personnalité, en tant que patron de presse, de Crelle. Sur les sept contributions de Poncelet, les deux premières sont extraites de ses leçons de mécanique appliquée aux machines professées en 1825 et 1826 à l’école spéciale de l’artillerie et du génie de Metz11. Pour le premier mémoire, Crelle rédige en allemand une note indiquant diverses publications de Poncelet sur le sujet. Les deux mémoires suivants ont été préalablement présentés à l’Académie royale de l’institut de France en 1824. Des rapports dont le rapporteur était Cauchy ont été publiés. Le premier de ces mémoires est accompagné d’une note de l’éditeur et d’une note de l’auteur. La note de l’éditeur stipule que : « Ce mémoire n’a été publié jusqu’ici » [Poncelet 1828] ; la note de l’auteur éclaire sur les conditions de transmission du texte : « Ce mémoire, redigé dans le courant de l’année 1823, a été remis à Mr Arago vers la fin de Novembre, même année, lors de son séjour à Metz. On peut voir à la page 349 du Tome XVI des Annales de Mathématiques, publié par M. Gergonne, le rapport qui en a été fait à l’Académie royale des sciences, par M. Cauchy »12 [Poncelet 1828]. L’autre mémoire contient également une note du rédacteur qui éclaire le contexte : Il s’est élevé sur ce mémoire une discussion entre son Auteur et Mr. Gergonne, redacteur des annales de mathématiques à Montpellier, qu’on trouve dans le tome XVIII de ces annales. Le redacteur du présent recueil est absolument étranger à cette discussion, et il déclare hautement qu’il n’a pas le but d’y entrer d’aucune manière par la publication du mémoire en question que son respectable auteur lui a bien voulu confier. [Poncelet 1829]
30 À la différence de Gergonne, Crelle n’entre pas en polémique. Il s’efface derrière son Journal. La querelle entre Gergonne et Poncelet est une querelle fondée sur une
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différence de conception entre l’analytique pour Gergonne et le synthétique pour Poncelet. Elle est analysée dans [Dhombres & Otero 1993, 56-57] et surtout analysée, dans le texte, par Christian Gérini [Gérini 2002, 238-246]. Il faut toutefois noter que Crelle, implicitement, prend parti puisqu’il ne publie que la version de Poncelet. Ces deux mémoires de Poncelet présentés initialement à l’Académie seront développés dans le cinquième mémoire de Poncelet, un long mémoire de 124 pages publié en quatre fois au cours de l’année 1832. Son sixième mémoire provient lui aussi d’une lecture effectuée à l’Académie en 1833. Son dernier mémoire semble être un inédit.
31 Ces quelques exemples d’échanges entre les Annales de Gergonne et le Journal de Crelle mériteraient d’être approfondis. L’étude la plus complète du Journal de Crelle reste encore celle d’Eccarius, en 1976 [Eccarius 1976]. Elle informe sur le réseau des auteurs et sur les aspects matériels liés à la fabrication (Eccarius a eu accès aux archives de l’éditeur). Le traitement du contenu du Journal resterait à effectuer pour étudier les lignes de fond et leur(s) évolution(s).
À Vienne en 1826 avec le Zeitschrift für Physik und Mathematik
32 Gergonne, dans ses lettres, ne mentionne pas une autre publication qui, elle aussi, a puisé certaines de ses sources dans ses Annales, le Zeitschrift für Physik und Mathematik. Cette revue a été lancée le 15 mars 1826 par Andreas Baumgartner et Andreas Freiherr von Ettinghausen. Ils sont présentés sur la page de couverture du Journal comme étant « Ordentliche Professoren an der k.k.Universität zu Wien ».
33 Plus précisément, Andreas Baumgartner est né en 1793 à Friedberg (dans l’actuelle République Tchèque). En 1823, il est nommé professeur de physique et de mathématiques appliquées à l’université de Vienne13. Andreas Freiherr von Ettingshausen (1796-1878) est natif de Heidelberg. Il a étudié à Vienne la philosophie et le droit. En 1817, il est nommé adjoint de mathématiques et de physique à l’université de Vienne, puis, en 1819, professeur de physique à Innsbruck, avant de devenir professeur de mathématiques à l’université de Vienne en 1821. En 1826, la même année que l’année de lancement du Zeitschrift, il a aussi publié Die kombinatorische Analysis. Ettingshausen se situe à ce titre dans la lignée de C. F. Hindenburg (1741-1808), le bâtisseur de l’école combinatoire – une école particulièrement étudiée par Philippe Séguin [Séguin 2005] – et l’initiateur de plusieurs projets éditoriaux. Il avait lancé avec Jean Bernoulli de 1786 à 1788 le Leipziger Magazin für reine und angewandte Mathematik puis un peu plus tard, toujours à Leipzig, de 1794 à 1800, seul cette fois, les Archiv der reinen und angewandten Mathematik.
34 Le volume de lancement du Zeitschrift für Physik und Mathematik est constitué de quatre fascicules (Heft) de 120 pages environ. Chaque fascicule commence par une section de physique (« Physikalische Abtheilung ») occupant les trois quarts du fascicule environ et d’une section de mathématiques (« Mathematische Abtheilung ») occupant le quart restant. Le Journal est donc d’abord un journal de physique comme l’indique explicitement son titre en commençant par mentionner la physique avant les mathématiques.
35 Dans la section des mathématiques, presque tous les articles sont en fait extraits d’autres revues : deux proviennent des publications de la société philomatique, deux des Annales de Gergonne, un des Annales de chimie et de physique, un d’une publication italienne publiée à Modène en 1813 ou de manuels (deux proviennent des Exercices de
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mathématiques de Cauchy). Presque tous ces articles proviennent donc de publications françaises. Cette omniprésence française n’est pas aussi forte dans la section physique14.
36 En ce qui concerne les deux articles issus des Annales de Gergonne, nous notons la présence d’un article de Poisson. L’article en question est intitulé « Auflösung einiger Aufgaben aus dem Gebiete der WahrscheinlichkeitsRechnung » [Poisson 1826]. Dès la première page, est précisée la source : « Aus Poisson’s Mémoire sur l’avantage du banquier au jeu de trente et quarante. (Annales des Mathématiques pures et appliquées par Gergonne Tome 16. (1825-1826) pag. 173 etc.) » [Poisson 1826]. Il s’agit en effet d’une retranscription de l’article sur les probabilités, mais elle est classée dans la rubrique « analise appliquée » [Poisson 1825-1826].
37 Le second article est intitulé « Ueber ein Kennzeichen der Anwesenheit imaginärer Wurzeln in einer gegebenen Gleichung » [Zeitschrift für Physik und Mathematik, I, 1826, 379-382]15. Il est indiqué que l’article est extrait des « Annales de Mathématiques pures et appliquées par M. Gergonne. Tome 16. 1825-1826. p. 382. » sans mention de titre d’auteur. L’article provient en fait d’une question posée dans le tome 15, en 1824-1825, des Annales : Si, dans une équation de degré quelconque, les coefficiens p, q, r, s, de quatre termes consécutifs quelconques, pris avec leurs signes, sont tels qu’on ait (q2 – pr)(r2 – qs) < 0, cette équation aura nécessairement deux racines imaginaires au moins ; et si une pareille relation a lieu pour plusieurs séries de quatre termes consécutifs, l’équation aura autant de couples de racines imaginaires au moins qu’elle offrira de pareilles séries. [Annales de mathématiques pures et appliquées, 15, 1824-1825, 164]
38 Dans le numéro suivant des Annales, une réponse est apportée à cette question. L’article est ainsi intitulé : « Démonstration du théorème d’analise énoncé à la page 164 du précédent volume » [Annales de mathématiques pures et appliquées, 16, 1825-1826, 382-385]16. La dernière note de bas de page, signée Gergonne, figurant en fin d’article est intéressante pour comprendre la source de cette question reprise par le Zeitschrift : « Le théorème qui vient d’être démontré, et beaucoup d’autres du même genre, font le sujet d’un mémoire de M. de Lavernède, dont on trouve l’extrait dans le volume de l’Académie du Gard, pour 1809. J. D. G. » [Annales de mathématiques pures et appliquées, 16, 1825-1826, 382-385]. Cette note éclaire sur la provenance de certaines questions dans les Annales. En l’espèce, il ne s’agit pas d’une question ouverte dont on ne connaît pas la réponse mais d’une question bâtie à partir d’un texte source. En l’occurrence, il s’agit d’un texte de Lavernède – le co-fondateur des Annales de mathématiques pures et appliquées – publié, en 1810, dans les Notices des travaux de l’Académie du Gard pendant l’année 180917.
39 Notre étude a seulement porté ici sur le premier volume. Elle a permis de mesurer une certaine impulsion éditoriale. Il resterait à confirmer ou infirmer l’hypothèse selon laquelle la partie mathématique du Zeitschrift est essentiellement importée de journaux ou manuels français. Il resterait également à mesurer plus précisément l’impact de ce Journal qui semble avoir essentiellement été diffusé dans le monde germanophone. Il n’a apparemment pas été diffusé en France. Aucune bibliothèque ne le possède18. Gergonne était-il au courant des emprunts faits à Vienne de ses Annales ?
40 En France, la géographie éditoriale est bouleversée au cours des années trente. En 1832, Gergonne stoppe sa publication pour prendre de nouvelles responsabilités de recteur [Gérini 2008] :
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La ponctualité que nous avons apportée pendant vingt ans dans la publication de nos livraisons doit assez avertir nos lecteurs que, si elles ont éprouvé, dans ces derniers temps, un retard assez notable, il n’a pas tenu à nous qu’il en fût autrement. Chargés depuis dix mois de la direction de l’enseignement dans quatre départements, il nous a fallu faire trêve à nos douces et paisibles occupations pour nous livrer à de fastidieux détails administratifs. Présentement que la machine est à peu près montée, nous ne négligerons rien pour nous remettre bientôt au courant? [Gérini 2002, 528]19
41 L’arrêt des Annales de Gergonne en 1832, combiné avec la faillite du Bulletin de Férussac en 1831, qui n’était pas exclusivement destiné à la publication mathématique mais qui publiait de nombreux articles de mathématiques [Taton 1947], [Bru & Martin 2005], a créé une sorte de vide éditorial en France. Il ne restait plus désormais que le Journal de l’Ecole polytechnique, un journal que nous avons qualifié d’institutionnel dans la mesure où il n’ouvrait ses colonnes qu’aux auteurs « très liés » à l’École polytechnique.
1835-1845 : l’organisation de l’espace éditorial
42 En 1835, le paysage éditorial est considérablement modifié avec le lancement des Comptes rendus hebdomadaires de l’Académie des sciences. Le contexte de ce lancement a déjà été très largement étudié. Maurice Crosland a montré les règles de fonctionnement des Comptes rendus [Crosland 1992] et Bruno Bel-hoste a montré le rôle joué par Arago pour ouvrir les débats académiques vers l’extérieur [Belhoste 2006]. D’emblée, les Comptes rendus deviennent le lieu où un savant prend date en annonçant un résumé de ses travaux. C’est, pour reprendre une expression d’Hugues Chabot, un « tribunal pour la science » [Chabot 2000]. Pour pouvoir développer les notes publiées dans les Comptes rendus, il faut des journaux spécialisés. Il y avait les Annales de chimie et de physique ; il manquait un journal de mathématiques. C’est un jeune homme de 27 ans, très proche d’Arago, qui lance en 1836 le Journal de mathématiques pures et appliquées : Joseph Liouville.
À Paris en 1836 avec le Journal de mathématiques pures et appliquées
43 Avant d’être un rédacteur, Liouville a été un auteur qui a connu diverses désillusions éditoriales, expliquant, sans doute en partie, son parcours. À la fin des années vingt, Liouville est un jeune polytechnicien qui décide d’arrêter sa carrière d’ingénieur pour vivre des mathématiques (comme beaucoup d’autres avant lui), en les enseignant et en les faisant progresser. À la fin des années trente et tout au long des années quarante, il connaît une ascension institutionnelle fulgurante (à l’École polytechnique en devenant professeur de la très convoitée chaire d’analyse, à l’Académie des sciences en devenant membre, etc.). Il a déjà publié plusieurs mémoires avec des expériences mitigées.
44 En 1830-1831 – dans le cahier de novembre 1830 des Annales de Gergonne – paraît son second mémoire (publié). Son article, un extrait d’une cinquantaine de pages, concernant ses Recherches sur la théorie physico-mathématique de la chaleur [Liouville 1830-1831] est très fraîchement accueilli par Gergonne lui-même, par le biais d’une note adjointe à l’article.
45 Cette façon de faire était courante chez Gergonne, qui portait souvent un jugement sur les articles qu’il publiait. À propos de l’article de Liouville, Gergonne croit « devoir s’excuser, vis-à-vis du lecteur, de lui livrer un mémoire aussi maussadement, je puis
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même dire, aussi inintelligiblement rédigé » [Liouville 1830-1831, 181]. Il précise ensuite qu’il n’a pas vérifié scrupuleusement l’article car à ce moment là Gergonne est en fin de règne et tout à ses nouvelles activités rectorales. De Liouville, présenté comme « élève ingénieur des ponts et chaussées », il explique : « je crus trouver dans le double titre d’ingénieur et d’ancien élève de l’École polytechnique une garantie suffisante du talent de rédaction de l’auteur, et j’envoyai de suite l’ouvrage à l’impression » [Liouville 1830-1831], avant de poursuivre : « Je ne prétends contester aucunement la capacité mathématique de M. Liouville ; mais à quoi sert cette capacité, si elle n’est accompagnée de l’art de disposer, de l’art de se faire lire, entendre et goûter » [Liouville 1830-1831], et de terminer par un verdict sans appel : Je désire bien sincèrement que M. Liouville se venge prochainement des reproches un peu sévères peut-être que, bien à regret, sans doute, je me trouve contraint de lui adresser aujourd’hui, en publiant quelque Mémoire que l’on puisse lire à peu près comme on lit un roman ; mais la vérité est que je le désire beaucoup plus que je ne l’espère. Une longue expérience m’a prouvé que le mal dont il est atteint est un mal à peu près incurable. [Liouville 1830-1831]
46 Il connaît un autre type de désillusions avec ses publications dans le Journal de Crelle. Liouville, malgré les points critiques développés par Gergonne, était un auteur très scrupuleux quant à l’écriture de ses articles. Nous possédons deux brouillons de lettres écrites en 1833 et 1834 à Crelle. Elles sont reproduites par Erwin Neuenschwander [Neuenschwander, 4, 58-60]20. Dans une première lettre, Liouville, avant qu’un de ses articles ne paraisse, souhaite relire son texte pour éviter d’avoir à insérer un erratum comme pour un article précédent. Il s’agit donc d’une critique implicite quant à la façon de gérer un journal. Dans la seconde lettre, il demande une confirmation de la part de Crelle à propos de la réception de plusieurs mémoires. « On est vraiment effrayé de penser qu’il faut cinq mois à un petit paquet pour arriver de Paris à Berlin » [Liouville, BIF, MS 36 40, Liasse 1841]. L’éloignement du centre de décision (Berlin) est d’autant plus regrettable que Liouville tient à ce que son mémoire « soit imprimé promptement » [Liouville, BIF, MS 36 40, Liasse 1841] parce qu’il « renferme (…) la méthode la plus étendue qu’on ait publiée jusqu’ici pour l’intégration des fonctions explicites d’une seule variable » [Liouville, BIF, MS 36 40, Liasse 1841].
47 Fort de ses expériences avec les deux grands patrons de presse (mathématique) de son temps (Gergonne et Crelle), Liouville, en janvier 1836, lance son Journal de mathématiques pures et appliquées, il profite d’une certaine vacance éditoriale21 décrite précédemment.
48 Liouville a dirigé seul son Journal pendant presque 40 ans (jusqu’en 1874). L’entreprise de Liouville était facilitée par la proximité parisienne des éditeurs et imprimeurs, contrairement à Gergonne qui fut obligé de correspondre avec ses collaborateurs par la seule et alors lente voie postale. Liouville bénéficia ainsi du savoir faire de la maison d’édition Bachelier qui devient entre la fin des années trente et au cours des années quarante, « la maison d’édition des mathématiques »22. Gergonne ne disposait pas de cet atout technique. Les conditions étaient donc tout à fait différentes, les enjeux d’un autre ordre, les moyens incomparables. La comparaison des lignes éditoriales des deux journaux a été présentée lors du Novembertagung de Paris en 2005. Le texte de l’exposé a partiellement été édité [Gérini & Verdier 2006]. Nous nous contentons ici de développer quelques points de continuité et de rupture.
49 Liouville rend hommage à Gergonne dans son avertissement :
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Toutes les personnes qui ont une teinture même légère des Mathématiques connaissent le succès mérité qu’ont obtenu les Annales fondées en 1810 par M. Gergonne, et continuées par lui pendant vingt ans avec un zèle qu’on ne peut trop louer, et un talent qui a triomphé des plus grands obstacles. [Liouville 1836]
50 L’héritage est donc d’abord sémantique avec un titre emprunté à Ger-gonne, le « journal de mathématiques pures et appliquées, ou recueil mensuel de mémoires sur les diverses parties des mathématiques ». L’emprunt est revendiqué par Liouville qui précise : « M. Gergonne ayant bien voulu nous dire lui-même qu’il verrait avec plaisir un nouveau journal succéder au sien, nous croyons avoir le droit de nous annoncer aujourd’hui comme ses continuateurs » [Liouville 1836]23. L’héritage s’inscrit également dans la forme : Notre journal sera mensuel comme celui de M. Gergonne. Le premier cahier paraîtra en janvier 1836, et les suivants de mois en mois, avec toute l’exactitude désirable. Ces cahiers seront de grandeur inégale, et varieront de 32 à 40 in 4°, suivant la nature des mémoires qu’ils renfermeront. Leur ensemble formera chaque année un fort volume contenant toutes les planches nécessaires pour l’intelligence du texte. [Liouville 1836]
51 Liouville s’inspirait ainsi directement du mode d’édition et de reliure annuelle des Annales de Gergonne constituées de fascicules (de 30 à 40 pages en moyenne, in-4°, reliés en volumes couvrant une année).
52 Liouville, sans citer explicitement Gergonne, met en avant quelques lignes de rupture. Son Journal est ouvert aux articles élémentaires concernant l’enseignement mathématique des collèges mais, assène-t-il : « nous tâcherons d’éviter les répétitions fastidieuses d’objets trop connus ; car s’il est bon de revenir de temps à autre sur les élémens des sciences, il faut que ce soit pour les perfectionner, et non pour y changer çà et là quelques mots et quelques phrases ; ce qui par malheur est arrivé trop souvent » [Liouville 1836].
53 Nous pensons que Liouville fait référence à Gergonne car plusieurs éléments d’archives (les carnets de Liouville à la Bibliothèque de l’Institut de France et les archives privées encore détenues dans la famille24) montrent que Liouville ne cessait d’annoter des articles publiés ici ou là. Pour les articles extraits des Annales de Gergonne, Liouville note souvent qu’il s’agit d’une approche élémentaire, qu’elle se trouve dans des textes classiques (les œuvres d’Euler sont souvent citées), qu’on aurait pu avoir le résultat plus simplement, que « ceci est bien connu », etc. Il se démarque également de Gergonne dans la gestion des critiques. Selon lui, le rédacteur doit mettre dans ses critiques « non-seulement de l’impartialité, mais encore de la bienveillance, et cherchant à faire ressortir le bien plutôt qu’à censurer le mal » [Liouville 1836]. Il est difficile de ne pas voir derrière cette phrase une critique sur la gestion « ingérante » de Gergonne.
54 Le Journal de Liouville est, dans son esprit, plus proche du Journal de Creile que des Annales de Gergonne. Alors que les Annales de Gergonne sont « c’est l’essentiel, au service d’une communauté naissante de professeurs » [Dhombres & Otero 1993, 68], le Journal de Liouville marque la naissance d’une nouvelle génération de mathématiciens. Avant Liouville, il y avait Fourier, Legendre, Lagrange, Cauchy en France, Gauss en Allemagne, Ivory et Wallace en Ecosse, etc. Avec Liouville, il y a Sturm et Chasles en France, Cay-ley en Angleterre, Lejeune-Dirichlet et Jacobi à Berlin, etc. ainsi qu’une multitude d’auteurs rattachés aux plus brillantes institutions européennes (Ecole polytechnique à Paris, université de Berlin, de Cambridge, etc.). Le Journal de Liouville a fait l’objet d’une étude à paraître [Verdier 2009]. Elle dresse une possible histoire du Journal de mathématiques
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pures et appliquées entre sa fondation, en 1836, et 1885. Cette étude, en un certain sens, est continuatrice d’une première étude, essentiellement statistique, effectuée par Sylvain Duvina [Duvina 1994].
À Cambridge en 1837 avec The Cambridge Mathematical Journal
55 Peu de temps après le lancement du Journal de Liouville, la situation se décante en Angleterre. Le vœu de Gergonne : « Il nous faudrait présentement un semblable recueil publié à Londres et un autre dans quelque grande ville d’Italie et alors rien ne nous manquerait pour être bien au courant » [Gergonne FTM, 1826] est partiellement comblé.
