I. Introduction
In 1901 Bertrand Russell wrote with great gusto and conviction that "the proof that all pure mathematics ... is nothing but formal logic, is a fatal blow to the Kantian philosophy."(1) In The Principles of Mathematics, which he was feverishly churning out at about the same time, he developed this radical thought further:
There was, until very lately, a special difficulty in the principles of mathematics. It seemed plain that mathematics consists of deductions and yet the orthodox accounts of deduction were largely or wholly inapplicable to existing mathematics. Not only the Aristotelian syllogistic theory, but also the modern doctrines of Symbolic Logic, were either theoretically inadequate to mathematical reasoning, or at any rate required such artificial forms of statements they could not be practically applied. In this fact lay the strength of the Kantian view, which asserted that mathematical reasoning is not purely formal, but always uses intuitions, i.e., the à priori knowledge of space and time. Thanks to the progress of Symbolic Logic, especially as treated by Professor Peano, this part of the Kantian philosophy is now capable of a final and irrevocable refutation... The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.(2)
Fifty years later, in My Philosophical Development, Russell
wrote retrospectively that "ever since I abandoned the philosophy of Kant
... I have sought solutions of philosophical problems by means of analysis;
and I remain firmly persuaded ... that only by analysing is progress possible."(3)
So in other words, Russell is saying that if logicism--the reduction of
mathematics to pure or symbolic logic--is feasible, then Kant's(4)
philosophy is false and should be replaced by the method of logical analysis.
If the origins of the analytic tradition could ever be traced to a single
defining event--a philosophical Big Bang--it is the logicist rejection
of Kant's philosophy of mathematics.
For all sorts of reasons I am deeply interested in the connection between Kant's Critical Philosophy and the historical and conceptual foundations of analytic philosophy. In earlier work(5) I have argued for three major points: that the analytic tradition emerged by struggling with some of the central doctrines of Kant's Critique of Pure Reason; that a careful examination of this foundational debate shows that Kant's doctrines were never refuted but instead only for various reasons rejected; and that ironically enough it is the foundations of analytic philosophy, not the Critical Philosophy, that are inherently shaky. This paper extends that line of argument by revisiting Kant's much-criticized views on the nature of arithmetic. This revisitation takes place against against the backdrop of the logicist thesis (defended of course by Russell, but originally due to Frege) that arithmetic is reducible to pure or symbolic logic. I want to make a case for the claim that Kant's theory of arithmetic is not in fact subject to the most familiar and forceful objection against it, namely that Kant's doctrine of the dependence of arithmetic on time is plainly false, or even worse, simply unintelligible; on the contrary, his doctrine about time and arithmetic is highly original, fully intelligible, and with qualifications due to the inherent limitations of his conceptions of arithmetic and logic, defensible to an important extent. My case has two stages. In the first stage, I reconstruct Kant's argument for the synthetic apriority of arithmetic (section II). And in the second stage, I develop a new account of his notorious doctrine of the dependence of arithmetic on time (section III).
II. Why Arithmetic is Synthetic A Priori
According to Kant, mathematics is the pure formal science of quantity or magnitude. In turn, quantities or magnitudes are of two fundamentally different kinds: numerical and spatial. Arithmetic is the pure science of numbers, and geometry is the pure science of space. Whether arithmetic or geometry, however, mathematics for Kant is synthetic a priori, not analytic a priori--which is to say that it is a substantive or world-dependent science, not a purely logical science. But how can mathematics be at once a priori (i.e., experience-independent and necessary) and also substantive or world-dependent? The answer is this: for Kant mathematics is possible because it presupposes the innate human cognitive capacity for pure temporal and spatial representation, the innate human cognitive capacity for pure intuition (CPR A38-39/B55-56) (P Ak. iv. 280-283). In turn, as the Transcendental Aesthetic shows, our pure intuitions of time and space are the non-empirical necessary subjective forms of inner and outer human sensibility.
By 'elementary arithmetic' I mean elementary logic (i.e., bivalent first-order
quantified polyadic predicate calculus including identity) plus the five
Peano axioms,
(1) 0 is a number.taken together with the primitive recursive functions over the natural numbers--the successor function, addition, multiplication, exponentiation, etc. According to Gödel's first incompleteness theorem, elementary arithmetic is incomplete, which is to say that there are sentences of elementary arithmetic that are true and unprovable. One way of making sense of this from a Kantian point of view is to say that elementary arithmetic is incomplete because it is synthetic and not analytic. Frege argued that elementary arithmetic is analytic because its truths are derivable from general logical laws together with "logical definitions."(6) But Fregean logicism foundered on Russell's set-theoretic paradox, the deep unclarity of Frege's notion of a logical definition,(7) and Gödel-incompleteness. Neo-logicists(8) argue that if we drop the Fregean identification of numbers with sets of equinumerous sets and adopt second-order logic (i.e., elementary logic plus quantification over properties and functions) plus Hume's principle,(9) then elementary arithmetic is after all analytic. But neo-logicists must appeal to a logic stronger than elementary logic in order to show this. And even in view of the provability of elementary arithmetic in second-order logic plus Hume's principle, still no one can deny that Gödel-incompleteness entails that elementary arithmetic is not analytic, on the assumption that the criterion for analyticity is provability in elementary logic.
