ńňđ. 2
(âńĺăî 10)



what is more primitive: conditions of presentation in sensibility.
However, I should caution that I do not think that the argument pro-
ceeds simply by retreating, or regressing, from forms of discursive
understanding (part one) to the syntheses of imagination as their pre-
condition (part two). The argument is more radical than this if, as I have
just suggested, it moves from forms of thought to forms of givenness.
Nevertheless, Allison is correct in pointing out that I talk of a ‘‘retreat’’
from the forms of discursive understanding to the syntheses of imagina-
tion. But when I use this expression, what I describe is the transition, in
my own book (KCJ), from part two (where I consider the logical forms of
judgment) to part three (where I consider the transcendental syntheses
of imagination, and thus not only xx24 and 26 of the Deduction, but also
the System of Principles of the Pure Understanding).23 When describing
Kant’s argument in Deduction B, what I say is that part two of the
argument is a revisiting, in light of the argument of part one, of ‘‘the
manner in which things are given,’’ namely the forms of intuition, space
and time, that were first expounded in the Transcendental Aesthetic.24
Kant’s point is that space and time themselves, which have been
described in the Transcendental Aesthetic as forms of intuition and
pure intuitions, are now revealed to be the product of the ‘‘affection of
sensibility by the understanding,’’ namely by the unity of apperception
as a capacity to judge. And so, by the mere fact of being given in space
and time, all appearances are such that they are a priori in accordance
with the categories, and thus eventually subsumable under them.
However, I am aware that I am not making my case any better in Allison’s
eyes by proposing to read the second part of the B Deduction in this way.
Our most fundamental disagreement bears precisely on this point.
So I now consider Deduction B, part two.

KCJ, p. 197.
See KCJ, pp. 212–16. I do not claim to be especially radical in my reading. It is Kant’s thesis
that I describe as radical, not my reading of it. What I hope on behalf of the latter is that it is
accurate. Nor do I make any claim to being the first to defend such an interpretation.
Predecessors include e.g. Hegel: see Glauben und Wissen, in G. W. F. Hegel, Gesammelte
Werke, Deutsche Forschungsgemeinschaft, ed. Rhein-Westfal. Akad. d.Wiss. (Hamburg:
F. Meiner, 1968–); Faith and Knowledge, trans. Walter Cerf and H. S. Harris (Albany: SUNY
` ´
Press, 1977). Pierre Lachieze-Rey: see L’Idealisme kantien, 3rd edn (Paris: Librairie philo-
sophique Vrin, 1972); Wayne Waxman: see Kant’s Model of the Mind. The originality I claim
for my view is my emphasizing the relation between the unity of apperception and the
logical functions of judgment, and my relating the unity of space and time to the ‘‘unity
that precedes the category of unity’’ (B131, in x15 of the B Deduction). More on this below.

Deduction B, part two
Allison objects to two main points in my interpretation of this second
part: my identifying the ‘‘formal intuitions’’ of x26 of the B Deduction
with the ‘‘forms of intuition and pure intuitions’’ of the Aesthetic, and my
claim that when Kant defines synthesis speciosa as an affection of sensibility
by the understanding, he means affection by the capacity to judge. Let
me consider each point in turn.
First, form of intuition and formal intuition.
I maintain that when Kant describes space and time as ‘‘formal
intuitions,’’ in the footnote to x26 of the Transcendental Deduction,
he is describing the very same space and time he characterized as
‘‘forms of intuition’’ or ‘‘pure intuitions’’ in the Transcendental
Aesthetic. I am not maintaining that the Transcendental Deduction
calls for a ‘‘revision’’ of the Transcendental Aesthetic. The term I use is
‘‘re-reading’’: what I think is that everything that was said in the
Transcendental Aesthetic about the nature of space and time stands,
but it is brought into new light by the argument of the Deduction.
Indeed, when Kant says, in x26, that space and time ‘‘are represented
with the determination of the unity of the manifold,’’ he immediately
adds: see the Transcendental Aesthetic (B160).25 And then he goes on:
this unity presupposes a synthesis by means of which ‘‘(in that the
understanding determines the sensibility), space and time are first
given as intuitions’’ (B161n). Here he refers us back to x24, where he
explained the ‘‘affection of sensibility by the understanding’’ as being a
synthesis speciosa, or the transcendental synthesis of imagination (see
B151–2). Space and time, then, are forms of sensibility, just as Kant
maintained in the Transcendental Aesthetic. But they are forms of a
sensibility affected by the understanding, and thus they are the product
of synthesis speciosa, the transcendental synthesis of imagination. And
I must say that it seems to me quite reasonable to maintain that the
unity, unicity (there is only one space and one time), and infinity of time
and space – all features attributed to them as pure intuitions, in the
Transcendental Aesthetic – are features we imagine or anticipate and
thus project as preconditions of the unity of experience. It strikes me as
quite reasonable to maintain that, on the one hand, the qualitative
features of spatiality and temporality depend on our sensibility, which

The same was said at B136n, B140, B137. But only now is the point brought into the
argument with full force.

thus provides ‘‘first formal grounds’’ of the ordering of sensations that
yields appearances; and that, on the other hand, the unity, unicity,
and given infinity of space and time – and thus space and time themselves,
as intuitions in which all appearances are combined and ordered –
are products of our imagination. This is no revision of the
Transcendental Aesthetic. The latter allowed for this further develop-
ment, indeed mentioned it explicitly in the B edition, where Kant intro-
duced the idea of a ‘‘self-affection’’ of the cognitive subject, in striking
parallel to the idea of synthesis speciosa introduced in the Transcendental
Deduction (cf. B68–9).
In support of my proposal that the ‘‘forms of intuition’’ of the
Transcendental Aesthetic turn out just to be the ‘‘formal intuitions’’
resulting from what Kant calls, in the B Deduction, the ‘‘affection of
sensibility by the understanding,’’ I observe that it would be a mistake to
suppose that ‘‘form of intuition’’ is universally opposed to ‘‘formal intui-
tion’’ as what is indeterminate to what is determinate. The reason this
would be a mistake, I maintain, is that Kant’s notion of form is a rela-
tional one, always paired with matter. And in this pairing, form means
‘‘determination,’’ matter ‘‘undetermined’’ (and determined by the form)
(see A266/B322). To this, Allison objects that the opposition between
form of intuition and formal intuition, in the footnote to B160, is an
opposition between what remains ‘‘indeterminate’’ and what is ‘‘deter-
minate.’’ Moreover, he points out that Kant also mentions ‘‘form of
intuition’’ as what is indeterminate elsewhere (B154). Of course I agree
with that. As Allison acknowledges, I myself insist that in the footnote to
B160 the ‘‘form of intuition’’ is indeterminate by comparison to the
‘‘formal intuition’’ which is determined by the ‘‘affection of inner sense
by the understanding.’’ What I add, however, is that the opposition so
understood cannot hold universally and cannot be an argument for
opposing formal intuition to form of intuition in all cases. My suggestion
is that ‘‘form’’ should always be understood in context, and in connection
with the specific matter for which it is the form. Thus the form of
intuition as mere ‘‘formal ground’’ (in On a Discovery) is a form for a
matter, sensations as mere affections of which we are not even conscious.
The formal intuition as providing ‘‘not only the manifold, but the unity
of the manifold’’ (B 160n) is a form for the matter of appearances. Recall
that, in the Transcendental Aesthetic, Kant says of the appearances that
their matter is ‘‘that which corresponds to sensation’’ and their form is
space and time as forms of our sensible intuition (and themselves pure
intuitions). In a footnote to the Transcendental Dialectic, Kant explicitly

equates ‘‘form of intuition’’ and ‘‘formal intuition’’: ‘‘Space is merely the
form of outer intuition (formal intuition)’’ (B457n).26
Second, affection of sensibility by the capacity to judge.
Allison thinks that in maintaining that the ‘‘affection of sensibility by
the understanding’’ is an affection of sensibility by the capacity to judge,
I am claiming that in synthesis speciosa, sensibility is affected by logical
functions of judgment rather than by the categories as full-fledged
concepts.27 But this is not exactly what I think. What I understand
Kant as saying is this: the unity of apperception, as a capacity to judge,
generates the representation of the unity and unicity of space and time,
as the condition for any specific act of judging at all, thus prior to any
specific synthesis according to the categories, let alone any subsumption
under the categories. This representation of unity (or, one might say, the
anticipation of an overall unity of appearances in one space and one
time), which is prior to any specific synthesis, was mentioned by Kant
without further explanation at the end of x15 of the Transcendental
Deduction. There he said that there is a unity which is not the category of
unity, but the higher unity that presides over all acts of judging. Applied
to the forms of intuition, we are now told, this unity generates the formal
unity of space and time within which any categorial synthesis at all
occurs. In my understanding, the formal intuitions thus generated are
the representations of space and time as ‘‘infinite given magnitudes’’
mentioned in the Transcendental Aesthetic, the ‘‘pure images of all
magnitudes’’ mentioned in the Schematism chapter, the entia imaginaria
mentioned in the table of nothing, and the ‘‘formal intuitions or forms of
intuition’’ mentioned in the Transcendental Dialectic as the original
intuitions in which the successive synthesis of appearances is achieved,
under the regulative idea of a world-whole.28

` ´ `
Cf. KCJ, pp. 222–3. See also my ‘‘Synthese et donation. Reponse a Michel Fichant,’’
Philosophie, no. 60 (1998), pp. 79–91, translated as ch. 3 in this volume.
Allison proposes that when Kant says, in the footnote to B160, that in the Transcendental
Aesthetic he has ‘‘ascribed the unity of space and time merely to sensibility, only in order to
note that it precedes any concept,’’ he means concepts of space and time, not the cate-
gories. He may be right on this point. But I do not think this can apply to the second
occurrence of ‘‘concepts’’ in the same footnote: ‘‘the unity of this a priori intuition belongs
to space and time, and not to the concept of the understanding.’’
The ‘‘infinite given magnitudes’’ of the Transcendental Aesthetic: A25/B39, A32/B48; the
‘‘pure images of all magnitudes’’ in the Schematism chapter: A142/B182; the entia imagi-
naria of the table of nothing: A292/B348; the ‘‘formal intuitions or forms of intuitions’’ of
the Transcendental Dialectic: A424/B457n.