56 En 1837, deux jeunes mathématiciens de Cambridge, Robert Leslie Ellis – né en 1817 – et Duncan Farquharson Gregory – né en 1813 et originaire d’Edinburgh – fondent The Cambridge Mathematical Journal. Ce journal est destiné aux étudiants en mathématiques qui ont du mal à se faire publier. Autant il était possible de se faire publier dans une publication académique pour quelqu’un qui disposait d’un réseau de sociabilité savant, autant il était difficile d’être publié pour un jeune étudiant désireux de diffuser ses mathématiques. Les deux rédacteurs écrivent : In bringing before the Public the First Number of the Cambridge Mathematical Journal, it will naturally be expected that we should say a few words, in the way of preface, on the objects we have in view, and the means we intend to adopt for carrying them into effect. [...] But we conceive that our Journal may likewise be rendered useful in another way—by publishing abstracts of important and interesting papers that have appeared in the Memoirs of foreign Academies, and in works not easily accessible to the generality of students. We hope in this way to keep out readers, as it were, on a level with the progressive state of Mathematical science, and so lead them to feel a greater interest in the study of it. [Ellis & Gregory 1837]
57 Le Journal rend effectivement compte de nombreux articles. Plusieurs ont été publiés dans le Journal de Liouville : deux articles de Terquem [Terquem 1839a, 1839b], un de Catalan [Boole 1843], un d’Abria [Abria 1843]25 et un de Liouville lui-même [Liouville 1841]. En réalité, l’article de Liouville est une transcription d’une courte note publiée par Liouville en 1840 [Liouville 1840]. En cette période, Liouville se passionne pour l’arithmétique. Dans sa note, il démontre que le nombre e n’est pas quadratique, c’est- à-dire n’est racine d’aucune équation algébrique du second degré à coefficients rationnels. En fait, Liouville étend la méthode « classique » démontrant l’irrationalité de e, la méthode – fondée sur ce que nous appelons aujourd’hui le développement en série entière de la fonction exponentielle – qui semble avoir été publiée pour la première fois par Janot de Stainville en 1815 [Stainville 1815, 339341]. Cette démonstration serait due à Fourier et a été transmise par Poinsot à de Stainville.
58 En 1844, le Cambridge Mathematical Journal perd un de ses rédacteurs en la personne de Gregory. Il a à peine trente ans. Nous ignorons dans quelle mesure la mort d’un des fondateurs a perturbé la gestion du Journal. Si nous nous en tenons à la nécrologie rédigée par Ellis [The Cambridge Mathematical Journal, IV, 1844, 145], [Ellis 1863], il semblerait que les responsabilités éditoriales étaient essentiellement entre les mains d’Ellis, même si Gregory participe activement. Entre 1837 et 1844, Gregory publie une trentaine de notes alors qu’Ellis en publie une vingtaine. En fait, beaucoup de notes, rédigées très probablement par Ellis (mais non systématiquement comptabilisées) sont
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de courtes notes mathématiques empruntées à d’autres auteurs (étrangers pour la plupart) et adaptées par Ellis.
1841-1842 : des journaux pour l’enseignement des mathématiques
59 En 1841 et 1842, le champ éditorial en France et en Allemagne se réorganise avec l’arrivée de journaux qui ne sont plus exclusivement centrés sur les recherches mathématiques mais sur l’enseignement des mathématiques. En France, en 1842, sont lancées les Nouvelles Annales de mathématiques, le « Journal des candidats aux écoles polytechnique et normale ». L’un des deux fondateurs (Olry Terquem), et sans doute la cheville ouvrière de la publication au moins dans la première décennie26, est l’un des acteurs principaux du Journal de Liouville. À partir de 1842, il n’écrit plus pour Liouville en tant qu’auteur mais continue à y publier des traductions ou à encourager des auteurs à y faire paraître des mémoires27. Dès lors, le champ mathématique français se restructure, d’un côté, le Journal de Liouville, se destine uniquement aux progrès de la science, de l’autre, les Nouvelles Annales se destinent au progrès de l’enseignement. Assez naturellement, Liouville prend l’habitude d’orienter les auteurs d’articles qu’il juge trop tournés vers l’enseignement « élémentaire », vers Terquem et ses Annales. Bien entendu, ce clivage mérite d’être nuancé. Certains articles publiés dans les Nouvelles annales de mathématiques ne sont pas destinés à l’enseignement mais sont des articles vulgarisant des recherches publiées ailleurs (par exemple dans le Journal de Liouville).
60 Le scénario est similaire en Allemagne. Après le Journal de Grelle lancé en 1826, Johann August Grünert, auteur d’une dizaine de mémoires dans le Journal de Grelle, fonde, en 1841, dans une petite université provinciale, à Greifswald, ses Archiv der Mathematik und Physik. Son Journal est au service d’une communauté enseignante [Schreiber 1996]. Grünert rend compte de ce qui se passe dans des journaux plus prestigieux. Ainsi, dès l’année de lancement, il prend parti dans un article faisant allusion à un réseau d’articles traitant de combinatoire paru dans le Journal de Liouville, dès la fin des années trente. Le titre de son mémoire est explicite : « Über die Bestimmung der Anzahl der verschiedenen Arten, auf welche sich ein neck durch Diagonalen in lauter mecke zerlegen lässt, mit Bezug auf einige Abhandlungen der Herren Lamé, Rodrigues, Binet, Catalan und Duhamel in dem Journal de Mathématiques pures et appliquées, publié par Joseph Liouville » [Grünert 1841]. Le contenu combinatoire des différents articles précédents a été spécifiquement analysé par Ulrich Tamm [Tamm 2005].
61 Il est tentant de rapprocher la situation allemande de la situation française, mais il semblerait qu’il faille apporter une distinction de taille. De nombreux documents d’archives (Carnets de Liouville à la Bibliothèque de l’Institut de France, lettres de Terquem à Catalan dans le fonds Catalan des archives de l’université de Liège, etc.) montrent que Liouville et Terquem travaillaient dans des directions communes en orientant les auteurs ou les textes vers la publication la plus judicieuse pour l’auteur. Les relations entre Crelle et Grünert étaient, semble-t-il, mauvaises. Il conviendrait de mesurer plus précisément les liens entre les deux hommes et leurs journaux. L’étude de Peter Schreiber mentionnée précédemment n’est qu’une première étape d’une étude des Archiv à approfondir. Quant aux Nouvelles annales de mathématiques, elles sont actuellement l’objet d’études d’un groupe de travail regroupant des chercheurs de Nancy et d’Orsay essentiellement28.
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La reprise en main en 1845 par Thomson du Cambridge Mathematical Journal
62 Au moment où la presse mathématique allemande et française se décline en deux pôles destinés, pour le premier, aux progrès des mathématiques avec les journaux de Crelle et de Liouville, et, pour le second, aux progrès de l’enseignement des mathématiques avec les Nouvelles annales de mathématiques à Paris et les Archiv der Mathematik à Greifswald, un jeune écossais, William Thomson, part à Paris étudier les mathématiques « continentales ». Son père, James Thomson, est professeur à l’université de Glasgow. Le fils a déjà publié, en 1841, dans le Cambridge Mathematical Journal, un article intitulé « On Fourier’s expansion of functions in trigonometrical series »29 [Thomson 1841]. À Paris, Thomson suit les cours de Liouville, de Sturm, etc. Dans une lettre faisant partie de son dossier personnel aux Archives de l’Académie des Sciences, à Paris, il décrit en 1877 son arrivée à Paris en 1845 : Then after Cambridge undergraduate course was finished my father and my Cambridge instructors sent me to the fountain-head, and I became an étudiant in the Quartier Latin for a few months of the year 1845 which live in my memory, full of happy recollections of the kindness and the valuable teaching I received from Sturm and Liouville and Regnault. [Thomson AAS, 1877]
63 Le père de Thomson a effectivement joué un rôle fondamental en incitant son fils à prendre Liouville comme un modèle en tant que mathématicien mais aussi en tant qu’éditeur. Le père écrit le 18 octobre 1843 : Think, therefore, of every fair, honorable, and practicable way of preparing for the contest: and when you hear of the bad health of others, use very precaution in your power regarding your own. Try to become known to persons of name and influence —Liouville for example. (Cité dans la biographie de Thomson rédigé par son homonyme, in [Thomson 1910, 63])
64 À son retour en Angleterre, Thomson reprend en main les destinées du Cambridge Mathematical Journal. Le 8 octobre 1845, Thomson écrit à Liou-ville, de Cambridge. La lettre concerne l’application du principe des images à la solution de quelques problèmes relatifs à la distribution d’électricité. Elle est publiée dans le Journal de Liouville [Thomson 1845]. À la fin de sa lettre, Thomson précise qu’il « étendra [ses] recherches dans le Cambridge and Dublin Mathematical Journal » [Thomson 1845]. Liouville s’empresse de préciser dans une note de bas de page : « C’est le titre que portera désormais le Cambridge Mathematical Journal, excellent recueil dans la direction duquel M. William Thomson vient de remplacer M. Robert Leslie Ellis » [Thomson 1845].
65 Thomson, sans doute beaucoup plus qu’Ellis, cherche à rapprocher son Journal de celui de Liouville, un Journal dont il était très proche. Dans la biographie de Thomson citée précédemment, est expliqué que lorsque le jeune William a gagné un prix de mathématiques d’une valeur de cinq livres, il s’est acheté un livre illustré de Shakespeare. Son père s’est alors offusqué ; il aurait préféré qu’il s’abonne au Journal de Liouville. Nous ne détaillons pas plus, ici, les relations entre Thomson et Liouville. Elles ont été amplement détaillées dans la biographie de Thomson. De nombreuses lettres éclairant la rencontre de Thomson avec le milieu académique parisien précisent le contexte [Thomson 1910, 113-160]. Un jeu de va-et-vient entre Thomson, Liouville et leurs publications s’instaure30. Thomson continue à publier plusieurs mémoires dans le Journal de Liouville, des mémoires originaux mais aussi des reprises, comme celle de
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Thomson, en 1852, d’une lecture faite à la Société royale d’Edinburgh en 1851 [Thomson 1852]. Dans le Cambridge and Dublin Mathematical Journal, les auteurs sont, pour moitié, comme le titre l’indique, issus de Cambridge ou de Trinity College à Dublin31.
66 Au cours de cette période, la presse scientifique subit de grosses restructurations de l’autre côté de la Manche. En 1832, les Annals of philosophy et The Edinburgh Journal of Science fusionnent pour devenir en 1850 le London, Edinburgh and Dublin Philosophical Magazine and Journal of Science. Le Cambridge (and Dublin) Mathematical Journal n’a fait l’objet d’aucune étude spécifique même s’il convient de faire référence à une étude de Marie-José Durand-Richard qui s’est intéressée à la diffusion des travaux de l’Ecole algébrique anglaise via les journaux scientifiques, dont les journaux anglais [Durand- Richard 1997].
67 Au cours des années cinquante et soixante, le paysage éditorial mathématique est radicalement modifié à l’échelle européenne : l’offre éditoriale s’étoffe et se diversifie en France, en Allemagne et en Angleterre et elle apparaît là où Gergonne l’attendait, en Italie. Nous entrons là dans une autre histoire, à faire, de la presse mathématique européenne au début de la deuxième moitié du XIXe siècle.
68 Dans cet article, nous avons cherché à poser, en nous appuyant et en renvoyant à des études pour la plupart récentes, les jalons d’une histoire des premiers journaux mathématiques en Europe. Les « premiers » journaux sont lancés dans le premier tiers du XIXe siècle. À l’exception du Journal de Crelle, tous périclitent à la fin des années vingt ou au cours des années trente. La « décennie 1835-1845 » est un temps de maturation et de mise en place d’un paysage éditorial organisé et tenu par quelques personnalités qui ont pignon sur rue à l’échelle européenne (Crelle et Liouville) ou à une échelle nationale (Terquem en France, Grünert en Allemagne, Ellis en Angleterre, etc.).
69 Notre écriture, délibérément « feuilletée » au sens défini par Michel de Certeau dans son Ecriture de l’histoire (1975) [de Certeau 2002], se voulait chargée de détails afin d’éclairer le fonctionnement de ces journaux et de leurs relations de filiation et/ou de concurrence, leurs politiques éditoriales (emprunts d’articles, critiques croisées, etc.), leur lectorat et leurs conditions de réalisation matérielle (typographie, impression, etc.).
70 Notre étude s’inscrit dans une démarche ayant pour objectif de dresser une cartographie des réseaux mathématiques et, dans une certaine mesure, une sociologie de la communauté mathématique au XIXe siècle. Nous n’avons donné ici que quelques éléments qualitatifs puisés dans des moments d’échanges choisis. D’autres éléments, s’appuyant sur des méthodes quantitatives que nous avons initiées en 2006 et 2009 [Verdier 2006], [Verdier 2009], sont à venir, en préparation ou à faire. Ces études s’inscrivent dans un moment d’intense numérisation32 permettant l’accès en ligne à de nombreuses publications jusqu’alors uniquement accessibles en bibliothèques. Les facilités d’accès d’aujourd’hui multiplient les points d’entrée de l’historien. Grâce aux nouvelles opportunités informatiques (recherches par mots clés, croisements de données sur une période donnée, etc.), il peut désormais mieux comprendre comment les mathématiques se sont écrites et diffusées sous l’impulsion de quelques dizaines d’hommes, un millier peut-être, au cours de la première moitié du XIXe siècle.
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VERDIER, NORBERT — 2006 Teaching and doing history of mathematics with Mathematica, actes du 8e International Mathematica Symposium (IMS’06). — 2009 Le Journal de Liouville et la presse de son temps : une entreprise d’édition et de circulation des mathématiques au XIXe siècle (1824-1885), thèse de doctorat de l’université Paris-Sud 11.
Archives « AAS » désigne les Archives de l’Académie des Sciences de Paris et « BIF » désigne la Bibliothèque de l’Institut de France à Paris. [Nom, Service d’archives, référence] désigne dans le fonds « Nom » du service d’archives en question le manuscrit référencé. Les lettres de Gergonne sont extraites de « Lacock Abbey Collection Lacock » appartenant au Fox Talbot Museum [FTM]. Nous avons utilisé les lettres N° 05580 (non datée), une lettre du 31 May 1824. (N° 01188) et une lettre du 16 déc. 1826 (N° 01188).34
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NOTES
1. Ce texte est extrait d’un exposé intitulé “Journals of Mathematics in the Early Ninteenth Century in France, in Belgium, in Germany and ... “here” in Scotland!” et donné dans le cadre des Novembertagung [17th, Edinburgh, Friday 3rd to Sunday 5th November 2006]. Nous remercions les deux rapporteurs pour leurs critiques fondées et constructives portées sur la version initialement soumise. Cette seconde version, en étoffant le texte de l’exposé d’Edinburgh, est largement construite sur leurs remarques et leurs questionnements. 2. Dans notre exposé intitulé « Questions, solutions et problèmes dans la presse mathématique au XIXe siècle » [Séminaire REHSEIS (Karine Chemla), Paris, 19 novembre 2007], nous avons évoqué les premiers journaux de mathématiques en Europe (Allemagne et Angleterre). Le fait qu’ils aient été éphémères explique pourquoi, généralement, est attribué aux Annales de Gergonne le qualificatif de « premier ». 3. Ferdinand François Désiré Budan de Boislaurent (1761-1840). 4. En 1814-1815, les puissances européennes décident de rattacher la Belgique aux Pays-Bas sous la souveraineté du roi Guillaume 1er d’Orange. Le 4 octobre 1830, la Belgique proclame son « indépendance ». La France fait un geste en refusant d’octroyer la couronne (proposée par Bruxelles) à l’un des fils de Louis-Philippe. L’Angleterre impose, en 1831, son candidat Léopold de Saxe-Cobourg. Devenue Léopold Ier (1790-1865), il épouse Louise-Marie d’Orléans, fille de Louis- Philippe. 5. Une note de bas de page est rajoutée : « Qui ne doit être considéré que comme un numéro d’annonce ». 6. Dans la Correspondance, cet article n’est pas attribué à un auteur, c’est pourquoi nous ne l’attribuons ni à Gergonne ni à Garnier. Son véritable auteur est la Correspondance elle-même. 7. Via différents catalogues en ligne (SUDOC, BNF, etc.) nous avons identifié seulement quatre bibliothèques en possession (pour certaines partielle) de la Correspondance mathématique et physique. Ce sont les bibliothèques de la faculté des sciences de Bordeaux, de l’Institut Henri Poincaré (IHP), de la Bibliothèque Nationale de France (BNF) et de l’Observatoire à Paris. 8. À l’exposé d’Edinburgh assistait Henrik Kragh Sorensen, de l’université d’Aarhus. Suite au colloque, il nous a envoyé les informations suivantes à propos de Schmidten : « Von Schmidten was a promising Danish mathematician in a very small community. He was born on February 11, 1799. He studied to become an artillerist because that was the education in Denmark with the most mathematics involved and he really wanted to study mathematics. In 1819 he won the silver medal of the Danish Academy of Sciences and Letters for a paper entitled “Recherches sur le calcul intégral aux équations linéaires”. The following year, 1820, he embarked on a European tour sponsored by the King. He spent three years in Paris but also visited Gauss in Göttingen and stopped in Bonn and Berlin. His travels were later influential in forming Niels Henrik Abel’s travel plans, in particular Abel’s meeting with Crelle in Berlin. In 1825, von Schmidten defended his doctoral dissertation “De seriebus et integralibus definitis”, left the service and became an associate professor at the University in Copenhagen and a member of the Danish Academy; in 1827, he became an extraordinary professor. In 1829, he made another short sojourn in Paris. In 1831, after contracting TB, he travelled to the Danish possessions in the Caribbean, where he died on July 16, 1831. » Nous remercions Henrik Kragh Sorensen pour ces informations sur cet auteur largement méconnu. 9. Cette question n’est pas attribuée à un auteur, aussi avons-nous choisi d’indiquer comme seule référence le journal. Toutes les références n’ayant comme indicateur que le nom des Annales sont dans le même cas de figure. 10. L’article n’est pas attribué, c’est pourquoi nous nous contentons de renvoyer à la référence Journal für die reine und angewandte Mathematik.
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11. Quelques-unes de ces leçons ont été développées par Constantinos Chatzis (LATTS, CNRS- Université Marne-la-Vallée-ENPC) dans son exposé : « Quelques contributions graphiques de Jean Victor Poncelet à l’art de l’ingénieur » [Séminaire REHSEIS, « Les pratiques de calcul des ingénieurs à l’ère industrielle » (sous la responsabilité de Marie-José Durand-Richard), 17 mars 2008]. 12. Comme dans le texte de Crelle, nous n’avons pas porté les accents et autres marques de ponctuation. 13. Après « l’intermède Zeitschrift », Baumgartner occupe d’importantes fonctions de direction : « Direktor der staatlichen Porzellanfabriken » en 1833, « Direktor der Tabakfabriken » de 1842 à 1848. À partir de 1848, il occupe plusieurs postes ministériels : ministre du Travail en 1848, ministre des Finances, etc. De 1851 à 1865, il est président de l’Académie des Sciences à Vienne. 14. Dans la section de physique, la plupart des articles sont extraits d’autres publications européennes : Annales de chimie et physique, Edinburgh Journal of Science, Edinburgh Philosophical Journal, Annals of Philosophy, Journal de pharmacie, Annales de l’industrie nationale et étrangère, Philosophical Transactions, Bibliothèque universelle, Abhandlungen der König Akademie der Wissenschaften zu Berlin, Mathematics for practica! Men, The philosophical Magazine and Journal, etc. Les articles cités sont souvent très récents. 15. Nous n’attribuons pas cette référence à un auteur car l’article en question n’est pas attribué. 16. Cet article est attribué à « un abonné ». Nous nous contentons d’indiquer la référence du journal. 17. Nous avons analysé cette pratique des problèmes dans la presse mathématique au cours de l’exposé intitulé « Questions, solutions et problèmes dans la presse mathématique au XIXe siècle » [Séminaire REHSEIS (Karine Chemla), Paris, 19 novembre 2007]. Nous avons en particulier étudié le texte de Lavernède. 18. La présente étude a été menée à partir d’un exemplaire personnel acheté dans une librairie spécialisée de Hambourg (Hamburger Antiquariat). 19. Cette indication de Gergonne clôt le dernier tome des Annales. Elle ne figure pas dans la version numérisée. 20. Erwin Neuenschwander précise également les références des articles auxquels renvoient ces deux lettres. 21. Il n’est pas le seul. En janvier 1836 a aussi été lancé Le Géomètre, un journal destiné à la préparation aux Grandes Écoles, et plus spécifiquement à l’École polytechnique et à l’École normale. Dirigé par un professeur de Louis-Le-Grand, Antoine-Philippe Guillard (1795-1870), ce périodique est éphémère puisqu’il disparaît quelques mois plus tard, en juin 1836. Le Géomètre a particulièrement été étudié lors de l’exposé : « Représentations des mathématiques au temps de Liouville (1809-1882) et de son Journal », Norbert Verdier, Institut Henri Poincaré, Paris, 25 mai 2005. 22. À l’époque de Gergonne, Bachelier était un « simple » libraire, successeur de Courcier, lui- même libraire des Annales de Gergonne depuis leur création. Dès 1830, Bachelier devient le libraire français officiel du Journal für die reine und angewandte Mathematik de Crelle. En 1832, Bachelier associe à sa librairie l’imprimerie du 12 rue du jardinet, fief de la maison Huzard-Courcier, devenant ainsi à la fois libraire et imprimeur. Il met un soin considérable à l’amélioration de la représentation matérielle des mathématiques (l’art typographique). 23. Avant d’être un « continuateur », Liouville a d’abord été un « lecteur » de Gergonne [Gérini & Verdier 2006]. 24. Nous remercions un des descendants directs de Liouville, Michel Drouineau, pour nous avoir expédié une partie des manuscrits de Liouville qu’il possède. 25. Il s’agit en fait d’un résumé d’une page de l’article initial d’Abria « in Liouville’s journal, tom. IV., p. 248 ». Cette note publiée dans The Cambridge Mathematical Journal n’est pas répertoriée dans le Catalogue of scientific papers.
