Philosophical aspects of proof theory

DAG PRAWITZ University of Stockholm

I. INTRODUCTION. DIFFERENT DIRECTIONS OF THE SUBJECT

1. The subject and its name

The term proof theory, in Gennan Beweistheorie, was introduced by Hilbert [1] in 1922 as the name of a subject which he had already described in a lecture [2] in 1917 as 'an important, new field of research' where 'we must make the concept of specific mathematical proof itself object of investigation, just as also the astronomer pays attention to his place of observation, the physicist must care about the theory of his instrument, and the philo sopher criticizes reason itsdf'. Hilbert had great expectations of this new discipline. In [2] he mentions a number of questions that belong to it: 'the problem of the solvability of every mathematical question, the problem of checking the result of a mathematical investigation, further the question of a criterion for the simplicity of a mathematical proof, the question about the relation between matter and form (lnhalt• lichkeit und Formalismus) in mathematics and logic, and finally, the problem of the decidability of a mathematical question by a finite number of operations.' He even thought he had found a solution of the continuum hypothesis by the help of his proof theory [3 J. But first of all, proof theory was meant to be the essential ingredient in Hilbert's program. The aim of this program was to obtain a finitary foundation of mathematics, and the way in which this was to be done has been variously formulated as showing that the use of transfinite principles leads to correct finitary results or that all transfinite elements can be eliminated in

Contemporary philosophy. A new survey. Vol. 1, pp. 235-277. ©1981. Martinus NiihoffPublishers. Tile Hague/Boston/London. 236 D. Prawitz principle from proofs of finitary results or, more simply, as demonstrating the consistency of mathematics, which problems are all eq uivalent under certain conditions. In spite of Hilbert's greater ambitions, to which also Bernays [4] has recently called attention, proof theory has to a large extent been dominated by Hilbert's program and has even come to be identified with work on it. Therefore, with growing interest in other problems concerning proofs, there has seemed to be a need to distinguish between a narrower and a broader use of the term proof theory. Prawitz [5, 6] suggests the name reductive proof theory for a study of proofs aimed at a reduction of trans• finite or in general abstract principles to more elementary ones along the lines suggested by Hilbert and the name general proof Theory for a broader investigation when proofs are taken as objects of study for their own intrinsic interest and not for reductive aims. This terminology is also followed e.g. in the recent monograph on proof theory by Cellucci [7]. Proof theory taken in a very broad sense is certainly not a creation of Hilbert. The subject can fairly be said to have started with Aristotle's works in logic. Indeed, logic as it was cultivated until quite recently was to a great extent identical with proof theory: it studied arguments, distinguished between valid and invalid forms of arguments, and tried to give a systematic presen• tation of the valid ones. In short, its general aim was to give a sys• tematic account of correct, strict reasoning or, in other words, of what constitutes a proof. In Hilbert's vision of the subject, however, proof theory was to be distinguished from the older tradition of logic in especially two ways: firstly, instead of proofs in a more absolute sense, he wanted to study formal proofs, i.e. derivations in given formal sys• tems, and, secondly, he thought of the subject as a strictly mathe• matical one. The first feature was made possible by Frege's sharpening of the axiomatic ideal by the creation of logistic or formal systems. The second feature was essentially Hilbert's own idea. Frege and his immediate successors such as Russell and Whitehead may be said to have studied proofs in the sense of having isolated different rules of inference and having investigated by what principles various mathematical proofs can be carried out, but the proofs themselves did not then occur as objects of a mathematical theory. , Philosophical aspects o/proof theory 237

Proof theory in Hilbert's sense thus makes its appearance at about the same time as model theory begins in the shape of Skolem's investigations and Tarski's semantics where also formal languages are objects of mathematical study. But in contrast to proof theory, model theory tries to study a notion of truth and logical consequence independently of the question of how we come to know that a sentence is true or a logical consequence of given premisses - it is true that most presentations of model theory contain proof-theoretical elements, but pure model theory may be presented without mentioning a deductive apparatus for the language studied (as most consistently done by Kreisel and Krivine [8]). If proof theory is to pay attention to the epistemological nature of proofs, i.e. to their raison d'etre as the means of getting to know truths and logical consequences, then, however, it is highly doubtful whether the two special features which charac• terize proof theory in Hilbert's sense are appropriate. Firstly, we know from G6del's first incompleteness proof that the notion of proof cannot in general be characterized in terms of formal systems: given any formal notion of proof, there is e.g. a universal numerical proposition which has no proof of the given kind but which nevertheless is not only true but can also be proved to be true:'by Godel's method. Secondly, one would expect that a study of proofs in their capacity as means of getting to know truths must take into account also the meanings of the sentences involved, and that it thus has to merge with a meaning theory as argued by e.g. Prawitz [9, 10]. It may be only a terminological question whether such a theory is called mathematical or not, but in any case, it has to attend to concepts quite different from the ones Hilbert had in mind. It is also to be noted (as in Prawitz [10]) that the meaning theory in question must be quite different from Tarskian seman• tics, and that there is no direct route from the notion of logical consequence to the notion of proof: a proof may correctly be said to consist of a number of valid inferences, but the condition that the conclusion is a logical consequence of the premisses is clearly only a necessary and not a sufficient condition for the appearance of an inference in a proof; nobody counts just some axioms immediately followed by a non-trivial theorem as a proof 238 D. Prawitz even if the theorem happens to follow logically from the axioms. Obviously, some kind of evidence is also required of the inferences in the proof; one must in some way take seriously that the con• clusion is seen to be true. In conclusion, three kinds of proof theory emerge: (1) reductive proof theory directed towards questions of consistency or, more generally, towards reductions of abstract principles to more elementary ones, (2) a more embracing, mathematical proof theory as envisaged by Hilbert (to be distinguished from his program), and (3) a more general proof theory not restricted by the features Hilbert had in mind but attending to the general epistemological nature of proofs and questions of validity of argu• ments. Kreisel has in some papers, e.g. [11, 12J , suggested the name theory of proofs for a study that concentrates on such things as the structural complexity of proofs but disregards questions about what makes a proof a proof, or in other words, what makes an argument valid, which has been the focus of papers such as Prawitz [10, 13J. What Kreisel has in mind here seems to be proof theory in the sense of (2) except that it is to be explicitly separated from the aims in (1) and (3). Kreisel [14] argues that there are reasonably adequate data for a 'natural history' of proofs (comparable to early mineralogy or biology) but that there is no hope at present for a rewarding, systematic science of proofs (by which he means a theoretical science that is rewarding in the same way as "success• ful" natural sciences like crystallography or genetics, to continue his examples above). We may here see a fourth suggested direction of proof theory if we want, viz. in the form of case studies of striking phenomena of proofs, exemplified by Kreisel [14 J (but in the terminology of that paper, also proof theory in the sense of (2) and (3) is to be classified as 'natural history'). There is thus no complete agreement in the literature about what direction the subject should or could take nor about the question what it should be called. As to the last question, the idea to use 'proof theory' and 'theory of proofs' as two non-synony• mous terms denoting different aims in the study of proofs does not strike one as very suggestive (and Kreisel need not be under• stood as proposing that but only as wanting to change the name of the subject to avoid associations with directions that according Philosophical aspects of proof theory 239 to him are better disregarded). One may ask whether it is proper to widen the term proof theory to cover also studies of kind (3), whose aims more or less coincide with that of traditional logic as noted above. But in view of the fact that modern logic embraces also other branches not concerned with proofs, it seems reasonable to use proof theory as a broad name of that part of modern logic which is concerned just with proofs and add the qualification 'general' when the need arises to distinguish it from proof theory in a more restricted sense. To the conflicting views about what directions proof theory should take (or could take at present), I shall return in several later sections.

