A Comparison with Frege

In Grundlagen der Arithmetik, §88, Frege wrote that, of all possible ways of forming concepts (Begriffsbildungen), the one that complies with the Aristotelian model is the least fruitful one. He illustrates the differ-ence between the Aristotelian model and the really fruitful definitions (such as his own definition of number or Weierstrass’ definition of the continuity of function) with the help of a geometrical analogy.²⁰ If we represent concepts or their extensions (in our sense, and not the traditional one) by means of what we would now call Venn-diagrams, i.e., by means of overlapping regions in a plane, then a concept de-fined by common characteristics is represented by an area common to all regions and enclosed by segments of their boundary lines. In this way the Aristotelian model of conceptual analysis – which, it will be recalled, amounts to representing a concept as a logical product of its partial concepts – always makes use of existing boundaries to de-marcate an area, and “[n]othing essentially new [...] emerges in this process” (ibid.)

Frege continues the passage by writing that “the more fruitful type of definition is a matter of drawing boundary lines that were not pre-viously given at all.” What can be inferred from this kind of definition

“cannot be inspected in advance; here, we are not simply taking out of the box again what we have just put into it”. There is a substantial similarity between the contrast that Frege draws in these passages and the way Kant saw the difference between the Aristotelian model, or the dissection of concepts, and the possibilities inherent in intuitive representation, or the “construction” of concepts. “What we shall be able to infer from it [sc. from a fruitful definition]”, Frege writes,

“cannot be inspected in advance; here, we are not simply taking out of the box again what we have just put into it. The conclusions we draw from it extend our knowledge” (ibid.) For Kant, the epistemic value of analytic judgments is merely “explicative” (A7/B11), since they enable us to make explicit what has “all along been thought” in a concept, whereas a synthetic judgment adds to a concept a

determi-20 The analogy is also given in Frege (1880/1, pp. 37-38).

nation which “has not been in any wise thought in it, and which no analysis could possibly extract from it” (ibid.) Frege also remarks in Grundlagen, §88, that Kant “seems to think of concepts as defined by giving a simple list of characteristics in no special order”. This is correct if limited to representations as they are dealt with in Kant’s

“formal logic”. However, if it is intended as a general criticism of Kant, Frege’s remark ignores the role of intuitive representations.

Indeed, if we follow Cassirer (and some recent scholars²¹), we can say that recognising the existence of the gap which Frege points out was the very reason that led Kant to his semantic or representation-theoretic innovations. Kant did accept the traditional notion of con-cept in its essentials.²² This is signalled by his definition of concept as a representatio per notas communes, a representation through common marks or characteristics (The Jäsche Logic, §1), which has its origin in the “logical actus of comparison, reflection, and abstraction” (id., §§9-11; 15-16), and it is further reflected in his characterisation of the task of “general logic”, according to which “it is to give an analytical ex-position of the form of knowledge [as expressed] in concepts, in judgments, and in inferences, and so to obtain formal rules for all employment of understanding” (A132-133/B171-172). On the other hand, Kant’s divergence from tradition becomes evident, when he introduces intuitions as a new species of representation in addition to concepts.²³ This is accompanied by the division of human cognitive faculties into understanding (cognition through concepts) and

sensi-21 See Michael Friedman (1992a, Ch. 1 Ch. 2) and J. Michael Young (1994).

22 For Kant’s appropriation of the traditional picture of concepts, see Wilson (1975, pp. 252-3) and Allison (1983, pp. 92-4).

23 In the Jäsche Logic, §1, Kant defines intuition as a individual representa-tion (representarepresenta-tion singularis). At A320/B376-377 he explains that subordi-nated to the genus representation stands an objective perception or cognition (cognition; Erkenntnis), which is divided into intuition and concept: “the former relates immediately to the object and is single”, whereas “the latter refers mediately by means of a feature which several things may have in common”.

bility (cognition grounded in intuition), which operate under distinct sets of rules.²⁴

2.1.4.4 Kant on Philosophical and Mathematical Method

That there is a fundamental difference between these two sorts of cognition is a claim which Kant argues for in the Transcendental Doc-trine of Method. There he draws a distinction between “philosophical”

and “mathematical” knowledge: the former is purely discursive knowledge, or “knowledge gained by reason from concepts”; the lat-ter he characlat-terises as “knowledge gained by reason from the con-struction of concepts”, to which he adds the preliminary explanation of “construction” as the apriori exhibition of an intuition which cor-responds the concept (A713/B741). This discussion shows, at the same time, that he was well aware of the differences between purely conceptual representation (in his sense) and the “really fruitful defini-tions in mathematics”, that is, of the inapplicability of the Aristotelian model of concept-formation to mathematics.

