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extraction of roots. See Kant and the Exact Sciences, 108“12.
Reason and the critical philosophy
294
A E
a


a'
b'
b c
B C D
Figure 11.1

through which general synthetic propositions must be constructed.”
Lisa Shabel explains his point.9 She argues that the empirical pro-
cedure is modeled in Christian Wolff™s “mechanical” demonstration
of the angle-sum theorem (that the sum of the angles of a triangle
equals 180o ), in his Mathematisches Lexicon. There Wolff constructs
the triangle ABC with angles a, b, c. (See Figure 11.1.)
He then uses a compass to “carry” the arcs describing angles a and
b along the line BD, creating angle a equal to a, and angle b equal to
b. He then concludes that the three interior angles equal 180o . As Sha-
bel explains, this “demonstration” amounts to a measurement of the
interior angles by fallible tools, and depends on visual inspection to
determine equality of the angles. The resulting judgment “is an empir-
ical assessment based on the features of the particular constructed
triangle; the skill of the geometer who ˜carries™ the arcs; and the pre-
cision of the tools used to do so.”10 In consequence, the conclusion
that the interior angles sum to two right angles is only a “metric judg-
ment” concerning a particular empirical triangle, lacking the univer-
sality and necessity required for a mathematical demonstration.
Euclid™s own demonstration, by contrast, represents the a priori
method establishing the necessity and universality of the angle-sum
theorem. In it, the geometer
extends one side of his triangle, and obtains two adjacent angles that together
are equal to two right ones. Now he divides the external one of these angles
by drawing a line parallel to the opposite side of the triangle, and sees that
here there arises an external adjacent angle which is equal to an internal one,
etc. (A716/B744)
9 See Shabel, “Kant™s ˜Argument from Geometry™,” 209“13.
10 Shabel, “Kant™s ˜Argument from Geometry™,” 211.
Reason and the critical philosophy 295
That is, Euclid proceeds by ¬rst extending line BC to D, then con-
structing line CE parallel to line AB. Since AC is a transversal, angle
a is equal to angle a, and since BD is a transversal, angle b is equal
to angle b. Thus the demonstration shows that the interior angles
of triangle ABC are equal to the three angles lying on BCD, and
consequently to two right angles or 180o . As Shabel points out, this
proof depends not on visual inspection or empirical procedures, but
only on the judgment of “containments among spatial regions” which
depends on “prior stipulations for constructing spatial regions,” avail-
able only through the pure intuition of space.11 Thus the diagram rep-
resents only the a priori act, which, Kant says, “considers the concept
in concreto, although not empirically” (A715/B743).
Owing to the constructibility of concepts in pure intuition, “Math-
ematics is thoroughly grounded on de¬nitions, axioms, and demon-
strations” (A726/B754). In all three respects it differs from philosophy,
which, as we have seen, cannot exhibit its objects a priori in intuition.
At A722/B750 Kant characterizes a transcendental proposition of phi-
losophy as “a synthetic rational cognition in accordance with mere
concepts, and thus discursive, since . . . no intuition is given by it
a priori.” Not only can philosophy not demonstrate its propositions
from the mere analysis of concepts, it cannot even provide clear def-
initions of its terms.
The most original part of Kant™s analysis is his theory of de¬nition.
At A728/B756 he contrasts real de¬nition, analyzing the concept of
a thing, with nominal de¬nition, de¬ning a word or “designation.”12
Now “to de¬ne properly means just to exhibit originally the exhaus-
tive concept of a thing within its boundaries” (A727/B755). By con-
trast, the “explication” or “exposition” of a concept merely identi¬es
some marks thought in the concept of a thing. It is no surprise to ¬nd
that empirical concepts cannot be de¬ned exhaustively; not only do
different persons think different marks with respect to the concept,
but an exhaustive analysis depends on experience:
Thus in the concept of gold one person might think, besides its weight, color,
and ductility, its property of not rusting, while another might know nothing
about this . . . And in any case what would be the point of de¬ning such a

11 Shabel, “Kant™s ˜Argument from Geometry™,” 212.
12 See Carson for a helpful discussion of real and nominal de¬nition, “Kant on the Method of
Mathematics,” 648.