(2) The successor of any number is a number.
(3) No two numbers have the same successor.
(4) 0 is not the successor of any number.
(5) Any property which belongs to 0, and also to the successor of every number which has the property, belongs to all numbers.,
Now to be sure Kant's logic is very different from the logics used by logicists and neo-logicists, for his logic is significantly weaker than elementary logic. Kant's logic includes only truth-functional logic, Aristotelian syllogistic, and a theory of (fine-grained, decomposable) monadic concepts, which is to say in more modern terms that it includes only monadic logic(10) and a partial anticipation of higher-order intensional logic. And this may lead us to think, as Alan Hazen has put it, that "Kant had a terrifyingly narrow-minded and mathematically trivial, conception of the province of logic."(11) Well, yes: Kant's conception of the province of logic does not include polyadic predicate logic. But on the other hand, Kant's logic certainly captures a fundamental fragment of elementary logic.(12) Furthermore, since we already know from Gödel-incompleteness that elementary arithmetic is not analytic on the assumption that the criterion for analyticity is provability in elementary logic, and since Kant's logic is weaker than elementary logic, there seems to be little or no reason to believe that Kant's argument for the syntheticity of arithmetic will be vitiated by the limited character of his logic alone.(13) For if a stronger logic shows that elementary arithmetic is not analytic, then Kant's thesis that elementary arithmetic is not analytic surely cannot depend on the relative weakness of his logic.
That having been said by way of finessing familiar worries about the limited scope of Kant's logic, in this section I will develop an argument for the synthetic apriority of arithmetic by unpacking Kant's argument for a slightly weaker thesis: that a fundamental fragment of elementary arithmetic is synthetic a priori. There are two reasons for this. First, since Kant's logic is only a monadic logic, and therefore contains no theory of either multiple first-order quantification or second-order quantification, he would not be able to formulate Peano's axioms (2) through (5). So he would not be able to formulate classical first-order Peano arithmetic or PA, much less a second-order reading of the principle of mathematical induction, axiom (5).(14) On the other hand however, even though lacking a general theory of quantification, presumably Kant would still be able to formulate primitive recursive arithmetic or PRA, the quantifier-free theory of the natural numbers and the primitive recursive functions.(15) But second, if a fundamental fragment of elementary arithmetic is synthetic a priori, then obviously elementary arithmetic as a whole is also synthetic a priori. For convenience, I will call the fundamental fragment of elementary arithmetic that was studied by Kant 'arithmetic*'. And to give my argument some theoretical bite, I assume that arithmetic* includes PRA and monadic logic but falls short of PA.
I turn now to Kant's argument for the synthetic apriority of arithmetic*. The argument has four crucial background assumptions that we need to make explicit before surveying it step-by-step. First, according to Kant a true proposition is analytic if and only if its denial leads to a logical contradiction (CPR A151/B191).Therefore the negative criterion of a synthetic proposition is that its denial is logically or analytically consistent. Second, for Kant the positive mark of the syntheticity of a proposition is its semantic dependence on intuition. In turn, an intuition for Kant is an immediate or non-descriptive, non-discursive or non-conceptual and non-propositional, essentially singular representation of some actual spatial or temporal object, or of the underlying spatial or temporal form of all such objects (CPR A19-22/B33-36, A68/B93, B132, A320/B377; P Ak. iv. 280-281). More precisely then, a proposition is synthetic if and only if its objective validity and truth require an intuition in this special sense (CPR A239-240/B299, A721/B749). Third, on this Kantian picture of syntheticity as semantic intuition-dependence, synthetic a posteriori propositions are dependent on empirical intuitions, and correspondingly synthetic a priori propositions are dependent on pure intuitions:
This principle [of syntheticity] is completely unambiguously presented in the whole Critique, from the chapter on the schematism on, though not in a specific formula. It is: All synthetic judgments of theoretical cognition are possible only through the reference of a given concept to an intuition. If the synthetic judgment is an experiential judgment, the intuition must be empirical; if the judgment is a priori synthetic, there must be a pure intuition to ground it. (PC Ak. xi. 38)Fourth and finally, according to Kant synthetic a priori propositions are necessary, but unlike analytic propositions they are not absolutely necessary or true in every logically possibly world. Rather they restrictedly necessary. This is to say that they are true in all and only the humanly experienceable worlds; in worlds that are not experienceable, they are objectively invalid or truth-valueless:
Here we now have one of the required pieces for the solution of the general problem of transcendental philosophy: how are synthetic a priori propositions possible?--namely, pure a priori intuitions, space and time, in which, if we want to go beyond the given concept in an a priori judgment, we encounter that which is to be discovered a priori and synthetically connected with it, not in the concept but in the intuition that corresponds to it: but on this ground such a judgment never extends beyond the objects of the senses and can hold only for objects of possible experience. (CPR B73)
Our theoretical cognition never transcends the field of experience.... [I]f there is synthetic cognition a priori there is no alternative but that it must contain the a priori conditions of the possibility of experience. (RP Ak. xx. 274)To be sure, this barely scratches the surface of an adequate discussion of Kant's analytic/synthetic distinction.(16) But for our purposes here, the bottom line is this: From these four points, it follows that a truth of arithmetic is synthetic a priori just in case it is (1) consistently deniable, (2) semantically dependent on pure intuition, and (3) necessarily true in the restricted sense that it is true in every experienceable world and never false otherwise because it lacks a truth-value in every unexperienceable world.