2. The philosophical relevance of the subject

The term mathematical logic was formerly used to denote logic treated by the mathematical method, i.e. using symbolic aids and proceeding ded uctively , see e.g. Church [15] . It has later also been used to refer to logic that is of special interest for mathematics, and today it often seems to stand for mathematical research into problems of only distantly logical origin. This shift in the meaning of the term from 'logic treated by the mathematical method' to 'mathematics originating from logical problems' is parallel to a certain general, current tendency in (what is called) logical research. The idea of a 'philosophical logic' may be understood as a reaction against this trend in the development oflogic. The term sometimes seems to be used in a general sense for logic of philo• sophical interest, but, more often, it seems to stand for the logic of concepts that are especially used by philosophers, such as various modalities, or are consid ered to be the special business of philosophers to discuss, such as the notion of preference. This terminological situation seems to me unfortunate. It is not necessarily the logic of concepts occurring in philosophical con• texts that are of the greatest philosophical interest. But, moreover, the terminology obscures the fact that logic (i.e. what is properly so called) as well as the closely related foundations of the deduc• tive sciences, regardless of whether these areas are studied with the help of mathematical techniques or not, are essentially philo- 240 D. Prawitz sophka! disciplines aimed at the understanding of certain aspects of reasoning and related problems. A discussion of the philosophical relevance of proof theory may illustrate the general situation. As is the case of today's mathe• matical logic, many parts of proof theory are not of any special philosophical interest. But, at the same time, proof theory con• tains many themes of an epistemological and meaning-theoretical nature. Therefore, in spite of its rather technical character, much of proof theory seems to touch more central themes of philosophy than many other branches of so-called mathematical or philo• sophical logic. This is true both for general proof theory with its stress on evidence and on how we get to know truths and for reductive proof theory with its attempt at a foundation of mathe• matics. In the rest of the discussion in this paper of philosophical aspects of proof theory with special attention to the development in the last decade, I shall especially deal with Hilbert's program and reductive proof theory (section II). Although the formulation of Hilbert's program belongs to the twenties, the proper way of understanding and evaluating it is something that has still been much discussed in the last decade, and technical works as well as monographs in this field have also been appearing very recently. The greater space given to reductive proof theory is not owing to the belief that this part of proof theory is philosophically the most interesting one - on the contrary, my conclusion is the opposite -- but merely to the fact that my discussion of reductive proof theory turned out to require most of the available space. In the next subsections, I shall therefore restrict myself to a quick sketch of some elements of the early history of the subject (1.3) and to a brief presentation of different directions in general proof theory (1.4). It should be stressed again that this paper is in no way intended to survey recent developments in proof theory but to present and discuss some philosophical aspects of the subject. Furthermore, since proof theory was not treated in the two pre• ceding series of philosophical chronicles, it did not always seem advisable to respect the time limit of the last decade. Philosophical aspects of proof theory 241

3. Early history of the subject

Frege's foundation of modern logic consisted firstly in the dis• covery of a way of analyzing logical forms (where, as stressed by Dummett [16], his way of analy zing universal quantification is especially crucial) and secondly in a related codification of logical inferences. f'rege is the first to analyze a significant enough area of proofs, and his work obviously constitutes also the foundation of modern proof theory in the wide sense of the term. Consecutive work by Russell, Zermelo and Hilbert, further developed and modified Frege's analysis. While they all analyze reasoning that partly dates very far back, partly emanates from Cantor, Heyting obtained, by a surprising slight formal modification of this work, a codification of some of the reasoning involved in the revised, so• called intuitionistic mathematics proposed by Brouwer. Also most logical work in the twenties and thirties, when meta• theoretical investigations began into the systems of inference rules that resulted from the analysis of Frege and his successors, was of proof-theoretical concern. This, of course, is especially true of Godel's completeness and incompleteness results about the scope and limitation of these systems. Most of this work including that by Godel does not directly deal with proofs, however, but rather with the derived notion of provability (in a formal system). The same is true for certain formulations of Hilbert's program, but, as will be discussed in section II when I return to this part of proof theory, the study of proofs played a part in the contemplated execution of the pro• gram. The fIrst ones to deal in an essential way with proofs as an object of study are Gentzen and - to a lesser extent - Herbrand. They are noteworthy both for their contributions to Hilbert's program, where their ideas have allowed a leap forward, and for their interest in the structure of proofs, i.e. not only the forms of the various local inferences of which the proofs are made up but also certain overall structures to which the various inferences may give rise. In addition, Gentzen [17] analyzed anew proofs in first order languages, which resulted in a new set of primitive inference rules that was significant in several ways. In particular, his choice of primitive inference rules can be seen as dictated by what it is 242 D. Prawitz in the end that makes an inference evident. This analysis forms the basis of Gentzen's other results and has been a main stepping-stone in present-day proof theory. Gentzen's main idea is to break down the inferences into two kinds of steps, called introductions and eliminations, where each step involves only one logical constant. Each rule is here such that one cannot imagine other more basic rules in terms of which the rule would be derivable. The rule for the introduction of sentences of a given form states a condition for inferring sentences of that form. The rule for the elimination of sentences of a given form states what direct conclusions can be drawn from premisses of that form. Furthermore, the rule for elimination is the inverse of the corresponding introduction rule in the sense that a proof of the conclusion of an elimination is, roughly speaking, already available if the premiss of the elimination is inferred by an intro• duction. This fact has later been called the inversion principle. As a typical example, an introduction of an implication A ~ B has the form

A

B

A~B where in other words the condition for inferring A ~ B is a proof of B from hypotheses A, which are said to be discharged or bound by this inference. An elimination of an implication A ~ B has the form

which is the inverse of the corresponding introduction in the sense that if the condition for inferring A ~ B stated by the introduction rule is satisfied, i.e. there is a proof of B from the hypothesis A, then a proof of B is aiready available by replacing every hy• pothesis A by the given proof of A. Philosophical aspects o/proo/theory 243

A collection of introduction and elimination rules is known as a system 0/ natural deduction. On the basis of such sy~tems, Gentzen formulated another somewhat more artificial kind of systems, called calculi of sequents, and for these systems he proved the so-called Hauptsatz according to which each proof can be put in such a form that nothing is introduced in the proof which is not also contained in the end result - the proof is in this way direct, containing no auxiliaIY elements. It was shown by Prawitz [18 J and, independently, to some extent by Raggio [19J that already the proofs in the systems of natural deduction can be put in a certain normal form and that this normal form theorem, which is equivalent to the Hauptsatz, can be seen as directly depending on the inversion principle. In fact, the inversion principle gives rise to certain reductions appli• cable when a sentence is obtained as the conclusion of an intro• duction and is used as the premiss of an elimination. The reduction removes such a round-about. As a typical example, we may consider a proof of the form shown to the left below. In accordance with the explanation of the inversion principle above, it reduces to the proof shown to the right

A A B A A-~ B B B where the conclusion B is obtained without the detour of first introducing A -+ B and then applying implication elimination. The normal form is arrived at by successively carrying out all such reductions, and its existence is proved by showing that a series of these reductions terminate. A normal proof can be seen as consisting of two parts, one upper, analytical part that proceeds from hypotheses by successive eliminations and one lower, syn• thetic part, where the obtained results are used to infer con• clusions by successive introductions. A normal proof proceeds in this way directly from the hypotheses to the conclusion of the 244 D. Prawitz proof without introducing any sentences in the proof that are not used in building up the conclusion of the proof.