Reason and the critical philosophy
296
concept? “ since when, e.g., water and its properties are under discussion,
one will not stop at what is intended by the word “water” but rather advance
to experiments. (A727“8/B755“6)
Despite their a priori origin, philosophical concepts are also not
de¬nable, because pure concepts of the understanding and reason
are “given” rather than made arbitrarily:
Strictly speaking no concept given a priori can be de¬ned, e.g., substance,
cause, right, equity, etc. . . . But since the concept . . . as it is given, can
contain many obscure representations, . . . the exhaustiveness of the analysis
of my concept is always doubtful, and . . . can only be made probably but
never apodictically certain. (A728“9/B756“7)
Pure concepts arise in the activity of judging, and are “given” as
concepts of synthetic functions. The concepts, like these functions,
are too indeterminate to specify their objects.
This leaves only arbitrary concepts that can be de¬ned, since “I
must know what I wanted to think, since I deliberately made it up,
and it was not given to me either through the nature of the under-
standing or through experience” (A729/B757). But even here there are
limitations, for “if the concept depends upon empirical conditions,”
one cannot be certain that it has an object. For example, my ability to
de¬ne the concept of a spiritual substance does not guarantee its exis-
tence. The only arbitrary concepts that guarantee the existence of their
objects are geometric, precisely because they can be constructed a pri-
ori, “and thus only mathematics has de¬nitions. For the object that it
thinks it also exhibits a priori in intuition, and this can surely contain
neither more nor less than the concept, since through the explana-
tion of the concept the object is originally given” (A729“30/B757“8).
Mathematical concepts are de¬nable because they are constructible
a priori in pure intuition. The form of intuition constrains the arbi-
trariness of the concept, while its construction ensures the existence
of the object. As Emily Carson points out, because construction is a
synthetic process, mathematical de¬nitions are synthetic rather than
analytic.13 On Kant™s view, de¬nition is the beginning point in mathe-
matics, whereas in philosophy, de¬nition “must conclude rather than
begin the work” (A730“1/B759“60).

13 See Carson, “Kant on the Method of Mathematics,” 648.
Reason and the critical philosophy 297
The constructibility of mathematical concepts also confers the sta-
tus of axioms on fundamental mathematical propositions. Axioms
“are synthetic a priori principles, insofar as they are immediately cer-
tain” (A732/B760). Now although philosophy has synthetic a priori
principles, these are discursive, i.e., “rational cognition in accordance
with concepts” (A732/B760). But synthetic judgments are always
based on a “third, mediating cognition,” since they cannot be obtained
from mere concepts. The principle that everything that happens has a
cause, for example, can be justi¬ed only in relation to “the condition of
time-determination in an experience” (A733/B761). For mathematics,
construction in pure intuition allows connecting the predicates both
a priori and immediately (A732/B761). The axioms of geometry are
just the fundamental principles of construction, such as “three points
always lie in a plane” (A733/B761). Kant also remarks that the princi-
ples of extensive measurement labeled the Axioms of Intuition are not
themselves axioms, but principles demonstrating the applicability of
mathematical axioms to objects of experience (A733/B761). Although
he does not say so here, Kant also claims in that section of the Analytic
that arithmetic and algebra lack axioms. There, at A163“4/B204, he
says “the self-evident propositions of numerical relation . . . are, to
be sure, synthetic, but not general, like those of geometry, and for
that reason also cannot be called axioms.” This is related to the view
discussed above, that arithmetic and algebraic formulas are rules for
calculating quantities in general.
Finally, only mathematical principles can be demonstrated. A
demonstration is “an apodictic proof, insofar as it is intuitive”
(A734/B762). Because mathematics derives its cognition from the
construction of concepts, “i.e., from the intuition that can be given a
priori corresponding to the concepts” (A734/B762), its non-axiomatic
principles deserve the title of theorems. As we saw above, philosophi-
cal principles such as the principle of causality cannot be presented in
intuition a priori, but require a transcendental deduction which must
appeal to the necessary conditions of experience. In consequence,
Kant says, philosophical principles should be labeled “dogmata” rather
than theorems (A736/B764). Despite this label, there is no room for
dogmatic methods in philosophy, since the attempt to prove spec-
ulative principles directly “merely masks mistakes and errors, and
deceives philosophy” (A737/B765).
