THE POINT OF KANT’S axioms of intuition.pdf

8/11/2019 THE POINT OF KANTS axioms of intuition.pdf

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Pacific Philosophical Quarterly

86 (2005) 135159

2005 University of Southern California and Blackwell Publishing Ltd. Published by

Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and

350 Main Street, Malden, MA 02148, USA.

135

THE POINT OF KANTS

AXIOMS OF INTUITIONby

DANIEL SUTHERLAND

Abstract:

Kants Critique of Pure Reason

makes important claims about

space, time and mathematics in both the Transcendental Aesthetic and

the Axioms of Intuition, claims that appear to overlap in some ways andcontradict in others. Various interpretations have been offered to resolve

these tensions; I argue for an interpretation that accords the Axioms of

Intuition a more important role in explaining mathematical cognition

than it is usually given. Appreciation for this larger role reveals that

magnitudes are central to Kants philosophy of mathematics and to the

part that intuition plays in it.

Introduction

The section of Kants Critique of Pure Reason called Axioms of Intuition

deserves more attention than it has received. Those who have considered

it agree that it is intended to justify the application of mathematics to the

objects of experience. Some have held that the Axioms also concern

a specific part of pure mathematics. I do not think, however, that the full

role of the Axioms has been appreciated. I will argue that the point of the

Axioms is twofold, concerning not only the applicability of mathematics

but the possibility of any mathematical cognition whatsoever, whetherpure or applied, general or specific.

1

The interpretation for which I argue

clears up some potential confusions concerning the treatment of space,

time, and mathematics in the Transcendental Aesthetic and the Axioms.

Most importantly, perhaps, it allows us to see that the Axioms contain a

substantial contribution to Kants theory of mathematical cognition.

I will argue that a proper understanding of the point of the Axioms opens

up a new avenue of research into Kants philosophy of mathematics and

the role of intuition in it.


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PACIFIC PHILOSOPHICAL QUARTERLY

2005 University of Southern California and Blackwell Publishing Ltd.

There are, I think, various reasons why the Axioms and the theory of

magnitudes appearing in it have not earned more attention. First, many

have held that Kants primary goal in the Critique of Pure Reason is to

respond to Humes skepticism about causation, a response that cul-

minates in the Second Analogy. Since the Axioms make little or no directcontribution to the argument leading up to the Second Analogy, many

have passed over them without serious investigation.

2

Second, and more

importantly, the point of the Axioms is obscured by misunderstandings

concerning, on the one hand, the Transcendental Aesthetic treatment of

mathematics and of space and time, and on the other, the Axioms treat-

ment of them. Third, there is a tension between two goals of the Axioms.

On the one hand, the Axioms are part of the System of Principles, whose

role in the Critique is to articulate the conditions for any possible experi-

ence. On the other hand, the argument establishing those conditionsgrounds a principle valid for all mathematical cognition whatsoever. This

result goes beyond the aims of the System of Principles and therefore

might be overlooked. Nevertheless, it is fundamental to Kants views on

mathematical cognition and cognition more generally, and he uses the

Axioms to present his views on mathematical cognition.

3

I will sort through these issues by focusing on two contrasts between

the Transcendental Aesthetic and the Axioms. Section 1 examines the role

of mathematics in the Aesthetic and the Axioms. I consider two accepted

strategies for resolving an apparent conflict between them, which providesthe background for the rest of the paper. Section 2 takes up the treatment

of space and time in the Aesthetic and the Axioms. Here, too, there is an

apparent contradiction, which is resolved by appealing to Kants notion

of determinate mathematical representations. Section 3 examines Kants

views on mathematical determination in more detail. The results are put

to use in Section 4, which returns to the role of mathematics in the Axi-

oms and the Aesthetic and provides a new strategy for resolving the

apparent contradiction between them.

1: Mathematics in the Aesthetic and the Axioms

The title Axioms of Intuition is misleading. The section immediately

following the Axioms of Intuition, the Anticipations of Perception, argues

for a principle concerning intensive magnitudes. Kant calls the principles

of the Axioms and Anticipations mathematical; he explains that these

principles are not themselves mathematical but are called mathematical

because they explain the possibility of mathematical principles (A162/B2012). The mathematical principles whose possibility is explained include

geometrical axioms, such as Euclids postulates. Nevertheless, the Axioms

of Intuition argue for just one principle that is not itself an axiom.


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THE POINT OF KANTS AXIOMS OF INTUITION 137


Kant altered the formulation of the principle from the first to the

second edition of the Critique

. The principle in the A-edition is: All

appearances, according to their intuition, are extensive magnitudes.

In the B-edition it is: All intuitions are extensive magnitudes (A162/

B202).

4

The significance of Kants reformulation of the principle willbe addressed in Section 4 below; for now, I will defer to the B-edition

formulation.

Kant states that the Transcendental Aesthetic is a science of all prin-

ciples of a priori sensibility, that is, of the a priori forms of space and

time (A21/B35). Since he also states that geometry is a science that

determines the properties of space synthetically and a priori

(B40), one

would naturally expect Kant to establish some claims of or about math-

ematics in the Aesthetic. Indeed, Kant attempts to establish important

features of space and time in the Aesthetic in addition to the fact thatthey are a priori

intuitions and mere forms of sensibility. For example,

he asserts that space and time can only be represented as single, that

many spaces or times can only be viewed as parts of the single unique

space or time and not as composing it, and that space and time are

represented as infinite.

Moreover, geometry figures importantly in the B-editions transcenden-

tal exposition of the concept of space (and the A-editions third argument

concerning space corresponding to it), where Kant explains the possibil-

ity of geometry as synthetic a priori cognition (B401; A24). Kant alsoappeals to geometry to establish with certainty and indubitability the

claim that space is an a priori

form of intuition and hence a merely sub-

jective condition of all outer objects (A46 9/B636).

Finally, Kant draws a contrast between his claim that space and time

are ideal and the Leibnizian and Wolffian claim that space and time are

mere relations. The latter can neither explain the possibility of a priori

mathematics nor bring the propositions of experience into necessary accord

with those assertions (A401/B57). Since space and time are ideal, and

hence all objects of the senses are mere appearances, the synthetic a priori

propositions of geometry apply to (and only to) appearances (A389/B55

6).

5

The transcendental ideality of space and time bring the propositions

of experience into necessary accord with the a priori propositions of

mathematics. The Aesthetic apparently concerns mathematics in various

ways: it argues for claims about the nature of space and time, explains the

possibility of geometry as synthetic a priori cognition, and explains the

applicability of geometry to appearances.

Kant seems to attribute some of the same roles to the Axioms of Intu-

ition, however. He states, for example, that the principle of the Axiomsmakes pure mathematics applicable to objects of experience in its com-

plete precision (A165/B206). Kant also asserts that the application of

mathematics to experience rests on the principles of the Axioms and the


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138



Anticipations, and that the objective validity of mathematical principles

(which requires their applicability to objects of experience) is grounded in

these principles (A160/B199).

Moreover, Kant also seems to hold that the Axioms and the Anticipa-

tions ground the possibility of mathematical principles themselves. Hesays, for example, that mathematical principles all acquire their pos-

sibility from the principles of the Axioms and Anticipations (A162/B202).