So much for the semantic framework. My reconstruction of Kant's argument for the synthetic apriority of arithmetic* is based on the notorious "finger-counting" passage at B14-16, the all-too-brief definition of the concept NUMBER at A142-143/B182, and §10 of the Prolegomena. For clarity's sake, I will spell out the argument step-by-step; and where it is relevant, I will also quote the texts upon which my reconstruction is based.
A Reconstruction of Kant's Argument for the Syntheticity of Arithmetic
(1) Assume the existence of arithmetic*. (Implicit premise.)
(2) It is a priori and thereby necessary that 7+5=12. (From (1) and the definition of apriority.)
(3) There is at least one logically possible world in which 7+5 12. (Premise.)
(4) So it is not necessary that 7+5=12. (From (3).)
(5) If and only if the pure intuition of time is invoked, can (2) and (4) be made consistent with one another. For in any possible world representable by our pure intuition of time, there is a sufficiently large and appropriately-structured supply of "stuff" (Stoff)--the total set of homogeneous temporal moments generated in the sempiternal successive synthesis of the sensory manifold--to constitute a truth-maker of the arithmetic proposition in question. That is, (2) is made true by all the experienceable worlds. And in some worlds that are not representable by our pure intuition of time, nothing suffices to be a truth-maker of the arithmetic proposition in question. That is, (4) is made true by some unexperienceable worlds. So (2) is consistent with (4), assuming pure intuition; otherwise they are inconsistent. (Premise.)
[N]umber [is] a representation that summarizes the successive addition of one homogeneous unit to another. Number is therefore nothing other than the unity of the synthesis of the manifold of a homogenous intuition in general, because I generate time itself in the apprehension of the intuition. (CPR A142-143/B182)Now, the intuitions which pure mathematics lays at the foundation of all its cognitions and judgments which appear at once apodeictic and necessary are space and time. For mathematics must first exhibit all its concepts in pure intuition... If it proceeded in any other way, it would be impossible to make a single step; for mathematics proceeds, not analytically by dissection of concepts, but synthetically, and if pure intuition is wanting there is nothing in which the stuff (Stoff) for synthetic a priori judgments can be given. (P Ak. iv. 283)
(6) Since the proposition that 7+5=12 is both necessary and
also consistently deniable, and its necessity is grounded on our pure intuitional
representation of time, it follows that it is synthetic a priori. (From
(2), (4), (5), and the definitions of syntheticity, apriority, and synthetic
apriority.)
(7) The argument applied to the proposition that 7+5=12 can be applied, mutatis mutandis, to any other truth of arithmetic* involving larger numbers. (Generalization of (6).)
(8) Therefore all truths of arithmetic* are synthetic a priori. (From
(7).)
Pretty obviously, the most controversial move in this argument occurs at step (3), in which the consistent deniability of truths of arithmetic* is asserted. But what sort of a logically possible world could be fail to be a truth-maker for something as apparently unexceptionable as '7+5=12'? We must find a possible world that lacks at least one of the underlying structural features of arithmetic.* One sort of world that will do the job is a radically finite world--in particular, a world containing no structure rich enough to include 12 or more elements--i.e., a world containing less than 12 objects--which thereby lacks the requisite supply of "stuff" for satisfying '7+5=12'.(17) Another different sort of world that would do the same job is a radically unrecursive world, that is, a world in which even if there is the requisite supply of "stuff" for assigning reference to all the number words, nevertheless there is no way of iteratively operating on that stuff. This would be a world with enough objects to satisfy arithmetic propositions but without any primitive recursive functions over those objects, including of course addition.(18)
Now because, according to Kant, our pure intuitional representation of time yields an infinite given whole (CPR A32/B47-48) in which the unidirectional series of homogeneous moments are successively summed up to the magnitude of any later moment, it follows that both the radically finite and radically unrecursive worlds just described will not conform to our pure intuition of time. Indeed, it seems plausible to believe that any countermodel to arithmetic* will also fail to conform to our pure intuition of time. And since our pure intuitional representation of the time-series is a necessary condition of the possibility of all sensory representation of objects (CPR A31/B46), it then follows that all the countermodels to arithmetic* will also be unexperienceable worlds. In turn, the recognition that all the countermodels to arithmetic* also violate the conditions of the possibility of human experience yields the further recognition that we must ground the necessary truth of arithmetic* directly on the innate human capacity for pure temporal intuition. For arithmetic* is true only in worlds that include either the time-structure itself or else something isomorphic to the time-structure. And in every temporally-structured world not only is arithmetic* true, but also its truth-maker is cognizable a priori, and furthermore it has a direct application to objects of human experience--neither of which is guaranteed in timeless worlds. So while there are some conceivable and therefore possible timeless or noumenal worlds that make arithmetic* true, none of them will count in favor of arithmetic*'s synthetic necessity, a priori cognizability, or applicability. Only temporally-structured worlds can count in favor of these; and only pure temporal intuition gives us direct cognitive and semantic access to all those special worlds.