At A716/B744 this contrast is applied to the special case of geo-metrical proof or inference. Since a purely logical proof agrees with a

“philosophical proof” in that both are analytical derivations or extrac-tions of what is contained in a complex concept, we can here substi-tute “logical proof” for Kant’s “philosophical proof”. Thus, one case in which the distinction between the two kinds of knowledge is con-spicuous is the contrast between purely logical reasoning and mathe-matical reasoning.

24 “We [...] distinguish the science of the rules of sensibility in general, that is, aesthetic, from the science of the rules of the understanding in gen-eral, that is, logic” (A52/B76). The latter science, i.e. “general” or “formal”

logic, is formal in the sense that it “treats of understanding without any re-gard to difference in the objects to which understanding may be directed”

(ibid.) It abstracts from sensibility and intuition – from the object-directedness of human cognition – and occupies itself solely with the rules which govern thought (judgment) qua conceptual unity or concept subordi-nation.

Assume, to begin with, that these two types of reasoning agree on their deductive strength or necessity or certainty (to revert to a more Kantian terminology). Assume, that is, that the degree of validity of a piece of mathematical reasoning is no less than that of an argument, like modus ponens, which is valid and whose validity is generally recog-nised to be a matter of logic – Kant would here speak about “pure general” logic, whose rules apply to all cognition irrespective of its content;²⁵ modus ponens is clearly sufficiently abstract to qualify as a rule of formal logic in Kant’s sense. In this sense, mathematical and logical reasoning are exactly on the same level.

Kant, for one, was in complete agreement with this assumption.

At A734-735/B762-763 he writes that the main difference between mathematical and philosophical/logical proofs or discursive proofs does not lie in their different degrees of certainty; both are species of what he calls apodeictic proof. Despite this similarity, Kant neverthe-less thinks that reasoning in mathematics is not purely logical in char-acter. To show this, he uses as an example Euclid’s proof of the proposition that the sum of the angles of a triangle equals two right angles (A716-717/B744-745).²⁶ When a philosopher attempts a proof of this proposition, all he has at his disposal is a set of given con-cepts, those that together form the concept of “a figure enclosed by three straight lines, and possessing three angles”. A purely discursive proof, or proof that is sensitive to logical structure only, consists in the activity of detecting and making explicit the constituents of given complex concepts. All that a philosopher or a logician can do, there-fore, is to analyse and clarify the concepts of straight line, angle and the number three: “however long he meditates on this concept [sc. of triangle], he will never produce anything new” (A716/B744). By

con-25 See the previous footnote. In what follows, when I speak about Kant’s views about logic, I mean his views about what he recognised as formal logic, that is roughly, everything that belongs to logic according to the Aris-totelian logical tradition. For further discussion of Kant’s views on logic, see section 5.2.

26 “In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles” (Elements, I.32).

trast, when a mathematician is presented with the problem of finding out the relation that the sum of the angles of a triangle bears to a right angle, he “at once begins by constructing a triangle” (ibid.). He then continues by carrying out a series of further constructions on the original figure and draws the relevant conclusions. In this way,

“through a chain of inferences guided throughout by intuition, he arrives at a fully evident and universally valid solution of the prob-lem” (A717/B745).