Kant refers to both roles when he states that he will include among the

synthetic principles of pure understanding those on which the possibility

and the objective validity of the latter [i.e. mathematical principles] are

grounded . . . (A160/B199). Kant has just finished explaining that math-

ematical principles rest on the synthetic principles of pure understanding

for their application to experience, thus their objective validity, indeed

the possibility of such synthetic a priori

cognition . . . (A159 60/B1989).The second role for the Axioms is reinforced by Kants claim that they

provide the principle of the possibility of axioms in general

, about

which he says: For even the possibility of mathematics must be shown in

transcendental philosophy (A733/B761).

One might think that Kants references to the possibility of mathemat-

ical principles are to the possibility of their applicability. In that case, the

Axioms would only concern the applicability of mathematical principles

after all. However, I think that the context of the passages from A15960/

B1989 show that by

such

synthetic a priori cognition Kant means thesynthetic a priori cognitions that belong to mathematics itself, not cog-

nition of their applicability. Furthermore, the discussion at A733/B761

fairly unambiguously refers to the possibility of mathematical principles

themselves.

6

Finally, The Axioms include a discussion of arithmetical and geo-

metrical propositions and the status of axioms, which concerns math-

ematics itself rather than its applicability.

7

Thus, the Axioms as well as the

Aesthetic seem to concern both the possibility of and the application of

the synthetic a priori

propositions of mathematics. Kants assertions cangive the impression that the Axioms are a redundant reworking of what

was established in the Aesthetic, and this has been the opinion of some

philosophers.

8

Faced with this difficulty, one naturally looks for a difference between

the Aesthetic and the Axioms that accords each its own role and corre-

sponds to some very general division in mathematics. One strategy claims

that the Aesthetic concerns the synthetic a priori

status of pure mathe-

matics, while the Axioms establish the validity of applying mathematics to

experience.

9

This suggestion has the virtue of corresponding to a distinc-tion in modern treatments of mathematics that also appears in Kant. In

fact, the issue of the applicability of mathematics was forced on Kant by

Christian Wolff and his followers. They wished to avoid conflict between


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the conclusions of mathematics and metaphysics, and in particular, the con-

flict between the geometrical proof of the infinite divisibility of extension

and the monadology. They did so by claiming the ideality of space and

denying that mathematics applies to metaphysical reality.

10

In the last para-

graph of the Axioms, Kant is at pains to refute the claim that mathematicsdoes not apply to the objects of experience. As just noted, however, the

Aesthetic already asserts the applicability of mathematics to experience;

furthermore, the Axioms seem to concern the possibility of mathematics

itself, not just its application.

11

To shore up this strategy, one might assign different dates of composi-

tion to the two sections. One could then claim that the Aesthetic reflects

a less mature view concerning the applicability of mathematics, a view

that Kant supplements and corrects in the Axioms.

12

This solution is

dubious, however. If the Axioms reflected Kants mature position andwere inconsistent with the Aesthetic, we should presume that he would

have altered the latter, or at the very least given us some indication that

the Axioms supercede the Aesthetic on this issue. Kant does neither

through two editions. Thus, the strategy of assigning pure mathematics

to the Aesthetic and applied mathematics to the Axioms is not as attrac-

tive as it might at first seem. This is not to say that Kant doesnt distin-

guish between pure and applied mathematics, only that this does not

distinguish the treatment of mathematics in the Aesthetic from that in the

Axioms.

13

A more promising strategy for distinguishing between the two treat-

ments is that the Aesthetic establishes some general topological features

of space and time, and hence of the objects that appear in them, while the

Axioms establish that a metric can be applied to space and time, and

hence to the objects that appear in them.

14

The intended features of space

and time do not correspond to the modern understanding of topological

properties they are general features of space and time such as their

singularity and infinity and the fact that the parts of space and time can

only be viewed as parts of a single unique space or time, not as composingthem. Kant claims in the Aesthetic that space and time have these general

features, lending support to this line of thought. Furthermore, it is plaus-

ible that the introduction of a metric into space and time explains the

applicability of mathematics in all its precision. This interpretation has

the virtue of corresponding, at least in rough, to a modern conception of

a possible division within mathematics between more general mathemat-

ical claims on the one hand and more specific sorts of mathematical

claims on the other.

There is, however, another tension between the Aesthetic and theAxioms. A resolution of that tension will lead to a different understanding

of the point of the Axioms and a better account of the difference between

the Aesthetic and Axioms treatments of mathematics.


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140



2: Space and time in the Aesthetic and the Axioms

The Aesthetic claims that space and time can only be represented as

single and all-encompassing and in such a way that their parts cannot

precede them. The Aesthetic argues that for this reason the original rep-resentations of space and time are intuitions (A245/B39, A312/B47).

The Axioms assert that all intuitions are extensive magnitudes and that

an extensive magnitude is one for which the representation of the parts

makes possible the representation of the whole (and therefore necessarily

precedes the latter) (A162/B203). But space and time are intuitions.

Hence, we have an apparent contradiction between the two texts: space

and time can only be represented in such a way that the whole precedes

the parts, and yet the representation of the parts makes possible and

precedes the representation of the whole.

15

I will call this the part-wholepriority problem.

One might think that the Aesthetics and Axioms accounts of the

representation of space are indeed contradictory and that this contradic-

tion is a sign of tensions in Kants views or even a deep flaw in Kants

philosophy.

16

It would be more sympathetic, however, to hold that the

account Kant offers in the Aesthetic suppresses the role of the under-

standing in order to focus on sensibility. In this reading, Kant provides an

incomplete account in the Aesthetic, which he elaborates in the Axioms

once he is in a position to discuss the contributions of the understandingto cognition.

17

There is good evidence that Kant gives a provisional account of space

and time in the Aesthetic. Kant asserts that in the Aesthetic he will iso-

late sensibility by separating off everything that the understanding thinks

through its concepts (A22/B36). A footnote to 26 of the B-Deduction

indicates what Kant has in mind. Kant draws a distinction between the

form of intuition and the formal intuition of space or time. The latter

contains a unity in the representation of space or time, and that unity pre-

supposes a synthesis by means of which the understanding determinessensibility. Yet that unity precedes all concepts and belongs to space and

time, not to the understanding. Kant states that in the Aesthetic I ascribed

this unity merely to sensibility, only in order to note that it precedes all

concepts . . . (B1601n). He thereby explicitly acknowledges that a full

account of our representations of space and time departs from that given

in the Aesthetic. At the same time, he indicates that the unity of space

and time presupposes the activity of the understanding, while claiming

that this unity is prior to concepts. Thus, understanding provides a unity

to our representations of space and time but does so without the use ofconcepts, and this is what Kant thinks allows him to isolate sensibility in

the Aesthetic and to abstract from what the understanding thinks through

concepts.


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One might react to this elaboration of his account by declaring that

Kant is papering over what is in fact a deep tension; he ought not to

assert that he can isolate sensibility from the understanding, and the only

recourse is to abandon Kants Aesthetic account of space in favor of that

outlined in the Axioms.

18

I think, however, that this is to give up tooeasily. Kant clearly thinks that the priority of the whole over parts is

not only a fundamental property of intuition, but one that can support a

crucial argument in the Aesthetic. The account of our representations of

space in the Transcendental Analytic is no doubt an elaboration of Kants

views in the Aesthetic in light of the role of the understanding, but we

should resist a repudiation of the Aesthetics important claims if there is

an alternative.