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26. C’est ce qu’affirme Camille Gerono, le co-fondateur dans une lettre écrite plus tard, le 21 avril 1869, à un des principaux acteurs de la presse mathématique au XIXe, Lebesgue : « [Terquem] écrivait avec une grande facilité ; peut-être aurait-il été mieux d’écrire un peu moins, et de réfléchir un peu plus. Enfin, il a tout seul, rédigé pendant 10 ans, les nouvelles Annales, je lui en suis reconnaissant » [Bertrand, BIF, n°139, Lettre du 21 avril 1869]. 27. Fin 2009, nous organisons, en partenariat avec la SABIX (sous la présidence d’Alexandre Moatti), un colloque à l’École polytechnique autour de Olry Terquem et de son œuvre. 28. Ce groupe d’études fait partie du projet ANR « Sources du savoir mathématiques au début du XXe siècle (2007-2010) ». Il est piloté par Philippe Nabonnand et Laurent Rollet (LHSP-Archives Poincaré) et réunit une dizaine de chercheurs de Nancy (LHSP-Archives Poincaré) et d’Orsay (GHDSO), essentiellement. Les Nouvelles Annales sont en cours de numérisation par NUMDAM. 29. Son article est une défense des conceptions de Fourier contre celles de Philipp Kelland, le successeur (anglais) de William Wallace à la chaire de mathématiques de l’université d’Edinburgh. 30. Nous n’avons pas détaillé ici les reprises entre le Journal de Liouville et le Cambridge and Dublin Mathematical Journal. Nous avions fait ce recensement lors d’un exposé donné à la maison française d’Oxford : « Éditer et publier des journaux de mathématiques à Paris et à Londres dans la première moitié du XIXe siècle », Science Capitals and Sociability (Research Programme Sciences and European Capitals, Responsable : Stephane Van Damme), 4-5 novembre 2005. 31. Lors de l’exposé donné à Edinburgh, nous avons insisté sur les contributions écossaises. Parmi la cinquantaine d’auteurs, nous dénombrons quatre auteurs écossais : Andrew Bell d’Edinburgh, et de Glasgow, Hugh Blackburn, James Thomson (le père) et William Thomson (le fils). 32. Nous pensons bien évidemment aux programmes institutionnels et organisés NUMDAM, Gallica, etc. mais aussi aux initiatives privées comme Google.books, qui sont certes produites de façon anarchique et désorganisée avec parfois une qualité de numérisation perfectible, mais qui permettent d’avoir accès à la totalité des textes et à faire des recherches sans les entraves posées par les programmes précédents (qui définissent et limitent d’emblée quels sont les types de recherches possibles). 33. Une note entre parenthèses précise : « Lu à la Société royale d’Edimbourg le 19 mars 1851, et extrait des Transactions de cette Société, vol. XX, deuxième partie. » L’article est en fait partiellement publié en 1853 [Edinb. Roy. Soc. Trans. XX, 1853, 261-288]. 34. « This letter attached, for unknown reasons, to Doc. No : 01188, at FTM, Lacock » précise Larry J Schaaf, le responsable de ce projet de publication de la correspondance de Talbot. Dans ce texte, nous avons rectifié les coquilles éditoriales (essentiellement les accents et apostrophes) de la traduction figurant dans le projet mentionné.
RÉSUMÉS
En 1826, Joseph Diez Gergonne, le fondateur des Annales de mathématiques pures et appliquées en 1810 à Nîmes (France), écrit à W.H. Talbot. Dans ses lettres, il évoque les autres journaux mathématiques. Dans cet article, nous étudierons ces « premiers » journaux de mathématiques de la première moitié du XIXe siècle, en Europe. Nous étudions leur dynamique éditoriale, leur public et le rôle des premiers patrons de presse : Gergonne, Garnier et Quetelet, Crelle, Liouville,
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Robert Leslie Ellis et Thomson, etc. Il s’agit donc d’étudier une Europe mathématique en construction par le biais de ses journaux.
In 1826, Joseph Diez Gergonne, who founded the Annales de mathématiques pures et appliquées in 1810 in Nîmes (France), wrote to W.H. Talbot. In his letters, he spoke about the others journals of mathematics. We will study these journals of mathematics in the first part of the nineteenth century in Europe. We will study their dynamic of transmission, their public, the leading role of the first editors of mathematics: Gergonne, Garnier and Quetelet, Crelle, Liouville, Robert Leslie Ellis and Thomson, etc. We will do an history of mathematical Europe through its first journals.
AUTEUR
NORBERT VERDIER IUT CACHAN & GHDSO (Université de Paris-Sud 11, Orsay)
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Varia
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Le concept d’espace chez Veronese Une comparaison avec la conception de Helmholtz et Poincaré
Paola Cantù
1 Introduction1
1 Veronese le premier a construit une géométrie non-archimédienne (une dizaine d’années avant David Hilbert et avec une méthode synthétique plutôt qu’analytique) et il a développé un concept de continu, qui contient des éléments infinis et infiniment petits. Veronese est très connu chez les mathématiciens pour ses études sur les espaces à plusieurs dimensions en utilisant la méthode par projections et sections de Luigi Cremona. Moins connus sont les écrits « philosophiques » qui concernent les fondements de la géométrie et des mathématiques et les traits les plus saillants de la conception épistémologique de Veronese.
2 L’article analysera le rapport entre géométrie et intuition spatiale, qui était une question très controversée dans la deuxième moitié du dix-neuvième siècle. Quel rapport y a-t-il entre l’espace physique dans lequel nous vivons, l’espace dans lequel nous situons les objets perçus et représentés et l’espace géométrique ? Dans les années soixante-dix du dix-neuvième siècle, Hermann Helmholtz a essayé de donner une justification philosophique de la géométrie riemannienne et de la géométrie hyperbolique, qui avaient été démontrées comme étant logiquement cohérentes. Helmholtz a fait une distinction entre l’espace physique, qui est euclidien, et l’espace intuitif ou représentatif, qui est compatible tant avec la géométrie euclidienne qu’avec les géométries non euclidiennes, mais pas avec d’autres théories de l’espace. Selon le point de vue de Helmholtz, on ne peut pas justifier des géométries de l’hyperespace, parce qu’une théorie géométrique doit inclure les propriétés de l’espace intuitif, et l’espace intuitif ne peut avoir que trois dimensions. Pour légitimer d’autres théories géométriques, par exemple la géométrie d’un espace à quatre dimensions, Giuseppe Veronese et Henri Poincaré proposent deux solutions différentes. Veronese soutient l’idée d’un accord nécessaire entre les propriétés de l’espace, que nous découvrons par observation de certains objets physiques, et les axiomes de la géométrie, mais il renonce à la restriction que l’espace représentatif ait trois dimensions. Poincaré, au
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contraire, ne présuppose pas que les axiomes de la géométrie s’accordent avec les propriétés de l’espace représentatif. Il affirme plutôt dans La science et l’hypothèse qu’on peut choisir conventionnellement les axiomes qui décrivent l’espace, pourvu qu’ils soient cohérents : le choix des axiomes reste libre, mais il est guidé par des faits expérimentaux.
2 La nature mixte de la géométrie : une science formelle et expérimentale
3 En reprenant une distinction de Hermann Grassmann entre sciences formelles et sciences réelles [Grassmann 1844, 22], Veronese partage les sciences en formelles (logique et mathématiques pures) et expérimentales (physique et mécanique). Les sciences formelles étudient les formes, c’est-à-dire les produits de la pensée qui n’ont pas nécessairement une existence réelle, ou comme le dit Veronese, « les choses pensées qui n’ont pas nécessairement une image existant dans un champ qui existe effectivement en dehors de la pensée » [Veronese 1891, vii]. Les sciences expérimentales étudient les objets qui existent effectivement en dehors de la pensée. Dans les sciences formelles, la vérité dépend de l’accord entre différents actes de la pensée, c’est-à-dire seulement du principe de contradiction ; dans les sciences expérimentales la vérité dépend aussi de l’accord entre la pensée et l’objet, ou mieux, de l’accord entre la pensée et la représentation mentale de l’objet. Les prémisses des sciences formelles sont des définitions ou hypothèses, parce qu’elles déterminent avec un acte créatif les propriétés des entités idéales dont elles parlent ; les prémisses des sciences expérimentales sont au contraire des axiomes, qui décrivent approximativement les propriétés des objets réels.
4 La géométrie n’est pas une science formelle et elle n’est pas une science expérimentale non plus, mais plutôt une science mixte : la nature de ses objets et de ses prémisses est double. Les objets de la géométrie sont en partie purement idéaux, parce qu’ils sont des produits de la pensée, et en partie obtenus par abstraction, c’est-à-dire que leurs propriétés sont déterminées à partir de l’observation de certains objets physiques concrets. Par l’application d’une loi générale de la pensée nous pouvons aussi sortir au dehors du champ de l’expérience et déterminer par exemple les propriétés d’une ligne illimitée. Le concept de ligne droite illimitée est en effet obtenu par quatre actes successifs d’abstraction : on observe un bâton très mince; premièrement, on fait abstraction des propriétés physiques du bâton et on retient seulement l’extension ; deuxièmement, on fait un passage à la limite en considérant la largeur du bâton comme étant nulle ; troisièmement, on fait abstraction de toutes les imperfections de l’objet ; quatrièmement, on sort du champ de l’expérience en imaginant que le bâton s’étend sans fin dans les deux directions.
5 La géométrie a une nature mixte non seulement parce que ses objets sont en partie idéaux et en partie obtenus par abstraction des objets réels, mais aussi parce que ses prémisses sont en partie empiriques et en partie semi-empiriques et purement abstraites 2. Les prémisses empiriques sont des vérités évidentes qu’on saisit par l’intuition quand on observe certains objets physiques : par exemple la proposition suivante, qui définit les propriétés d’un segment rectiligne : « la droite, dans le champ de notre observation, est déterminée par deux de ses points quelconques» [Veronese 1902, 435] 3. Les prémisses semi-empiriques ont une origine empirique mais elles ne
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sont pas vérifiables empiriquement, parce qu’elles sortent du champ d’une observation possible. Par exemple, la proposition « la ligne droite est déterminée par deux de ses points quelconques », où par ligne droite on entend une ligne illimitée. La distinction entre prémisses empiriques et prémisses semi-empiriques a été probablement inspirée par les Éléments d’Euclide, où il y a deux prémisses différentes pour le segment rectiligne et pour la droite illimitée. La géométrie contient enfin des prémisses purement abstraites, qui concernent des objets qui n’existent pas du tout dans le champ de l’observation : par exemple le postulat du continu formulé par Dedekind, le postulat de l’espace à quatre dimensions, l’hypothèse des parallèles, l’hypothèse de la divisibilité en parties d’une ligne droite, les hypothèses de l’infiniment petit et de l’infini actuel.
6 Est-ce qu’il y a alors un critère pour déterminer quelles prémisses géométriques sont possibles ? Les prémisses empiriques sont déterminées par certains phénomènes physiques (par exemple le mouvement d’un bâton) ; les prémisses semi-empiriques et celles qui sont purement abstraites peuvent être choisies librement à condition qu’elles ne soient pas elles-mêmes contradictoires et qu’elles ne contredisent pas les prémisses précédemment admises, c’est-à-dire les prémisses empiriques. Autrement dit, la condition pour que les prémisses géométriques soient possibles n’est pas seulement la cohérence logique, mais aussi la conformité à l’intuition de l’espace, c’est-à-dire aux propriétés intuitives spatiales exprimées dans les prémisses empiriques. Cette dernière condition, qui révèle que la géométrie est encore considérée comme science de l’espace, distingue la géométrie des mathématiques pures : une théorie, qui nie les propriétés élémentaires des objets physiques que nous connaissons par intuition, n’est pas selon Veronese une géométrie mais plutôt une théorie mathématique abstraite.
3 L’intuition géométrique
7 On a dit que selon Veronese une prémisse géométrique est possible, si elle est compatible avec l’intuition de l’espace, mais qu’est-ce qu’on doit entendre par intuition ? J’esquisserai brièvement les conceptions de Helmholtz, Klein, Veronese. L’intuition géométrique est-elle une intuition empirique immédiate ? Ou plutôt une intuition médiate à travers un concept ? Ou encore est-elle un acte immédiat qui permet de saisir des objets purement idéaux ? 1. Si par intuition on entend un acte immédiat et direct lié à la perception sensible, on doit admettre la perceptibilité immédiate comme condition de l’intuition d’un objet. Comme les entités géométriques ne sont pas perceptibles, ni les objets géométriques ni les propriétés décrites dans les axiomes ne peuvent être objet d’intuition (selon ce sens du mot intuition). 2. Hermann Helmholtz définit l’intuition géométrique comme une intuition médiate, dans laquelle un concept joue le rôle de médiateur. Nous pouvons avoir l’intuition d’un objet si nous pouvons formuler et spécifier d’une façon non ambiguë les impressions sensorielles déterminées par l’objet dans nos sens [Helmholtz 1878, 231]. 3. Felix Klein définit l’intuition géométrique comme un acte immédiat et pas médiat qui permet de saisir des objets purement idéaux. Il fait une distinction entre une intuition naïve et une intuition fine. L’intuition géométrique propre de tout homme adulte capable de former des images géométriques est considérée comme naïve : elle se développe avec des expériences mécaniques. Au contraire, l’intuition propre du géomètre, pour qui les objets idéaux de la géométrie sont familiers, est appelée fine [Klein 1893, 225].
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8 L’intuition géométrique est selon Veronese quelque chose de très proche, comme il le dit lui-même, de l’intuition de Klein. Elle est le résultat de la combinaison de l’intuition empirique et de l’abstraction [Veronese 1891, viii] et pas une forme a priori transcendantale de l’esprit, comme le démontrerait le fait que l’intuition des relations spatiales entre les objets n’est pas une qualité innée, mais se développe grâce à l’habitude et à l’expérience. C’est pour cela que l’intuition géométrique est mieux développée chez les géomètres et les peintres [Veronese 1906, 13-14].
4 L’espace géométrique chez Helmholtz
9 Je reviens maintenant à la distinction entre espace physique, espace représentatif et espace géométrique. J’esquisserai seulement quelques traits de la question : la présentation de certains problèmes relatifs aux fondements de la géométrie au dix- neuvième siècle sera peut-être simplifiée, mais nous permettra de souligner les points les plus importants pour comprendre la différence entre Veronese et Poincaré. Après la découverte des théories géométriques qui sont aussi cohérentes que la géométrie euclidienne mais qui ne sont pas compatibles avec elle, plusieurs mathématiciens (parmi lesquels Gauss) ont introduit une distinction entre espace physique et espace géométrique abstrait et ont considéré la géométrie comme une science expérimentale, qui décrit avec une certaine approximation les propriétés de l’espace physique. La prééminence de la géométrie euclidienne n’est pas expliquée par une nécessité logique, mais par la présomption qu’elle décrit mieux la réalité ; pourtant la géométrie euclidienne serait la vraie science de l’espace physique. La connaissance de l’espace que la géométrie fournit n’est pas nécessaire (parce qu’elle est le résultat d’une généralisation empirique) et ne peut pas être fondée purement a priori, parce que sa vérité dépend des propriétés de l’espace physique, c’est-à-dire « de quelque chose qui existe au dehors de nous et qui ne peut pas être seulement un produit de notre esprit » [Gauss 1806, 201].
10 Pour justifier la possibilité d’avoir l’intuition d’espaces non euclidiens et pour fonder philosophiquement la science géométrique, Helmholtz introduit une distinction entre espace physique et espace représentatif et essaie d’expliquer la raison de la cohérence des géométries non euclidiennes par l’accord avec la représentation intuitive que nous avons de l’espace. Si l’espace euclidien est la forme a priori de l’intuition et la géométrie est entendue comme science de l’espace physique, elle ne peut pas être autre qu’euclidienne. Si au contraire l’espace géométrique est considéré comme étant distinct de l’espace physique, on peut soutenir qu’il est compatible avec la représentation intuitive de l’espace. La forme a priori de l’intuition n’est pas l’espace décrit par la géométrie euclidienne, mais un espace sous-déterminé par rapport à celui-là. Pour affirmer que l’espace est la forme a priori de l’intuition sans soutenir que la géométrie euclidienne est la seule géométrie possible, Helmholtz détermine en effet un groupe minimal d’axiomes qui décrivent les propriétés a priori de l’espace et qui sont communs à la géométrie euclidienne, à la géométrie riemannienne et à la géométrie hyperbolique. D’une part, Helmholtz croit qu’il y a un espace (semblable à la forme transcendantale kantienne) qui est condition nécessaire de la connaissance spatiale, d’autre part, il nie que les déterminations de l’espace euclidien soient a priori [Helmholtz 1878, 700-701]. Pour expliquer la différence entre une forme de l’espace que nous connaissons a priori et l’espace euclidien que nous connaissons a posteriori,
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Helmholtz fait une distinction entre l’espace représentatif ou intuitif et l’espace géométrique. L’espace représentatif est unique et il est la condition de possibilité de l’expérience ; l’espace géométrique au contraire est une description abstraite du premier. Puisqu’il y a différentes descriptions de l’espace représentatif qui sont également cohérentes au point de vue logique, l’espace géométrique n’est pas unique ; il y a trois espaces géométriques différents : l’espace euclidien, l’espace riemannien et l’espace hyperbolique. Les trois théories géométriques sont toutes compatibles avec l’intuition de l’espace, parce que nous pouvons nous représenter un corps dans un espace non euclidien. La géométrie euclidienne est la meilleure description de l’espace physique parce que ses axiomes décrivent le mouvement des corps rigides en accord avec les lois de la mécanique.
11 Quel est l’avantage au point de vue philosophique de l’affirmation qu’il y a un espace intuitif qui est compatible avec les trois géométries nommées ? On peut tout aussi bien déterminer un objet commun aux trois géométries qu’expliquer pourquoi on ne peut pas démontrer a priori la vérité de la géométrie euclidienne. Toutefois, la possibilité d’un espace géométrique abstrait dépend de l’accord avec l’intuition et en particulier avec l’intuition tridimensionnelle. Selon Helmholtz, nous ne pouvons ni avoir l’expérience directe d’un objet dans un espace à quatre dimensions ni imaginer quelles impressions sensorielles il produirait en nous : donc, selon la définition d’intuition géométrique donnée par Helmholtz, nous ne pouvons pas avoir l’intuition d’un tel objet.
12 Je résume alors la conception de Helmholtz, que je prends ici comme point de départ de l’exposition de la différence entre Veronese et Poincaré. 1) D’une part, Helmholtz distingue la possibilité abstraite d’une géométrie de l’existence physique de l’espace qu’elle décrit et ainsi il justifie les géométries non euclidiennes. 2) D’autre part, il soumet la possibilité d’avoir une représentation d’une géométrie en accord avec l’intuition tridimensionnelle et c’est pour cela qu’il nie la possibilité de la quatrième dimension.
13 Comment est-ce qu’on peut justifier la géométrie de l’hyperespace ? Une solution à cette question a été donnée aussi bien par Poincaré que par Veronese.
5 Espace géométrique et espace représentatif chez Poincaré
14 Je présenterai d’abord la conception de l’espace de Poincaré ; ensuite je décrirai de façon plus détaillée la conception de Veronese, qui est beaucoup moins connue. Dans un article de 1894, qui a pour titre L’espace et la géométrie et qui a été publié comme quatrième chapitre de La science et l’hypothèse, Poincaré développe une analyse très précise de l’espace représentatif, pour montrer qu’il y a des différences essentielles entre l’espace représentatif et l’espace géométrique. L’espace géométrique, qui fait l’objet de la géométrie, est continu; il est infini; il a trois dimensions; il est homogène, c’est-à- dire que tous ses points sont identiques entre eux ; il est isotrope, c'est-à-dire qu'il n’y a pas une direction privilégiée. L’espace représentatif, qui « sert de cadre à nos sensations et à nos représentations », n’a pas toutes les propriétés nommées : « il n’est ni homogène, ni isotrope ; on ne peut même pas dire qu’il ait trois dimensions » [Poincaré 1902, 78-81].
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15 Par exemple, l’espace visuel pur, qui sert de cadre aux impressions purement visuelles dues « à une image qui se forme sur le fond de la rétine », a deux dimensions et n’est pas homogène, parce que les points qui sont au bord et ceux qui sont au centre de la rétine ne contribuent pas de façon égale à la formation des images visuelles d’un objet.
16 L’espace visuel complet, qui sert de cadre aux impressions purement visuelles et à certaines sensations musculaires qui nous donnent la notion de la troisième dimension, n est pas isotrope, parce que la troisième dimension ne nous apparaît pas comme jouant le même rôle que les deux autres. En outre il n est pas nécessaire qu'il ait trois dimensions : c'est au contraire « un fait d’expérience externe ». Rien n’empêche de supposer qu’un être ayant l’esprit fait comme nous, soit placé dans un monde où la lumière ne lui parviendrait qu’après avoir traversé des milieux réfringents de forme compliquée. (…) Un être qui ferait dans un pareil monde l’éducation des ses sens, attribuerait sans doute quatre dimensions à l’espace visuel complet. » [Poincaré 1902, 80]
17 L espace moteur a trois dimensions, mais ceci n’est pas nécessaire : il est plutôt le résultat d’une interaction entre nos organes sensoriels et le milieu dans lequel nous vivons. Le sentiment de la direction est le produit d’une association des sensations musculaires, laquelle est le résultat d’une habitude, qui elle-même résulte des nombreuses expériences. Sans aucun doute, si l’éducation de nos sens s’était faite dans un milieu différent, où nous aurions subi des impressions différentes, des habitudes contraires auraient pris naissance et nos sensations musculaires se seraient associées selon d’autres lois. [Poincaré 1902, 81]
18 L’espace géométrique a des propriétés différentes de celles de l’espace représentatif : donc il ne peut pas être considéré comme nécessaire en tant que forme a priori de l’intuition. Même l’espace représentatif, bien qu’il soit une condition de l’expérience, n est pas nécessaire : au contraire, rien n empêche de supposer que si notre milieu était différent, l’espace visuel complet ou l’espace moteur auraient pu avoir quatre dimensions plutôt que trois.
