Substance-Concepts and Function-Concepts

"Ernst Cassirer on 'Substance-concepts' and 'Function-concepts'" Jeremy Heis University of California, Irvine

DRAFT: 9 November 2010

Ernst Cassirer's Substanzbegriff und Funktionsbegriff, written in 1910 and translated misleadingly as "Substance and Function," gives off an odd impression to a contemporary reader. Though some of its assertions are quite suggestive, many of the book's central arguments and concepts appear elusive and unclear. Part of the reason is its historical style, which is foreign to analytic philosophers past and present.

Reichenbach famously rejected the kind of historical style employed in Cassirer's work and advocated that philosophy give a mathematical "logical analysis of science"

(Reichenbach 1924, p.xii). Although Cassirer talks about mathematics and mathematicians at great length in his book, he does not use mathematical methods in prosecuting his argument, as for instance Carnap did in the Aufbau or Reichenbach did in

Axiomatization of the Theory of Relativity. And he does not, as Frege did, try to import mathematical standards of clarity and rigorous argumentation into his writings. One looks in vain for an explicit explanation of the meaning of many of his key terms – including the words "substance" [Substanzbegriff] and "function" [Funktionsbegriff] themselves. In his famous review of Cassirer's book Einstein's Theory of Relativity,1

Schlick – with some understandable frustration – feared that there was no clear and nontrivial thesis that Cassirer was propounding and that his arguments – inasmuch as they were meant to support substantive and contentious conclusions – were simply straw

1 Schlick, Moritz. “Critical or Empiricist Interpretation of Modern Physics?” Heis "Ernst Cassirer on 'Substance-concepts' and 'Function-concepts'" men arguments. Indeed, one might wonder whether the same charges could be levied against Substance and Function as well.

An element of Substance and Function especially provocative for contemporary analytic philosophers is its apparent endorsement of what is often called "structuralism" in the philosophy of mathematics. The basic idea is that mathematical objects are just positions in structures: that is, all of the essential properties of, say, a particular number are irreducible relational properties between it and the other numbers (SF, 39). The view, rooted in the mathematician Richard Dedekind's work, has risen to prominence following

Paul Benacerraf's paper "What Numbers Could Not Be." That paper presents and develops ideas first bruited in Benacerraf's dissertation from 1960. There, in a footnote, he writes:

Cassirer says things closely allied to the views here expressed …[For instance, Cassirer writes (p.39)]: "What is here expressed is just this: that there is a system of ideal objects whose content is exhausted in their mutual relations. The 'essence' of the numbers is completely expressed in their positions." One might wish that Cassirer were somewhat clearer than he is in his mode of expression (and possibly thought). [Benacerraf 1960, p.162]

One might wish that indeed.

The difficulties in understanding the book begin with the most fundamental question: What is the book about? In the opening of the "Preface," Cassirer explains:

The investigations contained in this volume were first prompted by studies in the philosophy of mathematics. In the course of an attempt to comprehend the fundamental concepts of mathematics from the point of view of logic, it became necessary to analyse more closely the function of the concept itself and to trace it back to its presuppositions. Here, however, a peculiar difficulty arose: the traditional logic of the concept, in its well-known features, proved inadequate even to characterize completely the problems to which the theory of the principles of mathematics led. It became increasingly evident that exact science had here reached questions for which there existed no precise correlate in the formal language of traditional logic. The material content of mathematical knowledge pointed back to a fundamental form of the concept not clearly

27 September 2010 Page 2 of 40 Heis "Ernst Cassirer on 'Substance-concepts' and 'Function-concepts'"

characterized and recognized within logic. In particular, investigations concerning the concepts of the series and of the limit, the special results of which, however, could not be included in the general exposition of this book, confirmed this view and led to a renewed analysis of the principles of concept formation itself.

To readers familiar with the development of analytic philosophy, this sounds like an allusion to an argument propounded by Russell in Principles of Mathematics seven years earlier in 1903. Russell had argued that the development of mathematics had made it clear that its subject matter is not only number and magnitude, but any system of relations. Traditional logic was wholly inadequate for even characterizing the subject matter of modern mathematics, though, because it did not recognize that there are irreducible relations and so could not even represent asymmetric relations like arithmetic's x is the successor of y.

Moreover, the early chapters of Substance and Function are clearly a further development of the ideas Cassirer gave in his 1907 paper, "Kant und die moderne

Mathematik," which is a review of Russell's Principles. There he wrote:

It is, it appears to me, in fact a new and fruitful point of view, which is introduced by Russell in his treatment of formal logic. The entire "classical" logic has concerned itself with nothing but the subsumption of contents, with the super- and sub-ordination of the spheres of two concepts. […] Syllogistic appears overall as a particularly reactionary and inhibiting moment. Logic remains bound to the point of view of substance and thereby to the fundamental form of the judgment of predication, while the living scientific thought more clearly aims at the concept of function [Funktionsbegriff] as its own systematic middlepoint. One recognizes in this connection the value and necessity of the new foundation on which Russell is seeking to place logic. (Cassirer 1907, p.7)2

Here the opposition between "substance" and "function" picks out a logical point3 – there is a new branch of formal logic not reducible to syllogistic: the logic of relations – and a

2 In this passage, Cassirer footnotes Principles §214-5, where Russell argues that some of the most important basic judgments of mathematics do not fit into the traditional predication model. This is because asymmetric relations are not reducible to judgments of the form S is P. 3 See also SF, p.37, p.71; ETR, p.389.

