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experience to these principles. The understanding is the faculty of
concepts, and concepts are rules describing the nature of objects.
Even empirical laws of nature, although discovered a posteriori, express
necessary connections between features of objects. This necessity must
be grounded in principles governing the synthesis of all data given in
intuition. Thus pure principles are higher-order principles that govern
the application of speci¬c empirical concepts and laws to objects of
experience.
Following his distinction between mathematical and dynamical
categories in the Metaphysical Deduction, Kant distinguishes math-
ematical from dynamical principles. As we saw in chapter 4, at
B110 Kant labels the categories of quantity and quality mathematical
because they govern the operations that identify individual objects
and their properties in the data of intuition. The relational and modal
Schematism, Analytic of Principles I 145
categories are dynamical because they govern the relations of objects
to one another and to the subject. Kant now applies this distinction
to the principles. At A160/B199 he explains that mathematical princi-
ples pertain “merely to the intuition” of objects, whereas dynamical
principles pertain “to the existence of an appearance in general.” At
A178/B221 Kant also labels these constitutive vs. regulative principles.
In consequence, mathematical and dynamical principles differ in
their manner of evidence. The former are “unconditionally necessary”
and allow of intuitive certainty. By contrast, dynamical principles are
necessary “only under the condition of empirical thinking in an expe-
rience.” Therefore they lack “the immediate evidence that is character-
istic of the former” (A160/B199“200). As the Schematism points out,
the quantitative and qualitative features of objects governed by the
mathematical principles are exhibited in intuition. This is required
to intuit spatiotemporal objects at all. But the objective temporal
relations between states of affairs, and their relations to thinkers, are
merely thought. At A178“9/B221“2 Kant connects this point with the
fact that only the intuitions of objects, and not their existence, can be
constructed. That is, in intuition we are given a spatiotemporal array
to be discriminated into individual states of affairs, but we are not
given objective temporal positions and relations. Moreover, having
intuited an event, we can infer that it follows necessarily from some
prior state, but we cannot identify that state a priori. Now we turn
to the arguments for the mathematical principles in the Axioms of
Intuition and the Anticipations of Perception.

3 . th e a xioms of in tu i ti on
The synthetic a priori principles of quantity and quality govern the
mere intuition of objective states of affairs. The Axioms of Intu-
ition concern the synthesis of formal (spatiotemporal) properties;
they specify that to experience determinate objects, they must have
extensively measurable properties. The Anticipations of Perception
apply to the synthesis of the matter of intuition, the sensations corre-
lated with real properties of objects. They state that both sensations
and the properties corresponding to them must have some degree
of intensity. Although Kant refers to Axioms and Anticipations in
the plural, in fact there is only one principle for each heading. This
Schematism, Analytic of Principles I
146
is because the notions of extensive and intensive measurement each
incorporate all three categories under their respective headings.
Kant expresses the principle of the Axioms differently in the A
and B editions, but the point is the same.12 The A edition says, “All
appearances are, as regards their intuition, extensive magnitudes”
(A161). The B edition reads, “All intuitions are extensive magni-
tudes” (B201). Kant™s point is that appearances must have extensively
measurable spatiotemporal properties to be perceived as individual
objects or states of affairs. Thus the Axioms (and the Anticipations)
attempt to justify the application of pure mathematics to empiri-
cal objects. In the paragraph preceding the Axioms he explains that
these pure principles are not themselves mathematical principles, but
only principles “through which the former principles all acquire their
possibility” (A162/B202). This also explains the title “Axioms of Intu-
ition.” For although the Axioms are not themselves mathematical
axioms, they establish the validity of mathematical axioms for empir-
ical objects.13
The B edition contains a new argument in the ¬rst paragraph; the
A edition argument then follows in the second paragraph. Only the B
edition version refers to the schema of number, while the earlier ver-
sion focuses instead on the concept of extensive measurement. In both
cases, however, Kant argues that since the synthesis of space and time
underlies the synthesis of intuitions of objects in space and time, the
mathematical procedures governing the former must also apply to the
latter. The B edition argument explains that the synthetic processes
for representing determinate spaces and times require us to combine
the homogeneous spatiotemporal manifold into uni¬ed wholes. The
concepts governing this synthesis are the arithmetical concepts of
number. In other words, to perceive distinct empirical objects occu-
pying determinate spatiotemporal positions, the intuitions of these
objects must be extensively measurable. At B203 Kant concludes:
“appearances are all magnitudes, and indeed extensive magnitudes,
since as intuitions in space or time they must be represented through

12 I am indebted to Paton™s discussion, Kant™s Metaphysic of Experience, at 2:111“33.
13 Kant gives as axioms of geometry “space has only three dimensions” (B41), and “between
two points only one straight line is possible; two straight lines do not enclose a space, etc”
(A163/B204). For time: “It has only one dimension; different times are not simultaneous,
but successive” (B47).