We are presented with an alternative once we see that in Kants view, the

Aesthetic concerns space and time as given intuitions, while the Axiomsconcern determinate spaces and times that is, geometrical figures,

durations, and the spatial and temporal features of appearances. Con-

sideration of space and time as given intuitions treats them as representations

of our sensible faculty abstracted from the concepts of the understanding

as well as from all empirical content (A22). According to Kant, this

leaves us with a priori representations of space and time as single and

all-encompassing, representations in which their parts cannot precede the

whole. In contrast, concepts of the understanding are employed in the

cognition of determinate

spaces and times. In Kants view, the conditionsfor cognizing a determinate

space or time requires that the representation

of its parts precedes the representation of the whole. The conditions for

representing space and time as given intuitions are therefore quite differ-

ent from the conditions for cognizing determinate spaces and times.

This resolution of the part-whole priority problem rests upon Kants

account of the conditions of representing space and time as given intui-

tions and the conditions of cognizing determinate spaces and times, a dis-

tinction rooted in his theory of cognition. It is a resolution that has been

endorsed by various commentators.

19

Nevertheless, I think its implica-tions for the role of mathematics in the Aesthetic and the Axioms have

not been fully appreciated, for this resolution also allows us to distinguish

the role of mathematics in each without appeal to a distinction within

mathematics between either pure and applied mathematics or topological

and metrical mathematical properties.

20

I will argue that in Kants view,

mathematics consists of mathematical cognitions, which are grounded in

our cognition of determinate

spaces and times, a determination that is

carried out by the understanding. Thus, the Axioms explanation of the

possibility of the synthetic a priori cognitions of mathematics takes intoaccount the particular contributions of the understanding. Moreover, the

Axioms concern the possibility of all mathematical cognition, whether pure

or applied, topological or metrical. In contrast, the Aesthetic explanation


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of the possibility of synthetic a priori cognitions is more general in the

sense that it abstracts from the particular contributions of the under-

standing. Furthermore, the Aesthetic concerns the possibility of synthetic

a priori

cognitions more generally, not just the synthetic a priori cognitions

of mathematics, although the latter serve as paradigm examples. Theremainder of the paper will flesh out these claims.

3: Determination, construction, and mathematics

Kants notion of a determinate space or time rests upon a broad theory

of determination that has its roots in the philosophy of Christian Wolff

and Alexander Baumgarten, which is in turn based on Leibniz.

21

In that

theory, something is determined with respect to a property if it either hasor does not have that property, and the property is called a determina-

tion. A predicate is a concept that belongs to the concept of a thing when

the determination designated by the predicate belongs to the thing.

22

For

example, the predicate-concepts blue and rectangular belong to the

concept of the book on my table when the determinations blue and rectan-

gular belong to the book on my table.

Wolff and Baumgarten incorporate Leibnizs doctrine of the complete

concept of a thing in their principle of thoroughgoing determination:

for a thing to be an individual is to be determined with respect to everypossible pair of contradictory predicates.

23

In Baumgarten, Wolff, and

Leibniz, the thoroughgoing determination of an individual with respect

to concepts exhausts the possible determinations of a thing.

Kant says of both a concept and a thing that they are determined with

respect to contradictorily opposed predicate-concept pairs. According to

Kants principle of the determinability of concepts, at most one of two

contradictorily opposed concepts can apply to a concept, which follows

from the principle of contradiction. According to Kants version of the

principle of thoroughgoing determination, at least

one of every pair ofcontradictorily opposed concepts must apply to a thing. Kant calls this

the principle of the synthesis of all predicates which are to make up the

complete concept of a thing . . . (A572/B600).

Kant has both an ontological and an epistemological sense of determi-

nation. In the ontological sense, the concept of a thing is determined with

respect to a predicate-pair when the thing has one of the properties expressed

by the predicate-pair. In the epistemological sense, the concept of a thing

is determined in my cognition of the thing when I attribute a predicate

concept to it in a judgment; for example, when I judge that the book onmy table is rectangular.

24

Although Kant adopts the general framework of the theory of determi-

nation, he fundamentally alters it. In his view, the Leibnizian-Wolffian


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tradition treats intuitive representations of sensibility as merely confused

conceptual representations of the understanding. It therefore overlooks

the fundamental difference between concepts and intuitions and the

unique contribution intuitions make to cognition (A2701/B3267).

Kant nevertheless retains the view that determination is a matter ofapplying concepts. In cognizing an object, we determine the concept of

the object through an act of the understanding. Sensibility plays the pas-

sive role of being determinable.

25

Kant builds on this fundamental departure from the Wolffian theory of

determination to give a radically different account of the determination

of spaces and times. Kants discussions in the B-Deduction, the Axioms

of Intuition, and the Discipline of Pure Reason show that Kant thinks of

determinate spaces as lines and surfaces that we are capable of drawing in

thought (B1378, A1623/B203, and A715/B743). A line, circle, and coneconstructed in Euclidean geometry are paradigm examples. Determinate

spaces also constitute the spatial properties of objects. In 26 of the B-

Deduction, Kant states that we, as it were, draw the shape of a house when

we apprehend it [zeichne gleichsam seine Gestalt] (B162), suggesting that

a determinate space constitutes part of our representation of the house.

The Axioms confirm this; they argue that the apprehension of an appear-

ance is possible only by means of a synthesis through which the represen-

tations of a determinate space or time are generated (B202). Determinate

spaces therefore comprise any spatial figure that can be drawn inthought as well as Euclidean figures constructible in accordance with a

straight-edge and compass.

Following Baumgartens metaphysics, Kant divides determinations into

inner and outer determinations; that is, determinations that belong to

the object alone, and relations between the object and other objects. The

determinate space of a house is an inner determination. A full determination

of the spatial properties of an object will include its outer determinations

as well and hence its spatial relations to other objects. The inner determina-

tions of an object, however, are primary and include a determinate space.Kant says less in this context about determinate times, but an analo-

gous account would construe a determinate time as the relative temporal

ordering and duration of states of an object, to the extent that they can

be determined independently of other objects. In contrast, the outer tem-

poral determinations would be the temporal duration and ordering of an

object and its states relative to other objects. The conditions for the deter-

mination of these relations are further specified in the Analogies.

Determination involves a combination or synthesis, and the determina-

tion that generates a determinate space or time is quite special. The act ofunderstanding involved is called figurative synthesis and is carried out on

the manifold of space and time in the productive imagination. It is only

through this synthesis that a determinate intuition is possible:


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form of intuition (space and time) can be cognized and determined com-

pletely a priori. He then adds that with regard to space and time

. . . we can determine our concepts a prioriin intuition, for we create the objects themselves

in space and time through homogeneous synthesis, considering them merely as quanta. . .[this] is the use of reason through the construction of concepts . . . (A723/B751).

The role of determination in construction is brought out especially well in

38 of Prolegomena, where Kant discusses the mathematical basis of the

laws of nature:

Space is something so uniform [gleichfrmiges] and as to all particular properties so

indeterminate that we should certainly not seek a store of laws of nature in it. On the other

hand, that which determines space to assume the form of a circle, or the figures of a coneand a sphere, is the understanding, so far as it contains the ground of the unity of their

constructions.

The mere universal form of intuition, called space, must therefore be the substratum of

all intuitions determinable to particular objects; and in it, of course, the condition of

the possibility and of the variety of these intuitions lies (Ak.4:321).