Reason and the critical philosophy
298
This last point becomes the focus of the second section of the Dis-
cipline of Pure Reason, where Kant argues eloquently for the critical
method based on the autonomy of reason. Just as citizens of a free state
legislate for themselves, the “very existence of reason depends upon
this freedom” (A738/B766), since any external constraint effectively
negates the function of reason. Although reason cannot establish its
claims dogmatically, it can use polemics to defend itself against dog-
matic claims to the contrary. Kant brie¬‚y returns to the worry that
reason could be divided against itself, reminding us that even the
antithetical claims of the Antinomies are not genuine contradicto-
ries, since the transcendental distinction between appearances and
things in themselves dissolves the apparent contradiction. Similarly,
the illusory arguments concerning God and the soul violate the con-
clusion that knowledge is only of appearances. Thus the debates of
dogmatic metaphysics are resolved by the critical power of reason
to correct itself. As for skepticism, Kant reiterates many of his crit-
icisms of Hume, and particularly Hume™s failure to recognize the a
priori contributions of the sensibility and the understanding. Any
parent will appreciate Kant™s clever comparison of dogmatism and
skepticism to the psychological stages of childhood and adolescence:
“The ¬rst step in matters of pure reason, which characterizes its
childhood, is dogmatic. The just mentioned second step is skep-
tical, and gives evidence of the caution of the power of judgment
sharpened by experience.” The critical method characterizes mature
reason, which “subjects to evaluation not the facta of reason but rea-
son itself, as concerns its entire capacity and suitability for pure a
priori cognition” (A761/B789). The ¬nal two sections apply Kant™s
conclusions on the power of reason to the use of hypotheses and
proofs.

3. th e d octrine of me th od : th e ca non
of pu re re a s on
The last section deserving discussion is the Canon, originally intended
as a metaphysics of morals. A canon is “the sum total of the a priori
principles of the correct use” of cognitive faculties (A796/B824). Here
he places his analysis of theoretical reason in the context of reason
in general, emphasizing the primacy of practical reason. Through a
Reason and the critical philosophy 299
discussion of the interests of reason, Kant sketches his conception
of the highest good, explaining the relations between morality, hap-
piness, and the ideas of God and the immortality of the soul. This
account presupposes the role of transcendental idealism in making
the realm of nature compatible with the demands of the moral law.
In the ¬rst section Kant inquires about the origin of the natural
tendency of reason “to venture to the outermost bounds of all cogni-
tion by means of mere ideas in a pure use” (A797/B825). Assuming
a uni¬ed function and purpose of natural faculties, the highest ends
of reason must be practical, and its theoretical use subordinated to
its practical use. The ¬nal aim of speculation, he says “concerns three
objects: the freedom of the will, the immortality of the soul, and the
existence of God” (A798/B826). But as Kant has shown, theoretical
reason cannot achieve cognition of any of these objects. Empirical
investigation must proceed on the assumptions that all phenomena
are caused, that substances are material, and that the only neces-
sities are changes of phenomenal states in accordance with causal
laws. With respect to speculative reason, these three propositions
are transcendent, that is, “considered in themselves, entirely idle”
(A799/B827).
Only practical reason can produce “pure laws determined com-
pletely a priori,” having more than merely regulative status, “which
do not command under empirical conditions but absolutely”
(A800/B828). These, of course, are the moral laws, which “concern
our conduct in relation to the highest end.” Thus the ultimate aim
of reason concerns “what is to be done if the will is free, if there is a
God, and if there is a future world.” It follows that “the ultimate aim
of nature which provides for us wisely in the disposition of reason
is properly directed only to what is moral” (A801/B829). In the ¬nal
analysis, because reason itself is a unity, and its highest ends are prac-
tical, the value of theoretical reason resides in its service to practical
reason.
Kant next sketches the idea of practical freedom as the capacity to
choose independently of necessitation by sensible impulses or desires.
At A802/B830 he contrasts the animal will (arbitrium brutum), whose
power of choice is causally determined by “sensible impulses,” i.e.,
instincts or desires, with the free will (arbitrium liberum), which can
choose based on a concept of the good. Experience proves that humans
Reason and the critical philosophy
300
have free will, and can exercise practical freedom, since they can
conceive of an objective good, and evaluate their desires accordingly.
In recognizing the necessity of the moral law, in conceiving how one
ought to act, the human will demonstrates its independence from
natural necessitation and thus practical freedom.