III. The Meaning of the Concept NUMBER
At this point, you might wonder what precisely is going on. In arguing that arithmetic* is synthetic a priori, is Kant arguing that arithmetic* is the science of time as we represent it in pure intuition, just as he argues that geometry is the science of space as we represent it in pure intuition? No. In the "transcendental exposition" of the representation of space in §3 of the B edition version of the Transcendental Aesthetic, Kant argues explicitly that our pure intuition of space is necessary and sufficient for the objective validity of geometry. But in the corresponding transcendental exposition of the representation of time Kant very pointedly does not focus on arithmetic* but instead on the "general doctrine of motion" (CPR B49) or universal Newtonian mechanics. I think that we may take this to be an indication of an important asymmetry between Kant's theories of geometry and arithmetic, as Philip Kitcher points out: "Kant did not believe, as is often supposed, that arithmetic stands to time as geometry does to space."(19)
In his classic Commentary on the first Critique, Norman Kemp Smith also makes a pertinent remark in this connection:
Though Kant in the first edition of the Critique had spoken of the mathematical sciences as based on the intuition of space and time, he had not, despite his constant tendency to conceive space and time as parallel forms of experience, based any separate mathematical discipline upon time.(20)In one sense this is quite correct, but in another sense it is misleading. The problem lies in a certain ambiguity in Kemp Smith's phrase 'based on'. That phrase has both a logico-metaphysical sense and a semantic sense. According to the logico-metaphysical sense, X is based on Y if and only if Y is a necessary and sufficient condition of X (where this covers everything from identity to strong supervenience). But the semantic sense of 'based on' is different. According to the semantic sense, X is based on Y if and only if X is aboutY. In turn, X is about Y if and only if Y is the semantic value of X, i.e., Y determines the extension of X. So if X is a concept-term, and X is aboutY, then Y determines what objects X applies to; if X is a propositional term, and X is about Y, then Y determines the truth-maker(s) of X; if X is a theory, and X is about Y, then Y determines the model(s) of X. It should be noted that Y certainly can (although it does not necessarily have to) determine the extension of X by being identical with the extension of X. But the crucial point is that once we have isolated the semantic sense of 'based on' as aboutness, it is then quite correct to say that Kant does not conceive of arithmetic* as a science that is about time and its formal features in the way that geometry is about space and its formal features. Instead, arithmetic* is about the natural numbers and their formal features, not about time. Still, as Michael Friedman aptly puts it, "there is no doubt that [for Kant] arithmetic involves time."(21) So how can it be true that for Kant arithmetic is not about time, yet arithmetic still presupposes time as a necessary and sufficient condition of its objective validity?
The interpretation I favor is that our pure intuition of the infinite unidirectional successive time-series supplies a fundamental semantic condition for arithmetic*, but does not fully determine the semantics of arithmetic* until it is combined with a second representational factor--i.e., a purely logical factor. I will discuss this purely logical factor at the end of this section. But right now we need to see how pure intuition manages to supply a fundamental semantic condition for arithmetic*. The answer is revealed in these texts, the first of which we have seen already:
[N]umber [is] a representation that summarizes the successive addition of one homogeneous unit to another. Number is therefore nothing other than the unity of the synthesis of the manifold of a homogenous intuition in general, because I generate time itself in the apprehension of the intuition. (CPR A142-143/B182)
Time is in itself a series (and the formal condition of all series). (CPR A411/B438, emphasis added)
Arithmetic attains its concepts of numbers by the successive addition of units in time. (P Ak. iv. 283)
Time [is] the successive progression as form of all counting and of all counting and of all numerical quantities; for time is the basic condition of all this producing of quantities. (PC Ak. xi. 208, emphasis added)Here is what I think Kant is driving at: the pure intuition of time sharply constrains what can count as a model for arithmetic*, but does not itself determine the extension of number terms or arithmetic propositions. Arithmetic*, by means of number concepts, represents the natural numbers, their intrinsic and relational properties, and the recursive functions over them. But all models of arithmetic* are non-conceptually structurally restricted or limited by means of our pure intuitional representation of the infinite unidirectional successive time-series. So nothing will count as a model of arithmetic* unless it is at least isomorphic with the infinite unidirectional successive time series delivered by pure intuition. Pure intuition does not therefore tell us just what the intended or standard model of arithmetic* is--pure intuition does not tell us what the numbers are--but it does tell us what the numbers cannot be, and it lays down a basic condition for something's being a referent of numerical terms or a truth-maker for arithmetic* propositions.