Space as a given intuition is indeterminate; it is determined through a

construction of a concept such as the concept of a circle, cone, or

sphere.28

The fact that Kant rests construction on mathematical determination

has important consequences for his conception of mathematics. Since

Euclidean geometry requires construction of concepts, geometry depends

upon the determination of spaces. Arithmetic also depends upon a con-

struction of number-magnitudes in pure intuition, as does algebra.29

In fact, Kant delimits the field of mathematics by appeal to construc-

tion. In the Discipline of Pure Reason, Kant makes clear that mathematics

consists of mathematical cognitions. He states that mathematical and

philosophical cognitions are not distinguished by their objects, but by themethod according to which they are established, and that the method of

mathematical cognition is the construction of concepts, while the method

of philosophical cognition is rational cognition through concepts alone.30

Kant explicitly warns against delimiting mathematics and philosophy by

their objects. He gives examples of objects that are considered by both:

magnitudes, in particular their totality and infinity, lines, spaces, and the

continuity of extension (i.e., its infinite divisibility) (A7145/B7423).

Even a motion as a description [Beschreibung] of space, i.e., a figurative

synthesis in productive imagination exemplified by the drawing of a line

belongs not only to geometry but to philosophy (B155n). Mathematics

thus consists of those cognitions established through the construction of

concepts, and onlythose cognitions. Hence, mathematics onlyconsists of


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146 PACIFIC PHILOSOPHICAL QUARTERLY


cognitions which ultimately rest on a determination of intuition through

a figurative synthesis.

This may come as a surprise, for as we saw in Section 1, Kant calls

geometry the science that determines the properties of space synthetically

and a priori. This suggests that any synthetic a priori truth about thenature of space (at least those that are not clearly epistemological in

character, for example that space is ideal) should belong to mathematics.

Thus, one would think that the claims of the Aesthetic that space is single

and infinite belong to mathematics in virtue of the fact that they are gen-

eral truths about space. This line of thought, however, delimits geometry

by its object rather than the method through which its cognitions are

established, and hence violates Kants emphatic demarcation of mathe-

matics and philosophy. If, for example, the singularity and infinitude of

space can be established through construction, then these claims willbelong to geometry; otherwise not. The same condition holds for the

purported claims of any other field of mathematics.

This is not to say that we cannot consider whether either mathematics

as a whole or sub-fields of mathematics concern a particular sort of object.

The point is that one cannot start with the objects to demarcate mathe-

matics. Once we have delimited which cognitions count as mathematical,

we are free to consider whether there are objects particular to mathematics

(or whether geometry can be distinguished from arithmetic, for example).

Kant does identify a kind of object particular to mathematics, as we willsee below.

4: Determination, magnitudes, and intuition in the Axioms

Kants broadest division of the Critique is into the Doctrine of Elements

and the Doctrine of Method. Since Kant demarcates mathematics accord-

ing to its method, his account of construction appears in the latter section.

The former section includes the Transcendental Aesthetic the Transcen-dental Analytic, and the Transcendental Dialectic; the first two contain

most of the positive presentation of his views. As we have seen, in Kants

account the determination of our cognitions requires the application of

concepts of the understanding. Since the Aesthetic concerns sensibil-

ity in abstraction from what understanding thinks through its concepts, a

treatment of the determination of our cognitions does not belong in the

Aesthetic.31That treatment can only be given within the Analytic, whose

purpose is to explain the contributions of the understanding together

with the contributions of sensibility. Thus, the Analytic would be theproper home of a positive account of mathematical cognition.

The Analytic contains a systematic presentation of all the synthetic

principles that flow a priori from pure concepts of the understanding


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under the sensible conditions of the forms of space and time (A136/

B175). Kant holds that these principles are nothing other than rules for

the objective use of the categories. Given Kants classification of the cate-

gories, one would presume that a general treatment of mathematics would

fall under the categories of quantity. The Axioms of Intuition correspondto the categories of quantity. Furthermore, as noted earlier, the Axioms

and the Anticipations of Perception are called the mathematical prin-

ciples, and they concern extensive and intensive magnitudes, respectively.

Since the principle of the Axioms asserts that all intuitions are extensive

magnitudes, one would expect to find a treatment of mathematical deter-

mination, and hence of mathematical cognition, that depends on the con-

cept of magnitude.32 In fact, the concept of magnitude does play this

central role.

Kant distinguishes between two kinds of magnitude in the Axioms:quantaand quantitas. These notions and the distinction between them are

at the heart of Kants theory of magnitude; for our present purposes,

however, we can pass over the differences between them.33What is import-

ant at present is that Kant states the following:

Now of all intuition none is given a prioriexcept the mere form of appearances, space and

time, and a concept of these, asquanta, can be exhibited a priori in pure intuition, i.e.

constructed . . . (A720/B748).

Thus, in Kants view, concepts of space and time can be exhibited in a priori

intuition, and hence constructed, if they are constructed as quanta, that is,

under a concept of magnitude. In his discussion of algebra, it is quantitas

that is constructed; that is, algebra is also constructed under a concept of

magnitude (A717/B745). In a passage already cited above concerning the

difference between mathematical and philosophical cognition, Kant asserts

the centrality of the concept of magnitude in mathematical construction:

. . . the form of intuition (space and time) . . . can be cognized and determined completelya priori. . . With regard to the [form of intuition] we can determine our concepts a priori

in intuition, for we create the objects themselves in space and time through homogeneous

synthesis, considering them merely as quanta. . . . [this] is the use of reason through con-

struction of concepts . . . (A723/B751).

Kant holds that we can construct the concept of magnitude itself. He

goes much further, however. When he discusses the demarcation between

mathematics and philosophy, Kant states that mathematical cognition

only concerns magnitudes:

The form of mathematical cognition is the cause of its [i.e. mathematics] having solely

to do with quanta. For only the concept of magnitudes [Gren] can be constructed, i.e.

exhibited a priori in intuition (A7145/B7423).


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Kant states that mathematics concerns only magnitudes, because only

concepts of magnitudes can be constructed.34In other passages we have

examined, Kant refers to constructing concepts of Euclidean figures,

such as the concept of a circle or cone. We see here that whatever is

constructed must fall under the concept of magnitude; in other words, theconcept constructed must itself be the concept of magnitude or fall under

the concept of magnitude.35

As we saw in the previous section, Kant insists that we demarcate

mathematics not by its objects but by the method of establishing mathe-

matical cognitions. We can nevertheless identify the restricted range of

objects that result from the construction of concepts. All mathematical

cognition, and as a consequence all of mathematics, requires a determina-

tion through construction of a concept that falls under the concept of

magnitude.Kant says almost nothing about construction in the Axioms, but he

does discuss the role of the concept of magnitude and the conditions for

cognizing determinate spaces and times. Kant defines a magnitude as a

homogeneous manifold in intuition in general. In the course of his argu-

ment, Kant contends that determinate spaces or times are generated

through the synthesis of a homogeneous manifold in intuition, that is,

a synthesis through which space and time in general are determined

(B2023).36A determinate space or time are generated through the syn-

thesis of the homogeneous manifold of space or time, which falls under theconcept of magnitude, and the product of that synthesis is a determinate

magnitude.37Furthermore, the synthesis requires that the homogeneous

manifold be represented in intuition. Kant has articulated an important

condition of any mathematical cognition whatsoever.

Kants larger argument turns on the claim that the apprehension of an

appearance rests on the very same synthesis that generates a determinate

space or time for the appearance (B202, B203). It is this fact that guarantees

the applicability of mathematics to experience (A166/B207, A224/B271).