And this is why I described the second part of the Deduction as
making a more radical argument than is generally perceived. As I
understand him, Kant is claiming that the space and time represented
as one space and one time within which any object of experience is given,
are themselves, before any specific categorial activity (synthesis or ana-
lysis or subsumption under the categories) the product of the very same
unity of apperception that proceeds to generate syntheses according to
the categories and thus initiates the never-ending process of cognition.
So anything given in space and time, just by being given in space and
time, stands under the unity of apperception and thus the categories.
That this is the thrust of Kant’s argument seems to me to be confirmed
by what he says in xx21 and 26: the first part of the deduction considered
the categories as forms of thought. He states that we must now consider
the manner in which things are given. And he claims that he will show
that with, not in, the forms of intuition, a priori modes of ordering are
given (B161). Here at last Kant addresses the worry he expressed before
even beginning the Transcendental Deduction proper: it was relatively
easy, he said, to show that appearances must conform to forms of space
and time, because these forms just are forms according to which appear-
ances are given. The matter is quite different in the case of the
categories. For ‘‘appearances can certainly be given in intuition
independently of functions of the understanding’’ (A90/B123). Well,
this contrast loses much of its sting if space and time themselves, as
‘‘the manner in which things are given,’’ stand under the very same
unity of apperception that is the source of synthesis according to the
categories. This, I think, is the completion of the transcendental deduc-
tion Kant was announcing as early as x21.
A great deal more might be said in answer to Sedgwick’s and Allison’s
thoughtful comments. Within the limit of this response I will only men-
tion one last point. Both of them raise, only to withdraw it immediately,
the possibility that my reading of Kant’s argument might bring it into
some surprising proximity to later German Idealism. Allison makes, and
then withdraws, the suggestion that my view of the forms of intuition as
resulting from an ‘‘affection of sensibility by the understanding’’ might
bring Kant closer to Fichte’s view than either he or I would have
expected. Sedgwick makes, and then withdraws, the suggestion that
my talk of ‘‘generating’’ the categories might bring Kant closer than
either she or I would have thought to what she calls ‘‘Hegel’s attack on
the a priori.’’ I think this common pattern in their comments is due to the
fact that, in my reading, Kant’s notion of both ‘‘the a priori’’ and ‘‘the

given’’ is more complex than is generally supposed. This complexity was
certainly grasped by the German Idealists better than it has been in more
recent readings of Kant, even while they (especially Hegel) chastised
Kant for remaining adamant in distinguishing receptivity (passivity) and
spontaneity (activity) in our cognitive capacities. As for me, my view is
that Kant was right to insist on this distinction, and I do not think
anything in my reading of the Critique leads to loosening it in any
way.29 I do think, however, that one of the benefits of my interpretation
is its making clearer how Kant could remain true to this distinction while
radically challenging what we have come to call, after Sellars, ‘‘the Myth
of the Given.’’30
I have tried to show that this challenge, and Kant’s elucidation of the
reason-giving activity by way of which we relate our representations to
objects, was made possible by two extraordinary moves. The first is
Kant’s invention of the notion of a form of intuition – namely a form,
or forms, for ordering and individuating what is empirically given. The
second is his unprecedented use of a quite traditional logic of concept
combination, into which he introduces the reference to an x of judgment
that ultimately stands for the intuited individual’s thought under con-
cepts combined in judgment. Both inventions are essential to the argu-
ment of the Metaphysical and Transcendental Deductions of the
Categories. But the full measure of their pay-off can be gleamed not
there, but rather in the next section of the Critique: the System of
Principles of the Pure Understanding, where Kant expounds his con-
ception of mathematics and its application to the science of nature, the
meaning and use of the traditional metaphysical concepts of substance,
causality, and universal interaction, and the meaning and use of the
modal categories – possibility, actuality, necessity.31

On this point, see my ‘‘Point of view of man or knowledge of God: Kant and Hegel on
concept, judgment and reason,’’ in Sedgwick, Kant and German Idealism.
Cf. Wilfrid Sellars, ‘‘Empiricism and the philosophy of mind,’’ in Herbert Feigl and
Michael Scriven (eds.), Minnesota Studies in the Philosophy of Science, i (Minneapolis:
University of Minnesota Press, 1956), pp. 253–329; repr. with an introduction by
Richard Rorty and a study guide by Robert Brandom (Cambridge, Mass.: Harvard
University Press, 1997); John McDowell, Mind and World (Cambridge, Mass.: Harvard
University Press, 1994; 2nd edn 1996).
On the relation of space and time to the relational categories, see in this volume chs. 6
and 7.


Michael Friedman has offered a rich and stimulating discussion of my book,
KCJ. While giving a characteristically generous and clear-sighted account of
my views, he maintains that on the whole I fail to do justice to what is most
revolutionary about Kant’s natural philosophy, and instead attribute to
Kant a pre-Newtonian, Aristotelian philosophy of nature. The reason for
this distortion, according to Friedman, is that I put excessive weight on
Kant’s claim to have derived his categories from a set of logical forms of
judgment which he inherited, with some adjustments, from a traditional
Aristotelian logic. In taking Kant at his word on this point, I wrongly
attribute to him a traditional view of concepts and concept formation that
was shared by early modern empiricists and rationalists alike, but that Kant’s
lasting contribution is precisely to have rejected. And I fail to give their full
import to Kant’s remarkable insights into the newly discovered applications
of mathematical concepts and methods to the science of nature. According
to Friedman’s assessment, then, at worst my book ends up hurling back
Kant’s philosophy into the dark ages of Aristotelianism. At best, it reveals in
Kant a tension between Aristotelianism and Newtonianism that more
enlightened minds are now better able to identify and pry apart.1

See Michael Friedman, ‘‘Logical forms and the order of nature: comments on Beatrice
Longuenesse’s Kant and the Capacity to Judge,’’ Archiv fu Geschichte der Philosophie, vol. 82
(2000), pp. 202–15.


The questions Friedman raises are insightful and challenging.
However, my impression is that his assessment of my position suffers
from the relatively scarce attention he devotes to my views about the role
of synthesis in Kant’s Transcendental Analytic. I insist throughout the
book that this notion – not that of the ‘‘logical use of the understanding’’
according to the logical forms of judgment – carries the weight of Kant’s
conception of mathematics and its application in natural science. As early
as ch. 1 (‘‘Synthesis and judgment’’) I explain that Kant’s argument in the
metaphysical and transcendental deductions of the categories is built on
the consideration of two quite different, but related and complementary,
aspects of the understanding’s employment or use: its ‘‘logical use’’
according to the logical forms of judgment; and its use in ‘‘pure syn-
thesis,’’ that is, in the a priori ordering of manifolds in space and time,
the work of pure (productive) imagination. And I show how Kant’s
critical notion of synthesis is gradually developed in connection with
his epistemological insights into the concepts and methods of mathe-
matics. It ought to come as no surprise, then, if in neglecting what I say
about synthesis and focusing his discussion almost entirely on what I say
of the relationship between Kant’s categories and the logical forms of
judgment, Friedman should find my interpretation difficult to reconcile
with Kant’s avowed Newtonianism.
Still, I think Friedman is correct in stressing the disagreements
between our respective readings of Kant’s argument in the
Transcendental Analytic of the Critique of Pure Reason. In what follows I
shall try to clarify the grounds of this disagreement on each of the points
raised by Friedman.

Bottom up or top down?
Friedman sees me as defending an essentially ‘‘bottom-up’’ interpreta-
tion of the relation between Kant’s pure concepts of the understanding
and experience. In other words, he thinks I maintain that Kant’s cate-
gories are derived from experience by an inductive method relying on
procedures of comparison and abstraction performed upon what is
given to our senses. To this supposedly ‘‘bottom-up’’ view of Kant’s
categories and their application in natural science, he opposes his own
‘‘top-down’’ view, according to which the modern mathematical science
of nature relies on the instantiation of strictly a priori synthetic princi-
ples: Kant’s ‘‘Principles of the Pure Understanding,’’ expounded in the
Transcendental Analytic of the Critique of Pure Reason.