2.1.4.5 Summary

From the Doctrine of Method we can derive two lessons concerning mathematics, one negative, the other positive. Firstly, there is the ob-servation, backed up by reference to existing mathematical practices, in particular to Euclid’s geometry, that the classical model of concept-formation is inapplicable within the region of mathematics. Secondly, there is the positive proposal, to wit, Kant’s version of what was re-ferred to above as the apriorist program: in order to cast philosophi-cal light on current mathematics, and in particular, to explain the ap-riori character of the knowledge it delivers, an account must be given of the content of mathematical judgments that hinges on the notion of “constructing concepts”.²⁷ Accordingly, we face the following two questions of interpretation: (i) how does the construction of concepts generate (mathematical) contents which go beyond what can be ex-hibited purely logically? (ii) what are the implications of Kant’s an-swer to (i) for mathematical knowledge?

The first of these two questions is best approached by considering one of the more pressing issues for any interpreter of Kant’s theory of mathematics. This question has to do with the relation between the Doctrine of Method, on one hand, and Transcendental Aesthetic, on the other. The former explains the syntheticity of mathematics as

27 Here, again, I speak simply of the content of mathematical judgments, even though the full articulation of the semantic component of the apriorist program involves more than this.

a function of “constructibility” (and there are of course more than one way to understand this connection²⁸). The latter argues, roughly, that the syntheticity of mathematics can be understood only by the assumption that space and time are apriori intuitions, a claim that, apparently anyway, makes no mention of mathematics as construc-tive.²⁹

28 For example, Hintikka (1967, 1969a) argues, drawing on Kant’s defini-tion of intuidefini-tion as a singular representadefini-tion, that in Kant’s theory of mathematics construction amounts to the introduction of particulars to in-stantiate general concepts. What makes mathematics synthetic, according to Hintikka’s Kant, is the use of instantiation methods, i.e., the fact that mathematical reasoning typically involves the introduction of “new individu-als” (as when we use auxiliary constructions in a geometrical proof). A dif-ferent suggestion is to say that the syntheticity of mathematics is a function, not of the special character of mathematical reasoning, but of its axioms. Thus, Gordon Brittan writes that the reason why mathematics is synthetic, accord-ing to Kant, is that it involves propositions which make existential claims (1978, pp. 56-61). Here the contrast is with the propositions of logic, which are “purely formal” and “empty of content”, according to Kant. It should be noted, however, that the difference between a position like Hintikka’s and Brittan’s is really quite minimal (contrary to what Brittan argues). As regards geometry, the relevant existence assumptions are expressed by Euclid’s first three postulates. For Kant, however, postulates are “practical propositions”, which is to say that they have to do with a certain kind of action (in this case construction), and their import is that the action/construction can be carried out (cf. A234/B287). As regards their role in reasoning, they do not function as premises from which consequences are derived (purely logically or “ana-lytically”, as it were). Rather, a postulate is both a starting-point for proof (as in Euclid’s protasis) and that by means of which the proof (or part of it) is carried out. It follows that, for Kant, there is no distinction between axioms and rules of inference in anything like our sense. Brittan writes: “The syn-thetic character of the propositions of mathematics is a function of some feature of the propositions themselves, and not of the way in which they come to be established” (1978, pp. 55-56). But in the Euclidean context this contrast is really quite spurious.

29 Cf. Butts (1981, p. 267). It must be said, though, that this apparent discrepancy is likely to disappear once it is realised that the Aesthetic is not intended by Kant to stand on its own, but should be read in conjunction with the Analytic; cf. here Walsh (1975, §6).

2.2 Constructibility and Transcendental Aesthetic

The answer that I prefer to the above question concerning the rela-tion between constructibility and the Aesthetic (what I shall call the

“semantic interpretation” of Kant’s notion of construction) can be introduced by contrasting it with a line of thought that is found in many sophisticated expositions of Kant’s theory of mathematics. Ac-cording to this line of thought, the Aesthetic and its doctrine of space and time as apriori intuitions are related only “externally” to mathe-matics itself. That is to say, the constructibility of mathemathe-matics is one thing, to be understood in one way or another on its own; that mathematics is concerned with space and time as the forms of intui-tion is a claim that emerges only when Kant sets to himself the task of explaining the objective validity of mathematics, or its applicability to empirical objects or appearances.³⁰