These two results turn on the contribution of the understanding to deter-mining our cognition. Together they give content to Kants claim that the

Axioms of Intuition establish the applicability of mathematics while also

demonstrating the possibility of mathematical cognition itself (A15960/

B1989, A733/B761).38

5: Synthetic a prioricognition and mathematics

in the Aesthetic

We are now in a position to distinguish the Aesthetic treatment of math-

ematics from that in the Axioms. The Aesthetic is not concerned with

the contributions of the understanding or explaining the possibility of


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mathematical determination. Instead, the Aesthetic focuses on the contri-

bution of sensibility in making synthetic a priori cognition possible. The

question is how it is possible to have a priori cognition that goes beyond

what is analytically contained in concepts. The answer is that space and

time are a priori intuitions: that space and time are intuitions provides anextra-conceptual connection between concepts; the fact that they are a priori

makes it possible for a priori cognitions to arise from them (B73). It is this

sense of possibility that is at issue in the Aesthetic, and the explanation

extends beyond mathematical cognitions to all synthetic a priori cognitions,

although mathematics contains paradigm examples of such cognitions:

Time and space are accordingly two sources of cognition, from which different synthetic

cognitions can be drawn a priori, of which especially pure mathematics in regard to the

cognitions of space and its relations provides a splendid example (A389/B556; see also

A467/B64).

The possibility of mathematical determination, and hence the apprehen-

sion of appearances, presupposes that synthetic a priori cognitions are

possible in the sense explained in the Aesthetic, for without pure intu-

itions mathematical determination would not be possible. This sense of

possibility explained in the Aesthetic nevertheless omits the more par-

ticular conditions of mathematical cognition.

Similarly, the Aesthetic explanation of the applicability of mathematicsonly considers the contribution of sensibility. The ideality of space and

time explains the possibility of a prioriknowledge of their properties, but

it thereby also explains why mathematics applies (and only applies) to

those objects (A26/B42, B73, see also A149/B188).

I have appealed to mathematical determination to distinguish between

the treatment of mathematics in the Aesthetic and in the Axioms. One

might think that the exclusion of mathematical determination from the

Aesthetic is inconsistent with the transcendental exposition of the con-

cept of space, whose argument presupposes that we have determinategeometrical cognitions and seeks to explain its possibility (B40-1). In my

account, however, the Aesthetic is only precluded from explaining or

appealing to the cognitive conditions for the mathematical determination

of geometrical figures. The argument does not turn on how geometry

arrives at its synthetic a priori cognitions, only on the fact that geometry

consists of them. For evidence of this fact, Kant refers the reader to the

Introduction; the synthetic character of geometrical cognitions follows

from a consideration of the content of the concepts involved in a few rep-

resentative mathematical propositions, while their apriority is establishedby appeal to their necessity and strict universality (B4, B1417). At no

point does Kant appeal to the nature of mathematical determination or

explain how the synthetic a priori claims arise.


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A similar worry could be raised concerning the general topological

properties of space and time Kant argues for in the metaphysical exposi-

tions; one might think that Kant at least tacitly appeals to the deter-

minate cognitions of geometry when he argues that space is single and

infinite. Recall, however, that mathematics and philosophy are demar-cated by their method, not by their objects, and that philosophy and

mathematics concern some of the same objects, including space. Thus, the

same claim concerning space might be demonstrated both philosophically

and mathematically. Euclids third construction postulate, which asserts

that a finite straight line can always be continued in a straight line, entails

that space continues in every direction ad infinitum, a conclusion which

therefore belongs to geometry. That does not, however, preclude it from

also being established in philosophy as well. More importantly, even if

Kant tacitly appeals to mathematics to establish this claim, the Aestheticwould be relying on a result of mathematics, not on an account of the

mathematical determination required to establish it. Only the latter would

overstep the proper scope of the Aesthetic.39

It is in precisely this light that I think we should understand an altera-

tion in the metaphysical deduction between the A and B-editions. In the

A-edition, Kant makes a claim about the determination of magnitudes to

argue that space is an intuition:

5) Space is represented as a given infinite magnitude. A general concept of space (which is

common to a foot as well as an ell) can determine nothing in respect to magnitude. If there

were not boundlessness in the progress of intuition, no concept of relations could bring

with it a principle of their infinity (A25).

In the B-edition Kant replaces this with an argument that only appeals to

the nature of conceptual representation, and makes no mention of deter-

mination. Kant, I think, removed the first argument because it depended

on a claim about the determination of magnitude, a claim that he is not

in a position to either explain or defend in the Aesthetic if he is going toabstract from the contributions of the understanding.

6: Kants formulation of the Axioms principle

I would like to anticipate two further objections to my interpretation of

the distinction between the Aesthetics and Axioms treatment of mathe-

matics. It was inspired by the resolution of the part-whole problem. The

Axioms argue for the principle that all intuitions are extensive magni-tudes, and they define an extensive magnitude as one in which the repre-

sentation of the parts precedes and is presupposed by the representation

of the whole. Since space and time are intuitions, this principle applies to


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them. The Aesthetic, however, claims the contrary: space and time are

represented as single and all-encompassing and are represented in such

a way that their parts cannot precede them. We avoided this impasse by

restricting the Axioms to a claim about determinate spaces and times,

while the Aesthetic concerns space and time as given intuitions. Hence,the principle should be: all determinate intuitions are extensive magni-

tudes. Yet this is not how Kant formulates it; why should we not take

him at his word?

There is a two-part response to this worry. First, the context of the

Axioms implicitly restricts Kants claim in the Axioms to determinate

intuitions. As noted earlier, the System of Principles includes conditions

for the cognition of appearances in accordance with the categories (A149/

B188). Since such cognition is a determination, the principles are condi-

tions that govern our determination of appearances. The context of theAxioms principle is sufficient to restrict it to determinate intuitions.

Second, our analysis of the argument in 4 above confirms the restriction, for

the argument that the representation of the parts precedes the representa-

tion of the whole appeals to the successive synthesis of space and time

required for the representation of a line, for example, or a determinate

time. Thus, the argument can at best establish that determinatespaces and

times and appearances are extensive magnitudes; it cannot establish that

space and time as given intuitionsare extensive magnitudes.

Since the result of determining space or time as a given intuition isan extensive magnitude, it is tempting to think that space and time are

themselves extensive magnitudes, especially if all magnitudes are either

extensive or intensive. That is not Kants view, however, for only deter-

minate magnitudes are either extensive or intensive, as the arguments of

the Axioms and the Anticipations show. Space and time as given intuitions

are represented as magnitudes in virtue of containing a homogeneous

manifold in intuition. Nevertheless, they are not determined, and hence

are neither extensive nor intensive.

A final worry concerns the point of the Axioms. The System of Prin-ciples clearly focuses on the conditions for the cognition of objects of ex-

perience. Kant states, for example, in introducing the System of Principles,

that [e]xperience . . . has principles of its form which ground it a priori,

namely, general rules of unity in the synthesis of appearances . . . (A156

7/B195-6). Kant adds that the supreme principle of all synthetic a priori

judgments is that every object stands under the necessary conditions of

the synthetic unity of the manifold of intuition in a possible experience. The

focus on principles of experience would seem to restrict the role of the

Axioms to explaining the applicability of mathematics to the objects ofexperience. The possibility of mathematics itself is simply not on the table.