19 Si l’idée d’un espace géométrique ne s’impose pas à notre esprit et ne peut être fournie par aucune sensation isolée, comment est-elle née ? « Nous y sommes amenés, répond Poincaré, en étudiant les lois suivant lesquelles les sensations se succèdent » [Poincaré 1902, 83]. L’espace géométrique n est pas un cadre imposé à chacune de nos représentations, considérée individuellement, sinon il serait impossible de se représenter une image dépouillée de ce cadre. Nous ne nous représentons pas les corps extérieurs dans l’espace géométrique : nous nous les représentons dans l’espace représentatif. Nous pouvons néanmoins raisonner sur ces corps comme s’ils étaient dans l’espace géométrique, c’est-à-dire dans un espace qui est déterminé par les lois suivant lesquelles se succèdent nos images de ces corps. Nous pourrions aussi imaginer que de telles lois soient différentes et ainsi nous raisonnerions sur les objets comme s’ils étaient dans un espace différent. Affirmer que nous pouvons nous représenter un espace, par exemple l’espace hyperbolique, veut dire que nous pouvons raisonner comme si les corps étaient dans cet espace, c’est-à-dire que nous pouvons supposer que les lois selon lesquelles se succèdent nos images de ces corps soient différentes des lois selon lesquelles elles se succèdent effectivement. Mais pas trop différentes. Par exemple, si ces lois sont les lois qui déterminent une géométrie non-archimédienne, nous ne pouvons pas raisonner comme si les corps étaient dans un espace de ce genre.
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6 La conception de l’espace chez Veronese
20 Veronese distingue l’espace géométrique de l’espace physique et de l’espace représentatif en s’appuyant sur sa conception de l’intuition géométrique comme combinaison d’intuition et d’abstraction. Il maintient, comme Helmholtz, que la géométrie est une description de l’espace dont nous avons l’intuition, mais il n’attribue pas à cet espace les mêmes propriétés qu’il attribue à l’espace euclidien. L’ensemble minimal d’axiomes qui décrivent l’espace intuitif, ne coïncide pas avec celui qui est donné par Helmholtz : l’espace intuitif, comme l’espace géométrique, n a pas un nombre déterminé de dimensions.
21 L’espace physique ne peut pas être défini, parce qu’il n’est pas un concept : il est le milieu qui contient les corps qui existent en dehors de nous et il est déterminé par le champ de notre observation. Il est l’objet de la physique et il a trois dimensions [Veronese 1895-7, 33-34]. L’espace intuitif ne peut pas non plus être défini, parce qu’il est une intuition : il est une représentation idéalisée de l’espace physique et le milieu qui contient les représentations des corps que nous percevons avec les sens. Il est lié à la perception sensible, de laquelle il dépend. Quand nous percevons avec les sens (surtout le toucher et la vue) la présence des corps au dehors de nous, nous avons aussi l’idée qu’il y a un certain milieu qui contient ces corps et dans lequel chaque corps occupe une place déterminée [Veronese 1905, 210].
22 L’espace géométrique peut être défini, parce qu’il est un concept, un produit abstrait du raisonnement humain [Veronese 1905, 202]. L espace géométrique est obtenu par abstraction de l’espace intuitif : il ne contient ni les objets physiques réels ni les représentations sensibles correspondantes, mais des entités purement abstraites. L’espace géométrique est défini comme un système de points ayant la propriété suivante : pour chaque figure qu’on peut construire, il y a au moins un point en dehors. Cela veut dire que, si on considère une figure à n dimensions, on peut toujours construire une autre figure qui ait n + 1 dimensions. Donc il y a des espaces géométriques qui ont trois dimensions, des espaces qui en ont quatre et ainsi de suite.
23 L’espace géométrique euclidien est défini selon son concept non comme une description abstraite de l’espace physique, mais comme un cas particulier et à trois dimensions d’un espace géométrique général, qui n a pas un nombre déterminé de dimensions. (...) les propriétés [de cet espace] qu’on ne peut pas démontrer (et donc qu’on assume comme prémisses) dérivent en partie de l’observation externe et en partie de certains principes abstraits qui ne contredisent pas les propriétés empiriques ; les figures, autant que le point garde sa signification primitive, sont toujours accompagnées par l’intuition spatiale. [Veronese 1891, 211] 4
24 Les figures de l’espace général sont accompagnées par l’intuition. Donc on peut aussi avoir l’intuition d’espaces qui ont plus de trois dimensions et cela est vrai, parce que l’espace intuitif n’est pas défini comme ayant un nombre déterminé de dimensions. Cela n’est pas dit explicitement par Veronese, mais on peut le déduire des trois faits suivants.
25 1 Premièrement, puisque l’espace physique a trois dimensions, si nous voulons appliquer la géométrie à cet espace, nous devons poser comme axiome pratique que l’espace intuitif a trois dimensions : « L’espace intuitif est une figure avec trois dimensions par rapport à ses points » [Veronese 1891, 454]. Cela veut dire qu’on ne
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peut pas affirmer que l’espace intuitif ait trois dimensions si on ne pose pas explicitement cela comme axiome. En particulier, cet axiome est nécessaire seulement pour l’application de la géométrie à l’espace physique. D autres axiomes nécessaires pour cette application pratique de la géométrie sont l’axiome du mouvement des corps rigides et l’axiome des parallèles. On peut donc affirmer qu’entre les propriétés qui caractérisent l’espace intuitif, il n’y a pas ces caractéristiques de l’espace euclidien. Donc l’espace intuitif est compatible avec les géométries non euclidiennes et avec la géométrie de l’hyperespace ou, autrement dit, on peut avoir l’intuition de ces géométries.
26 2 Deuxièmement, dans un espace à quatre dimensions le point est tout à fait comme le point d’un espace à trois dimensions, c’est-à-dire que nous pouvons avoir l’intuition du premier exactement comme nous pouvons avoir l’intuition du deuxième : [Dans la définition d’espace général] le point n’est ni un système de nombres ni un objet quelconque, mais le point exactement comme nous nous le représentons dans l’espace ordinaire. Les objets composés de points sont des objets (figures) auxquels nous appliquons continuellement l’intuition spatiale combinée avec l’abstraction, et donc la méthode synthétique. [Veronese 1891, 610] 5
27 3 Troisièmement, Veronese attribue explicitement au concept d’une intuition de l’hyperespace un sens : ce concept peut être compris seulement si on considère l’intuition géométrique comme le résultat de la combinaison de l’intuition et de l’abstraction. Comment pouvons-nous avoir l’intuition d’un espace à quatre dimensions ? Veronese utilise un exemple qu’on peut trouver aussi chez Helmholtz [Helmholtz 1876, 668], Poincaré [Poincaré 1902, chap. 5] et qu’on retrouve dans un célèbre livre de Edwin Abbott, Flatland : Imaginons-nous qu’il existe dans le plan un être doué de raison qui a deux dimensions ; imaginons en outre que son monde est le plan et que cet être peut prouver par son expérience seulement l’existence de la partie du plan dans laquelle il fait ses observations. Pour mieux imaginer cet être hypothétique, imaginons notre ombre sur un plan ; à chacun de nos mouvements correspond un mouvement de l’ombre et si nous faisons abstraction de notre personne, cette ombre nous apparaît comme un être qui vit et se meut sur le plan. [Veronese 1891, 457] 6
28 Si nous supposons que notre ombre puisse imaginer avec sa raison qu’en dehors du plan existe un point, nous pourrions affirmer que l’ombre parvient à la représentation de l’espace tridimensionnel exactement comme nous parvenons à imaginer l’espace à quatre dimensions. Maintenant si nous nous mettons à la place de notre ombre, ou mieux de cet être hypothétique doué de raison, nous pouvons supposer logiquement qu’en dehors de notre espace il y ait un autre point et en conclure l’existence idéale de l’espace à quatre dimensions [Veronese 1891, 458].
29 Veronese ne propose pas seulement cette expérience mentale, mais aussi une manière de représenter une figure à quatre dimensions sur une feuille, c’est-à-dire sur un plan. Je ne présente pas en détail cette procédure, qui est très compliquée. J’ajoute seulement qu’elle est basée sur une méthode par projections et sections et je résume les observations conclusives de Veronese.
30 Si on recourt à l’abstraction, « il semble qu’on ait presque l’intuition d’une figure à quatre dimensions représentée sur le plan, tandis que ce fait est le résultat de cette abstraction combinée avec l’intuition spatiale » ; cette illusion n’est jamais complète mais elle permet d’imaginer la projection d’une figure à quatre dimensions dans l’espace ordinaire [Veronese 1891, 612] 7.
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31 Veronese croit donc que nous pouvons nous représenter un espace à quatre dimensions, bien que nous n’en puissions pas avoir une intuition complète. Là où l’intuition n’arrive pas, arrive l’abstraction, qui est très développée chez les géomètres comme chez les peintres : se représenter une figure à quatre dimensions dans l’espace ordinaire n est pas autre chose que peindre un corps tridimensionnel sur une toile. La possibilité de se représenter des hyperespaces dépend de l’introduction d’un concept d’intuition géométrique, qui est radicalement différente de l’intuition dont parle Euclide. Ce concept d’intuition géométrique est une extension du concept d’intuition spatiale, qui joue le rôle de critère de distinction entre théories mathématiques et théories géométriques. Le fondement intuitif, qui, selon Kant, était un trait commun à la géométrie et aux mathématiques, devient chez Veronese le critère de distinction entre les deux sciences. Comme on l’a dit en exposant sa conception de la géométrie, on ne peut pas poser comme prémisses des hypothèses qui contredisent les axiomes empiriques, qui expriment les propriétés de l’intuition spatiale. Par exemple on ne peut pas nier qu’un segment rectiligne est déterminé par deux de ses points ou qu’un cercle est déterminé par trois de ses points. Juste parce qu’il dérive d’une combinaison de l’intuition empirique et de l’intuition abstraite, l’espace géométrique doit garder les propriétés que nous attribuons intuitivement aux objets situés dans un espace tridimensionnel.
32 L’accord avec l’intuition spatiale sépare la géométrie d’autres théories mathématiques et limite la possibilité de choisir conventionnellement les prémisses géométriques : c’est là un des points où la position de Veronese diffère le plus de celle de Poincaré, qui nomme géométrie la théorie de l’hyperboloïde à une nappe, qui a pour axiome une proposition qui contredit selon Veronese notre intuition spatiale. La proposition est la suivante : « Il est impossible de faire coïncider une droite avec elle-même par une rotation réelle autour d’un de ses points, ainsi que cela a lieu dans la géométrie d’Euclide quand on fait tourner une droite de 180° autour d’un de ses points » [Poincaré 1887, 82].
33 Poincaré dit explicitement que la raison pour laquelle la géométrie de l’hyperboloïde à une nappe avait échappé aux théoriciens de son temps était précisément le fait qu’elle entraîne des propositions qui étaient considérées comme implicitement fausses parce que trop contraires aux habitudes de notre esprit. Ce qui est selon Veronese la raison pour laquelle elle ne peut pas être considérée comme une géométrie.
7 Poincaré et Veronese : quelle relation y a-t-il entre espace représentatif et espace géométrique ?
34 Je peux maintenant répondre à la question que j’avais posée auparavant, c’est-à-dire comment peut-on justifier la géométrie de l’hyperespace, en comparant les réponses de Poincaré et de Veronese.
35 Poincaré nie la condition que la possibilité abstraite d’une géométrie soit soumise à l’accord complet avec l’espace représentatif, qui sert de cadre à nos sensations et à nos représentations. Il admet pourtant la possibilité de choisir conventionnellement les axiomes de la géométrie. Il n y a pas d’autres conditions que la cohérence pour la possibilité abstraite d’un espace géométrique, mais il y a des conditions pour qu’on puisse se représenter cet espace : on peut se représenter un espace non euclidien ou un
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espace à quatre dimensions mais on ne peut pas se représenter un espace non- archimédien, parce que ceci présuppose un continu de troisième ordre et diffère donc trop profondément de l’espace ordinaire.
36 Veronese au contraire maintient que l’espace géométrique doit s’accorder avec l’intuition de l’espace, mais il nie que l’espace intuitif soit tridimensionnel ou archimédien. Il refuse la possibilité de choisir conventionnellement les axiomes d’une géométrie, mais il soumet les axiomes à l’accord avec l’intuition spatiale exprimée dans les prémisses empiriques : celles-ci ne peuvent pas être niées.
37 Toutes les théories géométriques compatibles avec ces prémisses sont représentables : la géométrie hyperbolique, la géométrie riemannienne, la géométrie de l’hyperespace aussi bien que la géométrie non-archimédienne. Il y’a aussi d’autres théories cohérentes, qui décrivent des propriétés de points et de droites, mais qui ne sont pas des géométries, parce qu’elles contredisent les prémisses empiriques : ce sont plutôt des théories mathématiques abstraites.
38 La condition selon laquelle les axiomes s’accordent avec l’intuition spatiale exprimée par les prémisses empiriques n’est pas une limitation superflue de la liberté de choisir les axiomes. Cette condition joue, selon moi, un rôle important dans l’épistémologie de Veronese : elle fournit un critère de distinction entre la géométrie comme science de l’espace et les mathématiques pures. C’est en effet par les prémisses empiriques que la géométrie est caractérisée comme science mixte et pourtant distincte des sciences formelles.
39 Poincaré au contraire ne soumet la possibilité géométrique à aucune condition supplémentaire par rapport aux mathématiques. Il n utilise donc pas ce critère pour faire une distinction entre possibilité géométrique et possibilité mathématique, bien qu’il dise que le choix des axiomes géométriques est guidé par des faits expérimentaux. Poincaré considère comme géométriquement possible beaucoup plus de théories que Veronese, mais il considère qu’il y a moins de théories géométriques qui soient représentables intuitivement. En particulier, bien qu’il admette que la géométrie de l’hyperespace pourrait être représentable si le milieu dans lequel nous vivons était différent, il affirme que nous ne pouvons pas imaginer un espace non-archimédien. Notre espace représentatif est selon lui incompatible avec une telle géométrie, parce qu’elle modifie radicalement notre notion habituelle de continu de deuxième ordre et diffère pourtant « beaucoup trop profondément de notre espace ordinaire » [Poincaré 1902, 93].
40 Veronese dit au contraire que la géométrie non-archimédienne contient des hypothèses purement abstraites, qu’on peut accepter ou refuser, mais qui ne sont pas en contraste avec l’intuition spatiale. Bien que je n’aille pas exposer de façon détaillée la notion du continu non-archimédien de Veronese ni la comparer avec les observations de Poincaré sur le continu mathématique, j’essayerai néanmoins d’esquisser brièvement la différence entre les deux auteurs à ce propos.
8 Le continu de deuxième ordre et le continu non- archimédien
41 Je présente ici la conception du continu qu’on trouve dans la Science et l’Hypothèse et je ne mentionne pas les élaborations successives de Poincaré sur la notion de continu. On
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peut ainsi mieux comprendre la différence essentielle qui sépare Veronese de Poincaré à cette époque.
42 Poincaré identifie le continu mathématique avec le continu des nombres réels. Cette notion de continu est obtenue par des accroissements successifs du domaine des nombres et implique pourtant que les éléments du continu soient extérieurs les uns aux autres. Le concept de continu mathématique, qui est un système ordonné de points, s’oppose au concept de continu physique, qui est le continu que nous percevons avec les sens. On a observé — affirme Poincaré — qu’on ne peut pas distinguer un poids de 10 grammes et un poids de 11 grammes et de même un poids de 11 grammes d’un poids de 12 grammes, mais qu’on peut facilement distinguer le premier du dernier. Ce qui est exprimé par les relations suivantes : A égale B, B égale C, A est plus petit que C (A = B, B = C, A < C), relations que Poincaré considère comme la formule du continu physique (SH, 51). Il y a là une contradiction, que les mathématiciens éliminent en créant un continu mathématique, qui puisse décrire de façon adéquate le continu physique. Le vrai continu mathématique est un ensemble d’individus extérieurs les uns aux autres : il s’oppose donc à la conception ordinaire de continu, « où l’on suppose entre les éléments du continu une sorte de lien intime qui en fait un tout, où le point ne préexiste pas à la ligne, mais la ligne au point » [Poincaré 1902, 48].
43 La conception de Veronese est radicalement différente. Premièrement le vrai continu mathématique n est pas le continu analytique des nombres réels, mais plutôt un continu synthétique, c’est-à-dire un système ordonné de segments et pas de points. La ligne en effet préexiste au point, qui n’est autre qu’un signe de séparation de segments sur une ligne. Les propriétés du continu synthétique ne correspondent pas nécessairement à celles du continu analytique : en particulier il n’est pas nécessairement vrai que, si A est plus petit que B (A < B), il y ait toujours un nombre naturel n tel que le produit de n par A sera plus grand que B (nA > B). C’est-à-dire, le continu synthétique peut être non-archimédien et contenir des segments infiniment petits (c’est-à-dire qui ne peuvent jamais devenir plus grands que d’autres segments donnés) et infiniment grands. Veronese associe ensuite des nombres à ces segments et il obtient un système ordonné non-archimédien. Poincaré ne comprend pas l’originalité de la construction synthétique de Veronese et dans un compte rendu de l’œuvre de Hilbert de 1904 il affirme que Veronese développe « une véritable arithmétique et une véritable géométrie non-archimédiennes où les nombres transfinis de Cantor jouent un rôle prépondérant » [Poincaré 1904]. Au contraire, les nombres de Veronese sont introduits avec un procédé synthétique radicalement différent de celui de Cantor et en conséquence ils ont des propriétés différentes. Veronese peut étendre aux nombres infiniment grands les propriétés des nombres réels, tandis que les transfinis de Cantor ne sont pas commutatifs par rapport à la somme. En outre, il n’y a pas dans le continu de Veronese un premier nombre infini par rapport aux nombres réels, tandis qu’il y a un premier nombre transfini dans le système de Cantor. Il y a donc une différence essentielle entre le continu analytique dont parle Poincaré et le continu synthétique de Veronese, bien que Poincaré ne semble pas capable de la comprendre.
44 Deuxièmement, Veronese diffère de Poincaré parce qu’il croit que le continu mathématique, s’il est conçu synthétiquement, donne une description adéquate du continu intuitif. Poincaré au contraire soutient que le continu mathématique s’oppose à la conception ordinaire du continu.
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45 Enfin, selon Veronese le continu non-archimédien ne contredit nullement les propriétés du continu intuitif, parce que les segments infiniment petits ne sont pas observables. Le continu non-archimédien est donc aussi représentable qu’un espace à quatre dimensions. C’est là une troisième différence : Poincaré au contraire soutient que nous ne pouvons pas imaginer un espace non-archimédien, parce qu’il diffère trop profondément de notre espace ordinaire.
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— 1889 Il continuo rettilineo e l’assioma V di Archimede, Memorie della Reale Accademia dei Lincei, Atti della Classe di scienze naturali, fisiche e matematiche, Série 4, 6, 603-624. — 1891 Fondamenti di geometria a più dimensioni e a più specie di unità rettilinee esposti in forma elementare. Lezioni per la Scuola di magistero in Matematica. Padova: Tipografia del Seminario. — 1893-94 Osservazioni sui principii della geometria, Atti della Reale Accademia di scienze, lettere ed arti di Padova, 10, 195-216. — 1895-97 Elementi di Geometria, ad uso dei licei e degli istituti tecnici (primo biennio), trattati con la collaborazione di P. Gazzaniga, Verona e Padova: Drucker. — 1902 Les postulats de la Géométrie dans l’enseignement, Compte Rendu du 2e Congrès international des mathématiciens (Paris 1900), Paris : Gauthier-Villars, 433-450. — 1905 La geometria non-Archimedea. Una questione di priorità, Rendiconti della Reale Accademia dei Lincei, 14, 347-351. — 1906 Il vero nella matematica. Discorso inaugurale dell’anno scolastico 1905-1906 letto nell Aula Magna della R. Università di Padova il giorno 6 novembre 1905, Roma: Forzani e C. — 1909 La geometria non-Archimedea, in G. Castelnuovo (ed.), Atti del 4° Congresso internazionale dei Matematici (Roma 1908), Roma: Tipografia dell’Accademia dei Lincei, I, 197-208. Traduction française in Bulletin des sciences mathématiques, 33, 186-204.
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NOTES
1. Le texte suivant est une nouvelle version d’une communication au Séminaire des Archives Poincaré sur le concept d’espace. Je remercie tous les participants pour les précieuses observations critiques dont j’ai tenu compte pendant la rédaction finale du texte. 2. Veronese n’utilise pas le terme semi-empirique, qui a été introduit par Aldo Brigaglia dans un article sur Giuseppe Veronese et la géométrie de l’hyperespace en Italie [Brigaglia 1994, 256]. 3. Les prémisses empiriques de la géométrie sont selon Veronese les suivantes : 1) Il y a des points distincts. Tous les points ne sont pas identiques. 2a) Il y a un système de points à une dimension identique par rapport à la position de ses parties et déterminé par deux de ses points et continu. Ce système s’appelle ligne droite. 2b) Il y a des points en dehors de la ligne droite. Chaque point qui n’appartient pas à la droite détermine avec chacun des points de celle-ci une autre ligne droite. Si deux lignes droites quelconques ont un point commun A, un segment (AB) de l’une est identique à un segment (AB’) de l’autre. 4) Si un côté d’un triangle quelconque devient indéfiniment petit, la différence entre les deux autres côtés devient aussi indéfiniment petite. 5) Sur deux droites quelconques on choisit deux couples de segments AB = XY et AC = XZ ; si le segment BC = YZ, alors les deux droites sont identiques. 4. “Lo spazio generale è dato da un sistema di punti tale che, data o costruita una figura qualunque vi è almeno un altro punto fuori di essa; le cui proprietà non dimostrabili derivano in parte dall’osservazione esterna e in parte da principi astratti che non contraddicono alle prime; e
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le figure, finché il punto conserva il suo primitivo significato sono sempre accompagnate dall’intuizione spaziale.” 5. “(...) il punto non è né un sistema di numeri, né un oggetto di natura qualsiasi, ma il punto tale e quale ce lo immaginiamo nello spazio ordinario; e gli oggetti composti di punti sono oggetti (figure) a cui applichiamo continuamente l’intuizione spaziale combinata coll’astrazione, e quindi il metodo sintetico. ” 6. “Immaginiamo che nel piano, esista un ente ragionevole a due dimensioni, il cui mondo non sia che il piano e che colla sua esperienza possa provare soltanto l’esistenza della parte del piano in cui esso può eseguire le sue osservazioni. Per immaginarci meglio questo essere ipotetico, figuriamoci la nostra ombra sopra un piano; ad ogni nostro movimento corrisponde un movimento di essa, e se noi facciamo astrazione dalla nostra persona, ci pare che quell’ombra sia un ente che abbia vita e si muova nel piano. ” 7. “ (...) pare quasi di avere l’intuizione di questo fatto, mentre esso è il risultato di questa astrazione combinata colla intuizione spaziale. Certo che la illusione non è completa come nella rappresentazione piana, perché dovremmo intuire la retta PX in modo da non avere che un solo punto con (ab), e ciò è impossibile, perché non intuiamo completamente che lo spazio a tre dimensioni. Ma come questa illusione non è necessaria per la costruzione della rappresentazione piana di una figura a tre dimensioni, non lo è neppure in quella delle figure a quattro o a più di quattro dimensioni nello spazio ordinario. ”
RÉSUMÉS
Giuseppe Veronese (1854-1917) est connu pour ses études sur les espaces à plusieurs dimensions ; moins connus sont les écrits « philosophiques », qui concernent les fondements de la géométrie et des mathématiques et qui expliquent les raisons pour la construction d’une géométrie non- archimédienne (une dizaine d’années avant David Hilbert) et la formulation d’un concept de continu, qui contient des éléments infinis et infiniment petits. L’article esquissera quelques traits saillants de son épistémologie et analysera le rapport entre géométrie et intuition spatiale, en comparant les conceptions de l’espace géométrique dans les œuvres de Veronese, Helmholtz et Poincaré. Selon Veronese l’espace intuitif, de même que l’espace géométrique, n’est pas euclidien et il n’a pas un nombre déterminé de dimensions : on peut donc avoir l’intuition aussi bien d’un espace à quatre dimensions que d’un continu non-archimédien.
