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PHL100Y1
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Peter King
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Philosophy

PHL100Y1

Peter King

Winter

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KANT: THE POSSIBILITY OF PURE MATHEMATICS
[1] The question at issue in the First Part of the Prolegomena is how pure mathematics is possible.
It will help enormously if we get two distinctions under our belt right away. The ﬁrst is that
Kant draws a sharp distinction between intuition and understanding. The former refers to our
capacity for direct (sensory) awareness, and that of which we are aware is “presented” in the
intuition; the latter refers to our capacity for thinking, which again may “present” an object
(these are very roughly parallel to Hume’s distinction of impressions and ideas). Second, Kant
talks of both the form and the matter of experience. In this case, any feature that an experience
has to satisfy in order to count as that kind of experience is part of its form, otherwise it is part
of its matter. Consider the following claim: “Any physical object is three-dimensional”. Well,
three-dimensionality is part of the form of our experience of physical objects, since what it is to
be a physical object is to occupy (three-dimensional Euclidean) space.
[2] Kant wants to establish four main claims in Part One:
(M1) Geometry and Arithmetic consist of a priori propositions.
(M2) Geometry and Arithmetic consist of synthetic propositions.
(M3) Geometry and Arithmetic require pure intuition.
(M4) Geometry and Arithmetic get their pure intuitions from the forms of sensibility.
It is very easy to lose the forest for the trees in Kant, so it’s worth reminding ourselves just what
he is driving at. Remember, Kant wants to establish how mathematics can be both necessary
and informative. The former he thinks is a matter of its being a priori, the latter of its being
synthetic. So he has to convince us that arithmetic and geometry are somehow true in virtue of
including information that isn’t derived from experience.
[3] Kant simply cites the necessity of mathematics as grounds for (M1) [294]. His train of thought
seems to run as follows. (See Kant’s remarks in §15 (294): “But experience teaches us what exists
and how it exists, but never that it must necessarily exist so and not otherwise.”) Mathematical
propositions are necessary propositions: it is in no way contingent that (say) the interior angles
of a triangle are 180 . Now in experience we only ever meet with contingencies, not necessities,
as Kant takes Hume to have established. Even if we should encounter necessary truths, they
do not present themselves to us as such. But you can’t build necessary structures on contingent
foundations. Hence the necessity that undeniably attaches to mathematical truths cannot be
derived from or based on experience. Thus such truths are independent of experience, that is,
they are a priori.
[4] As regards (M2): To show that mathematical propositions are synthetic, Kant argues as
follows:
(M2a) All constructive connections are synthetic.
(M2b) All mathematical propositions are constructive.
The argument is contained in §7, in reverse order. Consider (M2a), the claim that all constructive
connections are synthetic. Recall that Kant initially deﬁnes a synthetic proposition as one in
which the concept corresponding to the predicate ampliﬁes the concept corresponding to the
subject. Now one way in which such an ‘ampliﬁcation’ can take place is when the subject and predicate are conjoined in a single object of experience [281]:
Empirical intuition enables us without difﬁculty to enlarge the concept which we frame
of an object of intuition by new predicates which intuition itself presents synthetically
in experience...
Experience, therefore, is one way to connect the concept of the subject with the concept of
the predicate. But there is another way to do so, namely by construction. In its most general
terms, a ‘construction’ is the introduction of something new, typically via a representation, that
serves to link the subject-concept with the predicate-concept. Kant obviously has geometrical
construction in mind here. We shall see how this process is to work in a moment, but this
characterization of a ‘constructive connection,’ general as it is, should serve to establish (M2a),
the claim that constructive connections are synthetic. For by the introduction of something
new we clearly go outside the subject-concept, and hence the proposition linking the subject to
the predicate is synthetic. Establishing (2b) is more difﬁcult. For geometry, think of geometrical
construction proofs. For arithmetic, Kant says that even in a simple equation like 7 + 5 = 12
we have recourse to some intuition, since we establish its truth by enumeration. Thus:
(1) Interpret numbers as quantities of units.
(2) To determine which quantity corresponds to a given number, one must enumerate the
units.
(3) Interpret the operation of addition as quantity-combination, or, equally, as successive
enumeration.
Let E() be the enumeration-function. Then the reason for the dual characterization in (3) is that
Kant takes the following to be the case:
E(5) + E(7) = E(5 + 7) = E(12)
and the fact that the function E() distributes over addition is a non-trivial res

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