As I have pointed out, however, Kant also describes the Analytic as

containing a systematic presentation of all the synthetic principles of pure


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understanding, that is, those synthetic principles that flow a priori from

pure concepts of the understanding under the sensible conditions of the

forms of space and time (A136/B175). This more general description of

the Analytic clearly encompasses principles of mathematical determina-

tion. Furthermore, as we saw in 4 above, the Axioms argument rests ona sub-argument that all determinate intuitions are extensive magnitudes.

Kant has argued for and is entitled to this claim, which goes beyond

establishing a principle of experience and has important consequences

for the nature of mathematical cognition. If Kant is going to provide an

account of mathematical cognition (other than that part which directly

concerns mathematical method), the Axioms are the natural place for it.

There is, however, a tension between the narrower and broader aims of

the Analytic that surfaces in the Axioms, which led Kant to alter his for-

mulation of the principle from the A- to the B-edition. In keeping withthe narrower aim of establishing a principle of possible experience, Kant

focuses on the taking up of appearances into empirical consciousness

and on the apprehension of appearances that makes experience possible.

The A-edition version, All appearances, according to their intuition,

are extensive magnitudes, does nothing to dispel the impression that the

principle is only about appearances whose apprehension constitutes experi-

ence. The B-edition version, All intuitions are extensive magnitudes,

more clearly states the broader conclusion, which includes all determinate

intuitions and not merely those which underlie appearances.40Hence, thepoint of the Axioms is twofold: they establish both a principle of experi-

ence and a principle concerning the nature of any mathematical cognition

whatsoever.

Conclusion

Kants critical philosophy, and above all his account of synthetic a priori

knowledge, is strongly influenced by his understanding of mathematicsand especially geometry. Despite its importance, however, Kant distri-

butes his account of mathematics across the Critique: the Introduction

provides reasons to think that mathematics consists of synthetic a priori

cognitions, the Aesthetic and Axioms articulate conditions of mathemat-

ical cognition, and the Discipline of Pure Reason presents its method.

Kants presentation and the larger aims of the Critiquecan easily lead to

misunderstandings of the point of the Axioms that obscure his views on

the nature of mathematical cognition. If we think that the Axioms are

restricted to arguing for either the applicability of mathematics to objectsof experience or the possibility of introducing a metric into space and

time, we may miss the fact that the principle of Axioms concerns any

mathematical cognition whatsoever. Correcting these misunderstandings


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clarifies the treatment of mathematics and space and time in the Aesthetic

and the Axioms. It also allows us to see that magnitudes are at the heart

of Kants philosophy of mathematics, that intuition plays an important

role in allowing us to represent magnitudes, and that Kants theory of

magnitudes warrants further investigation.41

Department of Philosophy

University of Illinois at Chicago

NOTES

1 I will use Axioms to refer to the section by that name, which argues for just one

synthetic principle of the pure understanding; for grammatical reasons I will henceforth

treat it as a plural.2 See Peter Strawson, (1989, p. 3) and Robert Paul Wolff (1963, pp. 228231, p. 238) for

explicit articulations of this view. More recently, Sebastian Gardner has stated that . . . the

section which is crucial for the Critiques legitimation of the metaphysics of experience is

the Analogies of Experience (supplemented by the Refutation of Idealism) . . . the Axioms

and Anticipations make a relatively slight (and uncontentious) contribution to Kants

objectivity argument . . . (1999, p. 166). Gardner accordingly gives the Axioms relatively

little attention.3 While neglect of the Axioms of Intuition is the norm, it is not universal. Michael

Friedman devotes attention to Kants notions of quantaand quantitas, which are discussed

in the Axioms and elsewhere (1992, Chapter 2). My own views have benefited greatly fromhis work. My approach differs from his, however, in focusing above all on the Axioms, and

in this paper, in explaining the point of the Axioms in order to highlight the importance of

Kants theory of magnitudes for Kants views on any mathematical cognition whatsoever.

It seems to me that there are important aspects of Kants theory of magnitude that only

emerge in the approach I suggest, although a full account of them is beyond the scope of

this paper. Beatrice Longuenesses careful and penetrating treatment of the Axioms also

does not fit the norm (Longuenesse, 1996). My interpretation of the role of the Aesthetic

and the Axioms is, I think, in broad agreement with hers, although we differ on the nature of

the determination underlying determinate spaces and times. I will return to this point below.4 References to the Critique of Pure Reason will be in the standard way to the original

pagination of the A and B editions, while all other references to Kants work will be to

volume and page number of Kants Gesammelte Schriften(Kant, 1902).In the main, I have

followed Paul Guyer and Allen Woods translation of the Critique of Pure Reason (Kant,

1998) and Gary Hatfields translation of the Prolegomena (Kant, 2002), with occasional

departures.5 Kants point at A389/B556 is the restrictionof the synthetic a priori cognition to

appearances, but he is thereby also clearly inferring that we havesynthetic a priori cogni-

tion of appearances, of which mathematical cognition is a shining example.6 There is a more sophisticated version of this worry: one might concede that Kant is

discussing the possibility of mathematical principles themselves but maintain that he

means their real possibility, and that their real possibility is tantamount to their applicabil-

ity. I discuss this second concern in more detail below; see footnote 38.7 While both the Axioms and the Anticipations are called mathematical principles, the

Axioms are often singled out for their relation to mathematics. I will address the reasons

for this below.


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8 This view is most notably endorsed by Vaihinger. He thinks that Kant establishes the

a priority of pure mathematics as such, only because he confuses two notions of space

articulated in the Aesthetic. Vaihinger claims that Kants account of space as . . . an

unconscious, potential transcendental form of intuition, which is there before all experi-

ence is sufficient for the validity of the application of the propositions of mathematics to

all empirical objects . . . (1970, Vol. 2, p. 88). Vaihingers interpretation therefore leaves no

need for the Axioms of Intuition in establishing the applicability of mathematics to experi-

ence. Bennett states that the Axioms should not have been located in the Analytic at all,

since they are a direct consequence of the Aesthetic (1966, p. 170); see also Wolff, who

states According to the division of the Critique which Kant establishes in the Intro-

duction, [the Axioms] should not really exist at all, for mathematics is supposed to be

dealt with in the Aesthetic (1963, p. 228).9 For examples of this view, see Kitcher (1982, 5), and Walsh, (1975, p. 11011).

10 The Wolffian position was openly attacked at the Berlin Academy in 17451747.

Friedman gives an account of these issues in the Introduction to Kant and the Exact Sciences

(Friedman, 1992).11 Some have thought that the Axioms argue for substantive claims about mathematics

apart from its application to the objects of experience but are nevertheless committed to

the view that the Axioms should only concern the application of mathematics. Kemp

Smith, for example, thinks that the Axioms make substantive claims about mathematics,

but holds that these claims are only very externally connected to the main argument. He

sees little relation between these claims and the rest of the Axioms; see Kemp Smith (1979,

p. 348). Walsh states that he finds Kants attempt to connect his general principle [of the

Axioms] with the part played by mathematics baffling; see Walsh (1975, p. 116). At least

part of his bafflement can be traced to his consistent description of the Axioms as concern-

ing merely the applicability of mathematics to the objects of experience. Others have recog-nized that it is part of the purpose of the Axioms to argue for substantive claims about

mathematics apart from its application. See Brittan (1978, pp. 97ff), Guyer (1987, pp. 190ff),

and Longuenesse (1998, p. 274).