But actually, I do not defend a ‘‘bottom-up’’ interpretation of Kant’s
categories, their acquisition, and their use. On the contrary, I insist that
according to Kant, categories are a priori concepts that originate in the
understanding alone: this is precisely what their agreement with a table
of logical functions of judgment is supposed to show. And I agree with
Friedman that according to Kant the modern mathematical science of
nature rests on the instantiation, in connection with the empirical con-
cept of matter, of the synthetic a priori principles which predicate the
categories of all appearances. But Kant’s question in the Critique of Pure
Reason is: how is such application of pure concepts of the understanding
possible, that is, what makes it legitimate to presuppose that the cate-
gories are universally true of objects given to our senses? In answering
this question, Kant lays out two main aspects of the human intellect and
its use: what he calls the ‘‘logical use of the understanding’’; and what he
calls the ‘‘transcendental synthesis of imagination’’ which is, he says, the
‘‘first application of the understanding (and the one that grounds all
others’’ (B152)). The relationship between the ‘‘logical use of the under-
standing’’ and the transcendental synthesis of imagination is at the core
of Kant’s metaphysical deduction of the categories, namely his laying out
of their complete table according to the leading thread provided by a few
elementary logical functions of judgment.
In its logical use, says Kant, the understanding orders various repre-
sentations – intuitions or concepts – under a common representation (a
generic concept). The forms according to which such ordering takes
place (the logical forms of judgment), then, are not only forms according
to which concepts are combined (subordinated to one another) in judg-
ments. They are forms (modes of combination of concepts) that guide
the very acquisition of concepts from the sensible given in the first place:
empirical concepts are formed for use in judgment (A68/B93). In KCJ
I argue that this aspect of the logical use of the understanding is also
what Kant calls, in the Critique of the Power of Judgment, ‘‘reflection’’ (the
‘‘bottom-up’’ process of forming empirical concepts from the represen-
tation of particular objects). And I examine in great detail the ways in
which each logical form of judgment guides this reflective process of
concept formation. In doing this, I rely on the important appendix to the
Transcendental Analytic, the Amphiboly of Concepts of Reflection
Now, immediately after expounding the ‘‘logical use of the under-
standing’’ and the table of logical forms according to which it is exer-
cised, Kant goes on to argue that for this logical, reflective use of the

understanding to take place, synthesis must have occurred. By synthesis,
he means the combination of sensible manifolds in intuition. This com-
bination has a ‘‘pure’’ aspect: for any empirical manifold to be synthe-
sized, the forms of space and time in which intuited manifolds are given
and ordered must themselves be combined in such ways that the mani-
folds in them can be reflected under concepts according to logical forms
of judgment. Categories, says Kant, are just the pure concepts that guide
these syntheses or combinations: they are concepts of the unity of syn-
thesis of the spatiotemporal manifolds. As such, they guide the synthesis
of manifolds in very much the same way in which, for instance, a concept
of number guides the enumeration of a collection (A78/B104).
So considered (as ‘‘concepts of the necessary unity of synthesis’’),
categories are quite different in kind from the generic concepts formed
by comparison and abstraction. In KCJ I explain in detail how Kant’s
discovery of the categories under this aspect is related to his under-
standing of mathematical concepts as opposed to concepts of natural
kinds acquired by empirical inductive processes. However, I also main-
tain that the categories, which, as ‘‘pure concepts of the unity of synth-
esis’’ guide synthesis and, as such, are necessarily at work before any
analysis or reflection takes place, are themselves reflected as ‘‘clear’’
concepts only after empirical concepts have been formed under their
guidance. Indeed, Kant is quite explicit about this twofold status of the
categories when he describes his method of investigation at the begin-
ning of the Transcendental Analytic:

We shall follow the pure concepts all the way to their initial germs and
foundations in the human understanding, in which they lie prepared [in
denen sie vorbereitet liegen], until finally they are developed under the spur
of experience and are presented by this same understanding, freed from
the empirical conditions that attach to them, in their purity. (A66/B91)

In my view, the ‘‘initial germs and foundations’’ of the categories are the
logical functions of judgment as a priori forms of discursive thought;
their ‘‘development under the spur of experience’’ is their emergence as
concepts of the unity of synthesis (namely, a priori rules for the unity of
synthesis, guiding it toward analysis according to the logical forms of
judgment); their ‘‘presentation, freed from the empirical conditions’’ is
their reflection as clear concepts under which appearances are sub-
sumed, for instance when we form causal judgments or when we apply
concepts of extensive or intensive magnitudes to objects of experience.
According to such an account, then, when Newtonian science appears in

the history of human knowledge, it inherits this long process of develop-
ment and clarification of the pure concepts of the understanding. Does
this make my account of the categories a ‘‘bottom-up’’ account rather than
a ‘‘top-down’’ one? I do not think so, for the following two reasons.
First, in my account, the categories are a priori concepts that guide
‘‘from the top down’’ the syntheses of sensible intuitions so that our
representations are related to objects susceptible to being conceptua-
lized by means of reflection, and thereby related to other concepts in
judgments. The top-down procedure thus precedes and makes possible
the bottom-up. The role of categories as logical functions of judgment
governing reflection capable of yielding concepts of objects presupposes
their role as synthesis determiners (concepts of the unity of synthesis).
Second, this is why we can be confident that when Newton presup-
poses – as he does, according to Kant – the truth of the synthetic a priori
principles instantiated in the laws of motion of the Principia, he is war-
ranted in doing so: the categories, and thus the principles that predicate
them of appearances, are indeed true of the objects of perceptual
experience, the middle-sized objects of the modern mathematical
science of nature. This being said, it remains the task of empirical science
to determine which specific combinations and connections of appear-
ances instantiate the pure principles of the understanding. The answer
to this question can be given only by considering any empirically
discovered combination and connection in the context of the totality
of (endlessly revisable) experience.
In order further to substantiate this view, let me now consider the two
cases Friedman discusses more particularly: quantity and causality.

Friedman focuses his discussion on the issue of the respective primacy of
continuous and discrete magnitudes in Kant’s treatment of the cate-
gories of quantity. He contends that I give undue privilege to the latter
over the former, whereas in Kant’s treatment, continuous magnitudes
are primary. Number itself is ‘‘conceived in terms of the addition of line
segments with an arbitrarily chosen unit, say, rather than in the Fregean
style in terms of the extensions of concepts.’’2 In failing to perceive this
primacy, says Friedman, I remain insufficiently aware of the relationship

Ibid., p. 206.

between Kant’s critical philosophy and the modern mathematical
science of nature.
Let me first recall the three main questions Kant addresses, concern-
ing the categories in general: (1) what is thought in them, as ‘‘pure
concepts of the understanding’’? (2) How do they relate to sensible
intuition? (3) How does the account of their relation to sensible intuition
justify the synthetic a priori judgments that state their universal applic-
ability to appearances? With respect to the categories of quantity (unity,
plurality, totality), if we follow the metaphysical deduction of the cate-
gories, Kant’s answer to the first question is that they are pure concepts
of just those syntheses necessary so that particulars are subsumed under
concepts in singular, particular, and universal judgments.3
Kant’s account of number occurs in the course of his answer to the
second question: how do categories relate to sensible intuition? Number,
says Kant, is the schema of quantity, namely a ‘‘representation that gathers
together the successive addition of unit to (homogeneous) unit [eine
Vorstellung, die die sukzessive Addition von Einem zu Einem (gleichartigen)
zusammenfaĂźt]’’ (A142/B182). I argue that ‘‘homogeneous’’ should be
understood as ‘‘of the same kind,’’ i.e. ‘‘falling under the same concept.’’4
In relating number to the pure concept of quantity and the latter to the
logical quantity of judgments, I maintain that Kant thus appears strikingly
close to Frege’s view that numbers are properties of concepts, namely that
they attach to collections of individuals falling under the same concept.5
Now, Friedman urges that I ‘‘slide without any real argument’’ from
this notion of number as attaching to sets of objects thought under a
concept (the proto-Fregean notion of number), to number as assigning
to individual objects particular sizes or magnitudes (the pre-Fregean,
Euclidean notion of number, where number is defined in relation to the
measurement of line segments in space). I find the charge surprising: in
fact I take pains to explain the transition from the first to the second use
of number in some detail, and then conclude that according to Kant,
‘‘when we measure a line by adding units of measurement, what we do is
in effect recognize in the line a plurality of elements thought under the
same concept: ‘segment equal to segment s’.’’6 In my view, the notion of

In KCJ I defend the view that the correspondence between logical forms and categories
is: singular judgment/unity, particular/plurality, universal/totality. Friedman challenges
this view. I discuss this point below, pp. 45–6.
KCJ, p. 250.
KCJ, p. 257.
KCJ, p. 265.