So to repeat, I am saying that Kant's thesis about the role of our pure intuition of time in arithmetic* is that our pure intuition of the infinite unidirectional successive time-series supplies a fundamental semantic condition for the objective validity or meaningfulness of the concept NUMBER by partially determining what will count as a referent for numerical terms or a truth-maker for arithmetic* propositions: such terms and propositions cannot be about the natural numbers unless their extensions are isomorphic with time. A similar point is made by Charles Parsons:
Time provides a universal source of models for the numbers..... What would give time a special role in our concept of number which it does not have in general is not its necessity, since time is in some way necessary for all concepts, nor an explicit reference to time in numerical statements, which does not exist, but its sufficiency, because the temporal order provides a representative of the number which is present to our consciousness if any is present at all.(22)Of course the pure intuition of time does more than merely constraining the class of models for arithmetic*. By virtue of the threefold fact that our pure intuition of the infinite unidirectional time-series (i) is built dispositionally into human representational capacities, (ii) is a necessary condition of all sensory experience of objects, and also (iii) picks out time, which is partially constitutive of the empirical world, it follows then that Kant can neatly explain not only (i*) how arithmetic* is synthetically necessary, or true in all experienceable worlds and never false otherwise (i.e., because time is included in every experienceable world and every model of arithmetic* is isomorphic with time), but also (ii*) how arithmetic* is cognizable a priori for creatures like us (i.e., because our capacity for pure temporal intuition is innate), and (iii*) how arithmetic* is guaranteed to have empirical application (i.e., because the representation of time is guaranteed to have empirical application). The dimension of applicability, moreover, is a particularly crucial factor, as Frege points out in Basic Laws of Arithmetic:
It is applicability that raises arithmetic from the rank of a game to that of a science. Applicability therefore belongs to it of necessity.(23)All of this adds up to an important point. As Michael Potter has observed, two fundamental and intimately-related problems in the philosophy of arithmetic are (1) how to explain arithmetic's necessity?, and (2) how to explain arithmetic's empirical applicability?(24) But there is also a second pair of similarly fundamental and intimately-related problems: (3) on the one hand a uniform semantics of natural language implies that numbers are humanly-knowable truth-makers of arithmetic truths, but on the other hand the reasonable assumption that numbers are causally inert abstract objects implies that they are unknowable when combined with the equally reasonable assumption that mathematical intuition is analogous to sense perception,(25) and (4) what are the numbers?(26) Paul Benacerraf has famously argued that the third and fourth problems arise in a particularly acute way for logicism.(27) If Benacerraf is right about these troubles with logicism, and if I am right about Kant's theory of arithmetic, then the deep significance of Kant's philosophy of arithmetic lies in the fact that he adumbrates a joint solution to the four problems--a joint solution that appears to be inaccessible to logicism.
Now I want to wrap up the paper by taking a look at a very puzzling letter that Kant wrote to his friend and disciple Johann Schultz in 1788, i.e., one year after the publication of the second or B edition of the first Critique. Schultz was then working on the manuscript of a book entitled Prüfung der kantischen Kritik der reinen Vernunft ('Examination of the Kantian Critique of Pure Reason'), which he had shown to Kant. In that manuscript, Schultz had anticipated Frege by defending the idea that all the truths of arithmetic are purely logical or analytic.(28) Here is the key part of Kant's response to the manuscript:
Time, you correctly notice, has no influence on the properties of numbers (considered as pure determinations of quantity), as it may have on the character of those alterations (of quantity) that are possible only relative to a specific state of inner sense and its form (time). The science of numbers, notwithstanding the succession that every construction of quantity requires, is a pure intellectual synthesis, which we represent to ourselves in thought. But insofar as specific quantities (quanta) are to be determined according [to this science of numbers], they must be given to us in such a way that we can grasp their intuition successively; and thus this grasping is subjected to the time condition. (PC Ak. x. 556-557)Part of what Kant is doing here is simply reiterating his view that while arithmetic* presupposes our pure intuition of the time-series, arithmetic* is not itself the science of "alterations" (Veränderungen) or events--that is, arithmetic* is not about time. But for our purposes the crucial question is, what does Kant means by his remark that "the science of numbers ... is a pure intellectual synthesis, which we represent to ourselves in thought"? What he seems to be saying is that arithmetic* is grounded on pure conceptualization, which of course in his terms would make it purely logical or analytic in nature. So by 1788 has Kant quietly switched over to some version of logicism?
No. One way of seeing this is to return to a subtle point he makes in the B edition of the Critique of Pure Reason. There Kant explicitly commits himself to the thesis that all mathematics is strictly constrained by pure logic in that "the inferences of the mathematician all proceed in accordance with the principle of contradiction" (CPR B14). But this constraint on inference and proof "is required by the nature of any apodictic certainty" (CPR 14), so it is not special to mathematics. More generally, it does not follow that mathematics is essentially logic just because its proofs must meet some minimal pure logical requirements:
Since one found that the inferences of the mathematician all proceed in accordance with the principle of contradiction ..., one was persuaded that the principles could also be cognized from the principle of contradiction, in which, however, they erred; for a synthetic proposition can of course be comprehended in accordance with the principle of contradiction, but only insofar as another synthetic proposition is presupposed from which it can be deduced, never in itself. (CPR 14)
This of course sets Kant's view on mathematics sharply apart from
the Leibnizian view, according to which all necessary truth is ultimately
reducible to the logical principle of identity or non-contradiction (Leibniz
regarded these as equivalent). But the crucial point is that it is a mistake
to think that the admitted fact of strict logical constraints on mathematics
entails a reduction of mathematics to logic. Kant's view, on the contrary,
is that mathematics can essentially include logical elements without in
any way undermining its syntheticity.