In their translation of the Critique, Guyer and Wood cite R5585 (18, pp. 2412), one of

Kants reflections dated 17791781, which suggests to them that the Axioms and the

Anticipations are called mathematical principles because they are the conditions of the

possibility of applied mathematics (Kant, 1998, p. 284 fn. 58). The passage reads:

1. Possibility of (pure) mathematics.

2. Application [struck out]. Possibility of applied. For all things (as appearances) have

a magnitude: extensive and intensive. (Through it mathematics receives objective

reality. It does not pertain to entia rationes.)

The first item listed supports the idea that the Axioms concern both pure and applied

mathematics.12 Walsh, for example, endorses this view. He states that as Kant writes the Analytic, he

. . . now realises that the notion of a space-time world, i.e. of a set of objects in space and

time, is far more difficult than was at first suggested. . . . It was only in the Analytic that

Kant made any attempt to work out this story in detail, and it accordingly follows that the

discussions of the latter must take precedence over those of the Aesthetic (1975, pp. 112113).13 The modern distinction between pure and applied mathematics can also lead to

anachronistic misunderstandings of Kant. In contemporary accounts of pure mathematics,

even geometry is given a formalization that abstracts from a spatial interpretation. In

Kant, on the other hand, all pure mathematics is intimately related to pure space and time.

The contemporary distinction between pure and applied mathematics may suggest that in


21/25



Kants view non-Euclidean geometries are logically possible, while only Euclidean geometry

is really possible a real possibility determined by the form of our intuition of space. Kant,

however, has no place for non-Euclidean geometries as logical possibilities, since for him

any geometry will necessarily require going beyond logic to intuition, and the only form

of intuition available is Euclidean. For more on the anachronistic use of the distinction

between pure and applied mathematics, see Friedman (1992, pp. 556, p. 66, pp. 98104).

See also Longuenesse (1998, pp. 2745).14 See, for example, Gordon Brittan (1978, pp. 9294) and Paul Guyer (1987, p. 191).

Guyer holds that Kant establishes that certain topological features of shape hold for all

objects of empirical intuition, while the Axioms establish that a metric can be applied to

those objects. It is important to note, however, that Guyer holds that this is a description

of what Kant actually accomplishes rather than what he thought he was establishing. See

Guyer (1987, p. 191, especially fn. 6).

Brittan holds that the principle of the Axioms serves to rule out merely possible worlds

that are not really possible worlds worlds in which there are objects that do not stand

in relations that allow the introduction of a metric (1978, pp. 9294). Note that the claim

that the purpose of the Axioms is to establish the possibility of introducing a metric is not

incompatible with the view that the Axioms merely concern the applicability of mathe-

matics to experience, if the purpose of the Axioms is restricted to establishing the ability to

introduce a metric in the field of experience. Walsh, for example, appears to hold this view

(1975, p. 111). Brittan, in contrast, holds that the metrical properties are established for

space and time, which in turn explains the applicability of mathematical properties to the

objects presented in them.15 More carefully, Kant states in the Aesthetic that the parts of space cannot precede

the single all-encompassing space as its components [Bestandteile] (from which its com-

position would be possible) (A245/B39). In the case of time, Kant does not mentioncomposition (A312/B47). Nevertheless, one might attempt to dodge the contradiction by

claiming that the Aesthetic only asserts that the parts of space (and time) cannot precede

the whole in such a way that allows the parts tocompose the whole. That will not solve the

difficulty, however, since the Axioms claim that the whole presupposes the parts explicitly

concerns parts that compose the whole (B201n). The solution to the contradiction lies

elsewhere.16 For examples of this view, see Vaihinger (1970, Vol. 2, pp. 2248), Kemp-Smith (1979,

pp. 949), and Wolff (1963, pp. 2289).17 Cassirer holds that many of the statements of the Aesthetic are incompatible with

claims Kant makes later in the Critique because they reflect precritical views articulatedmore then a decade before in the Inaugural Dissertation. He holds that the arguments of

the Aesthetic must therefore be taken provisionally (1954, pp. 234). See also Walsh (1975,

pp. 1125), Kitcher (1982, p. 237ff), and, in particular, Longuenesse (1998, pp. 2123, and

p. 274 fn. 70).18 As does Wolff (1963, pp. 22930).19 Melnick describes the position particularly clearly (1973, pp. 189), as does Brittan

(1978, p. 97). See also Walsh (1975, p. 114), Allison (1983, pp. 946), Guyer (1987, p. 191),

and Longuenesse (1998, p. 274).20 Melnick, for example, distinguishes between space and time as given intuitions and

determinate spaces and times to differentiate the Aesthetics and Axioms treatments of

them. Nevertheless, he does not explain why Kant discusses geometry in the Aesthetic or

why he claims to have established the applicability of mathematics to experience there. A

similar lack of appeal to determinate spaces and times holds for Walsh (1975, pp. 1145),

Brittan (1978, pp. 959), Allison (1983, pp. 9498), and Guyer (1987, p. 191, especially fn. 6).


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Longuenesse, in contrast, highlights the importance of determinate spaces and times and

their determination in Kants account of mathematical cognition in her penetrating account

of the Axioms. Longuenesse goes further in calling for a radical rereading of the Aesthetic

(1998, pp. 21228).21 Baumgarten was a follower of Christian Wolff; Kant used Baumgartens Metaphysica

for years in his lectures on metaphysics; Baumgartens text is presented with Kants mar-

ginalia in volume 17 of Kants Gesammelte Schriften.22 See Baumgartens Metaphysica 3458 (17, pp. 3338) for the notion of determina-

tion, and 133 (17, pp. 2333) for the view of predicates as concepts. The differences

between the subject-predicate distinction, subject-conceptpredicate-concept distinction,

and thing-determination distinction are not always clear.23 Baumgarten, Metaphysica148154 (17, pp. 5658).24 See Wood, Kants Rational Theology, p. 37ff for a helpful discussion of these matters.25 Kant explains the categories by stating: They are concepts of an object in general, by

means of which its intuition is regarded as determined with regard to one of the logical

functions for judgments (B128). He also says of understanding that it is generally speak-

ing, the faculty of cognitions. These consist in the determinate relation of given representa-

tions to an object (B137). This is in contrast to the contribution of sensibility, which

comes out in Kants description of the imagination:

Now since all of our intuition is sensible, the imagination, on account of the subjective condition under

which alone it can give a corresponding intuition to the concepts of understanding, belongs to sensibility;

but insofar as its synthesis is still an exercise of spontaneity, which is determining and not, like sense,

merely determinable, and can thus determine the form of sense a prioriin accordance with the unity of

apperception, the imagination is to this extent a faculty for determining the sensibility a priori. . .

(B151-2).

Kant not infrequently refers to sensibility and the forms of intuition as the determinable.

When Kant speaks of the contribution of the forms of space and time to our cognition of

objects, he almost always refers to the conditionsthese forms impose rather than to their

determination.26 See Michael Friedman for a detailed assessment of the role of figurative synthesis that

emphasizes its importance not only for mathematics but for physics (see Friedman 1992,

Chapters 1 and 2). See also Beatrice Longuenesse for a thorough but quite different ana-

lysis of figurative synthesis, which Kant also calls synthesis speciosa (Longuenesse 1998,

Part III).27 Note that here Kant is mentioning two contributions: the conditions of sensibility and

the determination of sensibility by the understanding. See footnote 25 above.28 See also A714/B742. Kants general definition of construction as exhibiting an a

prioriintuition corresponding to a concept may seem to leave open the possibility of con-

structions that are not effected through a mathematical determination. An example that

appears to exploit this possibility appears in The Metaphysics of Morals:

The law of the reciprocal coercion necessarily harmonizing with everyones freedom under the principle

of universal freedom is, as it were, a construction, that is, an a priori exhibition of that concept in pure

intuition a priori, by analogy with [exhibiting] the possibility of bodies moving freely under the law of

the equality of action and reaction (6 p. 233).