Now at A803/B831 Kant makes the claim discussed in chapter 9,
that the existence of practical freedom does not prove the reality of
transcendental freedom, the power to initiate a series spontaneously,
independent of all causal in¬‚uences. This is because the speculative
question remains, “whether in these actions, through which it pre-
scribes laws, reason is not itself determined by further in¬‚uences, and
whether that which with respect to sensory impulses is called freedom
might not in turn with regard to higher and more remote ef¬cient
causes be nature.” Here in the Canon Kant characterizes practical
freedom “as one of the natural causes, namely a causality of reason in
the determination of the will.” So although transcendental freedom
is “contrary to the law of nature,” it is a problem only for theoretical
reason. We saw in the Fourth Antinomy how transcendental idealism
provides the solution.
In the second section Kant lays out his conception of the highest
good, and the relation between morality and happiness. He begins at
A804“5/B832“3 with the three questions addressing the interests of
reason:
1. What can I know?
2. What should I do?
3. What may I hope?
The ¬rst question concerns only speculative reason, and is answered
in the critical theory of knowledge. The second is a question for prac-
tical reason; the third, which “is simultaneously practical and theoret-
ical,” introduces the notion of happiness. De¬ning happiness as “the
satisfaction of all our inclinations,” Kant distinguishes between “prag-
matic” laws aiming at happiness and the moral law, which is motivated
by “the worthiness to be happy” (A806/B834). Whereas pragmatic
laws are empirically based, depending on both the agent™s subjec-
tive inclinations and experience of causal connections, the moral law
“abstracts from inclinations and natural means of satisfying them,
and considers only the freedom of a rational being in general and the
Reason and the critical philosophy 301
necessary conditions under which alone it is in agreement with the
distribution of happiness in accordance with principles.” Thus only
the moral law can be known a priori and commands absolutely.
In the remainder of this section Kant introduces the foundation of
a “moral theology” in order to solve two problems. First is the general
problem of systematic unity mentioned in chapter 10: what guarantees
that the world of nature will permit moral action? The other is how
to provide an incentive to the rational agent to act morally: what
guarantees that doing the right thing will result in happiness? The
solution requires postulating the existence of a morally perfect being,
whose divine wisdom and benevolence ensure the ef¬cacy of moral
action as well as a morally just distribution of happiness in a future
world.
In the ideal moral world, free rational agents all act in conformity
to the moral law. Each action has a “thoroughgoing systematic unity
in itself as well as with the freedom of everyone else” (A808/B836).
Although this intelligible notion abstracts entirely from empirical
conditions, it nonetheless has objective reality as a standard for human
action in the sensible world. Thus it answers the question, “What
should I do?” and so provides a model for worthiness to be happy.
But it does not explain what guarantees that moral choices will be
effective or why one should hope to be happy. The solution to these
problems lies in the “ideal of the highest good” (A810/B838). This
is the idea of a divine intelligence, God, who ensures that the natural
order will be consistent with the moral order, and that those who are
worthy attain the happiness they deserve. Kant says that moral ends
would not be attainable unless some ef¬cient cause determined for
moral conduct “an outcome precisely corresponding to our highest
ends, whether in this or in another life. Thus without a God and
a world that is now not visible to us but is hoped for, the majestic
ideas of morality are, to be sure, objects of approbation and admira-
tion but not incentives for resolve and realization” (A812“13/B840“1).
On Kant™s view, the “complete” good for rational agents requires both
moral conduct and the happiness a worthy agent deserves: neither the
happy immoral agent nor the unhappy moral agent satis¬es our con-
ception of a just world. And, obviously, happiness is not distributed
according to moral worth in the world of appearances. But the moral
law is absolutely necessary. Consequently, to avoid regarding moral
Reason and the critical philosophy
302
laws “as empty ¬gments of the brain” (A811/B839), the rational agent
must presuppose ¬rst, that moral action is ef¬cacious in the present,
and second, that it will be rewarded in a future life, if not in this
one.
In this moral theology, God is the “single, most perfect, and ratio-
nal primordial being” whose supreme will is the source of natural as
well as moral laws. God has the traditional attributes of omnipotence,
omniscience, omnipresence, and is eternal. Because the systematic
unity of ends requires us to regard the laws of nature as if they were
commands of this divine will, we are also justi¬ed in representing
nature as “a system of ends,” having a purposiveness “inseparably
connected a priori to the inner possibility of things” (A816/B844).