number as attaching to arbitrarily chosen units of measurement is thus
to be understood in the light of the notion of number attaching to
extensions of concepts, which itself is referred back to our capacity to
form judgments determined as to their logical quantity (that is, to our
capacity to subsume individuals under concepts, and thus to represent
them as homogeneous units). This does not mean that measuring a line
segment, a surface, or a volume, is forming a discursive judgment in
which a generic concept is subordinated to another. All it means is that
the capacity to recognize homogeneous units, susceptible to being gone
through and synthesized as units of measurement, depends on the
discursive capacity to judge according to the logical form of quantity.
Of course, the discursive capacity is not the only faculty in play here.
Number, as the schema of quantity, or as a ‘‘representation that gathers
together the addition of unit to (homogeneous) unit’’ also depends on
the intuitive capacity to ‘‘go through and keep together’’ collections of
(homogeneous) units through time, and thus on our pure intuition of
time (our capacity to keep track of our representations in one time).
A related issue is that of the way we should understand the relation-
ship between the order in which Kant lists the logical forms of quantity in
judgment (universal, particular, singular), and the categories of quantity
(unity, plurality, totality). In my book I maintain, with Michael Frede
and Lorenz Kruger, that in listing the categories of quantity Kant
reverses the order in which he lists the logical forms of quantity in
judgment. Friedman maintains, with Manley Thompson, that there is
no good reason for attributing to Kant such a reversal.7 On the contrary,
he says, close scrutiny of Kant’s texts shows that Kant does intend the
category of unity to correspond to the logical form of universal judg-
ment, that of totality to the logical form of singular judgment. According
to Thompson, we can understand the correspondence in the right way if
we keep in mind that Kant’s categories of quantity are defined in con-
nection with the measurement of quanta, magnitudes. Because of this,
determining units (Einheiten in the sense I have advocated in connection
with number) depends on forming universal judgments such as: ‘‘Every
line of exactly this length is to be counted as a unit.’’ Plurality is uncon-
troversially connected with particular judgments. Totality is related to

See Michael Frede and Lorenz Kruger, ‘‘Uber die Zuordnung der Quantitaten des Urteils
und der Kategorien der Große bei Kant’’, Kant-Studien, vol. 61 (1970), pp. 28–49; KCJ,
pp. 247–9; Manley Thompson, ‘‘Unity, plurality, and totality as Kantian categories,’’ The
Monist, vol. 72 (1989), pp. 168–89; Friedman, ‘‘Logical forms,’’ p. 205. I am grateful to
Michael Friedman for having brought Manley Thompson’s article to my attention.

singular judgment: a judgment that asserts a predicate of a singular thing,
which as an empirical object is a quantum, namely something that is
quantitatively determined as a totality of parts (a totality of the arbitrarily
chosen units by which it is measured). This is an attractive explanation. If
Thompson is right, as I think he is in this case, I have to revise my view
concerning the correspondence between forms of judgments and cate-
gories, at least in the case of the application of the logical forms of quantity
to the determination of spatial quanta (magnitudes), and thus concerning
the generation of the categories of quantity grounding pure mathematics
and its application in natural science. This revision notwithstanding, I
would still suggest that Thompson’s analysis in no way contradicts, but
rather confirms my thesis that thinking a unit of measurement is in effect
thinking intuited individuals (the units of measurement) under a concept,
‘‘segment equal to segment s.’’ The example of universal judgment
Thompson proposes to justify the parallelism he defends says precisely
the same thing: ‘‘Every line of exactly this length is to be counted as a unit.’’
We think or recognize (by virtue of our having stipulated) units of
measurement under the concept: ‘‘line of exactly this length’’ and we
thus obtain homogeneous units that allow us to determine the measure-
ment of any line or any spatial magnitude.
As for the case of the quantitative determination of discrete collections
of individual elements, and especially the case of individual empirical
things, I am less convinced by the complex argument Thompson also
offers in support of the correspondence between logical form of singular
judgment and category of totality, logical form of universal judgment
and category of unity. I will not attempt to discuss his view here.
Whatever the case may be on this last point, I would maintain that
Kant’s groundbreaking move is to trace back to the logical function of
quantity in judgment our capacity to determine or pick out homo-
geneous units (ÂĽ units thought under the same concept), and to ground
on this capacity the generation of categories of quantity.
The transition from (1) the quantitative determination (quantitas) of
discrete magnitudes or aggregates (collections of homogeneous units:
apples, points, strokes . . . ), to (2) the quantitative determination of con-
tinuous magnitudes (quanta, namely objects immediately intuited as one
rather than many, in which units of measurement may nevertheless be
arbitrarily delineated and added to one another), and even more to (3)
the quantitative determination of continuous magnitudes not by way of
arbitrarily chosen discrete units, but by way of the representation of their
continuous generation through time – as in Newton’s calculus of

fluxions – this transition is made possible not by the categories of quan-
tity alone, but by their application to space and time as intuitions or more
precisely, as the intentional correlates of intuition (in imagination). So, it
will be useful here to consider separately the two related issues: (1) the
role of space and time as pure intuitions in the representations of infinity
and continuity; and (2) the respective primacy of discrete or continuous
magnitudes in Kant’s account of the categories of quantity and their

Space, time, infinity, and continuity
Kant defines space and time as ‘‘infinite given magnitudes’’ in the
Transcendental Aesthetic, namely before either the metaphysical or
the transcendental deduction of the categories. Similarly, in the chapter
on the Schematism of the Pure Concepts of the Understanding, he
defines space and time as ‘‘pure images of all magnitudes (quanta)’’
before defining number as the ‘‘schema of magnitude (quantitas).’’8
Thus it is from their being intuitions, not concepts or conceptually
determined, that space and time derive their property of infinity,
namely their property of being (represented in imagination as) larger
than any magnitude represented in them. Nevertheless, I argue in KCJ
that these intuitions (singular representations that are immediately pre-
sent to the mind in the way perceptions are) are themselves the result of
the ‘‘affection of sensibility by the understanding,’’ or synthesis speciosa, or
transcendental synthesis of imagination.9 In other words, representing
space and time as one (as intuitions) and as one whole within which all
appearances ought to be situated and ordered, depends on the original
effort of the mind that eventually makes it possible to synthesize parti-
cular manifolds under the guidance of the categories – and in the first
place, the categories of quantity. This does not mean that the pure
intuitions of space and time are themselves generated by a successive
synthesis of homogeneous units (space and time themselves cannot be
measured). But they are the one formal whole within which any collec-
tion of homogeneous individuals can be recognized, any spatiotemporal
magnitude can be delineated, any arbitrary choice of unit can be made,
or any measurement can be taken.

See A25/B39–40; A142/B182.
See KCJ, p. 220.

To sum up: as I understand Kant’s view, according to him the repre-
sentation of space and time as infinite does not follow from the application
of the categories of quantity. Rather, it is the precondition of any applica-
tion of the categories of quantity. As such, it depends on the same act of the
mind (the original effort to judge, applied to the pure forms of intuition)
that generates the categories of quantity in their various applications.
What about the representation of space and time as continuous mag-
nitudes? Kant defines continuity as ‘‘the property of magnitudes accord-
ing to which no part is the smallest’’ (A169/B211). And he adds: ‘‘Space
and time are continuous magnitudes, for none of their parts can be given
without enclosing it within limits (points and instants), and thus only in
such a way that this part is again a space or a time’’ (ibid.). The property
of continuity, then, cannot be defined without appealing to the repre-
sentation of parts and whole, and to the unity of the synthesis (whole,
unity of a plurality) of arbitrarily chosen units (parts), namely the schema
for the category of quantity. There is no representation of continuous or
discrete magnitude without making use of the category of quantity and
its schema. Nevertheless, just as in the case of infinity, the fact that space
and time have the property of continuity does not depend on the
category itself, as a pure concept of the understanding, but on space
and time’s being pure intuitions, where the whole precedes the parts
and the delimitation of further parts can be pursued indefinitely. This is
why I wrote that applying the categories of quantity to space and time as
original quanta
provides them with a meaning they would not have by being merely
related to the logical forms of quantity in judgment, and number is given
a relation to infinity and continuity that could not be obtained by its mere
definition as ‘‘a representation that gathers together the successive addi-
tion of homogeneous units’’.10

So, Friedman is certainly correct in stating that
it simply does not follow from the idea that space and time provide the
‘‘places’’ for the extensions of concepts, and thereby secure the applica-
tion of discrete quantity or number to . . . objects (qua items falling under
a concept), that space and time are also infinite and continuous magni-
tudes which thereby secure the application of the mathematics of
continuous quantity to these same objects.11

KCJ, p. 267.
See Friedman, ‘‘Logical forms,’’ p. 206.

The latter properties (that they are themselves infinite and continuous
magnitudes and thus secure the application of continuous quantity to
objects) follow from their being intuitions, pre-conceptually represented
(in imagination) as ‘‘infinite given magnitudes’’ (B39–40) in which any
spatial or temporal magnitude can be generated by a continuous synth-
esis through time (as in the drawing of a line).
Now, in my view, the respective primacy of discrete or continuous
magnitude should be understood in light of this cooperation between
the intuitions of space and time and the pure concepts of quantity in
Kant’s account of the application of the latter to appearances (and thus
his answer to the third question mentioned above: how do categories of
quantity apply to appearances?).

