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[For the geometer to establish synthetic truths about circles, Kant holds, he must not
only have Ÿthe concept circle but must also have Ÿa pure intuition of a circle. This pure
intuition, he sometimes says, exhibits the concept; it illustrates or exemplifies it; it shows
the geometrician what a circle is, taking him from the merely conceptual thought of circles
to a kind of abstract non-sensory view of a circle.
[The same story can be re-told about the perceptions of events: strip off everything
empirical, and everything conceptual, and you are left with a mere, bare, pure intuition of
time. As Ÿspace is a form of your sensibility in experiencing things outside yourself, Ÿtime
- Kant thinks - is a form of your sensibility in relation not only to things outside you but
also to the flow of your mental history. Just as geometry is based on pure intuitions of
space (or of spatial figures), Kant says, arithmetic is based on pure intuitions of time; see
section 10. End of explanation. We now return to Kant’s text.]
Nor is any principle of pure geometry analytic. That a straight line is the shortest path
between two points is a synthetic proposition. For my concept of straightness contains
nothing having to do with Ÿquantity - it is purely a Ÿqualitative concept - so it cannot
contain the thought of what is shortest, ·because that is quantitative·. Here again, we need
help from intuition if we are to have a basis for putting shortest together with straight.
Why are we so prone to believe that in such a necessary judgment the predicate is
already contained in our concept so that the judgment is analytic? The source of this
mistake is a certain ambiguity. We ought to join in thought a certain predicate (’shortest’)
to a given concept (’straight’), and this requirement is inherent in the concepts themselves.
But the question is not what we Ÿought to think along with the given concept but what we
Ÿdo think in it, even if unclearly. Once we distinguish those two ideas, we can see that
while the predicate is indeed attached to the subject concept necessarily, it is attached only
by means of an intuition that must also be present; it is not to be found in the subject
Some other principles that geometers use are indeed really analytic and rest on the
law of contradiction: for example ‘Everything is equal to itself’, and ‘The whole is greater
than its part’. These identical propositions can be useful in setting out arguments, but they
don’t actually say anything; they can be useful methodologically, but they don’t contribute
to the content of what is said. Furthermore, even these analytic propositions, though they
are indeed validated purely by our concepts, wouldn’t be allowed into mathematics if they
couldn’t be illustrated by propositions that are connected with intuition. ·For example,
‘The whole is greater than its part’ is allowed into mathematics because it can be applied
to numbers, areas and lengths, which are given to us in intuition·. 12
Pure mathematical knowledge differs from all other a priori knowledge in this: it
never proceeds f rom concepts, but is always achieved by construction of concepts.
Mathematical propositions must therefore go beyond the c oncept t o what the
corresponding intuition contains, ·because this intuition guides the construction of the
concept·; hence they cannot and should not come from the analysis of concepts, and are
therefore one and all synthetic.
This may seem a small and unimportant point; but the neglect of it has done harm to
philosophy. Hume had the worthy philosophical aim of surveying the whole field of pure a
priori k nowledge - a field in which the human understanding lays claim to great
possessions - but he carelessly sliced off a large part of the territory, its most considerable
province, namely pure mathematics. He thought that mathematics rested on the law of
contradiction alone. Although he did not classify propositions in quite the way that I do
here, or with the same names, he in effect said: Pure mathematics contains only analytic
propositions, but metaphysics contains a priori synthetic propositions. Now this was a
great mistake, which infected his whole system of thought. If he had not made this
mistake, he would have taken his question about the origin of our ·a priori· synthetic
judgments to cover not only Ÿmetaphysics (e.g. the concept of causality) but also
Ÿmathematics. He had too much insight to base mathematics on mere experience, so ·if he
had likened metaphysics to mathematics in the way I have been defending· he would have
spared metaphysics from that fate. Metaphysics, by being in the good company of
mathematics, would have been saved from the danger of the vile mistreatment that Hume
actually gave to it; and then, fine thinker that he was, he would have been drawn into lines
of thought like those that I am now offering - though he would have presented them in his
own uniquely elegant style.
(3) Natural science also...
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