Kant only states, however, that there is an as it were construction of the concept that is

analogous to presenting the possibility of bodies in accordance with Newtons third law.

Kant adds that the doctrine of right wantsa mathematical exactitude in its determination

but that this exactitude cannot be expected (6, p. 232, my italics). Kants tentativeness is

clarified in the only other mention of construction in The Metaphysics of Morals: . . . any


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moral proof, as philosophical, can be drawn only by means of rational cognition from

conceptsand not, as in mathematics, by the construction of concepts (6, p. 403). Every other

mention of construction in Kant is mathematical, and passages in which Kant describes

construction in any detail refer to determinate spaces or times or some determinate object.29 For arithmetic, see A1789/B221, A240/B299, A717/B745, A720/B748, and 4, p. 283;

for algebra, which relies on a symbolic construction, see A717/B745, A734/B762. I will not

attempt to fully explain the nature of construction in this paper; my purpose here is to

explicate the concept sufficiently to draw conclusions about Kants demarcation of math-

ematics. For an analysis of symbolic construction in particular, see Shabel (1998); for a

discussion of Kants arithmetic and algebra, see my Kant on Arithmetic, Algebra, and the

Theory of Proportions, Journal of the History of Philosophy, forthcoming.30 See Emily Carsons Kant on Method in Mathematics for a helpful discussion

(Carson, 1999).31 This is not to say that the claims of the Aesthetic are established throughabstraction,

only that its object of study is intuition apart from the activity of the understanding.32 Although Kant calls both the Axioms and Anticipations mathematical principles, and

the Anticipations principle also concerns magnitude, Kant thinks that the Axioms, which

correspond to the categories of quantity, are more fundamental to mathematical cognition.

The reason is that our cognition of intensive magnitudes is wholly dependent upon our

cognition of extensive magnitudes. Indeed, Kant holds that without extensive magnitudes,

we would not be able to represent an intensive magnitude as a magnitude at all. That is,

I think, what leads Kant to say at the end of the Axioms that it is the Axioms prin-

ciple alonethat makes pure mathematics in its complete precision applicable to objects

of experience (A165/B206). For a fuller explanation of the asymmetry between intensive

and extensive magnitudes, see my Kants Philosophy of Mathematics and the Greek

Mathematical Tradition, Philosophical Review, forthcoming.33 For more on the distinction between quantaand quantitas, see my The Role of Magnitude

in Kants Critical Philosophy, Canadian Journal of Philosophy, 34(30), Sept. 2004, pp. 411442.34 See also A178/B221, quoted above, which suggests that the product of mathematical

synthesis is the representation of a magnitude.35 See also 8, p. 391, where Kant speaks of Platos admiration for the fitness of geometrical

figures to resolve a variety of problems just as if the requirements for constructing certain

magnitude-concepts [Grenbegriffe] were laid down in them on purpose. . .36 Furthermore, the representation of a whole space or time presupposes the representation

of the parts which compose it I cannot represent a horizontal line, for example, without

representing the left and right halves. Kant defines extensive magnitudes as those with thisproperty, so determinate spaces and times are extensive magnitudes (A1613/B2034). See

footnote 32 above.37 Kant uses the expression determinate magnitude at B203.38 We are now in a position to respond to an objection raised in footnote 6 above, which

interpreted the Axioms references to demonstrating the possibility of mathematics as

demonstrating its real possibility, where that amounts to the applicability of mathematics

to the objects of experience. (Friedman suggests such a view of the matter when he states

that only appliedmathematics yields a substantive body of truths (Friedman, 1992, p. 94).)

This interpretation would undercut the claim that Kant aims to establish the possibility of

pure mathematics in addition to its applicability. It is apparently supported by Kants

insistence at B1467 and A224/B271 that mathematical cognitions are only of the forms of

objects as appearances, and that we have mathematical cognition only on the assumption

that there are things that can be presented in accordance with that form; that is, that math-

ematics is only possible insofar as it applies to things given in intuition.


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We can now see, however, that what guarantees the applicability of mathematics is

exactly that which explains the possibility of cognizing determinate spaces and times. This

is apparent in our analysis of the argument of the Axioms and at A224/B271. At the same

time, the distance between explaining the possibility of pure mathematics and establishing

its applicability is quite small; the premise that the apprehension of appearances requires

the generation of a determinate space or time is all that is needed. Thus, Kants under-

standing of the difference between pure and applied mathematics is quite different from

our modern understanding. Pure mathematics is only possible in connection with space

and time, and it is the same space or time in which appearances are given to us. There is no

possibility of describing geometry in a set of axioms and then asking whether the space of

experience satisfies those axioms. This is a point Friedman makes in his discussion of real

possibility: There is only one way even to think such properties [i.e. topological proper-

ties of space]: in the space and time of our(Euclidean) intuition (Friedman, 1992, p. 93).

It is also made very clearly by Longuenesse:

For Kant, explaining the possibility of pure mathematics and deducing the possibility of its applicationto appearances in a pure (mathematical) natural science are one and the same procedure . . . because

explaining the possibility of pure mathematics is also explaining the possibility of its use in empirical

science that is, the transcendental basis of its application to appearances (Longuenesse, 1998, p. 275).

39Friedman endorses the view that Kant appeals to mathematical truths to establish the

claims of the Aesthetic. See (1992, pp. 635) for a discussion of the infinite extent and

infinite divisibility of space. Friedman also argues that the claim in the Aesthetic for the

priority of the singular intuition spacerests on our knowledge of geometry. Our cognitive

grasp of the notion of space is manifested, above all, in our geometrical knowledge (1992,

pp. 6970). Friedman further refines his views in a later article (Friedman, 2000).

I am not myself taking a position on the role of geometry in the Transcendental Exposition;I am only explaining how it relates to the interpretation I have set out here.40Most commentators do not address the change in formulation of the principle. Guyer

is an exception; he states that the A-edition formulation suggests an ontological distinction

between intuitions and appearances that is not conveyed by the B-edition formulation.

However, he also thinks that Kant came to emphasize this distinction between intuitions

and appearances in the B-edition of the Critique, so the motivation for Kants change in

formulation is left unclear (1987, pp. 4412, fn. 3).41For an analysis of Kants theory of magnitudes, see my Kants Philosophy of Math-

ematics and the Greek Mathematical Tradition, Philosophical Review, forthcoming.

I would like to thank the Humanities Institute of the University of Illinois at Chicago for

a fellowship while working on this paper during 20012002; Robert Adams, Tyler Burge,

John Carriero, Emily Carson, Michael Friedman, Ofra Rechter, Lisa Shabel and Daniel

Warren for helpful discussions; Lisa Downing, Bill Hart, and Peter Hylton for comments

on a previous draft; and an anonymous reviewer for very helpful suggested improvements.

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Bennett, Jonathan (1966). Kants Analytic. London: Cambridge Univ. Press.

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27(4), pp. 489512.

Carson, Emily (1999). Kant on Method in Mathematics, Journal of the History of Philo-

sophy 37(4), pp. 629652.


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THE POINT OF KANT’S axioms of intuition.pdf