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thinkers saw that all the inferences of mathematicians proceed according to the law of
contradiction, and wrongly slipped into thinking that mathematical truths were known
from the law of contradiction. This was a great mistake. The law of contradiction can lead
one to a synthetic proposition, but only from another synthetic proposition. (Still, it must
be borne in mind that mathematical propositions are always a priori judgments, not
empirical ones. They carry necessity with them, and that cannot be learned about from
experience. If you disagree, I shan’t argue; I shall merely make this claim about the
propositions of pure  that is, nonempirical  mathematics!) 10
One might think that the proposition 7 + 5 = 12 is analytic, and that it follows
according to the law of contradiction from the concept of
the sum of 7 and 5.
But if we look more closely, we find that the concept of the sum of 7 and 5 contains only
the uniting of 7 and 5 into a single number;
and in thinking this we do not have the least thought of what this single number is in which
the two are combined. I can analyse my concept of the uniting of seven and five as long as
I please  I shall never find 12 in it. I have to go outside these concepts and  with the help
of an intuition that corresponds to one of them (my five fingers for instance)  add the 5
given in intuition to the concept of 7, adding them one by one. Thus we really amplify our
concept by this proposition 7 + 5 = 12, and add to the first concept a new one that was
not thought in it. That is to say, arithmetical propositions are always synthetic. It will be
easier to grasp this if we take larger numbers. It is obvious that however we might turn
and twist our concept of
the sum of 38976 and 45204
we could never find 84180 in it through mere analysis, without the help of intuition.
[Kant’s use of the term ‘intuition’ needs to be explained; the explanation will occupy five
paragraphs.Traditionally, the word has had two meanings. ŸIn one it contrasts with
‘demonstration’  you know something intuitively if it is immediately selfevident to you,
whereas demonstrative knowledge involves a series of deductive steps. ŸIn the other
meaning  which alone is relevant to Kant  our faculty of ‘intuition’ is our ability to be
mentally confronted by individual things, to have in our minds representations of the
things and not merely of certain features or properties of them. Kant uses ‘intuition’ to
stand not just for the faculty but also for the mental representations that it involves. Thus,
for example, when you see the Lincoln Memorial you have an intuition of it, and this is an
exercise of your faculty of intuition. That intuition is a ‘sensible’ one, meaning that you
get it through your senses. It stands in contrast with a concept of the Lincoln Memorial such as the concept or abstract thought of a large white memorial to a great American
statesman. Having in your mind a (conceptual) representation of a large white memorial
etc. is quite different from having in your mind an (intuitive) representation of the Lincoln
Memorial, that one particular individual object.
[Now, Kant holds that we are also capable of having in our minds intuitions that do
not come from the senses; he calls them ‘pure’ or ‘a priori’ intuitions. When in the
previous paragraph he speaks of the intuition of my five fingers, that is a Ÿsensible
intuition: I feel or look at the fingers. But he believes  as we shall see in section 7  that
pure mathematics involves Ÿpure intuitions: for example, a geometer works out the
properties of circles not by merely taking the abstract concept circle and analysing it, but
by somehow giving himself a pure intuition of a circle, and working out the properties of
all circles from that. This is something like imaginingacircle, but it is not ordinary
imagination, which is copied from sense experience.
[The basic idea is something like this: Every time you see or feel something circular,
various aspects of your mental state are contributed by the sensations that come from
outside you, and others are contributed by your understanding, i.e. the conceptusing
faculty. If all of that were somehow stripped off, what would be left is a very thin, abstract 11
intuition of the circular thing just as a circle. That is, nothing would be left of it but its
purely spatial or geometrical properties; they will be the same for every circular thing; so
the stripped down intuition will be the same in each case. That stripped down intuition is
what Kant calls a ‘pure intuition’ of a circle. According to him, this is not contributed by
sensation from outside you; rather, it is conferred on your mental state by your own mind,
specifically by your own faculty of sensible intuition. You are so built, he thinks, that you
have to experience the world outside yourself as spatial, not because the outer world is
spatial but because you impose spatiality on the intuitions you have of it. Kant puts this,
sometimes, by saying that what is represented in a pure intuition is the form of your
sensibility or of your sensible i...
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 Spring '13
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 Philosophy, Kant, Metaphysics, Critique of Pure Reason, ........., Analyticsynthetic distinction

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