4. Directions in general proof theory

4.1. The results mentioned above, Gentzen's Hauptsatz and the normal form theorem, were first obtained for first order logic. In recent years, they have been extended to a great number of other systems. Such an extension involves essentially two steps, whose exact formulation depends on wheth~!r it is formulated for a cal• culus of sequents or a system of natural deduction. In the latter case, they consist of, fin:tly, an analysis of proofs in the area in question in such a way that they are broken Jown into intro• ductions and eliminations with accompanying reductions, and, secondly, a proof of the fact that each proof can be put in normal form, i.e. in a form where all possible reductions have been carried out. Some of these extensions depend essentially on the analysis of the proofs involved, i.e. the right formulation of the intro• duction and elimination rules, while the proof of the normal form theorem raises no new difficulties; in other extensions, the latter has been the main obstacle. In Prawitz [18], where results in the natural deduction formu• lation were obtained for first order classical, intuitionistic, and minimal logic, extensions were made to some systems of modal logic (S4 and S5 in classical and intuitionistic variations), to certain non-standard systt'ms (like a relevance logic and a system similar to Fitch's set theory), and to ramified second order logic. An exttnsion to an infinite system of arithmetic of natural numbers (in the form of a system similar to a calculus of sequents) had already been obtained in 1951 by Schutte [20]. Later, Tait [21] and Martin-LOf [22] developed classical and intuitionistic systems, respectively, for infinite propositional logic. Among other various extensions essentially depending on a conceptual analysis of the proofs, I want to mention one by Stenlund [23] to an interesting theory of descriptions. For unramified (impredicative) second order logic, which has attracted special attention since it allows a formulation of classi• cal mathematical analysis, and, more generally, for higher order Philosophical aspects o/proo/ theory 245 logic (or simple type theory), the first step of the extension is straightforward (in analogy with first order), but the proof of the normal fonn theorem (for a long time known as Takeuti's con• jecture) turned out to be more difficult. It was solved by Tait [24], Takahashi [25, 26], and Prawitz [27, 28] by utilizing a certain correspondence between the c:llculi of sequents for classi• cal logic and Tarskian truth definitions (see further section 4.2 below). Extensions to intuitionistic logic of second and higher order were obtained by Prawitz [29] and Osswald [30], respec• tively. However, all these results for impredicative systems showed only the existence of a normal form for the proofs in question and did not establish the termination of any particular series of reduc• tions. This situation drew attention to the difference between such a normal form theorem and a normalization theorem that estab• lishes the termination of some reduction procedure; for the predi• cative systems, this latter and stronger result had usually been implicit in the proofs. For first order Peano arithmetic, Gentzen [31] had in fact (although he did not formulate it in these terms) obtained a normalization theorem for proofs of numerical equations. An extension to all proofs in the system was obtained by lervell (32]. More generally, Martin-Lor [33] analyzed intuitionh;tic proofs with ind uctive and iterated inductive definitions into introduction and elimination inferences and proved a corre• sponding normalization theorem. This result is of special interest for two reasons: it extends the idea of introductions and elimi• nations to apply not only to logical constants but also to atomic predicates, and it allows a constructive development of a con• siderable part of mathematics. The normalization theorem for this system depended among other things on two essential discoveries, namely (1) of the correspondence between proofs and certain functionals, which was to be expected in view of certain intuitionistic theones about proofs (see section 4.3 below), and (2) of the resulting possibility to carryover to proofs a certain notion of computability introduced earlier by Tait [34] for a system of functionals under the name convertibility. This allowed Martin-Lof to formulate a fairly general method of proving normalizability via proofs of computability. A considerable extension in a quite different direction but also 246 D. Prawitz depending on (1) above is made by Martin-Lof {35] who formu• lates an intuitionistic theory of types which is supposed to allow the formalization of existing, constructive mathematics. Intro• duction and elimination rules are here formulated for con• structions in general, of which proofs are only special cases. The correspondence between proofs and functionals made it also possible to strengthen the normalization theorem for many systems by showing that all reduction sequences (starting from a given proof) terminate in a unique normal form either by carrying over similar results from systems of functionals or by modifying Martin-Lors version of computability. Results of this kind, called strong normalization theorems, are surveyed in Prawitz [5] . The translation between systems of natural deduction and calculi of sequents established by Prawitz [18] to obtain the equivalence between the normal form theorem and the Hauptsatz was improved by Zucker [36] and later by Pottinger {37] to obtain something similar to the strong normalization theorem also for the calculi of sequents. Furthermore, Girard [38] was able to extend the computa• bility notion to second order systems of functionals, which enabled Girard [38,39], Martin-Lo[ [40,41], Prawitz [51, and Troelstra [42, 43] to obtain various normalization and strong normalization theorems for second and higher order intuitionistic logic. These results can also be extended to certain classical sys• tems, see Statman [44] and Prawitz [45] . Much before, however, Takeuti [46] had obtained something similar to a normalization theorem for a subsystem of classical second order arithmetic with quite different methods in his work on Hilbert's program, see further section 11.5.

4.2. Although Gen tzen [17] arrived at his calculi of seq uents by reflecting upon the systems of natural deduction and, in the case of classical logic, generalizing the deducibility relation by allowing several sentences taken disjunctively as conclusions (which to his surprise, as Gentzen notes in his paper [31], removed certain problems connected with classical negation), there is an equally interesting aspect of the calculus of sequents for classical logic, namely the close correspondence between its rules and the clauses of the Tarskian truth definition, which was perhaps first explicitly Philosophical aspects of proof theory 247 noted by Beth [47] in terms of his semantic tableaux. More precisely, the introduction and elimination rules in the systems of natural deduction become formally rules for introducing sentences in the succedent and antecedent, respectively, of a sequent but can also be read as the conditions for the truth and falsity, respec• tively, of sentences appearing as members of an iterated dis• junction. In this way, Gentzen's Hauptsatz may be looked upon as a generalization of the fact that the conditions for the truth and falsity of a sentence cannot be satisfied simultaneously. Furthermore, systematic applications of the rules of the cal• culus of sequents backwards, so to say, yield because of this correspondence all possible countermodels to the sequent that one started with, if such countermodels exist. When they do not, i.e. when the procedure of constructing countermodels breaks down at a finite stage, this will manifest itself in a demonstration of the fact that no assignment of truth values that refutes the sequent is consistent in the sense of not assigning both truth and falsity to the same sentence. Because of a certain property of duality, this demonstration constitutes at the same time a proof of the sequent in question. As a result, we get an immediate proof of Godel's completeness theorem for first order predicate logic, and as a further immediate corollary, we obtain Gentzen's Hauptsatz. Proofs of the completeness theorem have been given in this way by Beth [47] ,Hintikka [48] , Kanger [49] , and Schutte [50] . This aspect of Gentzen's work is discussed in more detail by Prawitz [51] and is further generalized by Kreisel, Mints, and Simpson [52]; for later comments on this generalization, see Kreisel [53]. It was extended to second and higher order logic to get Gentzen's Hauptsatz for those cases by the proofs of Tait, Takahashi, and Prawitz mentioned above, which are further discussed and reworked by Girard [54]; in this connection, see also Pappinghaus [55] .