Continuous and discrete magnitudes
Friedman urges that in Kant’s exposition of the categories of quantity,
the case of continuous magnitudes is primary, the case of discrete mag-
nitudes secondary. True, in the Axioms of Intuition the categories of
quantity are applied to continuous magnitudes, quanta given in space
and time and measurable either by choosing an arbitrary unit of
measurement and adding it successively (in which case the quantum
continuum is treated as a quantum discretum by virtue of its having
a determinate ratio to the chosen unit of measurement), or by using the
Newtonian method of fluxions, in which case the quantum is determined
not by the successive synthesis of discrete units, but by the successive
synthesis of continuously generated increments. Here, the combined
features of the intuition of time (a quantum continuum in which no part is
the smallest and thus any magnitude can be continuously generated)
and the intuition of space (itself a quantum continuum in which no part is
the smallest) are what determines the features of the quantitative deter-
mination of a quantum. And it is no surprise that the consideration of
continuous magnitudes should take such primacy in the Principles. For
Kant’s main concern there is to argue that mathematics is applicable to
appearances ‘‘in all its precision [in ihrer ganzen Pra
¨zision]’’ (A165/B206),
namely all the way down to the application of calculus and its notion of
the infinitesimal. It is worth noting, moreover, that the issue of continu-
ity is explicitly mentioned only in the Anticipations of Perception, when
Kant considers appearances not just as extensive magnitudes, but as
intensive magnitudes, namely with respect to the degree or instantaneous

magnitude of their reality. In the Axioms of Intuition, by contrast, appear-
ances are treated essentially as aggregates, namely discrete magnitudes,
although it does turn out, when Kant introduces the issue of continuity in
the Anticipations, that as extensive magnitudes appearances are also
continuous – infinitely divisible – by virtue of the continuity of space and
time themselves (see A169–70/B211–12).
In support of his thesis that for Kant the case of continuous magnitude
is prior and that of discrete magnitude parasitic upon it, Friedman
mentions a text from the Anticipations of Perception where Kant
explains in what sense 13 thalers (13 coins made of silver) can be called
a ‘‘quantum of silver.’’ According to Friedman, here Kant ‘‘asserts the
priority of continuous over discrete quantity (in counting a number of
coins).’’ If this were what Kant is asserting, it would be bizarre indeed.
For counting coins certainly seems like an unambiguous case of enumer-
ating a collection of discrete units. So what is going on here?
The example cited occurs at the end of a paragraph where Kant has
argued that since space and time are quanta continua (continuous mag-
nitudes), so are appearances with respect to their extensive as well as
their intensive magnitude (their reality). Then comes the obvious objec-
tion: is there nothing discrete in nature? Kant’s response: a discrete
collection, where ‘‘the synthesis of the manifold is interrupted,’’ is an
aggregate of appearances, not itself an appearance as a quantum (some-
thing that is itself one and can be quantitatively determined). This is
where the example of the 13 thalers comes into play. They can be called a
quantum only if I consider them as a given amount of silver (it is then a
quantum discretum, an amount of one and the same stuff [silver] that
nevertheless happens to be divided into parts). But as a collection of
coins, it is not a quantum but rather, an aggregate, that is, a number of
coins. Note that number is here associated with what is just an aggregate,
a discrete collection, and not a quantum, even presented as a discrete
collection of parts. However, Kant adds,
Since all numbers must have their ground in unity [Da nun bei aller Zahl
doch Einheit zum Grunde liegen muss], the appearance as unity must be the
ground, and as such, a continuum. (A171/B212)

The question is: what does Kant mean by ‘‘all numbers must have their
ground in unity’’? Does he mean that they presuppose a quantum to be
measured by way of number (as Friedman’s interpretation would
imply)? Or does he mean that they presuppose units that must be
successively synthesized? Although there are certainly arguments in

favor of the former interpretation,12 I think the latter is more plausible,
for at least two reasons. First, this reading agrees with Kant’s mention of
Einheit in connection with number and addition, in the Axioms of
Intuition. Kant writes:
Insofar as here [namely, in addition of numbers] one considers only the
synthesis of the homogeneous [of the units, Einheiten], the synthesis can
occur, in one way only, however universal the use of numbers can be.

(See the similar use of Einheit in reference to points and fingers, in the
Introduction to Critique of Pure Reason, B15/16.)
Second, understanding Einheit as the unit presupposed in number
rather than the unity of the quantum number would serve to measure,
seems essential to the argument Kant wants to make in the passage
where the example of the 13 thalers occurs. The idea is: of course 13
coins are a discrete collection, or aggregate. But any such collection
presupposes empirically given units which alone can be called appear-
ances (the collection is just an aggregate thereof); and they, the indivi-
dual appearances that serve as units, are quanta, and as such, continua.
So, there is no exception whatsoever to the statement: all appearances
are quanta continua. Friedman is mistaken, I think, in maintaining that
this is a statement about the mathematical primacy of continuous over
discrete magnitudes. Rather, it is a statement that emphatically stresses
the strict universality of the synthetic a priori judgment: ‘‘all appearances
are continuous magnitudes.’’
In the end, I would suggest that my disagreement with Friedman
about the primacy of continuous or discrete magnitudes in Kant’s treat-
ment of quantity boils down to this: Friedman’s concern is to show how
Kant’s categories of quantity are applied to appearances, first in the
Principles of the Pure Understanding (the Axioms of Intuition and
Anticipations of Perception, in the Critique of Pure Reason) and then in
their instantiation to the empirical concept of matter (in Kant’s
Metaphysical Foundations of Natural Science). My concern is with Kant’s
investigation into the origin of the categories of quantity (metaphysical
deduction), the justification of their application to appearances

‘‘Unity’’ can refer to the chunk of matter distributed into discrete pieces of silver as well as
to the discrete units (the coins). The same difficulty holds in the case of matter itself. Matter
is continuous, and thus one (in fact, the one and only substantia phaenomenon). But it is
distributed into discrete things, each of which is continuous, and can also be divided into
discrete parts, and so on.

(transcendental deduction), and the proof of the principles. As
Friedman correctly remarks, I say relatively little about the relationship
between the Principles of the first Critique and Kant’s views about natural
science. So, in a way, my story ends where Friedman’s begins. Now one
may wonder whether it would not be wiser to drop the side of the story I
have been trying to account for, and to start our reading of the Critique
with the System of Principles rather than with the metaphysical or even
the transcendental deduction of the categories. This is an option that has
been strongly advocated, in the history of post-Kantian philosophy, by
Cohen and his neo-Kantian followers, a tradition Friedman wants to
uphold. But I hold the contrary view. I think we have much to gain by
paying attention to what the neo-Kantians generally downplayed: Kant’s
claims about the nature of discursive understanding (and thus the role of
what he calls ‘‘general logic’’) and its relation to a priori forms of sensible
Let me now consider Michael Friedman’s second example, my treat-
ment of the relational categories: substance, causality, and universal

Substance, causality, interaction
Friedman maintains that by emphasizing as I do Kant’s metaphysical
deduction of the categories, I end up attributing to Kant an Aristotelian
metaphysics of nature that is clearly at odds with his avowed
Newtonianism. Friedman nevertheless credits me with recognizing in
crucial instances the non-Aristotelian features of Kant’s relational cate-
gories, e.g. Kant’s statement of the absolute permanence of substance in
the First Analogy of Experience; and his statement of the universal
reciprocal action of material substances in the Third Analogy.
However, according to Friedman all this means is that I have brought
to light some fundamental tensions in Kant’s metaphysics of nature,
without being myself sufficiently aware of these tensions. I thus fail to
raise the question that looms large in the wake of my book: does Kant’s
philosophy have the resources to resolve them?
Friedman is correct in stressing that I do not address the question of
the respective weight of Aristotelianism and Newtonianism in Kant’s
natural philosophy. This was not the object of my book. Rather, my
concern was with Kant’s theory of judgment, Kant’s explanation of the
relationship between logical forms and categories in the various stages of
the argument of the first Critique, and the light this sheds on Kant’s

critical system as a whole, especially the theory of judgment in the third
Critique. Still, it is true that if my account of these issues leads to the
deeply problematic conclusions that Friedman thinks it does where
Kant’s natural philosophy is concerned, the thesis I defend runs into
serious trouble. But I do not think my account leads to such problematic
conclusions. On the contrary, I think it alone can offer a satisfact-
ory explanation of what Friedman calls the ‘‘tension’’ between
Aristotelianism and Newtonianism in Kant’s natural philosophy.
To see this, one needs again to pay attention to the distinct and
complementary roles Kant assigns to the logical forms of judgment, on
the one hand, and to the pure forms of intuition and synthesis of
imagination, on the other hand. I will show this by briefly reviewing
my account of Kant’s argument in each Analogy, following the order of
Friedman’s comments. I will thus consider, first, substance and universal
interaction (the First and Third Analogies of Experience); second, causal
connection (the Second Analogy).