Giuseppe Veronese is known for his studies on spaces with more dimensions; less known are his “philosophical” writings, that concern the foundations of geometry and mathematics and explain the reasons for constructing a non-Archimedean geometry (several years before David Hilbert’s Grundlagen) and the formulation of a concept of continuity that admits infinitely big and small quantities. After sketching some relevant aspects of Veronese’s epis-temology, the article will analyse the relation between geometry and spatial intuition by a comparison of Veronese’s, Helmholtz’s and Poincaré’s conceptions of the geometrical space. According to Veronese, the representational space, and the geometrical space as well, are not Euclidean nor have a definite number of dimensions: that is why one can represent himself a space with four dimensions and even a non-Archimedean continuum.
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AUTEUR
PAOLA CANTÙ Laboratoire d’Histoire des Sciences et de Philosophie, Archives Henri Poincaré, Nancy-Université
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Sur le statut des diagrammes de Feynman en théorie quantique des champs
Alexis Rosenbaum
Introduction
1 Au fil d’un ouvrage magistralement documenté, David Kaiser [Kaiser 2005] a pu retracer une histoire des diagrammes de Feynman et montrer la fulgurante ascension de ces modestes objets graphiques durant la seconde moitié du vingtième siècle. Inventés par un jeune physicien des années 1940, ils se diffusèrent en effet avec succès en physique nucléaire et en physique des particules, mais également à travers la théorie de la gravitation et la physique des solides, jusqu’à devenir une composante prévalente de l’imagerie de la physique fondamentale. Diffusion dont l’une des caractéristiques les plus étonnantes est sans doute de s’être opérée à tous les niveaux de pratique de la physique, des manuels d’enseignement aux articles de recherche, en passant par les logiciels de calculs et l’évaluation heuristique des phénomènes.
2 Si leur histoire semble désormais avoir livré l’essentiel de ses secrets, ces diagrammes demeurent pourtant parfois l’objet d’un malentendu épistémologique. Richard Feynman rendit public leur usage dans le cadre de calculs de dispersion en électrodynamique quantique, une branche de la théorie quantique des champs. Tandis qu’il s’efforçait de calculer des amplitudes de probabilité de transition entre deux états d’un système, le physicien se trouvait en effet confronté à des développements perturbatifs complexes. Pour faciliter ses calculs, Feynman entreprit d’associer chaque terme mathématique à un graphe schématisant les événements intermédiaires ayant pu avoir lieu entre les deux états considérés. Les particules y étaient représentées par des lignes (qui pouvaient être dessinées de plusieurs façons en fonction du type de particule désigné) et leurs interactions par des vertex. Peu à peu, Feynman élabora ces graphes de façon plus formelle et étendit leur portée. Mais cet usage, dès son origine, semblait contrevenir à un principe philosophique de l’interprétation orthodoxe de la
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mécanique quantique, à savoir l’impossibilité de contempler ou de représenter des événements quantiques. Ce principe renvoie comme on le sait à l’interprétation courante d’une des relations d’incertitude de Heisenberg, selon laquelle on ne saurait en aucun cas connaître de façon simultanée et aussi précise que souhaité la position et l’impulsion d’une particule. Il faut rappeler que le principe philosophique qui est dérivé de cette interprétation revêt une acception radicale dans les écrits de Niels Bohr, pour qui aucune imagerie — qu’elle soit matérielle ou mentale — ne peut prétendre à une valeur réaliste dans le domaine quantique. Selon Bohr, la physique quantique doit en effet abandonner le projet de ressembler à ce qu’elle prend comme objet d’analyse et accepter que la réalité subatomique n’est appréhendée qu’indirectement, c’est-à-dire seulement à travers un formalisme mathématique qui permet de calculer les effets produits par des dispositifs expérimentaux précis. Parce que les systèmes étudiés sont impossibles à séparer du cadre expérimental dans lequel ils sont décrits, toute tentative pour se représenter ces systèmes « en eux-mêmes » est pour Bohr vouée à l’échec. Et parce qu’elles sont dérivées de la perception de notre environnement, les images que nous sommes tentés d’échafauder pour la représentation des phénomènes quantiques sont par nature susceptibles d’être trompeuses. À commencer par les images de trajectoires d’objets en mouvement, dont l’inadéquation est une caractéristique inférée des paradoxes auxquels conduit systématiquement leur usage.
3 Mais si une interprétation rigoureuse de la physique quantique requiert un abandon des images à vocation réaliste, que représentent donc les diagrammes de Feynman, abondamment utilisés en théorie quantique des champs ? Permettent-ils en un sens ou un autre de dépasser l’interdit bohrien de la représentation ? Il serait tentant de voir dans leur extraordinaire diffusion à travers plusieurs domaines qui relèvent de la mécanique quantique un démenti à 1’« iconoclase » de l’interprétation orthodoxe, voire à la conception anti-réaliste de la connaissance humaine promue par la majorité des défenseurs de cette interprétation. Certains physiciens, rappelle David Kaiser, en vinrent justement à considérer que les diagrammes de Feynman capturent dans une certaine mesure la réalité physique, en tout cas de façon plus directe que d’autres outils visuels [Kaiser 2005, 362, 368-9]. Cette tentation d’un réalisme d’« exception » réservé aux diagrammes de Feynman n’est pas surprenante. Après tout, ces diagrammes semblent bien prolonger — sur le papier et dans l’esprit des physiciens — les traces matérielles enregistrées dans les détecteurs de traces (chambres à bulles, chambres à brouillard, chambres à fils, émulsions photographiques...). Ces détecteurs, fort répandus en physique des particules, n’invitent-ils pas à représenter les processus quantiques invisibles sur le modèle des traces qu’ils fournissent [Harré 1988, 69] ? Incarnation graphique de cette représentation de d’invisible, les diagrammes de Feynman réintroduiraient ainsi la possibilité d’une connaissance visuelle indirecte du monde quantique, c’est-à-dire la possibilité d’offrir un aperçu sur « ce qui se passe réellement derrière les équations », en réinfusant des images conformes à nos intuitions classiques, et notamment en rétablissant d’idée de trajectoire dans ses droits. Dès lors, la formidable popularité des diagrammes de Feynman ne serait-elle pas d’indice d’un retour partiel à des conceptions plus réalistes ? Même si ces diagrammes ne représentent pas fidèlement les phénomènes quantiques, n’offriraient-ils pas à tout le moins un guide lorsque nous tentons de les imaginer [Meynell 2008] ?
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FIGURE 1: Deux diagrammes de Feynman pour un processus d’annihilation d’un électron et d’un positron suivi de la production d’une paire de fermions (quarks ou leptons).
4 Nous voudrions montrer ici en quoi il est inapproprié de conférer aux diagrammes de Feynman une valeur figurative si l’on se réfère à l’usage courant qui en est fait en théorie quantique des champs, et pourquoi, par conséquent, ces objets graphiques n’infligent aucun démenti à l’anti-réalisme de l’interprétation orthodoxe de la physique quantique. Pour ce faire, nous indiquerons tout d’abord pourquoi les diagrammes de Feynman ne sont nullement des images à vocation figurative. Puis nous présenterons le cadre conceptuel dans lequel il nous paraît plus légitime de situer leur statut.
1 Les diagrammes de Feynman ne représentent pas des événements physiques
5 Commençons par montrer que, dans l’usage standard de la théorie quantique des champs, les diagrammes de Feynman ne sont pas employés en tant que représentation figurative d’une réalité physique empirique1. Nous procéderons en distançant progressivement ces diagrammes de leurs apparentes prétentions iconiques à l’aide de trois arguments (A, B et C), puis nous rappellerons le point de vue de Feynman lui- même (D).
6 A) Tout d’abord, nul physicien ne songerait à considérer un diagramme de Feynman comme le tableau fidèle d’une scène subatomique, et ce au moins pour trois raisons simples : un diagramme de Feynman ne contient pas de données quant à la position des particules désignées, il est dépourvu d’échelles et ses lignes ne représentent aucunement des distances. Ces trois éléments renvoient tout simplement au fait qu’un tel diagramme est un graphe qui, en tant que tel, n’a pas vocation à la fidélité figurative. En outre, tout diagramme de Feynman correspond à une infinité d’événements possibles : même dans le cas le plus simple d’une particule libre, un diagramme renvoie à une somme infinie de « chemins spatio-temporels »2 que le calcul doit prendre en compte. En ce sens, un diagramme de Feynman ne peut être une image à vocation figurative, car celle-ci serait la représentation d’une scène unique et non d’une infinité de récits.
7 Néanmoins, ce constat n’est pas rédhibitoire car on pourrait envisager qu’un diagramme de Feynman constitue une forme de représentation « élaguée » ou « épurée » : il pourrait conserver des propriétés topologiques d’une scène physique, à défaut d’en conserver les propriétés géométriques, ou constituer une sorte de représentation idéalisée d’un ensemble de scènes physiques, à la manière d’un prototype. Après tout, la plupart des diagrammes que nous employons ne sont que des
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représentations schématiques destinées à présenter la disposition des parties d’un système ou son fonctionnement (on peut songer à la célèbre carte du métro londonien, ou à la représentation d’une installation électrique, par exemple). Leur objectif véritable n’est pas l’exactitude, mais l’illustration de traits ou de relations importantes que l’auteur du diagramme souhaite exhiber. Il convient donc de montrer pourquoi un diagramme de Feynman ne peut même pas prétendre à ce statut plus modeste.
8 B) Les diagrammes de Feynman employés en théorie quantique des champs correspondent à des éléments d’une somme conduite sur des amplitudes de probabilité et non des probabilités physiques. Les différents diagrammes correspondant aux termes d’une somme figurent donc non pas des événements physiques à proprement parler, mais des « subpossibilités » nécessaires au calcul quantique. Chaque graphe correspond à une subpossibilité qui se « superpose » aux autres ou qui « interfère » avec les autres. Pour un état initial et un état final physiques donnés peuvent ainsi être tracés et comptabilisés des graphes qui mettent en jeu un nombre différent de particules et même des types de particules différents [Weingard 1988].
9 Comme ces subpossibilités sont par définition inobservables, les physiciens nomment souvent les particules en question des « particules virtuelles ». Par opposition aux particules « réelles » qui désignent les états initial et final physiques, ces particules virtuelles ne figurent qu’en position intermédiaire dans les diagrammes et sont par définition exclues de toute mesure. Si les particules impliquées dans les parties intermédiaires des graphes possédaient une réalité empirique, les diagrammes de Feynman seraient effectivement incompatibles avec les principes formels de la mécanique quantique, car ils correspondraient à des termes mathématiques assignant des trajectoires à ces particules. Mais ce problème ne se pose pas avec les particules virtuelles, auquel le principe d’incertitude ne s’applique pas3.
10 Certains interprètes considèrent que la raison profonde de cet état de fait est que les diagrammes de Feynman employés en théorie quantique des champs correspondent seulement à des éléments d’un développement perturbatif — une approximation en série de puissances — de la solution des équations concernées. Un diagramme de Feynman serait alors le reflet non d’une solution exacte, mais d’un « détour » mathématique. Et l’on pourrait imaginer que la solution exacte échappe, elle, à la mise en diagramme : Lacki [Lacki 1998] suggère que si nous possédions cette solution exacte, nous la comprendrions peut-être moins bien que d’approximation perturbative qui nous est devenue familière. Formulé en termes plus radicaux, cela signifie que les particules virtuelles pourraient être de simples artefacts de d’expansion perturbative [Rohrlich 1999, 363]. Même si on ne souscrit pas à cette dernière formulation radicale, elle rappelle à quel point la figuration des particules virtuelles doit être objet de circonspection. Certes, tout physicien sait que les particules réelles occupent les entrées et sorties des diagrammes, tandis que les particules virtuelles, qui désignent les contributions des subpossibilités nécessaires au calcul, ne sont présentes que dans les parties intermédiaires. Mais il n’en reste pas moins que la coprésence de particules réelles et virtuelles sur un même graphe contribue à brouiller la différence de leur statut.
11 L’argument B) nous semble suffisant pour écarter l’interprétation réaliste des diagrammes de Feynman, en ce qui concerne d’usage standard qui en est fait dans la théorie quantique des champs, puisqu il permet de comprendre que ces diagrammes
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n’ont pas vocation à être des images, même épurées, de scènes physiques dans d’espace-temps classique [Fox 2008].
12 C) Nous ajouterons toutefois, à titre d’indice supplémentaire (ou d’élément à charge au dossier), que les diagrammes de Feynman utilisés le plus couramment pour le calcul perturbatif sont tracés dans un espace abstrait où chaque ligne correspond à une valeur possible de l’impulsion d’une particule. Cette complication supplémentaire peut sembler étrange, car les subpossibilités ou « pseudo-événements » intermédiaires pourraient tout simplement être dénombrés dans un diagramme d’espace-temps, à la manière des diagrammes de Feynman qui figurent dans les ouvrages de vulgarisation. Mais en pratique (pour des raisons que nous ne préciserons pas ici), les calculs s’avèrent en général plus faciles à conduire à partir de la « représentation en impulsion ». Ceci éloigne pour ainsi dire d’un degré supplémentaire les chemins figurés dans les diagrammes des tracés de trajectoire au sens visuel du terme, puisque la référence à d’espace physique est écartée des diagrammes dans cette représentation en impulsion.
13 Ceci nous suggère aussi qu’il existe une autre origine au malentendu réaliste que nous souhaitons dissiper. Est en effet source d’erreur le fait que certains diagrammes de Feynman reprennent la convention de représentation des diagrammes d’espace-temps, en particulier les diagrammes de Minkowski avec lesquels il arrive qu’ils soient confondus. Le problème tient parfois au fait que certains manuels introduisent d’abord les diagrammes de Feynman en les présentant comme des diagrammes d’espace-temps, puis passent aux diagrammes d’impulsion pour engager les calculs. Peut-être la ressemblance entre diagrammes de Minkowski et diagrammes de Feynman explique-t- elle d’ailleurs partiellement pourquoi les seconds ont été si facilement acceptés par tant de physiciens, dans la mesure où les premiers faisaient déjà partie de leur bagage culturel. Mais en réalité, les diagrammes de Feynman ne contiennent nulle indication de position spatiale ou temporelle (si d’on excepte la distinction triviale entre lignes entrantes et sortantes).
14 L ensemble de ces arguments permet de conclure que les diagrammes de Feynman ne figurent pas une réalité physique empirique, pas même de façon simplifiée. Ils ne représentent pas des événements qu’on pourrait voir où que ce soit, ou qu’un quelconque appareil de mesure pourrait donner à regarder, ou encore qu’une quelconque prise de vue pourrait surprendre. Cette conclusion a pour conséquence que même en tant que simple « image mentale », le caractère figuratif peut être selon nous dénié aux diagrammes de Feynman, dans la mesure où une telle image mentale ne serait que la version intérieure d’une image matérielle possible, alors qu’aucune procédure imaginable ne peut justement offrir l’image matérielle en question. Les diagrammes de Feynman ne peuvent même pas prétendre représenter ce qui se passe « dans le noir » hors de toute mesure ou perturbation extérieure.
15 D) Il ne nous semble pas sans utilité de rappeler que Feynman lui-même n’attribuait pas de fidélité figurative à « ses » diagrammes. Kaiser [Kaiser 2005, 176-7] rappelle certes que le jeune physicien des années 40 élabora ces graphes dans le cadre d’efforts personnels pour imaginer les phénomènes physiques dénotés par les équations de la théorie des champs. Il est également vrai que les pratiques de Feynman tranchaient sur d’austérité graphique entretenue par une partie des pères fondateurs de la mécanique quantique4. Et il est vraisemblable que le statut de ses diagrammes fut, même pour lui, sujet à réflexion, puisque, si d’on en croit son collaborateur Freeman Dyson, il les
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considéra d’abord comme des images ajoutant une dimension intuitive à la compréhension des processus physiques [Kaiser 2005, 189-90]5.
16 Toutefois, il n’existe à notre connaissance aucune déclaration écrite ou orale de Feynman selon laquelle les diagrammes constitueraient l’image figurative d’une réalité empirique. Au contraire, il mettait en garde contre d’illusion qui consisterait à le croire. Il insistait en effet sur la commodité de ce type d’images, qu’elles soient matérielles ou mentales, et prenait garde de ne pas leur attribuer une valeur plus fondamentale. Le comportement des objets subatomiques lui semblait énigmatique, bien éloigné de celui des objets macroscopiques, et seulement analysable dans les termes abstraits du formalisme. S’il était attaché à mieux « comprendre » les équations à d’aide d’images qu’il se constituait et qui faisaient partie intégrante de son travail de recherche, il ne conférait pas pour autant un statut théorique à cette démarche [Gleick 1994, 276-8]6.
17 En ce sens, d’invention de Feynman n’entrait nullement en contradiction avec les conceptions « orthodoxes ». Même si, contrairement à Bohr, le physicien américain n’était guère enclin à statuer en toute généralité sur la possibilité ou l’impossibilité de la représentation, il ne se méfiait probablement pas moins que Bohr des images classiques dans le domaine quantique. Après tout, il ne s’agissait pas non plus pour Bohr d’interdire en pratique tout recours aux images. Les physiciens étaient même invités par le physicien danois à faire usage d’images empruntées au monde macroscopique, mais seulement en tant qu’elles pouvaient servir d’analogies et d’auxiliaires pour la pensée. Comme on le sait, dans ses travaux épistémologiques, Bohr s’efforça en particulier de montrer d’importance de deux images empruntées à la physique classique, celle des objets matériels à trajectoire continue et celle des ondes, chacune pouvant être convoquée pour appréhender certaines caractéristiques des phénomènes quantiques en relation avec des instruments de mesure spécifiques. C’est seulement en tant qu’image à vocation réaliste que Bohr leur déniait toute valeur. On aura compris que l’usage courant des diagrammes de Feynman en théorie quantique des champs ne contrevient justement pas à cette conception.
2 Les diagrammes de Feynman constituent une notation graphique de termes mathématiques
18 Indiquons maintenant comment situer conceptuellement le statut des diagrammes de Feynman. Nous montrerons en quoi il est plus approprié de considérer ces diagrammes comme une notation graphique de termes mathématiques (A), puis tenterons de réfléchir sur le statut hybride du concept de notation graphique (B). Enfin, à la lumière de cette interprétation, nous reviendrons une dernière fois sur la tentation réaliste évoquée précédemment (C).
19 A) Quitte à inverser la direction de l’intuition, il nous semble préférable de considérer que les diagrammes de Feynman constituent avant tout une notation graphique de termes mathématiques. Par notation graphique, nous entendons un répertoire de formes spatiales, combinables selon des règles spécifiées et destinées à représenter symboliquement un ensemble d’éléments donnés ainsi que leurs relations. Le choix d’une notation graphique, en ce sens, est partiellement conventionnel, comme d’atteste
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la multiplicité des notations graphiques qui peuvent être employées en musique, par exemple, ou en généalogie (à travers la représentation du système de parenté).
20 Pour ce qui nous concerne ici, il convient de remarquer d’abord que le recours aux diagrammes de Feynman en théorie quantique des champs n’est pas indispensable : les calculs perturbatifs peuvent être menés en leur absence. Car il existe une équivalence entre termes algébriques et graphes. Une fois connues les conventions de construction des diagrammes, leur tracé se déduit de l’expression algébrique de la somme perturbative. Dans l’usage courant des diagrammes de Feynman en théorie quantique des champs, la série des termes permet de construire la série des graphes sans ambiguïté. Inversement, on peut reconstituer la série des termes algébriques à partir de la série des graphes grâce à des règles qui fournissent d’équivalent algébrique de chaque type de ligne et chaque type de vertex (ces règles sont d’ailleurs le plus souvent appelées « règles de Feynman »). Les termes mathématiques épuisent en ce sens le contenu des diagrammes de Feynman : ceux-ci ne disent rien de plus que d’expansion perturbative.