27 September 2010 Page 3 of 40 Heis "Ernst Cassirer on 'Substance-concepts' and 'Function-concepts'" metaphysical one – there are relations among objects not reducible to properties of their relata.4 That Aristotle's logic was interdependent with a metaphysics of Aristotelian substances was again a point Russell himself had made in 1900 in his famous book on

Leibniz.

However, if the reader goes into the book expecting just more elaboration of these

Russellian ideas, she will soon be shocked. For one thing, Cassirer regularly criticizes

Russell in the book: for his platonism, for his particular brand of logicism, and for his views on acquaintance.5 And inasmuch as Cassirer is expositing Richard Dedekind's

"structuralist" kind of logicism, he is attacking Russell's and Frege's definition of number.

For Dedekind's approach is – as Cassirer very strongly emphasizes6 – opposed to

Russell's and Frege's view. In fact, Russell himself had singled out Dedekind's view of the numbers for explicit criticism.7 Thus, Cassirer's relationship with Russell actually raises more questions than it answers. How did Cassirer begin with a criticism of the traditional logic – and so in agreement with Russell – and end with a view opposed to

Russell's? Moreover, Substance and Function begins in Ch.1 by criticizing the traditional Aristotelian logic, and at the beginning of chapter 2 he describes and endorses

Dedekind's view of the nature of mathematical objects. How does Cassirer get from his criticisms in Ch.1 to the controversial substantive conclusion in Ch.2? How is the argument supposed to go?

A starting point for an answer to these questions is given in the first paragraph from Cassirer's Preface, quoted earlier. There, Cassirer says that the precise target of his

4 See also KMM, p.6, SF p.8, p.56 for similarly metaphysical uses of "substance" and "function." 5 See, e.g., SF, p.316, and Heis, "'Critical Philosophy Begins at the Very Point Where Logistic Leaves Off.'" 6 See Ch.2, §III. 7 Russell POM, §242. I discuss this passage again in §III.

27 September 2010 Page 4 of 40 Heis "Ernst Cassirer on 'Substance-concepts' and 'Function-concepts'" attack on the traditional logic is its theory of concept formation [Begriffsbildung]. This was not a topic that Russell ever discussed explicitly; we don't now usually consider the topic a part of logic. This is partly a reflection of the fact that Cassirer's conception of logic is broader than ours, including topics that fall outside of the boundaries of "formal" logic.8 The subtitle of the book, after all, is Investigations into the Fundamental

Questions of the Critique of Knowledge.

This subtitle should indicate to us the Kantian elements in Cassirer's work, and it brings to mind the Kantian idea of "transcendental logic," at least as this was interpreted by Cassirer's teachers Hermann Cohen and Paul Natorp. As is well known, the fundamental idea of Marburg Neo-Kantianism was the method of transcendental logic, or just the transcendental method. On this view, philosophy's starting point is our current best mathematical natural science – not psychology or a metaphysics arrived at independently of reflecting on science. Philosophy takes this science as given – it does not try to justify it or revise it.9 Rather, it seeks to isolate the conditions of its possibility.

However, though the Neo-Kantian background to the book prepares us to expect to find different topics from what we would find in Russell's logical writings, this background simply raises another mystery. In Cohen's own book on logic, Logik der reinen

Erkenntnis, he was insistent that transcendental logic had to be a "logic of judgment," not of concepts.10 For this reason, he saw Cassirer's book – focusing as it did on the theory of concepts – as a falling away from Marburg orthodoxy.11

8 See Heis, "'Critical Philosophy Begins.'" 9 On the transcendental method, see [[[cites]]. 10 See Cohen, Logik der reinen Erkenntnis, 1st edition, p.499ff. 11 At least initially. For the history, see [[####.]]

The broad goal of the paper is to resolve these mysterious features of Cassirer's book.12 My narrow goal is to understand how Cassirer argues for a Dedekindian account of mathematical objects as "positions in structures" – as it were, how he takes us from

Ch.1 to Ch.2 of the book. But to answer that question will require addressing two preliminary questions.

1. Why the topic of concept formation?

2. What are substance-concepts and function-concepts?

My conclusions will be the following. Cassirer's choice of Begriffsbildung was an ingenious and dialectically subtle way to pull together many seemingly unrelated concerns of Cassirer's philosophy. His central distinction between substance-concepts and function-concepts is multi-faceted and breaks up into a series of interrelated contrasts between logical, metaphysical, and epistemological theses. Comprehending these contrasts under one umbrella as the contrast between "Substanzbegriff" and

"Funktionsbegriff" allows Cassirier to present his philosophy as the coming together together (in a rather complicated way) of distinct lines of reasoning from Russell, from

Kant, and from nineteenth century German philosopher-logicians. Last, there is, with some real digging, an argument that gets you from Ch. 1 to Ch.2. As will become clear, answering even my narrow question – what's Cassirer's argument for Dedekind-style structuralism – will require us getting clear on the main argument running through the book. I'll end in section IV by giving some brief remarks on how Cassirer's theory of the apriori is meant to follow from the reflections on concept formation I describe in sections

I and II of the paper.

12 Cf. Thomas Mormann, "Critical Idealism Revisited," p.305: "Often Cassirer's arguments do not meet the standards of modern analytic philosophy. I do not think, however, it would be a lost effort to attempt to clarify and improve them."