4.3. While the calculus of sequents for classical logic lends itself to establishing a link between proof theory and classical semantics (model theory), the system of natural deduction is especially suitable for certain discussions of the semantics of intuitionistic logic. Intuitionistically, the logical constants are often interpreted in 248 D. Prawitz terms of proofs or constructions as perhaps first explicitly done by Kreisel [56]. For instance, the implication A ~ B is taken to assert the existence of a construction c, which applied to a proof of A yields a proof of B, together with a proof of the fact that c has this property. In other words, proofs are not only thought of as demonstrating the truth of the sentences, but the sentences are un• derstood as speaking about proofs. There is then an obvious common ground between the semantics of intuitionistic logic and general proof theory. A correct intuitionistic interpretation of sentences of a certain form must, it seems, at the same time state what con• stitutes a proof of a sentence of that form. Kreisel's work in [56] has been elaborated by among others himself in [57]; see also Troelstra [58]. A similar but slightly different approach had already been vaguely suggested by Gentzen [17] . He looked upon the condition stated by an introduction rule for inferring sentences of a certain form as giving the meaning of the logical constant in question. For instance, an implication A ~ B would be understood as expressing the existence of a proof of B from the hypothesis A. The corre• sponding elimination rule, on the other hand, got its justification from this meaning since, as we saw in section 3, given that the condition for inferring the premisses of an elimination by intro• duction is satisfied, a proof of the conclusion is already available, or more precisely, is obtained by a reduction. The normalization theorem can be seen as a sharpening of this idea; it shows that the whole system of elimination rules is justi• fied in the sense that any proof in the system of a sentence (not depending on hypotheses) is transformed by a series of reductions to a non-reducible proof, which necessarily has to end in an intro• duction, i.e. the canonical way of arguing according to this point of view. But Gentzen's general idea has been further elaborated in two slightly different ways. One way is to note a similarity between natural deduction and an extended A-calculus for defining functionals. For instance, a derivation of B from an hypothesis A may be looked upon as an open term t(x) in which x is a variable ranging over proofs of the hypothesis A and which yields a proof t(p) of B whenever a proof p of A is substituted for the variable x. The proof of A ~ B obtained by implication introduction applied to the derivation of B from A may then be identified with the Philosophical aspects of pro of theory 249 function Xxt(x) which applied to a proof p of A yields the proof t(p) of B. To say as Gentzen does that implication intro• duction determines the meaning of implications may then be understood as saying that the meaning in question is determined by the requirement that a proof of A ~ B is to consist of a function that applied to a proof of A gives a proof of B and that can be written in the form Xxt(x), which corresponds to the first half of Kreisel's interpretation of implication. On the other hand, a proof obtained by applying implication elimination to a proof q of A ~ B and a proof p of A may be identified with the result of applying the function q to p. If q is written in the form AXt(X), the proof may then be written {Axt(x)}(p), which is identical with t(p). In the terminology of X• calculus, the latter term t(p) is said to come from {Xxt(x)}(p) by X-conversion. Implication introduction is in this way seen to correspond to X-abstraction, implication elimination to function application, and the reduction associated with implication to X-conversion. The formal analogy between implicational logic and X-calculus seems first to have been noted by Curry and was more systemati• cally developed and extended to full predicate logic by Howard [59] and Prawitz [5, 60]. Howard's formulation, the corre• spondence between propositions and types of functionals is formulated as an isomorphism (in Prawitz' formulation as an homomorphism). The correspondence has been utilized especially by Martin-Lor as mentioned above (section 4.1). This way of looking at proofs also leads to the identification of two proofs when one can be obtained by a reduction of the other as discussed by Martin-Lof [61], Kreisel [62], and Prawitz [5] . Feferman [63] has objected to this idea on the ground that a proof PI of a sentence A may reduce to a proof P2 but may con• tain much more information than P2; on its way of proving A, PI may establish the truth also of many other propositions not estab• lished by P2' This cannot be denied, but it must be observed that on the view presented here, a proof is not looked upon as a collection of sentences but as the result of applying certain oper• ations to obtain a certain end result; when understood in this way, it seems difficult to deny that e.g. the proof expressed by {Xxt(x)}(p) is identical to the one expressed by t(p). The main question 250 D. Prawitz must thus be to what extent the identification of proofs and con• structions such as those obtained in the A-calculus is relevant; in this connection, see especially Martin-Lof [35]. However, these questions about the identity of proofs cannot yet be considered settled in the literature. The other direction in which Gentzen's idea about justifications of inferences has been elaborated is as a suggestion for a criterion for the correctness of a proof or, rather (since if a proof is rightly called a proof, it has to be correct), for the validity of an argu• ment. Here, the starting point is thus a notion of argument, which is defined as a collection of sentences structured in a certain way. An attempt to work out such an idea has been made by Prawitz [ 13, 64]. It used a slight modification of Martin-LOf's notion of computability mentioned above, which was now worked out as a notion of validity of an argument. Roughly speaking, an argument is defined as valid if it either ends with an introduction whose premisses are proved by valid arguments or can be transformed to such a form by reductions. In other words, the introductions are considered as self-justifying, as constituting the canonical ways of arguing for the sentence forms in question, while the elimi• nations are justified by the very reductions that generate the normal form of proofs. The validity of a set of inference rules as defined by Prawitz [13 J is a general sufficient condition for the normalizability of the proofs obtained by applications of these rules. The question of the completeness of the logical constants from the point of view of such a semantics is discussed by Prawitz [ 10] and Zucker and Tragesser [65] . However, it is to be noted that this notion of valid inference suffers from the same weakness as discussed in section 1: to be permissible in a proof, it is not enough that an inference happens in fact to be justifiable, it must also be justified, i.e. it must be seen to be correct. Dummett [66, 67] has broadened the perspective above by suggesting that the meaning of any expression is determined by two features, on the one hand the condition for correctly uttering the expression, and on the other, the direct consequences of uttering it, or what the uttering of the expression commits the speaker to; which features have to be in harmony with each other. In the case of affirmative sentences of the usual logical form, these Philosophical aspects ofproof theory 251 two features are given by Gentzen's introduction and elimination rules, and the required harmony is expressed by the inversion prin• ciple (mentioned in section 3). In this way, following general ideas of Wittgenstein, Dummett wants to replace the truth condition of a sentence as central to its meaning by the condition for correctly asserting it. For this latter condition to be satisfied, the mere truth of the sentence is of course not sufficient - the speaker must also be in possession of a certain knowledge that warrants the assertion. In a similar way, the question of the correctness of an inference or of an argument may be related to the speaker's knowledge as discussed by Prawitz [9, 10]. It should be noted that none of the approaches discussed in this subsection is especially concerned with proofs in formal sys• tems or proofs restricted to the means expressible in a formal lan• guage. The main question that has been put here is what properties proofs of a sentence of a given form are to have - in this respect, the notion of proof is tied to questions of linguistic forms, but for well-known reasons, the proofs or sentences in a given language cannot in general be restricted to that language. So far we lack a general proof theory with a definition of an absolute notion of provability that is generally agreed upon com• parable to e.g. recursion theory with its definition of algorithm or model theory with its definition of logical consequence. Kreisel [12, 14] doubts that such a subject is now ripe for study or doubts the fruitfulness of such studies. Godel [68] in a paper reprinted in the series of chronicles preceding this one (Contem• porary Philosophy f) where he discusses provability instead of proofs, expresses the opposite conviction. Stressing that his incom• pleteness theorem excludes the possibility of identifying prova• bility in a calculus with the (absolute) notion of provability, he notes that this negative result only rules out certain ways of defining provability but does not make such a definition impossible. Similar remarks have been made by Myhill [69J, stressing the need of a general notion of proof. Some of the works referred to above constitute an attempt towards such a general notion of proof.

4.4. Since the discussions in this area have not yet resulted in any definite or generally accepted viewpoints, one cannot expect a 252 D. Prawi(z survey like the present one to do full justice to all the participants, i.e. I want to make the reservation that further developments will soon show that the accents should have been put in different places or that the whole picture should have been painted differ• ently. In conclusion, I also want to mention that other surveys of the topics discussed here have been made in papers by Kreisel [62], Prawitz [5], and Troelstra [43] in the last decade, that a mono• graph by Cellucci [7] dealing with certain parts of the subject has recently appeared, and that Gentzen's collected works have been available in an English translation by Szabo [70] for some time - for an extensive review of that volume, see Kreisel [71].

II. HILBERT'S PROGRAM. APPRAISAL. RECENT DEVELOPMENTS

I. The goal of the program

Hilbert often expressed himself as if the goal of his program was simply to establish the consistency of mathematical theories. This problem was stated by Hilbert [72] in 1900, was repeated in a number of articles in the twenties and is the basic question from which the exposition in the standard work Grundlagen der Mathe• matik by Hilbert and Bernays [73] starts. In view of Hilbert's emphasis on the consistency question, one may think that his position was that of strict formalism by which I mean the idea that mathematics may be understood as a number of games where formulas are derived according to prescribed formal rules, where no meaning is to be attached to the formulas (although, outside of mathematics, some of them may turn out to be useful) and where the only requirement to be put on the game is consistency. It should be clear, however, that Hilbert took a fonnalist position only towards sentences that contained reference to infinite totalities, which sentences he called ideal, while he considered the other sentences that did not make such reference, the so-called real sentences, completely meaningful. It should also be clear that strict formalism is not really coherent if one attaches importance to the consistency problem and considers it to be a mathematical problem, because then there are at least some mean- Philosophical aspects of proof theory 253 ingful mathematical sentences, viz. the ones expressing the consis• tency of certain formal systems. The question cannot be whether these sentences are provable in one or other formal system but whether they are true. As soon as this is realized, it is clear that there must be at least one realm of mathematics, viz. the one in which the proof of consistency is carried out, which cannot be viewed formalistically but must be understood as having a direct content; as is well-known, for Hilbert this realm was what he called finitary reasoning. Now, if one takes this position of moderate formalism, where a part of mathematics is considered as directly meaningful, con• stituting what mathematics is directly about, while another part, viz. the one containing reference to actual infinity, is taken as an ideal or formal part, added to smooth and round off the real part, then the important question must be whether proofs that use ideal elements in order to prove results in the real part always yield correct results. To establish this is just the goal of Hilbert's program. In some of Hilbert's publications, in particular in [3, 74], the program is formulated more or less in this way, but the one-sided emphasis on the consistency problem in many other publications has sometimes come to overshadow the proper for• mulation, to which in particular Kreisel has repeatedly called attention, among others in some recent publications [ 11, 75] . Although Hilbert's program as here formulated is equivalent to the consistency problem under certain conditions (see below), its proper formulation is important when it comes to evaluating its significance. Naively one may think that it is always of interest to establish the consistency of a mathematical theory. But as we have already seen, from the point of view of strict formalism, the question of consistency, let alone the question of how consistency can be convincingly proved, cannot even be put in mathematics. The consistency problem becomes significant in mathematics only when some part of mathematics is accepted as meaningful (with a definite content) and correct, while another part is put in more doubt. But then, as said above, the important question must be not only whether a mathematical theory is consistent but whether we can be sure that sentences belonging to the accepted part, i.e. real sentences to use Hilbert's terminology, which are proved using elements from the doubted part of the theory, are always true. 254 D. Prawitz

Starting from the consistency problem we thus arrive at the more mature Hilbert's program. But, as already remarked above, in his last publication [73] (together with Bernays) on this subject, the program is still formulated in terms of consistency. In conclusion, we may summarize Hilbert's general program as containing the following two steps

1) a separation of an immediately accepted part of mathematics, called the real part, which was identified by Hilbert with what he called finitary mathematics, from another not immediately accepted part, called the ideal part, which according to Hilbert typically contains references to the transfinite, and 2) a demonstration carried out in the real part of mathematics of the fact (if a fact) that every provable real sentence is true, i.e. that every sentence belonging to the real part which is proved by possible use of the ideal part is nevertheless true (according to the standards of the real part).