Schematism, Analytic of Principles I 147
the same synthesis as that through which space and time in general are
determined.” As H. J. Paton points out, this implies that intuitions are
measurable as extensive quantities only insofar as they are intuitions
of objects. Dream objects and other “pseudo-objects of our imagi-
nation” cannot be measured since they do not occupy determinate
locations in objective space-time.14
The earlier version analyzes the notion of extensive measurement
and shows how it applies to space and time. Kant begins by de¬ning
an extensive magnitude as one in which the representation of the
parts precedes and makes possible the representation of the whole.
The key to the notion of extensive measurement is the successive addi-
tion of parts to generate a whole. When one draws a line, for example,
one begins at some point in space and then generates its parts suc-
cessively. Similarly, in thinking of determinate (measurable) times,
one thinks “the successive progress from one moment to another,”
whose addition produces a determinate duration (A163/B203). Exten-
sive magnitudes are those produced by combining or adding previ-
ously delineated parts. In a long footnote at B202, Kant labels a whole
of extensive parts an aggregate, and a whole of intensive parts a coali-
tion. The feature essential to extensive measurement is the addition
process; as we shall see below, degrees of intensity are not constructed
in the same way.
This helps clarify Kant™s conception of the connection between
the pure concepts of quantity and the schema of number. Recall from
chapter 4 that the quantitative logical concepts express the forms of
universal, particular, and singular judgments. The three categories
that result from schematizing these concepts are (respectively) unity,
plurality, and totality. When applied to objects, these categories make
it possible to measure spaces and times. For example, to measure the
length of an object or a time period, one must ¬rst select a unit of
measurement (e.g., a foot, a minute). Then one applies it repeatedly
as required to obtain the resulting magnitude, which is a totality
composed of a plurality of units. Extensive measurement consists in
adding the independently de¬ned units successively to arrive at the
resulting sum. This is the sense in which the representation of the parts
precedes the representation of the whole. Kant correlates the category

14 Paton, Kant™s Metaphysic of Experience, 2:120“1.
Schematism, Analytic of Principles I
148
of totality with the singular judgment (rather than the universal as one
might expect) because to discriminate or refer to individual objects
in space-time presupposes identifying de¬nite spatiotemporal regions
that are totalities measurable in terms of a plurality of units.15
The last two paragraphs of the section merely reiterate some views
of mathematics expressed earlier in the Introduction and the Aes-
thetic. Kant points out that arithmetic does not have general axioms
as such, but rather numerical formulae, which are singular judg-
ments although they are synthetic a priori. He also emphasizes the
main point of the argument, namely to demonstrate the objective
validity of procedures for measuring objects. If extensive measuring
processes did not apply to appearances, we could not determine their
spatiotemporal properties.
Many commentators criticize Kant for inconsistency, claiming that
the Axioms view that in measurement the parts precede the whole
contradicts his position in the Aesthetic, that space and time are given
as wholes that precede their parts.16 Following Paton, Melnick shows
that this charge is unfounded. In the Aesthetic, Kant is analyzing
the (pre-synthesized) data given in the pure forms of inner and outer
sense. As we saw in chapter 3, he holds that this indeterminate mani-
fold is given as a whole in which parts are discriminated by drawing
boundaries. By contrast, the argument in the Axioms addresses the
necessary conditions for constructing determinate regions out of this
pure manifold. As Melnick puts it, “The original representation of
space is required as a background for any construction in space and
thus cannot itself be constructed. Any spatial construction is in the
context of an original representation of unlimited space.” He points
out the role of the productive imagination in this process. What
grounds the application of geometry to spatial appearances is not
perception of shapes, but rules for constructing ¬gures: “Through
perception we become aware of how the shape of an object looks
(or feels), but not the rule of construction of the shape. This rule
(and what is necessarily bound up with this rule) is something that is

15 As Falkenstein points out, the natures of space and time constrain the construction of spatial
and temporal parts. The subject can choose the order of construction, but not the topology
or metric of space and time. See Kant™s Intuitionism, 244“7.