I. Why Concept Formation?

Though it might seem odd to modern readers to begin a work in the philosophy of mathematics and science with a discussion of the nature and origin of concepts, this would not have been so strange to Cassirer's audience. The theory of concepts and concept formation was traditionally the first topic discussed in logic texts. Understood broadly, the question How are concepts formed? is a question in scientific methodology – what do scientists do? what are the methods that scientists use to form successful concepts?13 It can also be a question about the justification of scientific practices: What justifies the formation of new concepts? What constrains it? Indeed, the central question of modern mathematics can be put this way: what justifies mathematics' new found freedom to form concepts (as Cantor put it)?14 Cassirer was certainly not alone in interpreting the traditional logical question of concept formation in this broad way. For instance, in Wilhelm Wundt’s 1883 Logik, in addition to a discussion of the traditional doctrine of concept formation by abstraction, we find detailed analyses of the ways in which particular sciences form concepts—including 250 pages on mathematical methodology, with detailed discussions of the principle of duality, Steiner’s definition of a conic, the so-called principle of the permanence of form, and histories of the concepts

13 See SF, p.26. 14 Cassirer, KMM, p.47.

27 September 2010 Page 7 of 40 Heis "Ernst Cassirer on 'Substance-concepts' and 'Function-concepts'" of a mathematical function, and of the differential.15 Similarly detailed and original theories were given by Benno Erdmann16 and Benno Kerry.17

It is in this spirit that Cassirer approaches the question of concept formation in mathematics. His discussion begins, of course, with an extended polemic against the traditional theory of concepts. Cassirer sees it as containing two parts.

A. ‘Aristotelianism’ about conceptual structure: concepts are either simple or are composed of simple concepts by conjunction, addition, or exclusion.

A.1 The Sufficiency of the Syllogistic. Logic captures the relations between

concepts in judgments, paradigmatically in subject-predicate form, where

concepts are judged to exclude or include one another18; and the relations between

concepts in inference, paradigmatically in Aristotelian syllogisms, where new

15 The general account of mathematical method is given in volume II, second section, first chapter, “The general logical method of mathematics” (74-114 in the first edition (1883), and 101-147 in the second edition (1907)). A random selection will give the reader an idea of the scope of Wundt’s discussion. At the end of his general discussion of mathematical deduction, Wundt distinguishes different kinds of analogical reasoning in mathematics. A principle that he calls “the principle of the permanence of mathematical operations” allows one “to extend certain operations or fixed concepts beyond their region, by continuing a determinate logical process beyond the boundaries that have been set for valid norms according to an analogy with those norms.” This principle leads to new concepts and new laws, by transforming old concepts and thereby also the laws that hold of them (146-7). In the next chapter, he shows how this principle, which has great importance for the “development of mathematical thinking,” is at work in the development of the concept of number from natural numbers, to irrationals, complex numbers, quaternions, and transfinite numbers. Still later in the third chapter, he discusses its use in introducing spaces of more than three dimensions, and non-Euclidean spaces. 16 Erdmann (Logik, 1st ed, 1892, 2nd ed., 1907) argued that concepts are formed either in the traditional way from abstraction, or by unrestricted set-theoretic comprehension, tried to show that his theory of concept formation explained the formation of the concept of transfinite numbers, continuity, Dedekind’s Zahlkörper, and spaces of n dimensions. The psychology of concept formation by abstraction is presented on 65-92 of the second edition; the second level concepts are discussed on 158-175. 17 Kerry (System einer Theorie der Grenzbegriffe. Ein Beitrag zur Erkenntnistheorie (1890)), following Lotze, modeled conceptual structure on mathematical functions and thought that concepts could be formed by letting particular marks act like variable that pass off to the limit. He argued that his theory was adequate to explain the formation of the concept . 18 SF, 8: “The fundamental categorical relation of the thing to its properties remains henceforth [after Aristotle] the guiding point of view; while relational determinations are only considered in so far as they can be transformed, by some sort of mediation, into properties of a subject or a plurality of subjects. This view is in evidence in text-books of formal logic in that relations or connections, as a rule, are considered among the ‘non-essential’ properties of a concept.”

relations of inclusion or exclusion among concepts are arrived at from established

relations. 19

B. Abstractionism about conceptual formation: concepts are formed by noticing similarities or differences among particulars and abstracting the concept, as the common element, from these similarities or differences.20

B.1 Primacy of particulars: in forming new concepts, one must have grasped and

surveyed the particulars (or concepts) that fall under the concept (or at least a

good number of them) before abstracting the new concept from them. Grasp of

particulars is prior to grasp of concepts.21

From a post-Russellian view, this doctrine seems to run together properly logical notions with epistemological. But these two parts are not independent, and in the tradition they were thought to be two sides of one coin. On this traditional model, all concepts can be placed onto a Porphyrian genus/species tree. These conceptual structures lend themselves to Boolean treatment in terms of conjunction, disjunction, and negation.

Specification of a genus can be expressed as the conjunction of a genus and differentia, and the extension of a genus is disjoined into the extensions of its various species. Since the species of a genus are disjoint, each is then the negation of the rest. Concept formation by abstraction, as Kant himself emphasizes in the first Introduction to the

19 KMM, 7: “Thus syllogistic appears overall as a particularly reactionary and inhibiting moment. Logic remains bound to the viewpoint of substance and therefore to the fundamental form of the judgment of predication.” 20 SF, 5: “The essential functions of thought, in this connection, are merely those of comparing and differentiating a sensuously given manifold. Reflection, which passes hither and thither among the particular objects in order to determine the essential features in which they agree, leads of itself to abstraction. Abstraction lays hold upon and raises to clear consciousness these related features.” 21 See SF, 18-19, quoted above. See also SF, 8: “Only in given, existing substances are the various determinations of being thinkable. Only in a fixed thing-like substratum, which must first be given, can the logical and grammatical varieties of being in general find their ground and real application” (emphasis addded).