The question on the table is whether Kant in the letter to Schultz in 1788 is intentionally or unintentionally backsliding towards some sort of logicism about arithmetic*. And one reason for thinking that he is not backsliding , as we have just seen, is that in the B edition of the first Critique, published only a year before the letter to Schultz, he is explicitly committed to the idea that the presence of significant logical factors in mathematics is consistent with the denial of logicism. But the decisive reason for thinking that he is not has to do with his views on the role of logic in the semantic constitution of the concept NUMBER. In the letter to Schultz, Kant is saying, I think, that NUMBER does indeed have a purely logical source of representational content in our conceptual faculty, the understanding, but that this source of content does not exhaust the content of NUMBER.
So what, according to Kant, does NUMBER mean? Here is what he says explicitly in the first Critique:
No one can define the concept of a magnitude in general except by something like this: That it is the determination of a thing through which it can be thought how many units are posited in it. Only this how-many-times is grounded on successive repetition, thus on time and the synthesis of the homogeneous in it. (CPR A242/B300, emphasis added)And here is what I think he means by that remark, when we combine it with what he says in the letter to Schultz. Kant's view, it seems, is that NUMBER is necessarily partially based on the three "logical functions" of quantification in judgments:
Universal (e.g., all Fs are Gs),The logical functions of quantification, in turn, correlate one-to-one with the three categories of quantity:
Particular (e.g., some Fs are Gs),
Singular (e.g., the F is G, or this F is G) (CPR A70/B95).
TotalityNow in the Schematism, Kant says that "the pure image of all magnitudes (quantorum) ... for all objects of the senses ... is time" and that "the pure schema of magnitude (quantitatis), however, as a concept of the understanding, is number" (CPR A142/B182). As I understand it, what he means is that the concept NUMBER is what results if one takes the basic logical constants of quantity (all, some, the/this), maps them onto the corresponding metaphysical categories of quantity (totality, plurality, unity), and then systematically interprets those quantitative categories in terms of the pure intuition of time, as follows: (1) the logical function of universality in judgments, corresponding to 'all Fs', goes over into the infinite totality of successive moments of time, and so yields an exemplar or paradigm of the whole series of the natural numbers; (2) the logical function of particularity in judgments, corresponding to 'some Fs', goes over into any finite plurality of successive moments of time (i.e., a duration), and so yields exemplars or paradigms of any finite natural number; and (3) the logical function of singularity in judgments, corresponding to 'the F' or 'this F', goes over into any arbitrarily chosen single moment or unit of time, and so yields an exemplar or paradigm of the number 1. Now for Kant all empirical magnitudes or quantities are finite or infinite (CPR A430/B458), discrete or continuous (CPR A526-527/B554-555), and extensive or intensive (CPR A162-163/B203-204, A165-171/B208-212).(30) And as we have just seen, in the Schematism Kant tells us that all appearances, as magnitudes or quantities, fall under the schematized concept NUMBER (CPR A161-176/B202-218). So NUMBER is directly applicable to all sorts of empirical magnitudes by virtue of its construal in terms of the pure intuition of time.
Plurality
Unity (CPR A80/B106).(29)
In other words, according to Kant the concept NUMBER has a purely logical source of representational content; but that logical input does not exhaust its semantic content, since it also has a complementary non-logical source of its representational content--the pure intuition of the infinite unidirectional successive time-series. So the concept NUMBER is a partially logical but not wholly logical concept: it represents numbers in purely logical terms, but these logical terms alone do not suffice to fix its meaning or objective validity adequately. Its meaning is adequately fixed, however, when we supplement its purely logical content by combining it with a certain non-logical structure. That is, when we represent natural numbers by using and specifying the concept NUMBER, we must also invoke a supplementary pure intuition of the infinite unidirectional successive time-series, which supplies the other fundamental semantic condition for the objective validity of NUMBER in particular and for arithmetic* more generally, by sharply constraining what will count as a model for the latter, and by securing the empirical applicability of the former.
This returns us to the contrast between logicism about arithmetic and Kant's theory of arithmetic. Kant's view about the numbers, by contrast to that of any theory attempting to give a reduction of arithmetic to logic, is that something is a natural number if and only if it satisfies the purely logical categories of quantity and is isomorphic to some part of the infinite unidirectional successive time-series picked out by pure intuition. So a given number is how we collect or colligate all Fs, or some Fs, or the/this F, in a way that formally mimics the unidirectional successive synthesis of moments in time. The number 5, e.g., is how we collect or colligate whatever falls under the concept F (say, all the fingers on one hand including the thumb) in exactly the same way that we representationally generate just that many moments of time. And the number zero is how we collect or colligate no Fs at all in exactly the same way that we representationally generate no moments of time by representing the specious present in which nothing has yet happened--the "beginning, the pure intuition = 0" (CPR A165/B208). Otherwise put, the representational generation of numbers by counting is the logical representation of all objects, some objects, the/this object, or even no objects (which is represented in terms of negation and the particular quantifier), under some first-order (typically, empirical) concept C, taken together with the representation of time.