16 Three such commentators are Vaihinger, Kemp Smith, and Robert Wolff. See Melnick,
Kant™s Analogies of Experience, 18, for citations.
Schematism, Analytic of Principles I 149
brought to perception.”17 As Falkenstein says, the function of synthe-
sis is to turn a spatiotemporal array of representations (the data given
in intuition) into the representation of a spatiotemporal array.18 Far
from being incompatible, the doctrine of the Axioms completes the
analysis of spatial-temporal cognition begun in the Aesthetic.

4. th e a nticipations o f pe rce pt ion
This section is without doubt one of the most puzzling of the
Critique, for several reasons. First, Kant™s exposition does little to
explain the central notion of intensive magnitude. Second, the con-
clusion of the argument is not easy to identify, partly because of
changes in the two editions, and partly because of the terminology. In
particular, it is not clear whether the Anticipations principle concerns
sensations, or appearances, or both. Third, the arguments in both edi-
tions apparently depend on unjusti¬ed assumptions “ namely, that
sensations are caused by bodies outside us, and that physical interac-
tions between bodies are caused by intensive forces. In both cases the
argument would beg the question. And ¬nally, even if granted, these
assumptions are not suf¬ciently strong to demonstrate that either
sensations or properties of objects must be subject to a continuum
of degrees of intensity. No wonder most commentaries give the argu-
ment short shrift. Here I shall try to resolve some of these issues. As
we shall see, even by the most charitable reading, Kant cannot escape
some of these objections.19
One approach is to sketch the argument Kant ought to make at this
stage. Based on the Schematism and the Axioms, Kant needs to show
that only insofar as sensations are subject to procedures for measuring
intensive magnitudes can they be taken to give us information about
real properties of objects. Since the notion of an intensive magnitude
is the schema of the qualitative concepts, the argument will demon-
strate that these schematized concepts are necessary for “objectifying”
sensations. In short, the objectivity of the measuring procedure is
necessary to establish the objective reference of sensations. From this

17 See Melnick, Kant™s Analogies of Experience, 17“22.
18 Falkenstein, Kant™s Intuitionism, 249.
19 My interpretation has been greatly aided by discussions with Falkenstein.
Schematism, Analytic of Principles I
150
standpoint the argument parallels the Axioms argument, that the pro-
cedures of extensive measurement are necessary to confer objective
reference on the spatiotemporal features of appearances.
As with the Axioms, Kant revises the principle in the second edi-
tion, and adds a new proof at the beginning of the section. The A
edition principle states, “In all appearances the sensation, and the real,
which corresponds to it in the object (realitas phaenomenon), has an
intensive magnitude, i.e., a degree.” The B edition version reads:
“In all appearances the real, which is an object of the sensation,
has intensive magnitude, i.e., a degree” (A166/B207). Whereas the A
version claims that both sensations and real properties of objects must
have intensive magnitude, the B version appears to concern only the
objects of sensation. As suggested above, however, the point should
be that sensations must have a degree of intensity corresponding to an
intensive magnitude in the real properties of the object being sensed.
Thus the A edition version of the principle appears more precise.
Kant recognizes the paradox of an a priori principle “anticipating”
the nature of perception: “it seems strange to anticipate experience
precisely in what concerns its matter,” since this is given a posteriori
(A166“7). He does not directly answer the point until A175“6/B217“
18, where he distinguishes the quality of a sensation from its degree
of intensity. It is true that we cannot know a priori what qualities we
will sense; knowledge of sense qualities is contingent on the actual
experiences. What we can know a priori, however, is the “form” of
any sensation, namely that it must have a degree of intensity in order
to ¬ll space-time. As Paton puts it, the principle of the Anticipations
deals with the “form of the matter of appearance.”20
Before we examine the proofs we need to review some key terms,
previously discussed in chapter 3. Sensations are de¬ned in the Aes-
thetic as “the effect of an object on the capacity for representa-
tion, insofar as we are affected by it” (A19“20/B34). In chapter 3
we accepted Falkenstein™s view that sensations are modi¬cations of
the sense organs, and thus physical states. What Kant calls real of
sensation is the consciously represented sense quality, that which ¬lls
space and time. Color, sound, taste, odor, and warmth are examples of
sense qualities. Because qualities are the way we apprehend sensations,