Substance, and universal interaction
As I understand him, Kant argues in the First Analogy that we experi-
ence objective succession or simultaneity only as the succession or simul-
taneity of the accidental states of empirical substances, namely empirical
objects that we recognize under their essential properties – the proper-
ties they could not cease to have without ceasing to be the objects they
are.13 Now, according to the metaphysical and transcendental deduc-
tions of the categories, what makes us capable of so ordering our repre-
sentations in time is the ‘‘effect of the understanding on sensibility’’
(B152), guiding the syntheses of manifolds in sensibility in such a way
that empirical objects can eventually be reflected under concepts accord-
ing to the form of categorical judgments (completed by those of
hypothetical and disjunctive judgments, as the arguments for the second
and third analogies will show).
Up to this point in the argument, we have grounds sufficient only to
infer the relative permanence of substances, substances that might
appear and disappear, but that throughout their existence have some
essential features by which we recognize them as the (relatively perma-
nent) substances they are (Descartes’ piece of wax, say, or the moon and

See KCJ, pp. 334–7.

earth in Kant’s Third Analogy). So the question is: how does Kant’s
argument progress from this merely relative permanence to affirming
the absolute permanence of substances? My answer is that he makes this
move by appealing to our a priori intuition of time as the condition of
possibility of experience, and therefore (according to the transcendental
deduction of the categories) as the condition of possibility of all objects of
experience. Time itself is permanent: we intuit a priori (i.e. imagine a
priori) one and the same time in which all objects of experience are
ordered. But, as Kant affirms in each of the Analogies and in the general
principle of the Analogies, time cannot itself be perceived. Therefore,
the unity and unicity of time (the representation of all time relations as
unified and existing in one and the same time) can have empirical reality
only if all changes, without exception – including the coming into exis-
tence and going out of existence of what I have called the ‘‘relatively
permanent substances,’’ e.g. the coalescing and melting of Descartes’
piece of wax or perhaps the aggregation or disintegration of Kant’s
moon and earth in the Third Analogy – all changes are changes of states
of some absolutely permanent substance. And of course it is this abso-
lutely permanent substance that is instantiated, in Kant’s Metaphysical
Foundations of Natural Science, in the empirical concept of matter, the
object of Newtonian science.
What is interesting here is that if my reading is correct, Kant’s argu-
ment is an attempt to account both for the pull of Aristotelianism in our
ordinary perceptual world and for the truth of Newtonianism. But
grounding the truth of Newtonianism is also determining the limits of
its application, since affirming the absolute permanence of material
substances is premised on our pure intuition (in imagination) of one
unified time as the condition of possibility of our experiencing any
independently existing objects at all, and thus of there being any such
objects for us.
Kant’s argument in the Third Analogy rests on a similar appeal to the
threefold source of our representation of objective time determinations:
(1) the discursive source (our logical forms of judgment); (2) the intuitive
source (space and time as the pure forms of our sensible intuition); and
(3) the a priori syntheses of imagination that bring it about that appear-
ances are combined in such ways that they can be reflected under
concepts according to the logical forms of our judgments. I will not
attempt here to rehearse Kant’s argument in all its complexity. I will
recall only enough to help me answer Friedman’s principled objection to
my interpretation.

Friedman maintains that because of the Aristotelian view of nature I
supposedly attribute to Kant, I make it incomprehensible why it should
be an a priori law of nature that substances are in relations of universal
interaction. Only on Newton’s concept of gravitational force, Friedman
urges, is it the case that every action must have an equal and opposite
reaction. No such necessity exists on an Aristotelian view of substance.14
Now, I have suggested above that Kant’s argument in the First Analogy
of Experience accounts both for the pull of Aristotelianism in our ordin-
ary perception and for the progress from what we might call this ‘‘mani-
fest image’’ to the Newtonian ‘‘scientific image’’ of the world.15 In other
words, in my understanding of Kant’s argument, the mental capacities at
work in generating the Aristotelian image of the world also explain why
it was both possible and necessary that this image be eventually super-
seded by a Newtonian (mathematical) worldview. The same is true of
Kant’s argument in the Third Analogy. Here what we need to under-
stand is why Kant thinks that the same capacities that generate the
representation of objective simultaneity among the objects of our ordin-
ary perceptual world also provide us with the a priori knowledge that
they exist in relations of universal reciprocal action.
As I understand it, Kant’s argument is along the following lines: we
experience individual material things as existing simultaneously in space
only if we combine (synthesize) our perceptions of things present to our
senses with our representations (in imagination) of things not present to
our senses, in such a way that they can be reflected under concepts
combined in reciprocal hypothetical judgments, such as: ‘‘If A is present
to my senses at time t at point p1 relative to my own body, then B (which

See Friedman, ‘‘Logical forms,’’ p. 209:
On the Aristotelian conception of the community of substances in space, there is no
particular need for reciprocal interaction. The sun influences changing objects of the
earth, for example, but since the sun undergoes no actual change itself, neither the
earth nor objects upon it influence the sun in turn. In Kant’s Newtonian conception,
by contrast, every action must have an equal and opposite reaction, and so the earth
does necessarily influence the sun in turn – through its own (relatively small)
gravitational force.
See above, p. 54. What I mean is that in my reading, Kant accounts both for the fact that the
world appears to us as a world of only relatively permanent, qualitatively determined
things, subject to generation and corruption (this is what I call the ‘‘pull of Aristotelianism
in our ordinary perceptual world’’), and for the fact that ultimately, the unity of the time-
determinations of appearances depends on our recognizing the existence of one sub-
stance, matter, whose states change according to universal mathematical laws (the
Newtonian view of nature).

is not present to my senses) is also present at this same time t, at point p2
relative to my own body; and conversely, if B is present to my senses at
time t at point p2 relative to my own body, then A (which is not present to
my senses) is also present at this same time t, at point p1 relative to my
own body.’’ This is of course far from a representation of causal interac-
tion. However, if we generalize these statements to the reciprocal con-
ditioning of all things with respect to their places and changes of place,
states and changes of states, in one space and one time (in which our own
body is also situated), then we obtain the idea that all material substances,
insofar as they are perceived (experienced) as existing simultaneously,
stand in relations of universal reciprocal determination such that each
and every substance’s being at a certain place, in a given state, at a
given time, is a determining ground for each and every other sub-
stance’s being in a given place, in a given state, at that same time.
This (as yet indeterminate) notion of reciprocal determination of
position and state, when instantiated to the empirical concept of
material substance as something movable in space, is presupposed
in Newton’s Third Law of Motion. And as Friedman has shown, the
latter, when instantiated to the Keplerian regularities in the motions
of celestial bodies, leads to Newton’s formulation of the empirical law
of universal gravitation.16
If this explanation is correct, then Kant holds that although it is
only with Newton’s mathematical science of nature (prepared by
Kepler, Copernicus, Galileo, Descartes, and others) that a determi-
nate notion of universal interaction is formulated and expressed in a
mathematical law, an implicit, indeterminate notion of the reciprocal
determination of things and their states has to be at work in each and
every one of our experiences of the objective simultaneity of things
(where ‘‘experience’’ does not mean mere sense perception, but the
synthesis of perceptions by means of which they are related to inde-
pendently existing objects that they are taken to be the perception
of). This indeterminate notion is a far cry from Newton’s law of the
equality of action and reaction of moving forces (the Third Law of
Motion in the Principia) and even further from the mathematical law
of universal gravitation. But Kant’s point is that in order to formulate
these laws we need to have an a priori principle stating that all

See Michael Friedman, ‘‘Causal laws and the foundations of natural science,’’ in Paul Guyer
(ed.), The Cambridge Companion to Kant (Cambridge: Cambridge University Press, 1992),
pp. 161–99.

appearances stand in universal reciprocal determination. For – as
Kant argues in the Metaphysical Foundations of Natural Science – this
principle is presupposed in Newton’s third law,17 which is itself pre-
supposed in Newton’s proof of the inverse square law. The justifica-
tion of this principle is provided by the argument of the Third
Analogy: we would have no experience of identifiable and re-identifiable
objects existing simultaneously in space unless we presupposed the
universal reciprocal determination of their positions and states; now,
according to the Transcendental Deduction of the Categories, the
conditions of possibility of experience are the conditions of possibility
of the object of experience; so, there would be no objects of experi-
ence simultaneously existing in space unless they were in universal
reciprocal determination of their positions and states.
Because they are thus individuated in one space and one time by way
of the universal mutual determination of their positions and states,
appearances can be known under concepts according to a universal
subordination of genera and species for which the discursive form is
the form of disjunctive judgment. This, I argue, is the explanation for
the correspondence, in the Metaphysical Deduction of the Categories,
between the logical form of disjunctive judgment and the category of
community (universal interaction): the category of community is the
concept guiding the syntheses of appearances so that they can be
reflected under concepts according to the logical form of disjunctive
judgment. Now, Friedman objects that he simply does not see how only a
Newtonian conception of interaction makes possible concept formation
generating a universal subordination of genera and species.18 But that is
not what I say. Certainly an Aristotelian view of nature does represent it
according to universal subordinations of genera and species. Indeed
Kant explicitly acknowledges the Aristotelian ancestry of this discursive
form of systematicity.19 What I maintain is that according to Kant, since
material things are individuated in space and recognized as existing at
the same time only by way of the presupposition, which eventually
becomes the determinate knowledge, of their universal reciprocal
action, the concepts of natural kinds under which they are recognized
and combined according to the form of disjunctive judgment – which is

Immanuel Kant, Metaphysical Foundations of Natural Science, trans. Michael Friedman
(Cambridge: Cambridge University Press, 2004), AAiv, p. 544.
See Friedman, ‘‘Logical forms,’’ p. 210.
See Critique of the Power of Judgment, First Introduction, AAxx, pp. 214–15.

the form according to which our concepts of natural kinds are coordi-
nated and subordinated to one another – these concepts of natural kinds
are concepts of relational properties: forces. This is confirmed by the
appendix to the Transcendental Dialectic, where Kant insists that the
highest goal of natural science is to order all its concepts of force under
that of ‘‘one and the same moving force’’ (A663/B681).