Number concepts, in other words, are schematized concepts. That is, they are concepts whose meaningful content is partially determined by a non-logical structure--the structure of total infinite unidirectional time, as delivered by pure intuition. So the natural numbers are in effect nothing but positions in a logically-conceptually constrained intuitional time-structure, which is to say that Kant's theory of the numbers is a highly original (and specifically non-platonistic) version of ante rem mathematical structuralism.(31) But that is not to say that the natural numbers are really something other than the natural numbers. On the contrary, the numbers are what they are, and not some other things. Numbers are sui generis entities because they are fully determined by logical concepts with a sui generis semantic content, which is the same as to say that numbers are nothing but positions in an empirically applicable and humanly graspable (because intuitional) time-structure under special logical-conceptual constraints.
If my interpretation of Kant's response to Schultz is correct, then Kant is saying that pure logic on its own underdetermines the meaning of the concept NUMBER. But not only that: Kant is saying that only our pure temporal intuition can do the further semantic job that logic fails to do on its own, and also saying that numbers are sui generis entities--therefore radically irreducible entities--with sui generis properties and relations. Arithmetic* is about the natural numbers and their formal features, and requires both pure logic and the pure intuitional representation of time in order to be about such things. The natural numbers are the semantic values of numerical terms and among the semantic values of arithmetic* propositions. But the natural numbers, in turn, are natural precisely because their special structuralist ontology is primitive and essentially bound up with human nature. So in a twist on Leopold Kronecker's famous quip about number theory to the effect that God made the integers and everything else was done by humans,(32) we might say that for Kant human nature made the natural numbers and everything else was done by logic.
IV. Conclusion
As I see it, Kant is not asserting that arithmetic* is the pure science of time. Rather, Kant is asserting a highly original two-part doctrine about the cognitive semantics of the concept NUMBER: (a) that the content of the concept NUMBER requires our pure formal intuition of the sempiternal (or infinite unidirectional) series of successive moments of time as a non-logical necessary condition of that concept's objective representational content; and (b) that the content of the concept NUMBER equally requires the logical functions of quantity and their corresponding categories. If Kant is right about this, then arithmetic* is essentially the result of combining the formal ontology of our human intuitional representation of time with the conceptual resources of logic in Kant's sense. That Kant's own conception of arithmetic comprehends at most the primitive recursive fragment of elementary arithmetic, and that Kant's own conception of logic comprehends at most the monadic fragment of elementary logic, are ultimately far less important than his deep insight into the essentially two-sided temporal/intuitional and logical/conceptual structure of the pure science of numbers. This dual structure is at once irreducibly anthropocentric and also strictly constrained. So arithmetic for Kant is not only an exact science, but also and perhaps most fundamentally, a human science.
This Kantian doctrine directly challenges the logicist reduction of arithmetic to logic, and it offers a joint solution to a closely interlinked set of fundamental problems that logicism apparently cannot solve. But perhaps even more importantly, since if I am right Kant's theory is basically a doctrine about the cognitive semantics of the concept NUMBER, its truth or falsity is logically independent of the truth of Kant's transcendental idealism. So it may turn out that logicism's failure to solve the fundamental problems in the philosophy of mathematics depends largely on the failure of logicists to understand the deeper significance of Kant's Copernican Revolution.(33)
Robert Hanna
Department of Philosophy
University of Colorado at Boulder
Boulder, CO 80309-0232
rhanna@spot.colorado.edu
4. For convenience, I cite Kant's works infratextually in parentheses. The citations normally include both an abbreviation of the English title and the corresponding volume and page numbers in the standard "Akademie" (Ak) edition of Kant's works: Kants gesammelte Schriften, 29 vols, hrsg. Königlich Preussischen (now Deutschen) Akademie der Wissenschaften (Berlin: G. Reimer [now de Gruyter], 1902-). For references to the first Critique, however, I follow the common practice of giving page numbers from the A (1781) and B (1787) German editions only. Here is a list of the relevant abbreviations and translations:
CPR Critique of Pure Reason. Trans. P. Guyer and A. Wood. Cambridge: Cambridge Univ. Press, 1997.
JL "Jäsche Logic." In Immanuel Kant: Lectures on Logic. Trans. J.M. Young. Cambridge: Cambridge Univ. Press, 1992. Pp. 519-640.
P Prolegomena to Any Future Metaphysics. Trans. J. Ellington. Indianapolis: Hackett, 1977.
PC Immanuel Kant: Philosophical Correspondence, 1759-99. Trans. A. Zweig. Chicago: Univ. of Chicago Press, 1967.
RP What Real Progress has Metaphysics made in Germany since the Time of Leibniz and Wolff?. Trans. T. Humphrey. New York: Arabis, 1983.