21 Cette équivalence peut surprendre car si les diagrammes de Feynman ne « disent » rien de plus, ils sont pourtant beaucoup plus « parlants ». Une image vaut mille mots, dit l’adage, et un diagramme fait souvent apparaître de façon plus immédiate ou plus saillante certaines informations contenues dans un terme algébrique, jusqu à rendre parfois explicite ce que celui-ci masque [Larkin & Simon 1987]. Plus généralement, la spatialisation des informations peut donner lieu pour le physicien à des inférences plus rapides ou plus aisées, ce que chacun constate d’ailleurs en passant de d’expression analytique d’une fonction à sa courbe [Brown 1996]. Même lorsque le terme mathématique et l’objet graphique sont équivalents d’un point de vue informatif, leur efficacité inférentielle et leur pouvoir expressif peuvent donc être différents pour la cognition humaine [Barwise & Etchemendy 1990], [Recanati 2005]. En l’occurrence, dans le cas des diagrammes de Feynman, le praticien est vite séduit par d’aide visuelle que ceux-ci apportent à la compréhension et au calcul de la somme perturbative, ainsi que par la facilité avec laquelle on peut les manipuler, que ce soit sur le papier ou, dans une certaine mesure, mentalement. Pour être plus précis :
22 - d’une part, les diagrammes de Feynman facilitent le dénombrement et la classification des subpossibilités sur lesquelles porte le calcul perturbatif, autorisant un inventaire visuel sans lequel un terme risquerait par exemple d’être oublié ou deux termes confondus. Les diagrammes de Feynman constituent en ce sens un livre de comptes des termes de la somme algébrique, ce qui leur confère un avantage à la fois mnémonique (en tant qu’« aide-mémoire » pour le calcul) et heuristique. À partir du dénombrement des diagrammes, le physicien peut passer au calcul proprement dit ;
23 - d’autre part, les diagrammes de Feynman permettent de donner une valeur narrative à la somme algébrique, en présentant un scénario correspondant à chacun des termes. Une fois traduit en diagramme, le terme mathématique acquiert une lisibilité particulière non seulement parce qu’il s’offre plus facilement à l’inspection directe, mais aussi parce qu’il peut être lu à la manière d’une histoire : une particule virtuelle est « émise », puis elle « transite » jusqu à une seconde particule qui l’« absorbe », pendant qu’une troisième se « transforme » en une paire de particules nouvelles, etc. Comment d’aridité du développement algébrique serait-elle aussi parlante7 ? On peut ainsi comprendre pourquoi tant de physiciens ont choisi de conserver cet outil graphique, capable de faciliter et améliorer grandement l’efficacité des raisonnements.
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Mais on peut comprendre aussi pourquoi jusqu’à la fin de sa vie, le grand rival intellectuel de Feynman — Julian Schwinger — y voyait une invention plus pédagogique que physique.
24 Ces diagrammes nous paraissent donc constituer une excellente notation graphique, dans la mesure où ils permettent de traduire sans ambiguïté des termes algébriques et de bénéficier des commodités d’une représentation spatiale dont les éléments sont facilement manipulables et combinables. En somme, les diagrammes de Feynman possèdent bien un statut pictural (puisqu ils sont dessinés) et narratif (puisqu ils évoquent une série d’événements), mais non une valeur figurative. Remarquons que le premier scientifique à en avoir formalisé l’usage de façon rigoureuse, le mathématicien Freeman Dyson [Dyson 1949], ne s’y était pas trompé : dès qu’il s’attela à élaborer une dérivation des diagrammes à partir des équations de la théorie quantique des champs, il choisit de les considérer comme un outil de comptabilité perturbative et fit abstraction de leurs prétentions figuratives [Kaiser 2005, 188-95]8.
25 B) Reste que nos formulations semblent s’appuyer sur une distinction contestable en donnant d’impression que tracer un diagramme de Feynman consiste à traduire un objet mathématique d’un format vers un autre, de nature radicalement différente. Si le graphe est la version spatialisée, partes extra partes, d’un membre de la série algébrique, on pourrait en effet conclure qu’il existe d’une part une représentation algébrique qui relèverait du format « linguistique », et d’autre part une représentation graphique qui relèverait du format « spatial ». Pour éviter de laisser entendre que nous souscrivons à cette bipolarisation abrupte, il nous paraît important de préciser notre propos en rappelant qu’il est en réalité fort difficile de tracer une frontière stricte entre ces deux types de représentation, et que les diagrammes de Feynman en sont justement, à leur manière, une illustration.
26 Remarquons d’abord, au risque du truisme, que le recours à d’espace n’est pas propre aux images. Toute notation logique ou algébrique recourt elle aussi à un certain type d’occupation de l’espace. Dans la notation standard des fractions, par exemple, le numérateur est au-dessus du dénominateur, ce qui est bien entendu conventionnel : d’autres dispositions auraient pu être choisies, d’autres relations spatiales mises en jeu. Il arrive même que certains symboles logiques aient une valeur iconique (on peut penser notamment aux parenthèses [Recanati 2005]). Pour prendre un autre exemple, les formules employées en chimie retiennent certaines propriétés spatiales des molécules qu’elles désignent. Dans d’histoire de la chimie, le choix des notations a largement évolué, mais d’idée d’une homomorphie minimale entre les formules et les objets désignés a souvent été maintenue, ne serait-ce qu’à travers le principe de la notation des composés à partir de leurs éléments [Klein 2001]. La distinction pure et simple entre « écriture digitale » et « image analogique » serait ici simpliste : les pratiques des chimistes ont bien plutôt dévoilé une riche gamme de solutions entre ces deux pôles de représentation, une combinatoire de signes et d’images qui est d’indice du fonctionnement pragmatique et ouvert de leurs notations. Inversement, plusieurs types d’objets apparemment graphiques peuvent prétendre à un statut intermédiaire entre image et signe, dans la mesure où ils intègrent des éléments linguistiques dans une structure globalement homomorphique [Hall 1996], [Galison 1998]. Remarquons enfin que la distinction que nous sommes spontanément tentés de tracer entre le signe linguistique et d’image est peut-être un préjugé issu de notre écriture alphabétique (qui, après tout, est notre notation-mère). Une forme d’hybridation existe au contraire
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d’emblée dans d’aire culturelle sinisée à travers la notation utilisée par la langue chinoise, si bien que l’idée d’une synonymie entre termes symboliques et termes graphiques y est peut-être plus facile à accepter.
27 Quant aux images mentales, il est fréquent qu’elles mettent justement en jeu des déplacements et des combinaisons entre les deux « formats » évoqués, selon le tempérament cognitif des physiciens. De ce point de vue, Feynman n’était pas en reste puisque, dans ses représentations mentales, les symboles mathématiques se mêlaient de son propre aveu aux images picturales, sans qu’il parvienne à savoir précisément comment [Feynman 1979, 360-1]. La grande créativité de son imagerie semble avoir puisé dans des ressources très variées : l’intuition physique, affirment ceux qui connurent le physicien américain, n’était pas chez lui seulement d’ordre visuel mais également d’ordre auditif et cénesthésique [Gleick 1994, 276]. Ses images mentales paraissent avoir été des objets hybrides, faits de formes, de symboles, de sons ou de couleurs, accompagnés de sentiments physiques comme ceux du mouvement ou de d’accélération. Il y a tout lieu de penser que les célèbres diagrammes étaient issus de cette imagerie toute personnelle : The diagram is really, in a certain sense, the picture that comes from trying to clarify visualization, which is half-assed kind of vague, mixed with symbols. It is very difficult to explain, because it is not clear (…). It is hard to believe, but I see these things not as mathematical expressions, but a mixture of mathematical expressions wrapped into and around, in a vague way, around the object. I see all the time visual things associated with what I am trying to do. (cité in [Schweber 1986, 504-5])
28 C) Enfin, l’interprétation préconisée ici peut susciter une certaine frustration. Affirmer que les diagrammes de Feynman sont des représentations de termes mathématiques donne d’impression de les éloigner doublement du réel (en tant que représentations de représentations), alors qu’ils paraissent ressembler davantage à notre perception de la réalité physique que les termes algébriques. En les disjoignant de toute valeur figurative, ne risque-t-on pas de se replier sur un pragmatisme qui échoue à éclairer leur fécondité ? Faut-il voir une coïncidence dans le fait qu’une simple notation corresponde à une image d’apparence figurative ? Comme l’indiquait Richard Mattuck [Mattuck 1967, 72], “The diagrams are so vividly “physical-looking” that it seems a bit extreme to completely reject any sort of physical interpretation whatsoever”. Bref, en choisissant la prudence contre d’intuition, ne se prive-t-on pas d’un versant essentiel de leur signification ?
29 Il est indéniable que nous sommes habitués à concevoir ce qui relève du symbolisme mathématique ou linguistique comme une représentation dérivée de nos représentations visuelles : réfléchir à quelque chose que l’on connaît, c’est souvent se référer à une image mentale « première », les mots et les équations paraissant « seconds ». Toutefois, l’objectif n’est pas ici de discuter de l’ordre chronologique de nos représentations, mais de conférer aux diagrammes de Feynman un statut épistémologique approprié à leur usage. Or, leur usage est avant tout d’ordre calculatoire. Le physicien réfléchit et progresse dans ses calculs en se déplaçant entre le plan des équations et celui des graphes : cela n’exige nullement de supposer que d’un des deux plans relève plus intimement du « réel » que d’autre, quel que soit le sens prêté à ce dernier comparatif. Nous pouvons en revanche affirmer avec confiance que le physicien peut tantôt s’appuyer sur les équations pour tracer des diagrammes, tantôt à l’inverse s’appuyer sur des diagrammes pour poursuivre son travail algébrique. En ce
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sens, il nous semble suffisant autant qu’approprié de considérer termes algébriques et diagrammes comme des outils de calcul de forme différente.
30 Le fait que les diagrammes offrent une lisibilité supérieure, aussi troublant soit-il, n’impose aucune conclusion métaphysique. Le choix d’une notation judicieuse peut parfaitement être décisif pour la résolution de problèmes (des exemples de l’influence des notations abondent par exemple dans l’histoire des mathématiques). Et le statut de notation n’empêche nullement les physiciens de faire comme si les diagrammes de Feynman avaient une valeur de représentation. Pour ses calculs, chacun est libre de recourir aux images (matérielles ou mentales) de son choix pour autant qu’elles lui paraissent fructueuses. Le praticien du domaine quantique sait très bien ce qu’il a le droit ou pas de faire lorsqu’il manipule des représentations qui n’ont pas de vocation figurative. À chacun de se bricoler des images-outils variées pour guider ses pensées à propos d’espaces différents de l’espace perçu, sans pour autant prétendre figurer la réalité physique « elle-même ». Cette liberté a évidemment une grande valeur heuristique. Est-ce d’une des raisons pour lesquelles la conception que nous esquissons ici a depuis longtemps les faveurs de nombreux physiciens ? Elle semble en tout cas les protéger contre un réalisme naïf, tout en leur garantissant la pleine liberté de se représenter (mentalement ou matériellement) les systèmes étudiés dans la voie qui leur paraît la plus utile9.
Conclusion
31 Nous avons essayé d’expliquer pourquoi d’usage qui est fait des diagrammes de Feynman en théorie quantique des champs n’implique aucun retour à un réalisme pré- quantique. d’aspect dynamique et narratif de ces diagrammes exerce certes une véritable force de séduction, en paraissant offrir une contrepartie physique aux calculs mathématiques. Mais une fois la distinction entre particules réelles et virtuelles correctement caractérisée, il apparaît que ces diagrammes ne sont pas destinés à constituer une représentation naturaliste — pas même simplifiée ou caricaturale — de la réalité.
32 Nous avons ensuite tenté de montrer en quoi il nous semble plus approprié de considérer les diagrammes de Feynman comme une notation graphique de termes algébriques, une forme de sténographie hybride et adaptée aux calculs perturbatifs. N’oublions pas que ces calculs peuvent théoriquement être menés dans leur intégralité sans diagrammes. Au fond, la théorie quantique des champs ne requiert a priori pas plus d’image pour être pratiquée de façon correcte et complète que la mécanique quantique des pères fondateurs. Il s’agit d’un ensemble formel qu’un aveugle de naissance peut théoriquement apprendre et employer pour mener à bien ses calculs. En tant qu’auxiliaires graphiques de ces calculs, les diagrammes de Feynman nous paraissent donc finalement s’inscrire sans rupture épistémologique dans l’histoire du rapport de la théorie quantique aux représentations visuelles.
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NOTES
1. Ce qui n’empêche pas que des usages différents des diagrammes de Feynman aient pu exister ou existent, ce que Kaiser [Kaiser 1999] a pu notamment montrer en ce qui concerne les adeptes du programme de la Matrice S. 2. Nous employons ici une expression associée à la formulation de la théorie quantique des champs en termes d’intégrales de chemin, qui avait les faveurs de Feynman. 3. À l’écoute de la première présentation par Feynman de sa théorie à Pocono (1948), Bohr se leva en protestant contre l’attribution de trajectoires aux particules dans les diagrammes, usage qui lui semblait violer directement le principe d’incertitude. À tort. Bohr vint s’excuser peu après, suite aux « éclaircissements » de son fils Aage [Schweber 1986, 495]. 4. Même si les diagrammes de Feynman s’inscrivent dans une tradition visuelle plus ancienne, durant les années où Feynman est étudiant, la mécanique quantique ne possédait pas véritablement d’imagerie constituée. Pour se confronter aux comportements « contre-intuitifs » des objets quantiques, les représentations matérielles utilisées étaient assez pauvres : parmi elles figure en particulier le diagramme des transitions entre niveaux d’énergie, emprunté à la spectroscopie. Le premier texte influent qui tranche sur cet état de fait semble être un manuel de 1943 rédigé par Gregor Wentzel (Einführung in die Quantentheorie der Wellenfelder), qui contient une
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description d’un processus d’échange de particules préfigurant les diagrammes de Feynman [Miller 1984, 166]. 5. Il existe également quelques notes prises par des étudiants de Feynman qui témoignent en ce sens [Kaiser 2005, 186—7 & note 27], mais elles restent sujettes à caution. 6. On peut sentir cette démarche « pragmatique » notamment dans une lettre de février 1947 : “I dislike all this talk of others [of there] no being a picture possible, but we only need know how to go about calculating phenomenon. True we only need calculate. But a picture is certainly a convenience & one is not doing anything wrong in making one up. It may prove to be entirely haywire while the equations are nearly right—yet for a while it helps” (cité in [Schweber 1986]). 7. Les « histoires » dont il s’agit sont toutefois en un sens contenues dans le développement algébrique si on adopte la formulation en termes d’intégrales de chemin. l’apport narratif saute davantage aux yeux lorsqu’on passe de la formulation canonique de la théorie aux diagrammes. 8. Kaiser [Kaiser 2005, 189] exhibe un extrait de notes prises par des étudiants lors de conférences données en 1951. Dyson y affirme qu’il est préférable d’utiliser les diagrammes seulement comme aide pour visualiser les formules (“only as a help in visualising the formulae”). Visualiser les formules, donc, et non les processus physiques. Même si une note de cours est sujette à caution, la formulation est éloquente. Kaiser exploite d’ailleurs le décalage entre la conception « intuitive » de Feynman et la conception « déductive » de Dyson pour structurer une partie de sa propre présentation [Kaiser 2005, 175-95, 263-70]. 9. Attribuer aux diagrammes de Feynman le statut de notation graphique n’invite à aucun engagement ontologique particulier. Rappelons qu’il n’existe pas aujourd’hui d’accord sur l’ontologie fondamentale de la théorie quantique des champs. Par « ontologie fondamentale » nous entendons l’ensemble des éléments irréductibles que suppose une théorie pour rendre compte du réel. On peut soutenir que l’ontologie fondamentale la plus appropriée pour la théorie quantique des champs est une ontologie des champs, des événements, des processus ou encore des tropes [Kuhlmann, Lyre & Wayne 2002]. Il n’existe pas non plus d’accord sur le fait que les différentes formulations mathématiques de cette théorie (représentation dans l’espace de Fock, représentation en termes d’intégrales de chemin, formulation algébrique…) renvoient à la même ontologie fondamentale. Quand bien même un tel accord existerait, il ne nous procurerait de toute façon que des réponses de portée provisoire. Car en des temps où recherches et controverses sont si ouvertes au cœur de la physique fondamentale, le seul consensus que nous pourrions recueillir est que la théorie quantique des champs sera tôt ou tard supplantée. Si c’est le cas, que deviendront les diagrammes de Feynman ? Apparaîtront-ils comme de simples artefacts des calculs perturbatifs propres à un schéma théorique spécifique et obsolète ? Comme des approximations d’élaborations plus « profondes » ? Comme la préfiguration d’une réalité encore insoupçonnée ? Il semblerait bien téméraire de répondre a priori à ces questions.
RÉSUMÉS
Cet article s’efforce de présenter une conception du statut des diagrammes de Feynman appropriée à l’usage courant qui en est fait en théorie quantique des champs. Son objectif est de montrer que l’usage de ces diagrammes n’est pas susceptible de remettre en cause l’interprétation orthodoxe anti-réaliste de la mécanique quantique. Dans un premier temps est indiqué pourquoi les diagrammes de Feynman ne peuvent être considérés comme des représentations figuratives d’événements physiques. Dans un second temps, l’auteur explique
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pourquoi il semble préférable de concevoir ces diagrammes comme une notation graphique de termes algébriques. Le propos fournit l’occasion de réfléchir sur la capacité de la cognition humaine à intégrer des combinaisons hybrides de signes et d’images.
This article is about the conceptual status of Feynman diagrams in Quantum Field Theory. It is aimed at showing that the standard use of these diagrams does not threaten the bohrian instrumentalist interpretation of quantum mechanics. Firstly, it is explained why Feynman diagrams cannot be considered as naturalist representations of physical events. Secondly, it is argued that these diagrams should better be conceived as a graphical notation of algebraic terms.This conception gives an opportunity to reflect upon the capacity of human cognition to integrate hybrid combinations of symbols and images.
AUTEUR
ALEXIS ROSENBAUM École Nationale Supérieure de Techniques Avancées
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Why Quarks Are Unobservable
Tobias Fox
1 Introduction
1 Observation and observability are key concepts when philosophy of science and epistemology meet. Philosophy cannot predict which objects might be observable and which are not. It has to define “observability” as an epistemological term used to give meaning to scientists’ sentences when they claim to have observed something.
2 In this paper, I shall discuss the observability of a special kind of elementary particle— quarks. Today, it is common in physics to deal with entities that are barely observable, entities which are in principle only indirectly observable or unobservable. Now, with quarks, because of some physical aspects, it is not quite clear whether they are observable or not, even though they are fully-established in theory and sufficient experimental evidence of them exists. This has provoked some comments in philosophy of science, mainly by Kristin Shrader-Frechette and Michela Massimi. They came to the conclusion that quarks are—even directly—observable. Their argument is based on seminal works on the concept of observability by Grover Maxwell, Bas van Fraassen, and Dudley Shapere. My approach to this debate has two directions. First, to search for metaphors used by scientists to describe their findings in quark physics. And to address whether quarks can be seen or only “seen” in a certain experimental context. It is logical that if they could only be “seen”—and this is my presupposition— then quarks are only “observable”, i.e., metaphorically observable but not literally observable. Something only metaphorically observable could be anything from an entity that is directly observable in another context to unobservable in principle. The idea, that a physical entity is only “observable”, leads us to question its real relation to the epistemological concept of observability.
3 Second, I would like to focus on the exact role jet-events play in experimental particle physics. I will show in detail what jet-events are, and what their unique decay pattern in particle detectors looks like. However, my working hypothesis is that jet-events should not to be used as simple images to predict what might go on at a microphysical scale. The explanatory power of such events is more subtle, particularly when
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considering their relevance to the alleged observation of quarks. My general conclusion is that quarks are neither directly nor indirectly observable. This conclusion is not based on any genuine physical statement or theory in particle physics, but it might have consequence for the ongoing philosophical debate of scientific realism, particularly as the real classification of entities like quarks becomes more difficult when they cannot literally be observed or indirectly observed. Though, this final thought is not addressed in this paper.
4 So my chief statement is that quarks are not observable. To come to this conclusion I will try to develop, in the following Section 2, a concept of observation and draw a distinction between direct and indirect observation. Anything that is able to fulfil the criteria of direct or indirect observation has to be called directly or indirectly observable—regardless of whether this actually has been done or not. In Section 3, I will give an introduction to jet-events that occur in certain conditions when highly accelerated particles collide inside particle detectors. Then I shall apply the concept of observation to jet-events (Section 4). In this context at least, quarks are not observed in jet-events, and thus, we have strong indications to believe that they will not be observed under any other experimental conditions either.
2 What is observation of objects in physics?
2.1 Earlier proposals
5 The challenge to define “observation” has a lengthy history in philosophy. It is also a common subject in the philosophy of science and has particular relevance to the debate on scientific realism. I would like to focus on the meaning of observation in relation to physical objects. The most famous thoughts on this subject came from Grover Maxwell, Bas van Fraassen, and Dudley Shapere. And these authors are repeatedly quoted by later philosophers writing on quarks. Maxwell argues for the broadest possible concept of observation in a 1962-paper The Ontological Status of Theoretical Entities [Maxwell 1962]. In this work, Maxwell calls some objects theoretical while we would describe them as physically motivated and call them sometimes indirectly observable. He is not interested in a distinction between direct and indirect observation, but in a distinction between observability and unobservability, or, in his terms, between observation and theory. His inquiry looks at microbes, molecules, and elementary particles. These are objects that existed in scientific thought long before any tools were invented to observe them. Maxwell’s main thesis is that there can be no clear distinction between observation and theory: The point I am making is that there is, in principle, a continuous series [of observations] beginning with looking through a vacuum and containing these as members: looking through a window pane, looking through glasses, looking through binoculars, looking through a low-power microscope, looking through a high-power microscope, etc., in the order given. The important consequence is that, so far, we are left without criteria which would enable us to draw a non-arbitrary line between “observation” and “theory”. [Maxwell 1962, 7]
6 By Maxwell’s reasoning, deciding that an observation through a microscope is not an observation leads us to try and establish a reliable boundary between observation and non-observation. Maxwell argues that trying to fix this boundary at the point where tools like microscopes are used is a dead end, because window panes and glasses are in
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some way tools too and nobody would realistically claim that the observation of an object through glasses is an indirect observation. His conclusion is that everything called observation in science, is actually an observation. And that observability is changeable and depends on the scientific tools and theories available at the time. I conclude that our drawing of the observational-theoretical line at any given point is an accident and a function of our physiological make-up, our current state of knowledge, and the instruments we happen to have available and, therefore, that it has no ontological significance. [Maxwell 1962, 14-15]
7 To Maxwell, observation is more or less any reasonable inference that can be made from experimental data or examined entities. This concept is based on the obvious absence of a dividing line between observation and non-observation (or theory). According to Maxwell, we are even able to observe the gravitational field or it’s metric tensor GM„ “by sitting on a chair—and we do it directly” [Maxwell 1962, 14]. To me the advantage of Maxwell’s concept is that it is intuitive and carefully avoids the problem of theoryladen observation and other epistemological inferences. It openly states that any observation is laden by presupposed knowledge, whether scientific or knowledge of other kinds. The problem of theoryladeness in science (i.e., that nature and result of an observation depends on certain presupposed theories) is discussed elsewhere [Hanson 1958], [Kuhn 1962], [Goodman 1978].