Critique of Judgment is the inverse operation to specification. In abstracting, we move

"up the tree."22

Since theories of conceptual structure were intertwined with theories of concept formation in the tradition, Cassirer's discussion of concept formation allows him to consider, as Frege and Russell had done, the expressive and inferential limitations of traditional syllogistic as compared to the new logic of relations. But it also allows him to discuss methodological questions in mathematical practice that were off stage in Russell's writings. What's more – as I will show by the end of my talk – a chief conclusion of

Cassirer's polemic against abstractionism is that methodological issues within mathematics play a much larger role in questions of mathematical truth and existence than one might have thought.

Cassirer’s main argument in the first chapters of the book is that the traditional model does not accord with the function and structure of concepts as they are found in the exact sciences of his day: his question is “Is the theory of the concept, as here developed, an adequate and faithful picture of the procedure of the concrete sciences?” (SF, 11)23

Cassirer sees serious difficulties with the traditional model from the outset. When we form a concept by abstraction from our knowledge of a particular, we remove the elements that are not common with other particulars and thereby reduce the content of our knowledge of the particular. How could reducing the content of our knowledge be a

22 See, for instance, Ak 20:214-5. 23 See also: p.26: The totality and order of pure "serial forms" lies before us in the system of the sciences, especially in the structure of exact science. Here, therefore, the theory finds a rich and fruitful field, which can be investigated with respect to its logical import independently of any metaphysical or psychological presuppositions as to the "nature" of the concept

27 September 2010 Page 10 of 40 Heis "Ernst Cassirer on 'Substance-concepts' and 'Function-concepts'" cognitive achievement?24 This was a familiar argument among nineteenth century logicians. As the British Neo-Kantian T.H. Green put it,25

The process of abstraction, […] if it really took place, would consist in moving backwards. It would be a donkey-race. The man who had gone least way in it would have the advantage, in respect of fullness and definiteness of thinking, of the man who had gone the furthest.

There were two solutions extant in the literature at the end of the nineteenth century. The first, due to Friedrich Trendelenburg, argued that the theory of concept formation needed to be supplemented by an Aristotelian metaphysics of substantial forms. Abstraction then removes the accidental features in our representations of various particulars in order to isolate their essences; and surely the identification of essences is a cognitive achievement.26 The second approach, due principally to Hermann Lotze, argued that – if the structure of the concept is rich enough -- the formation of a successful concept can allow us to retain all our knowledge of particulars. Lotze's example is a curve in analytic geometry. If we first know all of the points that the curve occupies in space, and then ascend to a general formula – that is, a mathematical function describing the curve – , we retain all of our particular knowledge, since all of the positions of the curve in space can be derived from the functional formula itself. So given the right kind of concepts, concept formation does not impoverish our representation of particulars: it hits on

24 SF, 18: "If we merely follow the traditional rule for passing from the particular to the universal, we reach the paradoxical result that thought, in so far as it mounts from lower to higher and more inclusive concepts, moves in mere negations… If we adhere strictly to this conception, we reach the strange result that all the logical labor which we apply to a given sensuous intuition serves only to separate us more and more from it. Instead of reaching a deeper comprehension of its import and structure, we reach only a superficial schema from which all peculiar traits of the particular case have vanished." 25 Green, “Lectures on Formal Logicians,” 193, vol. 2 of his Collected Works. 26 Of course there needs also to be a story about why the procedure compare, reflect, abstract hits on essences and not unimportant common features. Again, the Aristotelian metaphysics will fill in the gap: perception is a causal relation between the object and the mind; in this causal interaction, the forms of perceived objects are transferred into the medium of the mind. [[See Trendelenburg [cites] and SF, p.7]].

27 September 2010 Page 11 of 40 Heis "Ernst Cassirer on 'Substance-concepts' and 'Function-concepts'" mathematical laws from which we can re-derive all of the properties of the various particulars that fall under the newly formed concept.27

This simple example is a paradigm for Cassirer: instead of supplementing a logical doctrine with metaphysics, observe the actual concepts that are employed in our most successful exact science. And the example of mathematical functions contradicts the traditional theory of conceptual structure, since – as Russell showed definitively – mathematical functions can, in general, be expressed only using the new logic of relations.28

When Cassirer tests the traditional theory of concepts against the actual procedure of our best current science, he is applying the method of transcendental logic – first introduced by his teacher Cohen. According to this method, philosophy takes our best sciences as a fact and investigates the conditions of their possibility. However, the relationship between Cassirer's polemic against abstractionism and Marburg Neo-

Kantianism runs much deeper than the fact that Cassirer's investigation is an instantiation of the transcendental method. The reason is that the abstractionist theory impinges on a characteristically Marburg Neo-Kantian preoccupation: the distinction between the

Kantian faculties of sensibility and understanding. Recall that for Kant sensibility is passive (or "receptive") and the understanding – the faculty of concepts – is active (or

"spontaneous"). That all our concepts can be formed by abstraction from given representations requires that there be a strata of representations that are entirely independent of concepts that can be the material for abstraction. That is, there must be

27 See Lotze 1874, 1843. I discuss Lotze's and Trendelenburg's views in Heis, "Frege, Lotze, and Boole" and "The Priority Principle from Kant to Frege." 28 Lotze himself had already recognized that mathematical functions characterize concepts that are not reducible to the traditional conceptual forms of conjunction, disjunction, and exclusion of marks.