The program requires of course that we are able to describe the mathematics spoken about in some suitable way. A first necessary step is thus to represent the mathematics in question, say some more or less inclusive part of mathematical practice, in the form of a theory of some kind. The formulation of such a theory can be more or less articu• lated. To present it as a formalized system, i.e. to give the language and the principles of the theory in such a way that it can be decided whether a given expression belongs to the theory or not and whether something is a proof in the theory or not, is to be viewed as a way of articulating the theory and not to reflect a formalistic attitude, where sentences and proofs are considered as just formal objects. But (as will be seen in section 2 below) Hilbert's ideas about how the program was to be carried out req uired formalization.

2. Execution of the program. General method

The theory to which Hilbert wanted to apply his program was in particular arithmetic for real numbers by which he meant a theory sufficient for developing usual mathematical analysis. Formal Philosophical aspects o/proo/ theory 255 systems for such a theory were developed in the Hilbert school by modifying Whitehead's and Russell's system in Principia Mathe• matica, in particular by taking the notion of natural number as a primitive and adding well-known axioms for this notion. Such a system was believed to cover or represent most of mathematical practice; hence, a successful realization of Hilbert's program for this system would give a foundation of mathematics in general. To carry out stage I) of the program as formulated in the pre• ceding section, we have to make a separation both with respect to what is immediately accepted as meaningful and with respect to what is immediately accepted as evident, the fIrst being a linguistic or semantic division of sentences into real and ideal ones and the second being a division of principles, i.e. both axioms and rules of inference, into those that can be used as evident in a real proof and those that cannot be so used. According to Hilbert, the real sentences embrace firstly all sentences expressing decidable propositions about (finite) configurations. Thus sentences expres• sing that a certain sequence of formulas is a proof, in a certain formal system, of a certain formula belong typically here, and since natural numbers are understood by Hilbert as series of strokes, sentences like 2 + 3 = 5 or 10 10 . 10 10 = 1020 also belong here. Since truth-functional compositions of decidable propositions are again decidable, the application of the corresponding sen• tential connectives to decidable sentences gives rise to new real sentences. The crucial question is then what kind of sentences universal and existential quantifications over, for example, all natural numbers give rise to. Such quantifications when understood as infinite conjunctions and disjunctions, respectively, are of course prime examples of transfinite propositions, but on the other hand already the con• sistency statement asserts something about all possible proofs of the system in question. Hilbert's solution of this dilemma consists in considering another 'finitary' interpretation of the universal quantification over e.g. natural numbers, namely (as he says in [3]) to understand a corresponding assertion as 'a hypothetical judgement which asserts something for the case when a numerical example is given'. Similarly, Gtmtzen [76] explains the construc• tive meaning of the sentence 'all natural numbers have the property P' as saying: 'Regardless of how far we progress in the 256 D. Prawitz formation of new numbers, the property P continues to hold for these new numbers'. Although a similar finitary interpretation of an existential judgement 'there exists a natural number with the property P' as attesting that 'we have found a natural number with the property P' is possible as Gentzen [76] remarks, Hilbert refers the existen• tial sentences to the ideal ones, and hence, he does the same for sentences that are negations of universal ones or have the form of an implication with a universal sentence as antecedent. Since a sen• tence V Xl VX2 .. . Vxn A(X1,X2 , ... , Xn) with A quantifier-free can always be replaced by an equivalent sentence V xA'(x), where A' is again quantifier-free, we can say in summary that the real sen• tences comprise the decidable ones and the ones of the form VxA(x) where each instance A(t) is decidable; the rest are con• sidered to be ideal. Furthermore, all proofs that can be carried out in arithmetic using only real sentences are accepted as immediately evident as long as universal sentences do not appear as assumptions in the proof. Thus, in particular universal introductions and proof by induction, where the induction hypothesis is not universal, are accepted. (Since we never essentially operate on a universal formula in a real proof, the universal quantifier is dispensable in the real part, and universal sentences are usually replaced by free• variable formulas.) The demonstration in step 2) of the program that proofs of real sentences, even when using ideal elements, always lead to true sentences, could in principle be carried out in many ways. For instance, one could hope for some special interpretation of the ideal part under which all principles assumed in the ideal part become true (like the interpretation of complex numbers as pairs of real numbers). But, since the real part embraces only finitary reasoning and the ideal part contains transfinite concepts, such an interpretation seemed improbable, and a distinctive feature of Hilbert's program is not only its goal but also the method by which one hoped to reach the goal, namely by an investigation of the proofs of the theory in question. Hence the need of a proof theory. Furthermore, this method where proofs were to be objects of a finitary proof theory required a formalization of the investi• gated theory. Philosophical aspects of proof theory 257

Given a formalization of a theory T of arithmetic, we note that the result to be established for T can be expressed by the real sentence:

(1) For each proof pinT and for each real sentence A in T: if p is a proof of A in T, then A is true.

The sentence (I) is of course not a real sentence ofT, but it is not required that the demonstration of the truth of the provable real sentences of a theory T is carried out in the real part of T itself. By introducing a function g that assigns Gadel-numbers to the objects of T and by defining a relation Pr and a property Tr to hold for Gadel-numbers when the corresponding objects of T satisfy the relation 'proof of and 'true', respectively, sentence (I) can be written

(I') For each proof pinT and for each real sentence A in T: Pr(g(p), g(A)) ~ Tr(g(A».

(I') is of course still not a sentence of T. But it now implies for each real sentence An of T with Gadel-number n:

(2) Vx(Pr(x, ii)~ An} (where ii is the numeral denoting n) which indeed is a real sentence of T, when Pr is defined directly by primitive recursion (so as to coincide with Pr as defined above). Kreisel [75] takes (2), which is called the local reflection principle by Kreisel and Levy [77] , to be the statement that is to be demon• strated in Hilbert's program, but it seems more appropriate to note it as a consequence in the real part of T of the sentence (1'), called the global reflection principle by Kreisel and Levy [77] ; (1) or (1') is what really has to be established. As is well-known, the Hilbert school never succeeded in proving (I) (or (2» finitarily for a theory T containing usual arithmetic (for real numbers). The only result obtained was the one by Gentzen for a first order theory of natural numbers. I shall report on some further recent progress for subsystems of analysis in section [5 ] below. As regards the formulation of the result to be proved in 258 D. Prawitz

Hilbert's program, we should finally note that when the real sen• tences are delimited in Hilbert's way, the consistency problem happens to be equivalent to Hilbert's program. Trivially, (l) implies consistency. Conversely, assume that T is consistent and that a real sentence A has been proved in T. If A is a quantifier• free sentence and 4ence decidable, then A must be true because otherwise I A would be true and the completeness of T with respect to decidable sentences would imply that also IA was provable, contrary to the consistency of T. If A is of the form VxB(x), then B(n) would also be provable for each numeral n, and hence, by what has just been said, B(n) must be true for each n. But this isjust what is required for the truth ofVxB(x). This equivalence is in effect noted by Hilbert, e.g. in [74], but was not enough focalized. With Kreisel, who has especially stressed the equivalence, e.g. recently in [11], one must say that the reason for this is insufficient discussion of the significance of Hilbert's project, which was usually just stated as the task of proving consistency. In particular, it is important to note that Hilbert's way of drawing the demarcation line between the real and ideal sentences is crucial for the eq uivalence. As noted above, existential sentences 3xA(x), where A(x) is quantifier-free, can as well as the universal ones be interpreted constructively and hence counted as real sentences. But then the equivalence between the consistency problem and Hilbert's program fails: nothing prevents a sentence 3xA(x) to be provable in a consistent theory T although A(n) is false for each numeral n. And of course this is fatal regardless of whether 3xA(x) is read in a finitary or in a transfinite way; in either case, consistency is no guarantee that the theory does not contain provable sentences which are simply false. We may also note that the terminology 'finitary' and 'trans• finite' is not very felicitous since the finitary sentences contain sentences formed by universal quantification over all natural numbers. Thus, the infinite enters into both the finitary and trans• finite sentences but in the first case as potential infinity and in the second case as actual infinity as explained above. What this differ• ence amounts to is not too obvious; it is one of the main issues that has to be clarified or perhaps rather reformulated in the dispute between classical and intuitionistic logic. However, I shall here follow this terminology of the Hilbert-school. Philosophical aspects ofproof theory 259

3. Motives for the program

Hilbert's program is meant as a contribution to the foundations of mathematics, but one can be interested in the program from a foundational point of view for various reasons.