Although we must postulate a divine intelligence as the source of
both natural and moral orders, it is a mistake to regard moral obliga-
tion as grounded in God™s commands: “we will not hold actions to be
obligatory because they are God™s commands, but will rather regard
them as divine commands because we are internally obligated to
them” (A819/B847). Effectively responding to the dilemma in Plato™s
Euthyphro, Kant maintains that we must regard the goodness of moral
action as independent of God™s will, based rather in the conception
of a rational agent.
The Canon ends by characterizing the nature of belief in this moral
theology. First Kant classi¬es different ways of believing or “taking
to be true.” Conviction occurs when belief has objectively suf¬cient
grounds, persuasion when the grounds are only subjectively suf¬cient.
Whereas the former has public validity, allowing “the possibility of
communicating it and ¬nding it to be valid for the reason of every
human being,” the latter has only private validity (A820/B849). This
distinction gives rise to three stages in relation to conviction: “having
an opinion, believing, and knowing” (A822/B850). In having an
opinion, one is conscious that its grounds are both “subjectively as
well as objectively insuf¬cient”; i.e., one cannot defend the view.
Believing occurs when one has only subjectively suf¬cient grounds
which one recognizes as objectively insuf¬cient. Knowing, of course,
requires both subjectively and objectively suf¬cient grounds. Kant™s
point is to differentiate moral and theological belief from theoretical
claims to knowledge. Since theoretical claims allow of objectively
suf¬cient grounds, judgments of theoretical reason make claims to
Reason and the critical philosophy 303
knowledge. But despite Kant™s defense of his moral theology, “no one
will be able to boast that he knows that there is a God and a future
life; for if he knows that, then he is precisely the man I have long
sought” (A828“9/B856“7). Nevertheless, “the belief in a God and
another world is so interwoven with my moral disposition that I am
in as little danger of ever surrendering the former as I am worried that
the latter can ever be torn away from me” (A829/B857). In this way
Kant resists the temptation to con¬‚ate practical assumptions with the
cognitive claims of theoretical reason.

4 . su mm a ry
The ¬nal sections of the Critique “ the Appendix to the Critique of
Speculative Theology and the Transcendental Doctrine of Method “
highlight the positive role of the ideas of reason, and the relation
between theoretical and practical reason. Kant also elaborates his the-
ory of mathematics in his critique of philosophical methods. In the
Appendix, Kant explains the regulative function of theoretical rea-
son in providing both a stimulus and methodological guidelines for
empirical inquiry. This analysis completes his revolutionary theory of
the intellect, rejecting the traditional views that conceiving is prior to
judging, and judging prior to reasoning. In contrasting philosophical
and mathematical methods in the Discipline of Pure Reason, Kant
¬lls in the theory of mathematical construction sketched in the Aes-
thetic. Because mathematical concepts originate in pure intuition,
they allow of a priori construction. As a result, mathematics begins
with de¬nitions, contains axioms, and produces demonstrations of its
theorems. Philosophy, by contrast, operates with discursive concepts,
which cannot be constructed or even de¬ned. As a result, philosoph-
ical principles lack the character of axioms and theorems; they can
be justi¬ed only indirectly, through transcendental deductions. Thus
dogmatic metaphysicians who claim immediate certainty for their
principles are mistaken.
Kant originally intended the ¬nal substantive discussion, in part
II of the Canon, as a metaphysics of morals, parallel to the transcen-
dental doctrine of theoretical reason. Here he argues that experience
proves that humans have practical freedom, the ability to choose inde-
pendently of sensuous impulses and desires. Connecting morality
Reason and the critical philosophy
304
with the transcendental ideas of reason, he argues that practical free-
dom requires us to postulate the existence of God and the immortality
of the soul, to guarantee that moral action will be effective in the sen-
sible world, and that the morally worthy agent will ¬nd happiness
in a future life. Recognizing the shortcomings of this account, Kant
published the Groundwork of the Metaphysics of Morals in 1785, and
the Critique of Practical Reason in 1788, containing his mature theory
of the autonomy of practical freedom.
Conclusion: Kant™s transcendental idealism




To ¬nish, let us return to the questions raised in chapter 3 about
the coherence and defense of Kant™s idealism. Since in section 4 of

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