8 Possibly, one could say that theoryladeness is everywhere, and therefore harmless. But Maxwell’s definition of observation stops being intuitive if entities are categorically unobservable in principle, like natural constants or pure mathematical objects (e.g., virtual particles). And it seems to me that some data from particle experiments that are called observations rely too heavily on theory, for example jet-events on which I will comment in Section 3.
9 In Bas van Fraassen, we find a sceptic concerned by the breadth of the concept “observable”. He does not deal with the distinction between direct and indirect observation but tries to pin down the distinction between observable and unobservable. This is quite simple, especially from an empiricist’s point of view. Van Fraassen’s examples are archetypal—seeing a vapour trail in the sky is not the same as observing a jet, but the unaided perception of the jet itself is (we come back to this later). Similarly, the track of a particle is not to be treated as an observation of the particle itself [van Fraassen 1980, 17]. Van Fraassen challenges Maxwell’s view and suggests a concept of observation that is aware of its vagueness and its anthropocentric quality. The vagueness follows, according to van Fraassen, from the fact that drawing the line between what is observable and what is only detectable is not possible without arbitrariness in Maxwell’s sense [van Fraassen 1980, 16]. The anthropocentric quality derives from the “able”-part in “observable”. If something is x-able, someone has to have the ability to x. Van Fraassen asks further whether it is possible for technical aids such as microscopes and telescopes to increase the number of observable objects. Van Fraassen reasons: This strikes me as a trick, a change in the subject of discussion. I have a mortar and pestle made of copper and weighing about a kilo. Should I call it breakable because a giant could break it? Should I call the Empire State Building portable? Is there no distinction between a portable and a console record player? The human organism is, from the point of view of physics, a certain kind of measuring apparatus. As such it has certain inherent limitations (…) It is these limitations to which the “able” in “observable” refers—our limitations, qua human beings. [van Fraassen 1980, 17]
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10 Furthermore, van Fraassen disassociates the observable-question from questions about observable and unobservable objects. By ridding himself of the need to find a line between observable and unobservable, van Fraassen is able to discuss his specific view of scientific anti-realism without being hampered by examples of new pretended observations in microphysics.
11 This survey—although a reliable alternative—has much in common with Maxwell’s and so my suggestions run along similar lines as before. There is, as argued above, no compelling arbitrariness in the demarcation, or any necessary reason, to restrict observability to the unaided human eye. Van Fraassen’s rejection of technical aids to observe events leads to a statement like this: A look through a telescope at the moons of Jupiter seems to me a clear case of observation, since astronauts will no doubt be able to see them as well from close up. [van Fraassen 1980, 16]
12 Thus, as to van Fraassen, the observable-label depends on the circumstances of the observation. While I fully agree with this opinion, his example seems to be based on something still open to be demonstrated. Since nobody so far has been close to the moons of Jupiter, all we know about their existence and observability is based on research done from a great distance with the help of, e.g., optical means, namely telescopes. We have to discuss the view through a telescope before we travel to the discovered objects (if such journey will be undertaken at all). Van Fraassen seems implicitly to agree to the existence of the moons as solid spheres orbiting Jupiter and, therefore, he already uses a concept of observation being independent of astronauts facing an object in front of them.
13 Finally, it must be pointed out that van Fraassen, based on his treatment of “observation” and “observability”, would presumably deny the observability of quarks. For him, quarks would be a theoretical ingredient to make quark-theory adequate to deal with such events. While I would agree that quarks are unobservable, I come to that conclusion for different reasons.
14 Next on the list is Dudley Shapere. He investigates an experiment designed to study the core of the sun by monitoring neutrinos from it with a subtle apparatus placed deep underground [Shapere 1982]. The resulting data details the frequency of each radioactive substance found in the apparatus. Originally, the experiment was arranged to test a theory about atomic fusion in the sun’s centre, something the data successfully confirmed. But Shapere was more interested in deriving a concept of observation from the experiment. According to him, the experiment was a direct observation of the core of the sun. This direct observation was theoryladen, but the theoretical components could be systematised and explained. He reasoned that for there to be an observation situation there must be, in general, a flux of information from the observed object to the observer. There has to be: (a) a theory about the observed object that has to release some kind of physical signal as information; (b) a theory about the transmission of this information that makes sure that the information is not altered or tampered with; (c) a theory about the receptor of that information which must not be a biological or even human receptor. Any kind of measuring device can be taken as a receptor [Shapere 1982, 490ff.].
15 Shapere summarises: x is directly observed (observable) if: (1) information is received (can be received) by an appropriate receptor; and (2) that information is (can be) transmitted
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directly, i.e., without interference, to the receptor from the entity x (which is the source of the information). [Shapere 1982, 492]
16 He shows explicitly how, according to the theory of core fusion, the centre of the sun emits neutrinos radiation in all directions, even in the direction of the Earth. According to particle physics, most of the neutrinos pass the Earth unaltered, but some interact with matter. Physics can predict the frequency of interactions in a given amount of, say, perchloroethylene provided it is placed underground to protect it from other cosmic rays. An interaction of neutrinos with perchloroethylene produces a radioactive isotope which can be counted. The theories about the sun, the flux of neutrinos, and the events in perchloroethylene as well as in a device like a Geiger counter tell us, according to Shapere, that there is information being reliably transmitted from the sun’s core to us. Thus, we would be able to observe the sun’s core directly.
17 So far, this notion of direct observation is convincing and it has the advantage of making the search for a definition of indirect observation, obsolete. Nearly everything is directly observable if we only have enough physical knowledge. Shapere excludes consciously the realm of quantum physics where theories of the source and the transmission are not clearly separated. This and other “special complexities and difficulties of interpretation” prevent the application of Shapere’s concept of direct observation to today’s elementary particles [Shapere 1982, 512-513].
18 There is, however, another reason to abandon his proposal. In Shapere’s view the same situation can be interpreted as a direct observation of several different things. Counting the amount of radioactive decays in the Geiger counter can be a direct observation of the centre of the sun, or of neutrinos interacting with perchloroethylene, or of a radioactive substance, or simply the Geiger counter sat in front of the scientist providing data. In that case, the notion of observation once again broadens out to approach the views of Maxwell, although it is restricted to entities capable of acting as a source of information. To me, any reasoning on “observation” has to allow for directness, indirectness and complete unobservability, even when considering entities that are somehow hidden from view. The correct definition of “observation” should help to decide if these objects exist due to direct or indirect observation or if they will perpetually remain theoretical entities.
19 By now we should have some idea of how the protagonists of philosophy of science deal with the problem of scientific observation and how these proposals might be flawed. There are, of course, other proposals from Ian Hacking and Paul Feyerabend to name a few, but they are less important to us since their focus is slightly different from ours. For example, both Hacking and Feyerabend treat observation as an ability of scientists, something one is able to learn and improve upon [Hacking 1983, 180-181]; [Feyerabend 1978, 40-47]. They therefore are discussing the term “observation” from a different point of view. My approach in direct comparison to Hacking and Feyerabend is asking: What is the definition of “observation” independent of abilities and scientific theories; what term of observation did we already develop when thinking about learned scientists who seem to observe more than others?
20 Feyerabend, moreover, focuses on the theoryladeness of observation and concludes pretty fast that every observation, even in everyday life, is theory-laden. According to him, at least three theories are involved when we, e.g., observe a simple table: a theory of the human eye, a theory of light, and a theory of the physiology of the whole observer. Even a table would be a theoretical notion hence. Feyerabend believes that
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the dichotomy between observable and unobservable breaks down [Feyerabend 1978, 46]. I think this is too broad a concept of theory. We reasonably discuss whether we observe a simple table without any use and need for scientific theories. This fact goes to show that observation (everyday and in scientific laboratories) does not have to be theoryladen.
2.2 The approach to the problem of observation
21 Before fully entering the discussion, I want to develop two basic premises. First, a definition of observation should be independent from all physical knowledge. The meaning of “observation” should not be influenced by changes in physics. Observing something in science is an epistemological undertaking that originates from an object’s observation in daily life. It neither depends on contingent technical knowledge, nor on an exclusive ability possessed by only a few. Therefore, I would like to argue in another way, as Ian Hacking has done in Representing and Intervening [Hacking 1983, 180-181]. I think that statements like “in former times observation was to do x but today, in the era of quantum physics observation is to do y” or “an expert sees electrons in a certain picture while a layman just sees light lines in the dark” (no quotations!) fail to address the heart of the matter, although it must be emphasised that physical insights can alter the realm of the entity being observed. (I will return to this idea later when talking about telescopes and microscopes.)
22 My second premise is that a definition of observation should avoid metaphors. It is easy to classify something as observable, audible, sensible, calculable or whatever, if it is not meant literally. Surely we “observe” the law of gravity while watching the curve of a stone thrown in the sky or we “observe” the quantum physical property “spin” in a Stern-Gerlach experiment. Literally, though, we observe only the stone in the first example and a continuous beam splitting in a (non-observed) non-homogeneous magnet field in the second. This is unquestionably the case at least temporarily. Entities like gravity and spin cannot be observed, only their effects. This is admittedly a complete different path in comparison to Maxwell or Feyerabend. Furthermore, if metaphors are used to describe experiments in microphysics, quote marks should be used to alert the reader to the distinction.
23 Observation is always a kind of perception. But what criteria does a perception have to fulfil to be classed as an observation of an object? Every observation begins with the human senses. It may otherwise cause some philosophers to quibble whether an automatic camera was actually observing an object prior to a human being arriving and observing the photograph or the stored film. (One can consider the observation accomplished since the presence of a human being changes nothing of how the object and its environment are observed). However, since observation is related to human knowledge I would prefer to exclude such debates and exclude Shapere’s approach at the same time, too. Thus, the challenge to define “observation” is restricted to what observation may be for a human being interested in empirical knowledge about the external world.
24 Next I want to set the precondition that any object observed has to be perceived as an independent, individual entity. Think of somebody delivering a speech in front of an audience. The speaker watches the audience while talking. The object observed is the group of hearers, and not the individual hearers themselves. And it could happen that
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an old friend of the speaker was in the audience, and surprises the speaker by saying hello to him privately after the speech. At that point, the speaker is obviously observing his friend, but it would be wrong to say that the speaker had previously observed his friend while he was in the audience. Accepting the above distinction rids us of several problems. It allows us to recognise that watching the moon is not the same as seeing each of its craters, or that observing a sugar cube is not the same as observing its grains, molecules, atoms or electrons. Thus, it should be noted that parts of a group may not be observable even if the whole group is. In this way, some misleading arguments about the separation of direct and indirect observation can be avoided from the start.
25 There are many everyday examples that illustrate the distinction between direct and indirect observation. Take again an aeroplane in the sky. Sometimes it is directly observable, sometimes we observe it indirectly because we see only the vapour trail caused be the aeroplane. But what makes this distinction? If the aeroplane is close enough we are able to distinguish it in space and time, i.e., to separate it from other planes or other phenomena in the sky. While there may be a lot of physical things between the aeroplane and our perception, like air, light-waves and our eyes, epistemologically-speaking there is nothing between the observing person and the aeroplane if the person can see that plane. Moreover, we are able to identify some important properties of the aeroplane—its colour, size or construction. It is without doubt directly observed. By watching only the vapour trail of a plane, we can still gain some information about an individual plane travelling through space and time. Since the plane is beyond our human sensory reach, we do not observe the plane directly, we only see the vapour, but relying on our personal knowledge or experience of how vapour trails form, we can infer from the directly observed vapour trail that we are indirectly observing an aeroplane. Other examples of direct and indirect observation are fire and its smoke or a skier and his tracks in the snow.
26 Indirect observation depends on causality and some understanding of the origin of vapour trails, smoke, and ski tracks, fairly using the Humean concept of causality. It therefore has to be described as a knowledge- or experienceladen process. In science, indirect observation is, by analogy, theoryladen. Whereas direct observation, whether in daily life or in science, is not.
27 The problem of theoryladeness (see Section 2.1) is not relevant here, since it is strongly related to the experienceladeness of indirect observation in everyday life, and in this instance experienceladeness does not stop us from making reliable judgments. Remember the thoughts of Maxwell. We are, in contrast to him, able to suggest both an observation/non-observation line and a direct/indirect line for observations. These distinctions are neither arbitrary nor a function of human physiological or psychological attributes because they are dealing conceptually with the differences between directness and indirectness. They would also allow most observations undertaken with scientific tools to be described as direct observations.
28 The next problem is to separate a directly observed object from a mere illusion. Sometimes, especially when new tools of observation are invented, it is not clear whether the visible result is an object or just its image. The best way to solve this dilemma is to do the same thing as we do in everyday life to separate between object and illusion—we try to find out more about that object, we need achieve broader acquaintance with it.
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29 If we are, say, on a street, and uncertain whether a car is real or just a picture on an advertising hoarding, we change the angle of our view by moving our head or taking a few steps to the side because we know from experience that a 3D object will look different from a 2D image if our point of view changes.
30 Now, what is important for us here is that a difference between the phenomena is only gained by expanding our visual experience of the object in question. From a distance, a single view of a real car may not differ from a picture. Similarly, people who wear glasses are sometimes surprised by a new object, say, laying in front of them on a table. Is it an object or just a spot on the glasses? The answer is obvious when you turn your head. If the object moves as your head turns, then it is a spot on the glasses. If it does not move it must be an object independent of your movement and therefore lying on the table. This is what I mean by broader acquaintance and when observing an object. And this concept is inspired by preliminary thoughts of Berkeley and Leibniz.1 This everyday epistemology may sound irrelevant to the subtle methods of today’s physics but, at the end of this Section, I try to show otherwise.
31 There is one last important point about the definition of observation. To observe something directly you must observe the object individually, but an indirect observation needs the viewer only to witness a physical phenomenon caused by the hidden object. But what about the individuality of an indirectly observed object? Think of a skier-track in the snow, a single one yet caused and consolidated by a group of skiers. Considering this example I must emphasis that the individuality of an indirectly observed object must be preserved. Otherwise we only indirectly observe a cluster of objects.
32 Our definition of direct and indirect observation is now complete. To summarise: - An object is directly observed if it is perceived as an individual entity within broader acquaintance. The observation does not depend upon physically-caused phenomenon. And the observation is not theoryladen, although it can take place with the assistance of technical devices as long (temporal) as individuality is preserved and broader acquaintance remains possible. - An object is indirectly observed if the physical phenomenon created by the object is observed directly. The indirectly observed object has to retain its individuality. And an indirect observation is always either the-oryladen or experienceladen, whether it be in science or in everyday life.
33 This definition shows from my point of view some advantages over earlier concepts: Maxwell was forced by his approach to deny a clear distinction between observation and theory; he did not include something like our notion of broader acquaintance in his discussion. Shapere’s proposal lacks of determination: the same situation can be interpreted as a direct observation of several different things. It is once more the notion of broader acquaintance that enables us to get rid of such underdetermination since if we make acquaintance we make it only of one object at a time.
34 Van Fraassen alludes to the dependence of contingent human physiology to pin down what observable is. With our definition of direct and indirect observation, however, we are independent of human physiology as well as of contingent technical progress. In addition to microscopes and telescopes, other instruments could be invented without impairing the above definition.
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35 To end this Section I would like to recount a few examples found in science. A view through a telescope is a direct observation. By moving the telescope to various places in the sky or on earth, we get a reasonable change in view. In the same way, astronomical objects can be seen from different perspectives in winter and in summer. A single glance does not confirm an observation but repeated viewings over time do. This was crucial to Galileo’s discoveries and this could held to be the main reason why Galileo’s critics got silent in the long run. No understanding of the laws of optics or theoretical knowledge of the functionality of telescopes is needed to recognise the event as a direct observation.
36 Likewise, the view through a light-microscope is a direct observation. Though the microscope itself cannot be moved, the object observed can be with a pipette or other small tool. The moving of the pipette, seen in the ocular, is logically associated with the movements of the viewer’s fingertips. Anybody, even those unfamiliar with microscopes, can recognise cells and bacteria after a short period of experience and by expanding the context of the situation. Night-vision goggles or glasses work in a similar way. But images of atoms made by a scanning-tunnel microscope work differently however. They rely on collecting single measured points in a grid pattern. Currently, the devices for scanning the surface of a solid material on atomic scale are not fast enough to result in an exact visual image, instead the results are an approximate 3D graphical image. It is similar to a picture. But even though we can explore the context of the picture, it is not the same as getting into broader acquaintance with pictured objects.
37 A conventional photograph is not a direct observation. It is an indirect one. From our general knowledge of photography, we know that a picture is typically shot in a certain place at a certain, very short time. Therefore, the individuality of objects seen in the picture is preserved. In the same way, the picture of particle tracks is a two-fold indirect observation. We never see the particle, rather a photograph of the trail of the particle. Still, the individuality of the particle is carried over to the picture. And we can, for example, deduce the particle’s charge from studying the photograph. Particle tracks, particularly in relation to these observations will be discussed in the following Section.
3 Quarks and jet-events
3.1 Jet-events in physics
38 Before applying our concept of observation to quarks, we need to review some experimental methods in high energy physics. To investigate elementary particles you must use detectors to record particle tracks as a kind of photograph. Particle detectors are basically a particle-sensitive material that interacts with particles pointwise as they pass the detector. The points of interaction are collected, listed, and graphically displayed in relation to their order in space and time. The result: pictures of particle tracks, or to be exact, of rows of points of interactions. The aim is to develop a particle detector sensitive enough, that the rows are so dense that they look like continuous tracks to the naked eye. Or so dense that a computer can calculate a continuous track as if the particle was a tiny object, literally flying through the detector.
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39 This model may not be consistent with some other aspects of quantum physics, but it is still useful in determining some properties of particles. Particles identified in this way are typically electrically charged like electrons, several mesons (on which I will comment soon), and protons. There are other types of particle detectors such as for photons and neutrons, but they are beyond the scope of this discussion.
40 Whatever particles like electrons are, it is fair to say that they are indirectly observed when they make a particle track. One individual track belongs for a short time to a single electron and by studying that track it is possible to identify some of the electron’s permanent properties. The analogy of a skier’s trail in the snow or a plane’s trail in the sky holds true, though we have to accept that typical objects are directly observable in multiple situations whereas particles are not. You can shake hands with a skier at the bottom of a slope, but the closest you can get to an electron is its track. This might be worth philosophical consideration another time, but right now it is important to note that particle tracks are examples of indirect observation.
41 Now to quarks. The first thing to point out is that they cannot—in theory—cause single particle tracks like electrons or protons, despite the fact they are electrically charged. The physical explanation for this is based on the strong nuclear force between quarks. This force binds quarks together in quark-confinement. A phenomenon special to quarks that are thought to be the basic components of protons, neutrons (nucleons, or three- quark-systems), and mesons (two-quark-systems). Imagine the (slightly misleading) picture that quarks are small balls, quark-confinement is like a rubber band that holds them together and turns nucleons and mesons into stable particles. Trying to separate quarks from each other, by for instance shooting them through a particle detector, only serves to increase the force between them. If you try to split a bounded quark- system with energy it fails to break that rubber band.
42 Under the rules of quantum theory an unstable system of two quarks with a high level of energy converts to a system of four quarks with less energy. By adding energy to quark systems you do not isolate single quarks, but produce new ones that immediately combine to form two- or three-quark-systems. Although the initial system breaks up, the result is a new system of bounded, confined quarks. So quark-confinement stops any free, isolated quarks occurring in particle detectors. Ironically, the only place where quarks can move freely is inside the radius of its system that is bound by the energy force. (For other introductions to quarks, the confinement, and jet-events see, e.g., [Pickering 1984], [Galison 1991].) Nevertheless, the properties of quarks can be investigated either by probing a nucleus or by destroying it, and this is where jet- events come in.
43 Making particles collide at high energy and either destroy them and/or creating new particles is one way to test physical theories. Normally, collisions cause particles to scatter in all directions from the point of collision. A picture from a detector will show particle tracks spraying outwards from the point of collision. But under certain circumstances—defined by the type of colliding particles and their energy—a new kind of event can occur in particle detectors. On these occasions particle tracks tend to be bundled into two, three or four jets. The diagram below shows just such an event with two jets or jet-branches.2
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44 The centre of the picture is the point of initial scattering. What happens there is, in theory, too small to detect. The events that occur take place on a scale far smaller than the core of an atom. But all the lines from the centre to the periphery of the diagram are particle tracks. In this example, they seem to be bundled into two jets. Quark physics explains why this pattern frequently occurs. The story goes as follows. When particles with high energy meet, the provided system decays and produces new particles. In several cases two quarks are created that fly off from each other in opposite directions. To explain jet-events it is important to assume that quarks do exist as small chunks and have a state of motion. Due to the strong force, quarks cannot keep moving forward. When the deposited energy is great enough, the connection between the quarks breaks and some energy forms two new quarks. Each of the secondary quarks recombines with one of the primaries. Now we have two mesons flying away from each other, still in opposite directions. They are carrying too much energy to be stable and so decay into further particles— electrons, low energy mesons, protons, neutrons, whatever. These are the particles that reach the detector and make particle tracks. (Actually even neutrinos and photons can be produced and particle detectors can also record some particle’s velocity, energy, and charge. Such evaluations occur during jet-event research too but shall not be part of this paper.) So, although nothing is left of the initial quarks and their secondary mesons, the resultant tracks indicate a common source and suggest that all particles are part of one of the bundles, and both bundles form jets that move in opposite directions (as shown in the diagram). It may be that particle tracks are ordered in this way by chance, but jet-events occur more frequently than probability would suggest. It makes sense that all particles are bundled into a branch of a jet-event because they all come from a single meson which itself comes from a single quark. So it seems plausible that the causal effects of an invisible object can produce an observable phenomenon, and that the data from the particles involved in an individual jet leads to the recognition of an individual quark. And this would make jet-events, an example of indirect quark observation. Yet by looking at the role jet-events actually play in high energy physics other issues arise.