Like Friedman, I insist that according to Kant, Newtonian science rests
on the presupposition of the universal validity of the causal principle.
But precisely for this reason, I maintain that Newtonian science is of no
use at all to prove the causal principle: this would be circular. So what we
need to know is: how does Kant prove its validity with respect to all
appearances, namely all objects that appear to our senses? Here again
the answer I propose rests on my analysis of Kant’s account of the ways in
which figurative synthesis generates our representation of the objective
ordering of appearances in time. In this case, I argue that according to
Kant, we would not even experience a succession as an objective succes-
sion unless this succession appeared to ‘‘presuppose something upon
which it follows according to a rule’’ (A189). Because this is a necessary
presupposition of any experience of objective succession, experiencing
such a succession is also looking for the ‘‘something upon which it
follows, according to a rule.’’ That is to say, experiencing a succession
as objective is also looking for the event or state of affairs that might be
known as an instantiation of the antecedent of a hypothetical rule whose
consequent is instantiated by the given objective succession. What the
antecedent is, we can find out only empirically. But the principle accord-
ing to which there is such an antecedent is an a priori law, and thus
absolutely necessary. And the connection we have found empirically, if
true (namely, if we have correctly identified the relevant connection in
relation to the unity of experience) is a necessary connection.
I nowhere state, nor do I for a moment entertain, the view attributed
to me by Michael Friedman that Kant defends a strictly inductive
method for discovering causal connections. On the contrary, I argue
that for Kant what makes it possible for us to progress from the mere
hypothetical judgment ‘‘if the sun shines on the stone, the stone gets
warm’’ to the causal judgment, ‘‘the sun warms the stone,’’ is that we have
already presupposed the a priori validity of the causal principle. And
what makes such a presupposition legitimate is the argument I just

recalled: no objective succession would be experienced unless its per-
ception had been obtained (‘‘synthesized’’) in accordance with the causal
principle (namely under the presupposition that ‘‘something precedes,
upon which it follows according to a rule’’). This being granted, it
remains that in Kant’s own words, the sensible mark by which we
recognize the existence of a causal connection is the constant conjunc-
tion of similar events or states of affairs: in the chapter on the
Schematism of the Pure Concepts of the Understanding, Kant defines
the schema of cause as ‘‘the real which, whenever posited, something else
always follows. It consists therefore in the succession of the manifold,
insofar as it is subject to a rule’’ (A144/B183).
In the Second Analogy, after expounding his argument for the claim
that presupposing the truth of the causal principle is an a priori condi-
tion of all experience, Kant considers a possible objection to this view. It
might seem, he says, that this claim contradicts the observations we all
make, according to which it is only by witnessing repeated similar
sequences of events that we come up with a rule for these sequences,
and thus form a concept of causal connection: thus the concept would
appear to be empirical after all. Kant’s reply is that it is here as with all
other a priori representations: we draw them out of experience, as clear
concepts, only because we have put them there in the first place (see
Now, Friedman objects to my citing this passage in support of my
claim that for Kant, particular causal connections are known only from
experience. According to Friedman, in referring to the ‘‘common obser-
vation’’ according to which our knowledge of particular causal rules is
acquired empirically, Kant is not expressing his own view. Rather, he is
giving voice to a view he expressly opposes. But I think Friedman’s
interpretation is not supported by the text. He may be misled by
Kemp Smith’s translation, which says:
This [i.e. the statement that the truth of the causal principle is an a priori
presupposition of all experience] may seem to contradict all that has
hitherto been taught in regard to the procedure of our understanding.
The accepted view is that only through the perception and comparison of
events repeatedly following in a uniform manner upon preceding appear-
ances are we enabled to discover a rule according to which, etc. . . .

But the text really says:
This may seem to contradict all the remarks [Bermerkungen] that have
always been made about the way our understanding proceeds; according

to those remarks, only through the perception and comparison of events
repeatedly following in a uniform manner . . . are we enabled to discover
a rule, etc. . . . (A195–6/B240–1, my translation)20

Later in the same paragraph, Kant continues:

The case is the same here as with other pure a priori representations (e.g.
space and time) that we can extract as clear concepts from experience
only because we have put them into experience, and experience is hence
brought about through them. (ibid.)

What Kant is opposing, then, is the view that just as particular causal
connections are known empirically, so is the universal causal principle
itself. In other words, he opposes the inference from the empirical char-
acter of our knowledge of particular causal connections to the empirical
character of our knowledge of the causal principle itself. The same point is
made even more explicitly in the Transcendental Methodology:

If wax that was previously firm melts, I can cognize a priori that some-
thing must have preceded (e.g. the heat of the sun) on which this has
followed in accordance with constant law, though without experience, to
be sure, I could determinately cognize neither the cause from the effect
nor the effect from the cause a priori and without instruction from
experience. [Hume] therefore falsely inferred from the contingency of
our determination in accordance with the law the contingency of the law
itself . . . (A766/B794)

Friedman charges that in stressing as I do the empirical character of
our knowledge of particular causal connections, I make Kant a propo-
nent of a Baconian inductivist method in natural science. It is true that I
relate Kant’s analysis of the transition from judgments of perception to
judgments of experience, in the Prolegomena, to the striking reference
Kant makes to Bacon in the B Preface of the Critique of Pure Reason.
There Kant credits Bacon with having ‘‘partly occasioned, and partly
further stimulated, since one was already on its track, [a] discovery
[which] can . . . be explained by a sudden revolution in the way of

At the time of my discussion with Friedman, the Guyer and Wood translation had only
recently appeared, and Kemp Smith’s was still the most familiar. The translation I give
here is the one I offered at the time, against Kemp Smith’s. For all other citations I return
to Guyer and Wood’s translation, unless otherwise indicated. On the particular point at
hand, Guyer and Wood concur with me in not attributing to Kant Kemp Smith’s mislead-
ing disclaimer: ‘‘The accepted view is that . . . ’’

thinking’’ (Bxii).21 The ‘‘discovery’’ Kant refers to is that ‘‘in order to
know something securely a priori, [one] had to ascribe to the thing
nothing except what followed necessarily from what [one] had put into
it in accordance with one’s concept.’’ After thus crediting Bacon, Kant
adds: ‘‘Here I will consider natural science only insofar as it is grounded
on empirical principles’’ (this is in contrast to geometry and its construc-
tions according to a priori concepts, for which he cited earlier the
example of Thales). He then cites as examples of the ‘‘revolution in
the manner of thinking’’ in natural science, Galileo’s experiments
with the inclined plane, Torricelli’s experiments with atmospheric
pressure, and even (less felicitously) Stahl’s experiments in transform-
ing metal into calcar. The point appears to be, then, that even Bacon,
the inspirer of a strictly empirical method in natural science, really
participates in the ‘‘revolution in the way of thinking’’ characteristic of
modern science, for even he taught his contemporaries that ‘‘reason
must approach nature with its principles in one hand, according to
which alone the agreement among appearances can count as laws,
and, in the other hand, the experiments thought out in accordance
with these principles’’ (Bxiii). In KCJ I suggest that here Kant may
have in mind Bacon’s explanation of the method of ‘‘crucial experi-
ments,’’ which he admittedly interprets in strangely aprioristic
terms.22 If my reading is correct, Kant’s reference to Bacon is
meant to show that although all natural science (and therefore
Newtonian science itself) is empirical science, it would be impossible
unless it could rely on strictly a priori principles.
To sum up: the reason I do not take myself to attribute to Kant the
defense of a strictly inductivist method in natural science is twofold.
First, for Kant as I understand him, although all knowledge of any
particular empirical connection is empirical, it rests on a presupposition
that is not empirical, but a priori: that of the universal validity of the
causal principle. Second, even though the schema of causality, and so the
empirical feature of objects of experience by which we will be alerted to
the presence of a causal connection, is, as it was for Hume, the repetition
of generically identical sequences of events or states of affairs (‘‘the real
upon which, whenever it is posited, something else always follows’’
[A144/B183]), nevertheless what makes possible the universalization of

Cf. KCJ, p. 176.
See KCJ, pp. 176–7, nn. 22 and 23.

such observed correlations into causal connections is the use of mathe-
matical concepts and methods to formulate universal laws of nature.23

Concluding remarks
In his concluding remarks, Friedman quotes Cassirer’s charge (in his
1910 book, Substanzbegriff und Funktionsbegriff)24 according to which the
Aristotelian theory of concepts and concept formation is responsible for
the errors both of rational metaphysics and of traditional empiricism,
and is in fundamental tension with the modern mathematical science of
nature. According to Friedman, my book brought to unprecedented
light the fact that precisely this tension between Aristotelianism and
Newtonianism is at the core of Kant’s critical philosophy. For I am
supposed to have shown that in his metaphysical deduction of the
categories, Kant adopts an Aristotelian view of concepts, judgments,
and concept formation. But in the System of Principles of the Critique
of Pure Reason and in the Metaphysical Foundations of Natural Science he is a
clear proponent of Newtonianism and the mathematical method in
natural science. Because I have not perceived the tension my own
analyses thus revealed, I have not asked the question Friedman now
asks: does Kant have the resources to resolve it?
But does Kant defend an Aristotelian theory of concepts and concept
formation, in the metaphysical deduction of the categories? I do not
think he does. Indeed one of the grounding theses of my book is that
although the forms of Kant’s ‘‘general pure logic’’ are essentially
Aristotelian, the use he argues we make of them in cognition is a radical
break from the Aristotelian view of concept formation shared by his
predecessors in the German Schulphilosophie, for two reasons: (1) because
for him the form of judgment is prior to its matter (concepts and objects),
so that even the most strictly empirical concepts are formed under the
guidance of the acts of judging and their forms; and (2) because before
any such acts of empirical concept formation, synthesis, that is, combina-
tion in imagination of manifolds in space and time, must have occurred.
In insisting on these two aspects of Kant’s anti-abstractionist view of
concept formation, I am in agreement with Cassirer, whose main target,

