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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

¨
7
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.
REVISITING
46

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 (¼ 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
SYNTHESIS AND LOGICAL FORMS 47

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
application.


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.


8
See A25/B39“40; A142/B182.
9
See KCJ, p. 220.
REVISITING
48

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

10
KCJ, p. 267.
11
See Friedman, ˜˜Logical forms,™™ p. 206.
SYNTHESIS AND LOGICAL FORMS 49

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
REVISITING
50

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
SYNTHESIS AND LOGICAL FORMS 51

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.
(A164/B205)

(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

12
˜˜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.
REVISITING
52

(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
intuition.
Let me now consider Michael Friedman™s second example, my treat-
ment of the relational categories: substance, causality, and universal
interaction.


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
SYNTHESIS AND LOGICAL FORMS 53

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


13
See KCJ, pp. 334“7.
REVISITING
54

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

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