3.2 Physicists on jet-events
45 If you look at the main publications on jet-events, it was not the fact that jet-events occur (and that they point to single quarks) that excited physicists, but that an
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ensemble of jet-events was collected. For example, there is one publication about the confirmation of the jet-model in a statistical counting omitting any emphasis of a single jet-event [Hanson et al. 1975]. See also this typical statement about jet-events: We have observed that events with a large transverse energy (…) have a dominant two-jet structure(...) A sample of two-jet-events has been studied and observed to feature properties characteristic of hard scattering of partons. [Banner et al. 1982, 210]
46 No leading physical publication brought up the discovery of an individual jet-event or a single quark. The experimental evidence for the validity of quantum chromodynamics in the context of jet-events consists mainly in the measurement of values like “transverse momentum distribution”, “distribution of the Fox-Wolfram moment”, and certain cross-sections [Soding & Wolf 1981, 265ff.]. This is crucial because it diminishes the value of single events and enhances the value of them collectively. Therefore we have to reconsider, whether an individual causation of a jet by a quark can be confirmed. Moreover, as mentioned in the introductory physical section, the reliability of a single jet-event is not certain. A physical comment on three-jet-events where one branch seems to stem from a radiated gluon instead of a third quark says: The observation of three-jet-events in high energy e+e– annihilation qualitatively confirms the prediction of hard gluon bremsstrahlung by leading order QCD [i.e., quantum chromodynamics]. It does not, of course, rule out other mechanisms as possible sources of these events. Therefore, detailed and quantitative tests are called for. [Soding & Wolf 1981, 276]; the quotation contains a reference to an acceding publication: [Preparata & Valenti 1981]
47 Hence, physicists argue quite careful when considering branches of jet-events. As if they were aware that the branches do not necessarily have to be produced by quarks. The power of the theory describing quarks—quantum chromodynamics—depends upon accurate predictions of the frequency of jet-events during numerous scattering events occurring under certain conditions. Predicting the number of jet-events is based on our ability to separate jet-events from other events. We need to acknowledge that jet- events do not have to be as neat as in the diagram above. A reliable separation can be done by computer programs [Duinker & Luckey 1980]. We must also remember that events that look like jets but are not can occur statistically (and depending on the quality of particle detectors) without any link to quark physics.
48 Taking another look at the seminal works on jet-events it continuously seems that jet- events are always collected statistically. Results of experiments do not depend on the qualitative discovery that jet-events exist, followed by an investigation of the properties of a single jet-event and its branches. Although results from single jet- events are collected, they are evaluated quantitatively. The following comment makes this quite clear: The observation of three-jet-events in high energy e+e– annihilation into hadrons may be considered a major triumph of the theory. QCD had predicted such events to occur as a result of hard gluon bremsstrahlung. Close examination of the details of the three-jet process at [the particle accelerator called] PETRA shows consistency with QCD and comfirms the vector nature of the gluon. From the rate of three-jet-
events the events determine as = 0.17 ± 0.01 (stastistic) ± 0.03 (systematic) at Q2 ≈ 1000 GeV2. [Soding & Wolf 1981, 289]
49 In the literature we find only these typical statements. There is no publication among the leading jet-articles presenting a single jet-event as an observation-situation of one or more certain quarks.
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50 Figuring out what is actually a jet-event is a subtle process. The job falls to particle physicists. But they must rely to some extent on epistemology because it is not possible to single out an arbitrarily accurate jet-event from a jet branch and say it is caused by a single quark. This constraint is not important for quantum chromodynamics or the engineering of particle detectors. Quark theory and quark experiments are today breathtakingly well established.3 To me it seems that actually physics is not particularly interested in single jet-events but philosophy is. This view is supported by the fact that several of the initial publications on jet-events do not display jet-events at all [Hanson et al. 1975]. Plenty of debates already exist on observation and quarks. These arguments reference plenty of experiments but say little about jet-events. The following comments make an interesting contrast to our discussion and deal with quark- confinement and its limiting influence on our epistemological access to quarks. Both appraisals come to the conclusion that quarks are indeed observable.
3.3 The observability of quarks in Shrader-Frechette and Massimi
51 Although Andrew Pickering does not address the question of the observability of quarks in his monograph Constructing Quarks [Pickering 1984], other scientific philosophers have discussed this topic. The first debate took place around 1982 and was initiated by an essay of Kristin Shrader-Frechette [Shrader-Frechette 1982a]: It is widely accepted that scattering data are a means of observing particles directly. The difficulty with quarks, of course, is that they are not detected directly in any experiments, although particles said to be composed of quarks are directly “observed”. More precisely, although effects of some particles are directly observed, in scattering experiments quark effects are not directly observed as quark effects, but are said to affect the behaviour of particles which are directly observed. [Shrader-Frechette 1982a, 133-134]
52 Shrader-Frechette goes on to deal with experiments confirming the J/ψ-meson (1974) and decays of charmed hadrons (ca. 1976). Quantum chromo-dynamics contains six different types of quarks, including one called charm. Charmed quarks are thought to be constituents of mesons or hadrons, providing them with special properties that other mesons or hadrons do not have. Examples are the J/ψ (or charmonium) and charmed hadrons. Verifying these particles would also verify quantum chromodynamics, if no other theoretical explanation exists.
53 Shrader-Frechette infers that charmed quarks actually have been observed in tests on J/ψ-mesons and charmed hadrons. Of course, no single charmed quark interacts with a particle detector—both charmoniums and hadrons being too short-lived to reach detecting devices—but they do cause certain decay schemes which particle physics call detect. Shrader-Frechette points to an important difference between the J/ψ- and charmed hadrons. The decay pattern of the charmonium can be explained without charmed quark involvement but the decay of charmed hadrons cannot. This fact is related to the creation of K-mesons seen in the decay pattern of charmed hadrons. Although the “observation” of charmed particles was obviously theory-laden (…), this “observation” was not as theoretical, in a potentially damaging sense, as that of charmonium. This is because, while hypotheses other than charmonium could have explained the J/ψ detection, there was no other known means of accounting for the K meson. Thus, at least in this micro-physical instance, observations of hadron decay provided a more overt, a more direct means of observation of charm (…)
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precisely because the theoretical context in which they occurred was more unique and admitted of more specific predictions. [Shrader-Frechette 1982a, 139]
54 Shrader-Frechette’s investigation draws its weight from the detailed analysis of background and alternative theories explaining the data of particle scatterings. And it seems possible to distinguish between direct and indirect observation albeit that theoryladeness is ever present. At least the last statement of the quotation can be read this way. But in the end, the conclusion that charmed quarks are observable either follows Shapere’s unsatisfying ideas on observation or it is meant metaphorically and thereby allows for nearly any interpretation. Shrader-Frechette ends with a little disclaiming summary: It should not be surprising that our observations are understood in a very abstract way, or that quark “observations” are quite different from everyday observations. [Shrader-Frechette 1982a, 141]
55 Participants of the ensuing debate—all printed in Synthese in 1982—depart from quarks to consider the more general question of what scientific observation is [Albright 1982], [Shrader-Frechette 1982b], [Gruender 1982]. J. R. Albright reasons that since a theoryladen observation needs a community of members to agree on any presupposed knowledge, then observing is not dissimilar to simply believing. Albright writes that a consensus is necessary in scientific observation if the observation is to be thought reliable. This proposal— touching on the territory of relativism—is criticised by Shrader-Frechette and David Gruender and is too far removed from our discussion of the concept of observation to warrant space here.
56 The observability of quarks concerns Michela Massimi, too [Massimi 2004]. Her start point is experimental realism. It is a position that originates from Ian Hacking, who opted for entity realism backed up by experimental scientific methods. Hacking argues that since we are able to build devices that emit electrons of a certain energy which have particular properties, those emissions should be treated as part of our reality [Hacking 1982; 1983]. He puts it like this: “If you can spray them, they exist.” Alternatively, if you can use electrons as tools to probe other material, electrons must exist. Or if they play a causal role in our model of microscopic events, they must exist [Hacking 1982; 1983, 23, 36].
57 Massimi’s ideas combine Hacking’s realism and Shapere’s concept of observation and apply them to quarks. She particularly focuses on the experimental context of scaling violation found in the measurement of structure functions of nucleons. In this context the existence of quark physics and science’s present model of nucleons is strongly confirmed. We do not have to go into the details of structure functions, but her point is that a statistical collection of scattering data results in a graphical curve—the structure function—that has a certain property. It shows an inclination. Were there no inclination, then there would be no relevance to quark theory. But the quantitatively- predicted value of inclination of the curve, using the theory of coloured quarks inside nucleons and exchanging gluons, was confirmed, and the structure of the nucleus can be said to be physically understood.
58 While collecting scattering data, neither was the nuclei destroyed nor the quarks isolated, but they were probed and manipulated. Thus, Massimi brings Hacking into her argument: “Seeing” quarks is not licensed by strong notion of manipulation (spraying), but by a weaker notion of manipulation (probing) that nonetheless still satisfies the engineering criterion of experimental realism (interfering/intervening). This way
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of “seeing” the nucleon’s constituents presupposes a massive amount of theory. [Massimi 2004, 49]
59 The Shapere-part of her proposal acknowledges that such observation (or “observation”, as Massimi prefers to put it) is theoryladen and that our state of knowledge changes our realm of observation: I believe with Shapere that in certain cases what counts as “observation” depends ultimately upon our current state of knowledge; and, most importantly for the rest of my argument, what kind of entities physicists can arguably claim to “see” in a lab turns out to be dependent on what kind of theory about these very same entities they endorse. [Massimi 2004, 49]
60 Massimi interestingly allies Hacking’s experimental approach to Shapere’s theory- focused concept of observability in quark physics, although Hacking’s “if you can spray or manipulate them, they exist” raised suspicion in some people, and I am told it is a tautology. It is also possible that Massimi’s concept of observation is nothing more than a metaphor for something else. For our purposes though, it is evident that studying structure functions represented as graphical curves on paper is neither a direct nor an indirect observation of quarks. Not that this effects our conclusion too much. Jet-events seem to be more intuitive and better direct evidence for quarks than structure functions could ever be.
61 My review of ideas on the observation problem generally (Maxwell, van Fraassen, and Shapere) and more specifically in relation to quarks (Shrader-Frechette, Massimi) has hopefully given you an insight into how deeply interwoven the discussion of observation and theoryladeness are. And how quickly accepting the idea of theoryladeness leads to the statement that “observable is anything”. Jet-events could provide a viable go-between.
4 The interpretation of jet-events
62 First, from our look at direct and indirect observation, it follows that in particle physics only indirect observation applies. This is not because elementary particles are too small —many small things like cells and microbes are directly observable and can be seen with the help of a microscope and our knowledge of their context. But elementary particles withdraw from broader acquaintance. It well may be that the principles of quantum physics provide the constraints. Using indirect observation is nothing to worry about, many things in everyday life as well as in science and high energy physics are only indirectly observable, so we have no reason to exclude them from our ontology.
63 But the question remains are quarks indirectly observed in jet-events? To answer it, we have to take a single branch of a jet-event and try glean as much information from it as possible, just as we might if we took a single particle track to learn about the particle’s charge-mass ratio, its energy and other properties. Again, I do not see whether other experimental contexts come closer to quarks than jet-events. As we have seen in Section 3.2, however, physicists do not take a single jet-event to get information that is causally linked with one or more distinct quarks. The criterion of indirect observation given in Section 2.2, therefore, is not fulfilled.
64 You could argue that there is physical significance in looking at single jet-events, and that it will make a nice task for a particle physicist sometime in the future.
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Alternatively, you could reconsider the origin of a jet-event branch in the quark physics model.
65 The indirectly observed particles that make tracks in a bundle, come from a short- living meson. This meson does not cause a trail itself but every trail from a jet-branch carries some properties of the original meson, like its electrical charge. The sum of the electrical charge carried by the secondary particles is equal to the charge of the initial meson. Although, such a meson is not observed directly in a jet-event, it can be identified by its charge that is carried by secondary particles. The condition of indirect observation, i.e., observing directly a physical phenomenon created by the object “meson” is fulfilled. The directly observed phenomenon is particle tracks. In this way, it is possible to say that the meson is indirectly observed. It is indirectly observed because it is identified in a decay pattern that itself consists of indirectly observed secondary particles.
66 Although the meson is thought to consist of two quarks, the quarks play no part in the decision of whether the meson is indirectly observable. According to the quark model, one of the quarks in the indirectly observed meson comes from the very centre of the particle collision that causes the jet-event. However, we cannot identify that individual quark by looking at the single jet-branch.
67 The electrical charge of a quark (which is always –1/3 or +2/3) is never conserved in a jet-branch since all detectable particles carry a charge of +1, –1 or zero. The distinction of whether the original particle was a quark or an anti-quark is also lost. Such information is not derivable from one single jet-branch. So, if indirect observation is meant literally, we do not observe quarks indirectly in a single jet-event.
68 Obviously, images of jet-events give an idea of a good, working particle detector but they are not crucial for the confirmation of theoretically-determined measuring values. By comparison, think of the discovery of the positron by Carl D. Anderson in 1932. In his publication, we see images of positron tracks. Since these images show that a single track is caused by a single positron, they were key to his claim that positrons are indirectly observable [Anderson 1932]. Getting even one image of a certain particle track means the discovery of a positron by indirect observation. And the charge-mass ratio typical for a positron can be determined by evaluating the curvature of that track. Remember our definition in Section 2.2. Anderson observed indirectly positrons in the same way as we observe indirectly an aeroplane by watching a vapour trail in the sky.
69 This type of investigation is impossible for quarks, although quark physics and quantum chromodynamics is none the worse for this. That quarks cannot be observed has little impact on the successful verification of the physical entity “quarks”, if the term “entity” is understood in its most basic sense. Many entities are not observable— forces, natural constants, fields, virtual particles, resonance particles, as well as natural laws or terms of natural laws.
70 To end this investigation I would like to appraise the meaning of my conclusion. There is a strong consensus that some objects, which are not directly observable, might be indirectly observable. There are additionally entities that to us do not exist because, among other reasons, they are not indirectly observable.
71 In the case of quarks, we have seen that they cannot be indirectly observed, even in the comparatively evidential context of jet-events. This should be a strong hint that quarks are in principle unobservable, although, this is not yet proved. This brings us to the
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debate of scientific realism. And the indications are that quarks can only be treated as entities, similar to other unobservable entities in science.4
BIBLIOGRAPHY
ABACHI, S. et al. — 1995 Observation of the Top Quark, Physical Review Letters, 74, 2632-2637.
ABE, F. et al. — 1994 Evidence for Top Quark Production in pp Collisions at √s = 1.8 TeV, Physical Review Letters, 73, 225-231. — 1995 Existence of the Top Quark, Physical Review Letters, 74, 2626-2631.
ALBRIGHT, J. R. — 1982 Comments Concerning the Visual Acuity of Quark Hunters, Synthese, 50, 147-152.
ANDERSON, C. D. — 1932 The Positive Electron, Physical Review, 43, 491-494.
BANNER, M. et al. (UA-2 COLLABORATION) — 1982 Observation of Very Large Transverse Momentum Jets at the CERN pp Collider, Physics Letters B, 118, 203-210.
BERKELEY, G. — 1709 A New Theory of Vision, London: Dent, 1910.
DUINKER, P. & LUCKEY, D. — 1980 In Search of Gluons, Comments on Nuclear and Particle Physics, 9(4), 123-139.
FEYERABEND, P. — 1978 Das Problem der Existenz theoretischer Entitäten, in P. Feyerabend (Ed.), Der wissenschaftliche Realismus und die Autorität der Wissenschaften, Braunschweig: Vieweg, 40-73.
GALISON, P. — 1997 Image and Logic, Chicago: Chicago University Press.
GOODMAN, N. — 1978 Ways of Worldmaking, Indianapolis: Hacket.
GRUENDER, D. — 1982 On Observing Quarks, Synthese, 50, 157-162.
HACKING, I. — 1982 Experimentation and Scientific Realism, Philosophical Topics, 13, 71-87. — 1983 Representing and Intervening, Cambridge: Cambridge University Press.
HANSON, G. et al. — 1975 Evidence for Jet Structure in Hadron Production be e+e– Annihilation, Physical Review Letters, 35, 1609-1612.
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HANSON, N. R. — 1958 Patterns of Discovery: An Inquiry into the Conceptual Foundations of Science, Cambridge: Cambridge University Press.
KUHN, T. S. — 1962 The Structure of Scientific Revolutions, Chicago: University of Chicago Press.
LEIBNIZ, G. W. — 1704 Nouveaux essais sur l’entendement humain, Paris : Garnier-Flammarion, 1966.
MASSIMI, M. — 2004 Non-defensible Middle Ground for Experimental Realism: Why We Are Justified to Believe in Coloured Quarks, Philosophy of Science, 71, 36-60.
MAXWELL, G. — 1962 The Ontological Status of Theoretical Entities, Minnesota Studies in Philosophy of Science, 3, 3-27.
PICKERING, A. — 1984 Constructing Quarks, Edinburgh: Edinburgh University Press.
PREPARATA, G. & VALENTI, G. — 1981 Deciphering the Structure of Hadronic Final States, Physical Review Letters, 47, 891-894.
SHAPERE, D. — 1982 The Concept of Observation in Science and Philosophy, Philosophy of Science, 49, 485-525.
SHRADER-FREOHETTE, K. S. — 1982a The Problem of Microphysical Observation, Synthese, 50, 125-145. — 1982b Consensus and the Visual Acuity of Quark Hunters—A Response, Synthese, 50, 153-155.
SÖDING, P. & WÖLF, G. — 1981 Experimental Evidence on QCD, Annual Review of Nuclear and Particle Science, 31, 231-293.
SPITZER, H. H. & ALEXANDER, G. — 1979 Pluto, in S. Homma, M. Kawaguchi, H. Miyakawa (Eds.), Proceedings of the 19th International Conference on High Energy Physics, Tokyo, 23-30 August, 1978, Tokyo: Physical Society of Japan Publ., 255-260.
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NOTES
1. See Berkeley’s New Theory of Vision, sect. CIII ff., and his Principles of Human Knowledge, sec. XVIII ff. [Berkley 1910]; and see Leibniz Nouveaux Essais, 4. book, chap. 11 [Leibniz 1966]. 2. For other pictures of jet-events see [Duinker & Luckey 1980, 127, 136], [Soding & Wolf 1981], [Banner et al. 1982], [Pickering 1984, 329]. 3. The signature of jet-events played a crucial role, among others, in the confirmation of the sixth and heaviest quark, the top quark [Abe et al. 1994, 1995]; [Abachi et al. 1995]. 4. I would like to thank Hernán Pringe and Richard Dawid for useful comments and criticisms. I am also deeply grateful to Lucy Ashton and Pierre-Emmanuel Finzi for going over an early draft and numerous enlightening comments. The responsibility for what results above is entirely mine, of course.
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ABSTRACTS
This essay deals with the question whether quarks—the basic components of matter and one of the youngest confirmed particles in high energy physics—can be observed directly or indirectly. First, I shall discuss earlier definitions of “observation” in physics suggested by Grover Maxwell, Bas van Fraassen, and Dudely Shapere. Then, I shall compare their results with a new consideration of the idea of “observation” in physics and the distinction between direct and indirect observation. One of the ways that quarks appear in experimental evidence is in jet-events which are special decay patterns in particle detectors. These jet-events are the central case study for this essay since they are the most convincing evidence of quarks so far. My inquiry into the physics behind jet-events and their treatment by physicists leads me to conclude that quarks are neither directly nor indirectly observed. I compare this thesis to other views on the observability of quarks, suggested by Kristin Shrader-Frechette and Michela Massimi. But in short, it seems to me that if the notion “observation” is to have a reasonable relation to an everyday life’s concept of observation, and if it is meant literally instead of metaphorically, then quarks are not observable in any currently-known experimental context.
Cet article pose la question de savoir si les quarks — constituants élémentaires de la matière et dernières particules de la physique des hautes énergies à avoir été confirmées — peuvent être observés de manière directe ou indirecte. D’abord, des définitions antérieures de « l’observation » en physique seront examinées — en l’occurrence, celles proposées par Grover Maxwell, Bas van Fraassen et Dudley Shapere. Puis, leurs résultats seront comparés à une définition du concept d’observation et à une différenciation entre l’observation directe et indirecte. Une possibilité de mettre en évidence les quarks de manière expérimentale est le phénomène des jet-events, qui représentent un type spécifique de désagrégation dans les détecteurs de particules. Ces jet-events contribuent à l’étude de cas ici centrale, puisqu’ils sont la preuve la plus convaincante à ce jour des quarks. L’examen de jet-events et leurs évaluations par des physiciens mènent à la conclusion que les quarks ne peuvent être observés, ni directement ni indirectement. Cette thèse sera comparée aux points de vue de Kristin Shrader-Frechette et Michela Massimi sur l’observabilité des quarks. On en tire la conclusion que les quarks ne sont pas observables, si l’on entend employer le terme « observation » dans un sens non métaphorique et en conservant une relation avec ses usages dans la vie quotidienne.
AUTHOR
TOBIAS FOX Vienna, Austria
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