3.1. Moderate formalism taken in a very literal or extreme way would deny the ideal part of mathematics any meaning and would consider it just as a convenient formal machinery for producing results about the real part. Such a position makes a successful realization of Hilbert's program very necessary since pending this, there are no reasons, except purely empirical ones, to assume the machinery to work in the intended way. But the natural mathe• matical reaction towards such a machinery would be, I think, to ask why it works, to try to understand it, and probably nobody in the Hilbert school understood (moderate) formalism in a literal way. Gentzen certainly did not; indeed, as already remarked, in [76] he discusses the existential sentences of the ideal part and gives them both a finitary and a transfinite interpretation.

3.2. The position of Gentzen is rather that the ideal sentences have some kind of meaning, which agrees with the usual way of reason• ing about them, but that this meaning is somewhat shaky; the ideal sentences are not yet entirely understood, and therefore one has to give them a finitary foundation, making sure that they do not give rise to any inconsistencies. Gentzen [76] says e.g.: 'Indeed, it seems not entirely unreasonable to me to suppose that contradictions might possibly be concealed in classical analysis." Although something would certainly be gained by a consistency proof when one maintains an attitude like Gentzen's, the obvious weakness with this position is that consistency or, eq uivalentIy, truth of provable real sentences, is far too little. Since also the so-called ideal sentences are granted a meaning, even if somewhat shaky, one must demand of a foundation that it also shows the results about ideal sentences to be in agreement with this meaning. And as we have already seen, Hilbert's program does not rule out that a sentence3xA(x) is provable although A(n) is false for all n. 260 D. Prawitz

3.3. Hilbert's various statements are more ambiguous or inconsis• tent than Gentzen's. On one side, he declares in [3] that 'we must realize that the infinite in the sense of an infinite totality, where we still find it used in deductive methods, is something fictitious', which seems to show that he takes moderate formalism literally. On the other hand, unlike Gentzen he never expressed any doubts about the consistency of classical analysis and, as Kreisel [781 points out, never spoke about investigating whether analysis is consistent but only about establishing the consistency; indeed Hilbert [I] asserts that a complete and justified certainty about inferences prevails in analysis, and he accuses a critic like Weyl of seeing ghosts. Therefore, there are good reasons to think that Hilbert saw his program not as removing any doubt about the correctness of trans• finite reasoning that he had himself but as a continuation of an axio• matic tradition, which wants to build mathematics on a base as small and simple as possible (and which in addition should serve to convert the sceptics). Hilbert is perhaps best understood in the same way as Frege, who did not doubt ordinary mathematical reasoning but nevertheless wanted a rigorous foundation of mathe• matics and especially a reduction of mathematics to logic, except that Hilbert wanted a reduction not to logic but to what he con• sidered to be the most proper base, namely finitary mathematics. What Hilbert here means by a reduction is indicated in the lecture [3 J where, speaking at the 100th anniversary of Weierstrass, he wants to continue Weierstrass' work of eliminating the infinitely large and infinitely small from analysis by showing how also references to infinite totalities in universal and existential quanti• fication can be replaced by finite procedures which yield exactly the same results. When Hilbert's motives are understood in this way, the interest is shifted from the question of the truth of the provable real sen• tences, which is not in doubt, to showing that any proof of a real sentence carried out in the ideal part by use of transfinite reason• ing can be replaced by a proof of the same sentence in the real part using only finitary reasoning. When this is shown by finitary means, there is a sense in which we reduce the transfinite to the finite, not by an interpretation, but by showing how the use of transfinite notions can be eliminated when we are only interested in results about the finitary. Philosophical aspects o/proo/ theory 261

Although the interest of the program no longer resides in demonstrating the truth of the provable real sentences in itself, such a demonstration if finitary provides just the reduction wanted as pointed out e.g. by Kreisel [11] (and, in fact, before by Hilbert [74]). In other words, if PT is a fmitary proof theory in which the sentence (1) of section 2 above is demonstrated and PT contains the real part of T, there is a transformation r such that

(a) for every proof pinT of a real sentence A, r(p) is a proof in PTof A.

To see that this is so, we note that a proof in PT of (1) gives a finitary defined function I{) such that

(b) if a is a proof in PT of the fact that p is a proof of the real sentence A in T, then I{)(Q') is a proof in PT of the fact that A is true.

Furthermore, PT can be assumed to be complete enough to con• tain uniform proofs about what objects in T are proofs, i.e. there is a function 1/1 defined in PT such that

(c) if P is a proof of A in T, then I/I(p) is a proof in PT of the fact that p is a proofof A in T.

Furthermore, any proof in PT of the truth of a real sentence A in T can trivially be transformed to a proof of A in PT when it is assumed that PT contains the real part of T; i.e. there is a finitarily defined function X such that

(d) if a is a proof in PT of the truth of the real sentence A in T, then X(o:) is a proof in PT of A.

If we set rep) = X[I{)(I/I(p»], then (b) - (d) gives the desired result (a). 262 D. Prawitz

4. The impact of G6del 's incompleteness theorems

Of Godel's two incompleteness theorems, the second one has often been thought to affect Hilbert's program seriously. Accord• ing to it, the consistency of a theory T that satisfies certain con• ditions usually satisfied by most codifications of arithmetic such as containing a sufficient part of first order arithmetic for natural numbers, cannot be proved by using only principles of T. It may be true that Hilbert thought that the consistency of arithmetic, and hence, the sentence (2) of section 2 for any real sentence A of T could be proved in the real part of T itself, but this is just the possibility excluded by Godel's second incompleteness theorem. However, as already noted, the leading idea of Hilbert's program as formulated above only requires that the demonstration of the truth of the provable real sentences of T (or, equivalently, of the consistency of T) is carried out in real mathematics, not neces• sarily in the real part of T itself. As long as there exists finitary mathematics that is not codified in a given theory T, the possi• bility of carrying out the program for T thus remains in spite of Godel's result. Nevertheless, Godel's second incompleteness theorem is of course an important result about the form that the execution of Hilbert's program must take. For recent discussions of this theorem, see Kreisel and Takeuti [79] and Smorynski [80] . It should be noted however that in several ways Godel's first incompleteness theorem affects Hilbert's program just as much or more than the second one. The theorem (in the form given by Rosser) constructs for every consistent theory T that contains a sufficient part of first order arithmetic a real sentence V xG(x) such that

(i) For each numeral n, I-G(n); but neither I-VxG(x) nor 1--,VxG(x)

Two ways in which this result affects Hilbert's program are to be noted. Firstly, after the discovery of this theorem, it is not only an abstract possibility that there should exist a consistent theory with a provable false sentence of the form 3x-A(x) where A(x) is quantifier-free (as discussed at the end of section 2 above). Given (i) above, we need only add 3x-,G(x) as a new axiom to the ones Philosophical aspects of proof theory 263 of T; the new theory T' is consistent (since otherwise, classically, VxG(x) is provable in T) but 3x-,G(x) is false given that all the provable G(n) are true. The fatal effect for Hilbert's program of the existence of such so-called w-inconsistent theories was remarked upon in a short note by Godel [81] and has been stressed by Kreisel e.g. in [II] . Secondly, it is remarked by Kreisel [75] that the choice of codification, i.e. the formulation of the theory T of arithmetic now becomes a crucial problem since it will necessarily be incom• plete. It may be said that the whole idea of securing transfinite reasoning or reducing it to finitary reasoning (the two alternatives of section 3) now becomes less convincing when this cannot be done once and for all; it may seem that one needs insights into the transfinite in order to know how to extend a given codification. And there will always remain transfinite principles that are intuitively correct (from a transfinite point of view) but have not been secured or reduced. One could try to defend the program by noting that the studied theories are complete in the empirical sense of codifying the actual, existing mathematical practice. Kreisel's point is then that mathematical practice, if that is what one is interested in, can be codified by other systems that use less strong transfinite principles. Hence, in any case, the discovery of incompleteness must lead to greater attention to the choice of codifications.