20 See Kant™s Metaphysic of Experience, 2:134“5n5.
Schematism, Analytic of Principles I 151
there is a correspondence between the quality and the sensation. For
this reason Kant slides easily from talk about sensation to talk about
the quality of sensation, as at A175“6/B217“18.
Recall that appearance is the “undetermined object (Gegenstand)
of an empirical intuition” (A20/B34), where Gegenstand refers to an
existing thing. Thus appearances are whatever is given in intuition.
Now the term the real in appearance is ambiguous: it could refer
either to the represented quality or to the properties of objects. In the
Principles, Kant characterizes the real in appearance as that “which
corresponds to [the sensation] in the object” (A edition) and that
“which is an object of the sensation” (B edition). I agree with Paton
that for Kant the real in appearance are the properties of matter as
determined by empirical science.21 Twice Kant gives as an example of
a degree of reality the moment of gravity, certainly a scienti¬c notion
(A168“9/B210“11). These properties may or may not resemble the
consciously represented sense qualities.22
The A edition proof from A167“9/B209“10 proceeds as follows:
1. Apprehension by means of sensation is instantaneous, i.e., it does
not take time.
2. Therefore, apprehension in sensation is not a successive synthesis
proceeding from the parts to the whole.
3. Therefore, the thing apprehended in sensation does not have exten-
sive magnitude.
4. That in empirical intuition which corresponds to sensation is
reality; that which corresponds to its absence is negation.
5. Every sensation is capable of diminishing gradually until it disap-
pears.
6. Therefore, between reality in appearance and negation there is a
continuum of sensations, such that between any two sensations
there is always a sensation; there is no smallest possible sensation.
7. Thus the real in appearance always has a magnitude which is not
extensive.
21 See Kant™s Metaphysic of Experience, 2:137“8. I disagree, however, with Paton™s identi¬cation
of sensation with “the sensum considered as modi¬cation of the mind.”
22 Given the primary“secondary quality distinction, Kant believes that sense qualities do not
resemble the real properties in objects causing them. Moreover, MFNS presents a dynamical
theory of matter in which the ultimately real properties are fundamental forces of repulsion
and attraction.
Schematism, Analytic of Principles I
152
8. A magnitude which can only be apprehended as a unity, and in
which multiplicity can only be represented through approximation
to zero, is an intensive magnitude.
9. Therefore, every reality in appearance has intensive magnitude,
i.e., a degree.
This version begins with the premise that apprehension in sensation
is instantaneous. Although Kant does not defend it, it is plausible as
an account of the perception of something occupying space-time.
Either the senses are affected or they are not. In apprehension the
understanding “takes up” this sensory material into consciousness
and presents it as a sense quality. Thus it would seem to be an all-or-
nothing affair. Conclusions 2 and 3 follow from this and the Axioms
view that the synthesis required to represent extensive magnitudes
takes time because the whole is generated from the parts. Therefore
the instantaneous apprehension of sensation cannot take place by
means of such a synthesis. In consequence what is apprehended in
sensation cannot have extensive magnitude.
Premise 4 introduces the concepts of reality and negation by estab-
lishing the “fact” on which the Transcendental Deduction depends,
namely that we take sensations to represent real properties of objects
given in intuition. Reality is that in the object which corresponds
to the sensation; negation represents its correlate, the absence of a
property. The key to the argument is premise 5, the controversial
claim that every sensation can diminish gradually until it disappears.
Unfortunately Kant offers no support, and it is not obvious how to
defend it. Some commentators see it as based on Kant™s physics, which
explains the impenetrability of matter by intensive forces of repulsion
and attraction. This reading reverses the order of argument, however,
since the Principles provide necessary conditions for experience and
consequently are presupposed by empirical laws and theories. Kant
says this explicitly in both the MFNS and in a discussion of the pos-
sibility of empty space and time near the end of the Anticipations.
(We will look at this passage below.)