30 Hintikka, for example, explains that Kant’s “transcendental problem of the possibility of mathematics” was that of making sense of how con-structions the introduction of particulars as proxies of general concepts can yield apriori knowledge, i.e., how we can be justified in making certain existential assumptions in the absence of the relevant objects (Hintikka 1969, sec. 16-18; 1984, sec. 3). Kant’s answer to this, according to Hintikka, makes use of this transcendental method: “[s]ince ‘reason has insight only into that which it produces after a plan of its own’ [Bxiii], the explanation of the universal applicability of knowledge obtained by using instantiation methods, i.e., by anticipating certain properties and relations of particulars, can only lie in the fact that we have ourselves put those properties and relations into objects in the processes through which we come to know individuals (particu-lars). Then the knowledge gained in this way must reflect the structure of those processes, and is applicable to objects only in so far as they are poten-tial targets of such processes” (1984, p. 347; italics in the original). Accord-ing to Kant, Hintikka continues, for beAccord-ings like us the relevant process is sense-perception. It follows that the properties and relations that mathemat-ics deals with are something that we put into objects in sense-perception, and they are due to the structure of this faculty, i.e., space and time as the forms of intuition. As Hintikka sees it, then, the Aesthetic serves the role of justifying the application of mathematical method to reality.

Brittan comes very close to this view. According to him, Kant gives the existential claims made in mathematics (like the postulates of Euclid’s

There are good reasons to maintain, however, that in Kant’s the-ory of mathematics the connection between constructibility, on one hand, and space and time, on the other, is deeper than that suggested by the above line of thought. Consider first the following quotation from Butts’s exposition of Kant’s theory of mathematics:

To construct an apriori intuition means simply to produce an individual example (in empirical intuition) according to rules of construction that are given by our conceptual system (in this case mathematics). [...] In the case of mathematical constructions the examples are used as representa-tives of universal concepts whose meaning is given in the mathematical system at hand. (1981, p. 269)

If we put it this way, there remains for Kant as well as his interpreters a compulsory question: What is it for a mathematical system to give a meaning to certain universal concepts? The problem that mathemat-ics presents for Kant is not just the one about the empirical applica-bility of mathematical constructions (although that problem is there, too). Equally important – indeed, more fundamental – is the semantic or representation-theoretic question of how mathematical concepts and judgments receive their content or meaning in the first place.

Appreciating the synthetic character of mathematical judgments forces one to recognise an unbridgeable gap between cognition through concepts and cognition grounded in the construction of concepts. To put the point in another way, it forces one to recognise that the content of mathematical judgments cannot be captured by ometry, which Brittan presents as claims “asserting the existence of mathe-matical individuals” (1978, p. 57), an epistemological twist through the no-tion of construcno-tion (id., p. 66). What is “really possible” – what could be experienced by us – is co-extensive with what is mathematically construct-ible: “the existence assumptions in Euclidean geometry are justified because of the structure or form of sensibility that in some sense “conditions” physi-cal objects, the existence of which in general we come to know through per-ception. The same activity that constructs concepts “constructs” objects, and thereby guarantees a fit between them” (id., p. 83; italics added). In a footnote Brittan refers to Hintikka, thus apparently declaring himself to be in agreement with the latter on the import of Kant’s transcendental method.

dint of the simple logical forms recognised in traditional logic. And it is Kant’s eventual explanation of the meaning- or content-constitutive role of construction – mathematical concepts have con-tent only because they can be constructed in pure intuition, which is why these concepts can represent features of objects that are “neces-sarily implied in the concepts” (Bxii), though not contained in them – that creates the link between the Doctrine of Method and the Aes-thetic (plus other relevant portions of the Critique): the content-constitutive role is grounded, precisely, in space and time as the forms of intuition.³¹

2.3 The Semantics of Geometry Explained

2.3.1 Constructions in Geometry

To see in more detail what is involved in the semantic interpretation, we may start by considering the case of geometry, which is generally

31 I have formulated the semantic interpretation in terms of the notion of content. It should be noticed that in the Kantian context “content” and “hav-ing content” are naturally associated with “objective validity” (and this is how Kant himself uses the German Inhalt, at least on occasion; see, e.g.

A239/B298). If the term is used in the latter way, we should say that,

A239/B298). If the term is used in the latter way, we should say that,