27 September 2010 Page 12 of 40 Heis "Ernst Cassirer on 'Substance-concepts' and 'Function-concepts'" representations of sensibility whose epistemic efficacy is entirely independent of the understanding. Natorp and Cohen were famously hostile toward the idea that sensibility

– the capacity for objects to be given to us through their affecting us – makes a contribution to our knowledge that is independent of the understanding.29 As Cohen argued, there is not and cannot be something "given" [gegeben] in an experience (Cohen

1902b, pp.24-5, 48-51).

Cohen's rejection of the "given" will remind those of us familiar with the history of analytic epistemology over the last fifty years of Wilfrid Sellars's famous attack on the

"Myth of the Given." The comparison with Sellars is apt, since Sellars, like Cassirer, connected the Myth with abstractionism and with the distinction between sensibility and the understanding; and both found the roots of this polemic in Kant's philosophy.30

According to Sellars,

[A]ll [forms taken by the myth of the given] have in common the idea that the awareness of certain sorts – and by 'sorts' I have in mind, in the first instance, determinate sense repeatables – is a primordial, non-problematic feature of immediate experience. ("Empiricism and the Philosophy of Mind," p.157)

On the view that Sellars here rejects, a subject could have, for example, an experience of a red thing as red without the subject knowing any other facts, possessing any other concepts, or having any other similar non-inferential acts of awareness. The Myth, then, is a kind of epistemic atomism. Moreover, a paradigm of the myth is the traditional doctrine of concepts, since – Sellars claims –

the abstractive theory, as Kant saw, makes the mistake of supposing that the logical space of the concept simply transfers itself from the objects of direct perception to the intellectual order, or better, is transferred by the mind as Jack Horner transferred the plum.31 ("Phenomenalism," p.90)

29 See here the very helpful discussions in (Friedman 2000, ch.3) and (Kim 2003). 30 See, for instance, Science and Metaphysics, Ch.1.

On the abstractive view, the concept could be derived from a prior awareness of red things as red. But this abstraction is a way of forming concepts for the first time only if – as the Myth requires -- the primitive awarenesses are themselves possible prior to the subject's possessing any concepts.

Before writing Substance and Function, Cassirer had argued at length in

Erkenntnisproblem volume 2 that Kant's philosophy provided the refutation of abstractionism. Commenting on the Transcendental Deduction, Cassirer writes

If, according to the traditional logical doctrine, the concept is merely the result of "abstraction" from a plurality of sensory data, so has it now been shown that "similar" impressions must be placed under a determinate rule of judging, before they – as is necessary for the process of "abstraction" – can be cognized as similar and be comprehended in a common genus. The unity of a genus presupposes the unity of an ideal norm, and the abstractive comparison presupposes a constructive connection. In its proper fundamental meaning, a concept is nothing other than the consciousness of this unity of synthesis. (EP 2, 676; cf. 667)

Cassirer here presents the positive argument of Kant's Transcendental Analytic as beginning by locating a gap in abstractionism: that between the subject's having similar impressions and forming a concept from them, the subject must also cognize the impressions as similar. To arrive at an empirical concept, it is not enough that the objects sensed are similar, and it is not even enough that the sensations produced by those objects have similar sensory qualities. The subject must herself recognize the similarity. The failure to recognize this middle step in the process of concept formation, Cassirer argues, prevented pre-Kantian philosophers from seeing the necessity of two fundamental

Kantian ideas. The first idea is that there are preconditions of experience – categories that a subject must possess antecedently to any experience and principles that the subject must know prior to any experience. And so there are some concepts, the categories, that

31 Sellars, "Phenomenalism," p.90. (Sellars later had second thoughts: see Science and Metaphysics, p.5.)

27 September 2010 Page 14 of 40 Heis "Ernst Cassirer on 'Substance-concepts' and 'Function-concepts'" are presupposed in any cognitive representation – even in the awareness of a red thing as red -- and so must already be in place prior to abstracting.32 The second idea – which is of course related to the first – is that every intuition (for instance, the intuition of a red thing as red, as opposed to a mere impression of red) requires the application of a concept as the rule for ordering the manifold in that intuition into one representation. And so intuitions, which are the only plausible material basis for abstraction, already presuppose concepts.33

In Chapter 1 of Substance and Function, though Cassirer does not mention Kant by name, he repeats the argument, in his own voice, that the theory of abstraction presupposes (without explaining) the capacity to represent similar impressions as similar.34 To this argument, Cassirer adds a suite of supplementary arguments. In abstraction, we are told, we take some representations and abstract out the common marks; but this just presupposes that our representations already contain marks.35 In abstraction, we take some given grouping of representations and abstract the common element; but this just presupposes that – before the process of comparing, reflecting, and abstracting even begins – we have a principle for grouping together just these representations.36