On this point, see also ch. 6 in this volume, especially pp. 172–7.
Ernst Cassirer, Substanzbegriff und Funktionsbegriff. Untersuchungen u die Grundfragen der
Erkenntniskritik (Berlin 1910, repr. Darmstadt: Wissenschaftliche Buchgesellschaft, 1969).
Transl. W. C. Swabey and M. C. Swabey, Substance and Function (New York: Dover, 1953).
References to the text are given in the German edition.

when he attacks the modern, psychologistic version of Aristotelian
abstractionism, is quite explicitly Mill, not Kant.25
What Cassirer does challenge in Kant is the view that understanding
the nature of mathematical concepts is understanding the ways in which
they are grounded in pure acts of the mind. To this view and to the role
Kant assigns to the pure intuition of time in generating concepts of
number, Cassirer opposes Frege’s and Dedekind’s logicist program of
a purely logical derivation of arithmetical concepts and laws. Only this
program and the modern quantificational logic of relations that makes it
possible, says Cassirer, can reliably overcome the old Aristotelian view of
concepts and the ontological primacy of substance over relations.26 This
leaves us, it seems to me, with a question slightly different from the one
Friedman reproaches me for not having formulated. Friedman’s ques-
tion is: ‘‘Does Kant’s epistemology have the resources to resolve the
tension between Aristotelianism and Newtonianism in Kant’s natural
philosophy?’’ To this question I have argued that the answer is, yes,
Kant does have the resources: the pure forms of intuition as the pure
forms of manifoldness, distinct from and complementary to the forms of
discursive concepts and concept formation. But the next question is: if
these resources offer the means to understand the move from
Aristotelianism to Newtonianism, do they also offer the means to under-
stand the nineteenth- and twentieth-century superseding of Newtonian
natural science, with its reliance on Euclidean space and strictly deter-
ministic causal laws? Interestingly, Kant’s theory of space and time as
pure forms of intuition, which bears the brunt of Kant’s account for the
move from Aristotelianism to modern mathematical natural science, is
also what primarily needs to be revisited in order to come to terms with
later developments in mathematics, their relation to logic, and their
application in natural science. I suggest that a re-examination of Kant’s
theory of space and time should not neglect either of its aspects: neither
Kant’s view of the role of spatiotemporal intuition in representing the
middle-sized objects of our ordinary perceptual experience, nor the role
of spatiotemporal intuition in grounding our scientific worldview.
Under both aspects such a re-examination was beyond the scope and
ambition of my book.

For Aristotle’s view, see Posterior Analytics ii, 19, 99b15–100b18, where Aristotle explains
that universals are deposited in our minds by the repeated perception of sensible indivi-
duals. Nothing could be further from Kant’s view of concept formation. On Cassirer’s
criticism of Mill, see Substanzbegriff und Funktionsbegriff, ch. 1.
See Substanzbegriff und Funktionsbegriff, chs. 2 and 3.


In the essay that accompanies his translation of Kant’s ‘‘Uber Kastners
Abhandlungen’’, Michel Fichant discusses some of the analyses I pro-
posed in my book (KPJ).2 His discussion of my view centers on the
nature of space as put forth in the Critique of Pure Reason. More speci-
fically, it centers on the distinction between ‘‘form of intuition’’ and

´ ´ ´
Michel Fichant, ‘‘‘L’Espace est represente comme une grandeur infinie donnee’. La radi-
´ ´
calite de l’Esthetique’’ [‘‘‘Space is represented as an infinite given magnitude’: the radicality
of the Aesthetic’’], Philosophie, no. 56 (1997), pp. 20–48 (henceforth ‘‘‘L’Espace’’’). This
article follows Fichant’s presentation (pp. 3–12, henceforth ‘‘Presentation’’) and translation
into French of Kant’s essay ‘‘Uber Kastners Abhandlungen’’ (‘‘On Kastner’s articles’’) AAxx,
¨ ¨
pp. 410–23 (pp. 12–20). Kastner was a mathematician whose three articles (‘‘What does
possible mean in Euclid’s geometry?’’; ‘‘On the mathematical concept of space’’; ‘‘On the
axioms of geometry’’) were published in Eberhard’s Philosophisches Magasin, as part of
Eberhard’s attempt to prove the superiority of the Leibnizian view over the Kantian view
of mathematics. Kant counters Eberhard by claiming that Kastner’s view is in fact in
complete agreement with his own. Kant’s essay on Kastner’s articles contains some of his
most illuminating remarks on space as a pure intuition, and its relation to geometry.
Michel Fichant’s article is an analysis of Kant’s view of space in contrast to that of Kant’s
Leibnizian predecessors. Fichant’s discussion of my thesis concerning the relation between
‘‘form of intuition’’ and ‘‘formal intuition’’ according to Kant is only a subsidiary discussion,
occupying a few pages in the main article: see pp. 35–8 and passim. I found his discussion
insightful and challenging, I am grateful to him for giving me this occasion to attempt a
clarification of my view. In this English version of my response, when citing Fichant’s
discussion I will give his own references to KPJ, and then give the corresponding page in
KCJ. In the cases where references to my book are my own, I will refer only to KCJ.


‘‘formal intuition’’ (introduced by Kant in x26 of the Transcendental
Deduction in the second edition of the Critique), and on the phrase in
the Transcendental Aesthetic according to which space is ‘‘represented
as an infinite given magnitude’’ (B40). Michel Fichant thinks that the
explanation I propose for Kant’s distinction leads me to intellectualize
the forms of sensibility expounded in the Transcendental Aesthetic and,
true to a tradition begun by Fichte and represented in France by
Lachieze-Rey among others, leads me, in effect, to deny that Kant grants
any independence to sensibility with respect to the understanding.3
This last reproach surprises me. Recognizing the irreducible charac-
ter of sensibility in the Kantian conception of knowledge is of central
importance to the argument of my book. More particularly, I try to
elucidate the consequences of the irreducibly receptive character of
our sensibility for Kant’s conception of the logical-discursive forms
themselves, that is, of the forms of spontaneity. In fact, Michel Fichant
takes pains to make clear that his criticism concerns only a ‘‘side issue’’ in
my book, and in no way challenges its central thesis.4 But if he is right,
then this means that the thesis I am defending with respect to space and
time is incompatible with the theses I defend in the rest of my book. So I
still need to answer the charge that I might be taking back with one hand
what I had granted with the other.
Michel Fichant is certainly right to say that we disagree on the parti-
cular issues at hand (Kant’s distinction between ‘‘form of intuition’’ and
‘‘formal intuition,’’ and the role of imagination in our representation of
space and time). Yet I do not think that the position he attributes to me is
the one I defend. I think our disagreement concerns four main issues:
(1) Kant’s view of the relation between the functions of the understand-
ing and the forms of sensibility; (2) Kant’s footnote to x26 of the
Transcendental Deduction, where he introduces the distinction between
forms of intuition and formal intuition; (3) the meaning of the expres-
sion ens imaginarium, which Kant uses to describe space and time; (4) the
relation between space as quantum and the category of quantity (quanti-
tas). In what follows I will try to clarify my position on each of these points
and explain what I believe to be the nature of our disagreement.

Fichant, ‘‘‘L’Espace’,’’ p. 24, n. 11.
Ibid., p. 35, n. 30.

Understanding and sensibility
Michel Fichant thinks that my view is in some ways similar to that of
Cassirer, for whom ‘‘the functions of the understanding are the precon-
ditions for ‘sensibility’.’’5 Yet nowhere do I defend such a statement. On
the contrary, I expressly state that the radical distinction between sen-
sibility, endowed with forms specific to it, and the understanding, with its
logical forms or functions, is at the heart of Kant’s argument from one
end of the Critique to the other, and in particular in the sections that are
the main target of my investigation, the Transcendental Deduction of
the Categories and the System of Principles. Our real quarrel does not lie
here. Rather, it concerns the question whether I am right to maintain
that Kant’s presentation of space and time in the Transcendental
Aesthetic, while fully belonging in an aesthetic in Kant’s sense, that is,
in a science of the rules of sensibility or receptivity, is nevertheless seen in
a new light when the reference to the synthesis speciosa (i.e. the figurative
synthesis, or the transcendental synthesis of imagination), is introduced
into the argument of the Transcendental Deduction of the Categories.
In my account, it then appears that space and time, as described in the
Transcendental Aesthetic, certainly belong to sensibility, but to a sen-
sibility affected (and not generated, a point about which my position
differs from that of Fichte!) by spontaneity, that is, by the understanding.
In the Transcendental Aesthetic, Kant could not mention this ‘‘affection
by the understanding,’’ nor did he need to mention it. Indeed it appears
briefly only in the second edition, in an addition to the exposition of time
(B67–8). Kant did not need to mention it because what is important in
the Aesthetic is to show that space and time are originally intuitions
(‘‘singular and immediate representations’’) and not concepts (‘‘universal


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