5. Progress in the program

No one in the Hilbert-school succeeded in carrying out anything like the Hilbert program for a system that embraces classical analysis. A result in this direction for a subsystem of analysis, so• called TIl-analysis, was obtained by Takeuti [46] in 1967. His work builds on ideas in Gentzen's consistency proof for first order arith• metic 0 btained in the thirties. Gentzen published two versions of his proof, the first in 1936 [76] and the second in 1938 [31], both using transfinite induction up to the ordinal number denoted Eo. An earlier version of Gentzen's consistency proof became publicly known because of a paper by Bernays [82] and was recently published in the name of Gentzen [83]. It was submitted by Gentzen in 1935 but 264 D. Prawitz was withdrawn after criticism directed against the means of Gentzen's proof which were considered to be too strong. For a discussion and a criticism of the criticism, see Bemays [82] and Kreisel [84] . Takeuti's proof pursues the lines of Gentzen's second published proof, which may be described in summary as consisting of the following steps:

I) Certain ordinals, i.e. a certain system of notations which may intuitively be understood as denoting the ordinal numbers of a certain initial segment, viz those up to Eo are introduced and a decidable ordering relation < between them is defined; 2) it is shown that any descending sequence of the ordinals is finite, or rather, a demonstration is given of the principle of transfinite induction for < up to Eo applied to decidable sen• tences A(x), which can be formulated

(TO from the premiss 'if A(~) for all ordinals ~ < (3, then A({3)' infer the conclusion 'A( -y) for all ordinals -y' up to € 0;

3) a reduction procedure for proofs in first order arithmetic is introduced of the same kind as those discussed in section 1.3 with the property that an irreducible proof (i.e. one to which no further reduction is applicable) of a decidable sentence belongs to the real part of the theory, and it is shown by finitary means for a certain assignment of ordinals < € 0 to proofs of decidable sentences that the assigned ordinal is lowered by a reduction, which by (TI) implies that every proof of a decidable sentence in fIrst order arithmetic can be reduced to a proof in the real part of arithmetic.

The third step thus shows how to transform proofs in a theory T of decidable sentences to proofs in the real part of T. As we noted in (a) of section 3, success in carrying out Hilbert's program for a theory T implies a transformation for proofs in T of all real sentences in T, i.e. not only the decidable ones but also those of the form VxA(x), to real proofs, but the latter proofs do not necessarily belong to the real part of T; in fact, it can be shown that for certain V xA(x) provable in first order arithmetic there can P~ilosophical aspects o/proo/ theory 265 be no such proof in the real part of T. But the transformation obtained in step 3) is obviously sufficient for Hilbert's program since it shows that every instance A(n) of a provable real sentence YxA(x) is true from which follows that alsoYxA(x) is true. The only doubts that one can have from a finitary point of view about these three steps concern step 2), which is the only point where the finitary part of the object theory in question is tran• scended (as Godel's incompleteness result requires). To avoid confusions, it should be noted that so-called transfinite induction is used in 2) only for decidable sentence while the object theory in question contains ordinary induction used on any sentence, decidable or not. Takeuti follows the same pattern I) - 3) but uses a higher ordinal and considers a stronger theory, viz second order arith• metic where the comprehension axiom

3XYx(xEX ~A(x» is restricted to formulas A(x) that are n~ in-the-wider-sense, which means that A can be transformed in a simple way to an equivalent formula of the form VYB(y) where B(Y) is first order. This system is sufficient to codify an essential part of classical analysis but is weaker than the theory that Hilbert had in mind which had no restrictions on the comprehension axiom. Takeuti obtains his result by using ordinal notation for a much larger initial segment of the ordinals, and possible doubts about the finitary character of the demonstration in step 2) of the principle (TI) are accordingly strengthened. Two recent mono• graphs, one by Takeuti [85] and one by SchUtte [86], give detailed expositions of Takeuti's consistency proof.

6. The significance of the program. Recent discussions

As we have seen, Hilbert's own motives for his program were rather ambiguous. Certain of his statements seem to indicate that he takes moderate formalism literally while others express a com• plete trust in all the methods used in classical analysis, in par• ticular the use of the excluded third. A more representative position for most work on Hilbert's program seems to be the one 266 D. Prawitz taken by Gentzen, who ascribes different kinds of meaning also to the ideal sentences but takes an actual transfinite interpretation and the corresponding forms of inferences to be more doubtful and therefore to require a consistency proof. The most prominent, recent exponent of such a view is Takeuti. In his book [85], he discusses the standpoint of the 'infinite mind', who is supposed to be able to examine infinitely many objects one by one, to form arbitrary subsets of a given domain, to let these subsets form a new domain of objects and so on. He con• siders classical mathematics in the form of second order arithmetic or the theory of finite types to be quite clear from such a stand• point but not set theory, which even from this standpoint contains problematic parts. But regardless of how clear and transparent this world of the infmite mind may appear, it is still an imaginary world for us, and we need to be reassured of its existence in one way or another, Takeuti says. He admits that 'mathematicians have an extremely good intuition about the world of natural numbers as conceived by an infinite mind' and that therefore the consistency question is not particularly important there. But he insists that 'in con• trast, we can conceive of the world of sets only through our imagination and our mathematical experience' and that 'con• sequently, the problem of consistency of the comprehension axiom is a serious and important foundational question'. 'Con• sistency' is here used as 'a sort of foundational watchword', whose frrst implication is that no contradiction can be derived; Takeuti adds that this implication is the most important assurance about the imaginary world of the infinite mind, but that we would sometimes like to know more. The most outspoken recent critic of this position (which I have called moderate formalism) is Kreisel, e.g. in [ 11, 75, 77, and 84] . Some of his points are the following ones:

(a) Long, tedious proofs that use elementary means are often less reliable (because of the great risk of error) than short proofs that use more abstract means and make us understand why the result holds; hence, the whole idea of increasing the reliability of abstract principles by a combinatorically com• plex consistency proof is misconceived. Philosophical aspects of proof theory 267

(b) As a matter of fact, Gentzen's consistency proof has not increased our confidence in natural number theory. But if any• thing like the doctrine of moderate formalism is valid, a con• sistency proof should increase our confidence in the theory by 100%. Hence, empirical experience refutes the doctrine. (c) Since consistency does not exclude the possibility that an exis• tential sentence 3xA(x) is provable while all its instances A(n) are false, consistency does not ensure the existence of the particular concepts for which the axioms were intended. (d) The ideas and principles of classical mathematics, including set theory understood in terms of Zermelo's cumulative hierarchy, are 100% reliable and hence a proof of their con• sistency cannot increase our confidence in them.