32 EP 2, p.677, 698; see also KLT, p.167, 176. 33 In EP 2, p.676, Cassirer quotes A103 – a concept is the "one consciousness that unifies the manifold that has been successively intuited, and then also reproduced, into one representation" – to make this point. See also SF, p.16-7; PSF 3, p.315. Cassirer goes on to draw from these attacks on abstractionism the familiar Marburg conclusion that the "division between intuition and concept resolves itself ever more clearly into a purely logical correlation" (EP 2, p.698). 34 SF, p.15. “The concepts of the manifold species and genera are supposed to arise for us by the gradual predominance of the similarities of things over their differences – the similarities alone, by virtue of their many appearances, imprint themselves upon the mind, while the individual differences, which change from case to case, fail to attain fixity and permanence. The similarity of things, however, can manifestly only be effective and fruitful, if it is understood and judged as such. That the ‘unconscious’ traces left in us by an earlier perception are like a new impression in point of fact, is irrelevant to the process implied here as long as both elements are not cognized [erkannt] as similar.” See also SF 337. 35 SF, p.17. See also "Zur Theorie des Begriffs" (1928), p.161. 36 SF, p.15; PSF 1, p.278-9.

II. Substance-concepts and Function-concepts

The second interpretive question we posed concerned the meaning of Cassirer's key terms

"Substanzbegriff" and "Funktionsbegriff." Given Cassirer's historical method, we should not expect him to use his terms with the same meaning in every context, but to vary the meanings depending on which historical tradition he is discussing or appropriating. (And we should remember that Cassirer was not being sloppy: his historical method of philosophizing put a premium on presenting his view as the culmination of distinct lines of reasoning within the history of philosophy. Using the words "function" and "function- concept" in many ways advances his purpose, even if it would frustrate the purposes of a

Frege or a Carnap.) This is certainly true of the word "function" [Funktion]. In many instances, of course, he uses it in its ordinary mathematical meaning. But the word takes on three other meanings depending on which tradition in the theory of concept formation and structure he is engaging with. First, when speaking Russellian, Cassirer will use the word synonymously with "relation," since in the new logic a function can be defined as a one-one or many-one relation. So, for instance, he claims that Dedekind's Was sind und was sollen die Zahlen? shows that the concept is a "pure functional concept" [Funktionalbegriff], since the "presuppositions of the derivation of the concept of number are given in the general logic of relations."37 Second, when Cassirer is investigating the methodology of the various exact sciences, he'll make claims about the function, that is the "role" or "purpose," of particular mathematical or scientific concepts within this or that particular scientific field.38

37 SF, p.36. See also KMM, p.7 (cited above); SF, p.265. 38 See, for instance, SF, p.iv, where Cassirer says that one of the fundamental thoughts of his book is that

The third use of "function" derives from Kant. For Kant, a "function" is a rule- governed activity of the mind. He attributes them exclusively to the understanding, the faculty of concepts. As active or "spontaneous," they are contrasted with the

"affections" of sensibility, which are passive and "receptive" (A68/B93). Thus, Cassirer will talk of "functions" when he is speaking of various mental or intellectual "activities"

[Tätigkeiten] or "procedures." For instance, when Cassirer gives the objection to abstractionism that he attributes to Kant – that similar representations can give rise to a common concept only if the subject already judges them to be similar – he concludes that there are other "pure conscious functions" (SF, p.337) or "intellectual functions" (p.16; cf. p.14) that are at play in concept formation prior to the activities of comparison, reflection, and abstraction.

However, in his more careful moments, Cassirer claims that these "intellectual functions" are not really mental acts, because they are not psychological states at all.

According to the particular, strongly anti-psychologistic reading of Kant favored by

Cassirer and the other Marburg Neo-Kantians, philosophy is not directly concerned with

"subjective 'representation [Vorstellung]''" or with the causal and temporal relations among the states of "particular thinking subjects." Rather, the goal is to identify "certain axioms and norms of scientific knowledge" on which the rest of our scientific knowledge is "logically dependent."39 These "principles and universal laws" constitute the "general "What the concept is and means in its general function can only be shown by tracing this function through the most important fields of scientific investigation." Carrying out this project, he says that the "function" (or "role") of the irrational numbers for Dedekind is to "designate a possible division of the region of rational numbers and thereby a possible ‘position'" (KMM 14-5; cf. SF, p.61); the "function" (or "purpose") of imaginary points in "the system of geometrical propositions" is to provide "fruitful and thoroughly positive geometrical insight" into the "connection of real geometrical forms" (SF, p.83). 39 SF, p.298 (cf. p.315). Continuing this point on the same page, Cassirer writes: "The proposition, that being is a "product" of thought, thus contains no reference to any physical or metaphysical causal relation, but signifies merely a purely functional relation, a relation of superordination and subordination in the validity of certain judgments." Here, Cassirer is exploiting the semantic flexibility of the word "function"

'form' of experience" (SF, p.268). Kant, on this reading, successfully identified the categories and principles as preconditions of Newtonian science. Marburg Neo-

Kantianism departs from Kant, however, in recognizing that in periods of scientific revolution, even these "intellectual functions" can be modified.

Such principles as, for example, those on which Newton founds his mechanics, do not need to be taken as absolutely unchanging dogmas; they can rather be regarded as the temporarily simplest intellectual “hypotheses”, by which we establish the unity of experience. We do not relinquish the content of these hypotheses, as long as any less sweeping variation, concerning a derived moment, can reestablish the harmony between theory and experiment. But if this way has been closed, criticism is directed back to the presuppositions [Voraussetzungen] themselves and to the demand for their reshaping. Here it is the "functional form" [Funktionsform] itself, that changes into another.