8. See, e.g., Hale, (1987), Tennant (1997), and Wright (1983).
9. Hume's principle says that the number of Fs = the number of Gs if and only if there are just as many Fs as Gs.
10. Monadic logic is a restricted form of elementary logic that permits quantification into one-place predicates only. Interestingly, monadic logic is not only consistent and complete but also effectively decidable. See Boolos and Jeffrey (1989: chs. 22, 25).
12. And we certainly should not undervalue the fact that Kant's logic partially anticipates higher-order intensional logic; indeed, this may be part of the key to understanding his theory of analyticity. See Hanna (2001: 80-83).
13. This is a standard complaint against Kant's argument for the synthetic apriority of mathematics, going at least as far back as Russell (1996). See Friedman (1992: 55-135).
15. See Skolem (1967), and Troelstra and Dalen (1988: 120-126).
17. Similar points about finite countermodels for arithmetic are made by Parsons (1983: 131) and Shapiro (1998: 604). This is not to say that appeals to such countermodels are uncontroversial, however. Indeed, there are at least two big worries about radically finite worlds: (1) it is well known that there are inferential gaps between imaginability and conceivability, and also between conceivability and possibility; and (2) radically finite worlds fail to verify what seem to be obvious truths based on the extensional law of identity, e.g., 12 13. Obviously I cannot adequately rebut these objections in a footnote; but here are very brief indications of possible replies. First, the imaginability-conceivability and conceivability-possibility gaps are alike double-edged swords, in the sense that both the critics and the defenders of the radically finite worlds thesis must appeal to conceivability arguments. Indeed it seems to me that the resistance to the possibility of radically finite worlds depends mostly on the challengeable thesis--challengeable because, presumably, justified by the step from the inconceivability of its denial to its necessity--that the natural numbers exist necessarily. Second, on my interpretation of Kant's modal theory, possible worlds are formal constructions on concepts (see Hanna [2001: 85, 241-242]), so for Kant the step from conceivability to possibility is automatically guaranteed. And third, for Kant extensional identity is not a purely logical notion (see Hanna [2001;142, n. 57]), and if he is right then it is not surprising that the extensional law of identity fails in some logically possible worlds.
18. A third sort of world would be a quus-world, i.e., a world in which some sort of non-Peano addition-function holds. See Shapiro (2000: 89-90) and Kripke (1982: ch. 2).
21. Friedman (1992: 105, n. 16).
22. Parsons (1983: 140). Nevertheless Parsons thinks that "Kant did not reach a stable position on the place of the concept of number in relation to the categories and the forms of intuition" (1992: 152). If I am right about the interplay between intuitional and logical factors in Kant's analysis of the meaning of concept NUMBER, then Kant's account is in fact more stable and cogent than Parsons supposes.
23. Frege (1964: vol. II, section 91).
26. This problem arises in several ways. In its most general form it is Quine's problem of "what there is" (1953: 14-15); in the context of first-order logic it is Hacking's problem about categoricity (1979); and in the context of second-order logic it is Frege's Caesar problem (1953: 68) about identifying the numbers with objects and Benacerraf's problem of the indeterminacy of the reference of number terms (1965).
27. Benacerraf (1965), (1972), and (1996).
28. To be sure, Leibniz had already anticipated this idea. But Schultz was apparently the first philosopher to float it after the publication of the Critique of Pure Reason. Significantly, the published version of the Prüfung does not contain this thesis. This could simply be a matter of Schultz's deferring to his teacher and master. But it could also be a matter of Schultz's believing that Kant's reply adequately handled his objection.
29. Kant sometimes reverses "totality" and "unity"; but for a good defense of the claim that Kant's real intention is to put them in the order I have used in the text, see Longuenesse (1998: 248-249).
30. The extensive continuum has a magnitude equal to the natural numbers, and the intensive continuum has a magnitude equal to the real numbers. So, given Kant's conception of pure intuition, together with the schematized "mathematical" categories, it follows that the empirical world is both an extensive and intensive continuum. And in this way, it seems, Cantor's continuum hypothesis is determinately true in every experienceable world.
31. See Shapiro (2000: ch. 10); and Shapiro (1997). Unfortunately it is hard to find a clear or widely-accepted statement of what is meant by saying that something (e.g., a universal) is ante rem. In any case, for me something is ante rem if and only if it is not uniquely located in spacetime and its existence does not logically require the existence of actual things. So, roughly speaking, for me something is ante rem if and only if it is abstract and neither de re nor in re. And Kant's logically-constrained pure intuitional representation of the infinite unidirectional time-series is ante rem in precisely this sense (CPR A30-36/B46-53, A291-292/B347-348).
33. A longer and slightly differently focused version of this paper is forthcoming in European Journal of Philosophy. I am grateful to the participants in the Philosophy of Logic and Mathematics Workshop at Fitzwilliam College, Cambridge University, in March 2000 and March 2001, for very helpful comments on presentations of parts of this material. Many thanks also to Fitzwilliam College for visiting fellowships in the Lent terms of 00 and 01, during which several drafts of the paper were composed. I am particularly grateful to Michael Potter for organizing the workshops, for hosting the visiting fellowships, and for many informal discussions of these topics. And finally I would like to thank an anonymous referee at EJP for his/her copious and cogent critical comments on a much longer version.