Of these arguments, (a) and (b) seem to me rather weak. The prem• iss of the argument (a) about the need of abstract notions is an observation that most people in the Hilbert-school would certainly accept, at least partly, as far as the risk of errors in long combina• torial proofs is concerned (see e.g. Bernays [87]) - after all, they wanted to save classical mathematics just because of the ease with which we obtain short and elegant proofs when using abstract notions. But of course, the premiss is not a carte blanche for the use of any abstract principles, and if one really has doubts about some of them, the use of trusted principles in one consistency proof can be carefully checked once and for all, even if very long, and this would then certainly increase our confidence. As for the argument (b), Takeuti [88] explicitly asserts that his confidence in natural number theory is increased by Gentzen's consistency proof. But most people probably agree with Tarski's [89J often quoted statement in a discussion of Gentzen's use of Eo-induction: '1 cannot say, however, that the consistency of arithmetic is now ... more evident than by epsilon .. .'. However, the reasons why this proof has not increased our confidence in the theory are clear: firstly, as already pointed out, almost nobody takes moderate formalism literally, but at most certain classical concepts are doubted, and then in particular the impredicative ones of classical analysis; secondly, the doubts there can be about the use of tertium non datur in first order natural number theory are expelled by the double negation (or equivalent) interpretations 268 D. Prawitz of classical first order number theory into intuitionistic first order number theory (due to Kolmogorov, Code] and Gentzen) if one accepts the latter theory. Hence, Kreisel's argument (b) is hardly relevant against most proof theorists who seek a consistency proof for impredicative notions; at most, it is an open question what reactions such a proof would meet. As an (empirical) argument against those who take moderate formalism literally, however, Kreisel's argument (induding the talk about 100%) is most relevant, and the important conclusion to draw is the one already mentioned above (section 3.2): for those who doubt transfinite principles, it cannot be enough to establish con,!j~tency or the truth of the real sentences. A reason• able foundation of mathematics cannot treat the transfinite part of mathematics as an instrument, a black box, that happens to give correct results; the weakness of such an instrumentalistic position, i.e. the position of moderate formalism taken literally, is obvious since the foundational task must be to explain why the instrument works, i.e. to understand it. Of course, we do attach some meaning to the transfinite notions, which guides our formu• lation of transfinite principles, and in case this meaning is not sufficiently clear, the task must be to explicate it or extract its mathematically fruitful ingredients. In short, to make Hilbert's program at all credible, one must require that it yields an inter• pretation of also the ideal sentences. This is a point stressed e.g. by Kreisel [78) (see also the parenthetical remark in point (3) below). The same is true if the motive for the program is not any doubts about the transfinite part but a demand for a constructive foundation as discussed in section 3.3. A demonstration of con• sistency, of the truth of the provable real sentences, or of the eliminability of ideal elements from proofs of real sentences cannot be expected to yield in itself a reasonable interpretation of the ideal sentences. And as explained in more detail in earlier sections, Kreisel's argument (c), already adduced by Godel [81], is a counterexample to such an expectation. The need of something more than a consistency proof is also admitted by Takeuti [85) as already mentioned and in effect by most proof-theorists. Hilbert, Herbrand, Centzen, and Takeuti have all tried to find certain interpretations of the ideal sentences Philosophical aspects ofproof theory 269 that occur in proofs, namely, interpretations which depend on the proofs in question. The crucial foundational problem is thus not the truth of the provable real sentences or the eliminability of ideal elements from proofs of real sentences but the pro blem of the meaning of the so• called ideal sentences. The significance of Hilbert's program must therefore be discussed in such a broader perspective when the pro• gram is understood or modified so as to include this question of interpretation. Three or four main lines may be distinguished in this discussion.

(l a) In want of a complete trust in the usual understanding of the classical conceptions, one may seek a constructive interpre• tation in terms of which all principles of classical mathe• matics become valid, in other words, a constructive reduction of the entire classical mathematics. This is e.g. Gentzen's and Takeuti's position. (1 b) Without any distrust in classical mathematics, one may seek a constructive foundation of it as the fulfilment of the axio• matic tradition - this may be Hilbert's position (sec. 3.3). (2) Doubting the classical principles, one may seek a constructive foundation of mathematics, not by an interpretation of all classical concepts, but by developing mathematics construc• tively as far as possible at the possible expense of sacrificing certain classical principles. This is the line fIrst taken by Brouwer. (3) A third line is to maintain that classical mathematics can be made or is already intelligible in its own non-constructive terms and therefore is completely reliable. This line is repre• sented by Kreisel (argument (d) above), who refers to the cumulative hierarchy when it comes to interpreting set• theory and who seems to think that no more explicit mean• ing-theory is needed or that the subject is not ripe for such a foundational study. In a less negative or pessimistic version of this line, one would try to analyze the meaning of the classi• cal notions further in a non-reductive way. (However, speak• ing about Kreisel it should be remarked that although he argues that there is no need in the sense of (I a) or (I b) for a constructive interpretation, he has nevertheless already in 270 D. Prawi!z

[90J made a contribution to the project of finding such an interpretation, viz. the so-called no-counterexample-interpre• tation.)

Kreisel's attitude towards these matters has influenced several logicians, including e.g. Bernays [91 J. But others like Feferman [92 J maintain with Takeuti that we do not really know what set• theoretical statements mean in the same way that we know what arithmetical statements mean; we have a complete mental picture of the totality N of natural numbers but not of the supposed totality of the subsets of N. If this view about meaning in mathe• matics is granted, there is of course a question about consistency (although there can be different pragmatic considerations concern• ing the likelihood of an inconsistency). This controversy cannot be decided in an easy way, and it can hardly be said that line (3) has been supported by a careful analy• sis. The greater confidence in set theory today compared to the beginning of the century is rather 'based partly on experience partly on habit and simply not thinking about the subject' as Takeuti [88 J says. Kreisel's attitude seems to be that nothing more can be done in that direction, but otherwise one may regard line (3) as constituting an open problem, another program for the foundation of mathematics. However, the doubts that one may have about line (3) do not make line (la) much more reasonable. In a discussion of that position, one must consider in more detail what interpretations it offers. For nt-analysis Takeuti [85 J pays especial attention to what he calls modulations of formulas which are obtained by taking the closure of certain instances of quantified formulas, the details of which depend on the proof in question. (E.g., V XA(X) may be replaced by VX 1 VX2 ••. VXn VX 1 VX2 ... VxmA(T), where T is a second order term or formula such that ACT) has been inferred from V XA(X).) As a consequence of a consistency proof for (second order) arithmetic that proceeds by proving the Hauptsatz, Takeuti notes the following corollary: if p is a proof of a sequent r -+ ~ in (full) second order arithmetic, then there is a so-called modulation r' -+ ~' of r -+ ~ determined by p such that r' -+~' is provable in predicative second order arithmetic and logically implies r -+ ~. Philosophical aspects of proof theory 271

The reductive interpretations of classical theories have not been convincing, however. For instance, in Takeuti's [85] interpre• tation by modulations, two provable sequents A ~ Band B ~ e may be interpreted by their stronger and predicatively provable modulations A' ~ B' and B" ~ e", but from this we cannot con• clude that A' ~ crt is provable predicatively (although B' ~ B" holds, it is in general not provable predicatively), and hence, the role of ,~, is drastically changed by this interpretation. It cannot be excluded that other more convincing interpre• tations may be found along these lines. But if a foundation of mathematics is to be based on certain elementary principles such as finitary, intuitionistic or predicative ones, the natural procedure seems to be to develop mathematics with the help of such principles and to see how far one gets. Feferman [93], Friedman [94] and Martin-Lof [35] have recently developed intuitionistic theories along this line (2). (Friedman sometimes calls his work a contribution to Hilbert's program but from the point of view dis• cussed here it is clear that it is rather a contribution to Brouwer's program.) As Feferman [92] puts it, it is of interest to isolate that part of mathematics that can be developed predicatively; and as is wen-known, most classical analysis can be developed in this way. Takeuti [85] rejects what he calls the quasi-foundational ten• dency to replace the results of mathematics by weaker construc• tive theorems. Although he does not make very clear what is meant by the derogatory term 'quasi-foundation', it probably only denotes the attempts to replace theorems one by one by weaker constructive theorems and not the endeavors towards a construc• tive building of mathematics. The position (I b) stays aloof from discussions about the reli• ability of classical mathematics but suffers of course from the same weakness as (la) when it comes to provide a constructive interpre• tation. In conclusion, it must be said that there is a strange tension in foundational works along the lines of Hilbert's program. On one side, one highly appreciates and wants to justify the use of all classical notions. On the other side, one requires the justification to be constructive, and to obtain such a justification, one tends to consider the classical part of mathematics as a mere instrument. When this untenable, instrumentalistic or formalistic position is 272 D. Prawitz given up, and the so-called ideal sentences are admitted to have a meaning, one tries to give a constructive interpretation of all the classical notions but rejects what seems to be the more natural alternative of developing mathematics constructively as far as possible. Such a position is not very convincing, and it is hardly surprising that it has not been possible to arrive at a foundation of mathematics in this way. This does not mean of course that work on the program cannot be relevant for other aims. The eliminability of transfinite reason• ing from proofs of finitary results - to the extent that this can be demonstrated - is in itself of interest. In general, work along the lines of Hilbert's program has given us insight in axiomatic ques• tions about the relation between certain systems; for a survey of works in proof theory from this aspect, see Kreisel [75]. Indirectly, such results may contribute to a better understanding of certain foundational questions. But it seems fair to say that the philosophically most interesting problems of proof theory now concern certain general questions about proofs about which there is no general agreement today. The main philosophical significance of Hilbert's program may be the discussions that it has given rise to and the resulting doubts that a satisfying foundation of mathematics can be obtained in that direction.

Acknowledgement

I am greatly indebted to Carlo Cellucci, Georg Kreisel, Per Martin• Lof, and Goran Sundholm for detailed and valuable comments on earlier drafts of this paper and to Neil Tennant for checking my English.

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