Here, Cassirer speaks equivalently of "principles and universal laws," the "'form' of experience," "presuppositions," and "functional form."40 In this sense, then, the investigation into the "functions" of knowledge is really an investigation of the epistemic preconditions of our current best science.

Given these various uses of "function," it is not surprising, then, that there is no one, unambiguous contrast that Cassirer draws between "function-concept"

[Funktionsbegriff] and "substance-concept" [Substanzbegriff]. In some cases, Cassirer speaks of function-concepts as a particular kind of a concept – a concept whose proper explication requires mathematical functions or the Russellian logic of relations. In §III of to make his point. An idealism like Kant's does not assert that objects are dependent on "intellectual functions" as a product is dependent on an act of production. Rather, statements about objects are logically dependent on (and so "functions of") higher laws. This logical dependence is more like the dependence of a conclusion on its premises. (Though, as we will see, the dependence is actually semantic rather than inferential: these intellectual functions are conditions for the sentences of science having meaning at all.) 40 Kant himself – whose vocabulary is of course Cassirer's source here – uses "form" and "function" so closely in the Transcendental Analytic that some commentators have claimed that Kant – as Cassirer does in this passage -- uses the terms interchangeably. See, e.g., Paton Kant's Metaphysic of Experience, vol.1, p.246. (To be sure, Kant does not use "function" and "form" interchangeably when speaking of the forms of sensibility – space and time – since functions for Kant are always of the understanding, never of sensibility. But this is no block on Cassirer's equating form and function even here: for him, Kant's final view is that "the functions of pure understanding appear as preconditions of ‘sensibility’" (KMM, p.35).) There are also passages, however, where Cassirer distinguishes functions and forms (SF, p.315).

Chapter 1, for instance, Cassirer repeats the argument that the most successful concepts – like concepts expressible as mathematical functions – allow us to derive all of the particulars that fall under the concept – say by substituting various constants for the variables in the function's analytic expression. Cassirer calls such concepts "function- concepts", and contrasts them with "substance concepts", whose proper explication only requires giving a genus and a differentia (SF p.19-21).

However, the most significant contrasts that Cassirer draws are not between kinds of concepts, but between philosophical theories about concepts. As we saw earlier,

Cassirer will use the contrast to advocate for the new logic over the old. According to the logic of substance-concepts, there is no independent logic of relations and logic is exhausted by syllogistic. The contrasting view, as it were the logic of function concepts, is the opposite view, drawn by Russell. Again, Cassirer, like Russell, associates this logical contrast with the contrast between the two answers to the metaphysical question whether every object is an Aristotelian substance – an object with no essential, irreducibly relational properties.

The fundamental use that Cassirer makes of "Substanzbegriff" and

"Funktionsbegriff" is, though, not Russellian, but Kantian: it contrasts philosophical views that overlook the epistemic preconditions of various kinds of knowledge, with those that recognize the "functions" that make certain kinds of knowledge possible.

Cassirer's argument begins in Ch.1 with the attacks on abstractionist theory of concept formation I mentioned earlier and unfolds progressively throughout the book, moving from the theory of concept formation, to the role of sensations in perceptual knowledge, to confirmation in empirical science and the justification of scientific existence claims.

The upshot of Ch.1 is a refutation of atomism about concept formation – the view that it is possible to form a concept without possessing any other concepts or knowing any facts.

The appeal of this concept formation atomism derives from the similarly atomistic view that sensations are the "self-evident, given starting point" of all empirical knowledge (SF, p.279) – that their epistemic efficacy does not depend on the subject's having any concepts or knowing any facts.41

The thesis that sensations are sufficient of themselves to provide a subject elementary knowledge of empirical facts is then a species of the "substance-concept" worldview. In Ch.4, "Concept Formation in Natural Science," Cassirer further undermines this view by arguing – from a discussion of various physical theories – against an atomistic account of the basic, foundational facts of experimental science: measurements. He summarizes his conclusion:

All measurement, however, presupposes certain theoretical principles and in the

latter certain universal functions of connection [Funktionen der Verknüpfung], of

shaping and coordination. We never measure mere sensations, and we never

measure with mere sensations, but in general to gain any sort of relations of

measurement we must transcend the “given” of perception and replace it by a

conceptual symbol, which possesses no copy in what is immediately sensed.42

41 In Ch. VI of SF, Cassirer argues in particular against a popular kind of atomism about sensations: the view that subjects have certain knowledge of their own subjective states, prior to and independently of any knowledge of "outer" things, and then infer from these states to facts about outer things. Cassirer argues, on the contrary, that our knowledge of subjective states is not more certain than that of our knowledge of outer things; that we cannot know facts about our own states without also knowing facts about outer things; and that a sensation with no objective reference and with no conceptualization is not a datum of consciousness, but a posit introduced by psychology for theoretical purposes. 42 This quotation appears in Einstein's Theory of Relativity (1921, p.427; O 95-6; cf. SF, p.141-2)), which picks up the argument from SF (especially Ch.4 of SF) and extends it to a discussion of general relativity. Cf. also Cassirer's summary of Ch.4 at SF, p.267: "If we take as given the whole of experience, as it is represented in any definite stage of knowledge, the whole is never a mere aggregate of perceptual data [Wahrnehmungsdaten], but is articulated and brought to unity according to definite theoretical points of view. It has already been shown from all sides that, without such points of view, no single assertion