An alternative reading bases premise 5 on the phenomenology of
sensation and the notion of something ¬lling time. It is a fact that
at least some aspects of sense qualities can diminish gradually. The
brightness of light and the loudness of sound are two such aspects.
Premise 5 makes only the weak claim that it is possible for sensations
Schematism, Analytic of Principles I 153
to diminish gradually until they disappear. Admitting the slide from
sense qualities to sensations, this could be justi¬ed as a brute fact about
sensory consciousness. Based on these examples, one might argue that
the notion of something ¬lling time entails the possibility in principle
that it can diminish gradually to nothing. A problem with this defense
concerns a scope ambiguity. One reading would take the “possibility”
in a weak sense, so that it is possible for every sensation to diminish
gradually, although perhaps some in fact do not. Kant™s conclusion,
however, apparently requires the strong sense that every sensation
is such that it can diminish gradually. We shall return to this point
below. In any case, the phenomenological approach has the advantage
of not basing Kant™s view of sensation on a particular physical theory.
As we shall see, the B edition version supports this reading.
The rest of the proof follows from the analysis of intensive mag-
nitude. Statements 6 and 7 are conclusions from premise 5. In line 6
Kant states that both sensations and the “real in appearance” have a
magnitude such that they can diminish continually to nothing. He
then concludes in line 7 that the real in appearance has a non-extensive
magnitude. Premise 8 de¬nes intensive magnitudes as those appre-
hended only as unities, and in which the parts can be represented
only “through approximation to negation.” Kant then concludes that
every reality in appearance has some degree of intensity.
The gist of the argument is this: in order to represent objects ¬lling
time (and space), we must conceive of sensations and the real prop-
erties of objects as having some degree of intensity. This presupposes
a conceptual scheme that permits us to take sensations to correspond
to real properties of objects. On Kant™s view this is the function of
the schema of intensive measurement. It is the correspondence in
intensity that makes it possible for sensations to represent empirically
real objects.
The argument in the B edition is essentially the same, although
Kant uses the technical vocabulary of matter and form, and empha-
sizes the synthesis of the understanding in relating sensations to
objects. The proof is this:
1. Perception is empirical consciousness in which there is sensation.
2. As objects (Gegenst¨ nde) of perception, appearances are not pure
a
intuitions like space and time, but contain the matter for an object
in general.
Schematism, Analytic of Principles I
154
3. Through this matter something is represented as existing in space
and time.
4. This matter is the real of sensation, which is a merely subjective
representation by which one is aware of being affected.
5. This matter is related to an object in general (by a synthesis of
the understanding).
6. From empirical consciousness to pure consciousness a gradation
is possible in which the real disappears and a merely formal con-
sciousness of the spatiotemporal manifold remains.
7. Thus there is also possible a synthesis in representing the magni-
tude of sensation from 0 to any arbitrary magnitude.
8. Sensation is not in itself an objective representation, since neither
space nor time is found in it.
9. Therefore, it has no extensive magnitude.
10. But it does have a magnitude (such that through its apprehension,
empirical consciousness can grow in a certain time from 0 to a
given measure).
11. Therefore, it has an intensive magnitude.
12. Corresponding to this all objects of perception, insofar as it con-
tains sensation, must be ascribed an intensive magnitude, i.e., a
degree of in¬‚uence on sense (emphasis added).23
This version begins with the analysis of empirical consciousness.
Premises 1“4 are based on the Aesthetic and so should not be contro-
versial. Here Kant emphasizes that perceiving objects involves sensing
something ¬lling space-time, which he calls the matter of appearance.
This matter, which we consciously apprehend as sense qualities (the
real of sensation), makes possible our awareness of things existing in
space and time. Premise 5 restates this as the transcendental “fact”
that we relate this matter to an object in general. Since an object in
general is the judgmental notion of an object, Kant is here reminding
us that the understanding refers the matter to an object by means of
pure concepts.
At line 6 Kant introduces the notion of the gradual diminution in
the degree of reality through an analysis of empirical consciousness.
23 I emphasize “it” to indicate that I have altered Guyer and Wood™s translation. They take
the pronoun diese to refer to objects of perception rather than perception. But the verb for
“contain” is in the singular “ enth¨ lt “ rather than the plural; moreover, it makes no sense
a

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