The various cases of measurement43 together show that the most basic results of scientific experimentation presuppose not only concepts and laws of pure mathematics, but also laws of nature and natural scientific concepts. For example, a particular experience of a particular electric current using a galvanometer presupposes a host of other concepts – mathematical concepts like , , and , and physical concepts like , , , and . Interpreting the motion of a magnetic needle as the change in the electric current itself presupposes various physical laws. The concept , then, could not be formed in the atomistic way that traditional abstractionism requires. Similary, Poincaréan reflections on the measurement of time show that the concept could only be formed together with a system of other concepts, including , , , and . Moreover, as Cassirer learned from Duhem, no single empirical statement of natural science can be confirmed atomistically.44 This confirmation anatomism, then, reinforces the attacks on concerning facts, in particular no single concrete measurement, would be possible." 43 Here are some of the cases Cassirer discusses: assigning a real number to a temperature by measuring a volume of mercury presupposes laws of geometry and the law relating temperature to the expansion of volume of mercury (p.142-3); when Regnault measured the volume of a gas in his tests of Boyle's law, he used an instrument the design of which just presupposes the "abstract principles of general mechanics and celestial mechanics" (p.143; cf. Duhem 1906, 145-7); the measurement of time requires the identification of a unit, the choice of which presupposes the law of conservation of energy, or the principle of inertia (p.145; cf. Poincaré 1906, 210-22); the measurement of the curvature of space presupposes the choice of a "rigid body" (p.107; cf. Poincaré 1902); the determination of the position of a body requires constructing a Langean "inertial system" (p.182); Ampère's measurement of the intensity of an electric current required a galvanometer, whose operations presuppose various physical laws (SF, p.280). As even the example of temperature – and apparent sensory quality – shows, the most basic scientific measurements go well beyond noting the intensities of this or that sensory quality. 44 SF, p.147: "[T]he individual concept can never be measured and verified by experience for itself alone, but it gains this confirmation always only as a member of a theoretical complex. […] Here each element needs the other for its support and justification; no element can be separated from the total organism and be represented and proved in this isolation. We do not have the physical concepts and physical facts in pure separation, so that we could select a member of the first sphere and inquire whether it possessed a copy in the second; but we possess the 'facts' only by virtue of the totality of concepts, just as, on the other hand, we conceive [konzipieren] the concepts only with reference to the totality of possible experience." Cassirer alludes to Duhem 1906 frequently in SF: p. 143-7, 280. At PSF 3, he puts Duhem's thesis in the vocabulary of "substance" and "function": the words of exact science derive their "meaning not from transcendent objects which stand behind them and which they copy, but from their achievement, their function of objectivization” (421; cf. 416, 461).

27 September 2010 Page 21 of 40 Heis "Ernst Cassirer on 'Substance-concepts' and 'Function-concepts'" abstractionism, since if an empirical concept could be derived from given experiences independently of other concepts or beliefs, then an empirical judgment like

"This is an F" could be confirmed directly by comparing the concept with the particular representation from which it was abstracted.

Since, Cassirer argues, the formation of concepts, the epistemic efficacy of sensations, and the confirmation of empirical judgments all have epistemic preconditions, the dream of direct, presuppositionless knowledge of objects falls away. (An epistemology of acquaintance is just as hopeless as the Aristotelian metaphysics that it requires.45) Against this atomistic view of existence claims, Cassirer claims that we could not know that an object exists outside of a system of concepts and judgments. For example, we are justified in claiming that certain kinds of numbers or points exist only when we can show that they perform "an indispensable function" in our system of mathematical propositions (SF, p.83). Similarly, when discussing Macquord Rankine's writings on energetics, Cassirer argues that a "substantial interpretation of energy" would require that energy is a determination of things that "belongs to them directly in their sensuous appearance", that it is an "isolated sensuous property to be perceived for itself."

On the opposed "functional theory of the concept" of energy, asserting the "being" of

45 Cassirer thinks that an Aristotelian metaphysics of substantial forms tries to avoid recognizing the epistemic preconditions of our knowledge by claiming that it is the nature of the mind to be able to take on the form of the substances that affect it in perception (see SF, p.7). On Cassirer's view, this does not explain the possibility of experience: it just restates the problem in metaphysical terms. (Using Kant's apt phrase, the perceived object cannot just "migrate" into our mind: Ak 4:282; EP2, 690.) Here, of course, we see a connection between the epistemological use of "function" (as epistemic precondition) and the metaphysical use of "substance." Cassirer elsewhere uses the mathematical notion of a function to oppose atomism about existence claims. In physics, where the objects are spatio-temporally located, the existence of an object cannot be experienced "save in connection with other spatial and temporal, near or remote elements; and this kind of connection presupposes a system of spatial and temporal positions, as well as a unitary whole of causal coordinations. The fact a is only accessible to us in a functional form as f(), (), (), in which f, ,  represent the different forms of spatial-temporal and causal connection" (SF, p.248; O 329).

27 September 2010 Page 22 of 40 Heis "Ernst Cassirer on 'Substance-concepts' and 'Function-concepts'" energy "would obviously lose its meaning, if we wished to separate it from the whole system of judgments, in which it has arisen" (SF, p.197).

Recognizing these epistemic preconditions necessitates, Cassirer argues, inverting the conceptual dependencies among the key concepts of epistemology – the concepts