Kant, Prolegomena to any Future Metaphysic

Pure mathematics must construct its concepts on the

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Unformatted text preview: l reflect the form of my sensibility; which is to say that· I can know a priori that I can intuit objects of the senses only in accordance with this form of sensibility. It follows that there can be ·and we can know· propositions that concern merely this form of sensibility, that such propositions are valid for objects of the senses, and that they cannot be applied to anything except objects of our senses. Section 10 Thus it is only through the form of sensible intuition [= ‘form of sensibility’] that we can intuit things a priori. Such a priori knowledge, however, concerns objects only as they appear to us through our senses, and not as they may be in themselves. If a priori knowledge of synthetic propositions is to be possible, and we are to understand how it is possible, it must be subject to this ·limitation to how things appear as distinct from how they are in themselves·. 20 Now, space and time are the two intuitions on which pure mathematics grounds all its judgments that present themselves as certain and necessary. Pure mathematics must construct its concepts on the basis of pure intuition, ·that is, the kind of intuition that is conducted a priori , with no reliance on the senses·. Mathematics cannot proceed analytically by dissecting concepts, but only synthetically; so without pure intuition it cannot take a single step, since only pure intuition provides the material for synthetic a priori judgments. Geometry is based on the pure intuition of space. Arithmetic forms its own concepts of numbers by successively adding units in time. Our representations of space and time are merely intuitions, however, ·rather than concepts·; and here is why. Start with empirical intuitions of bodies and their changes, and strip them of everything empirical - that is, everything you know about through sensation - and what you are left with is space and time. These are therefore pure intuitions. They must be involved in all empirical intuitions, and can never be omitted, because they underlie everything empirical. But just because they are themselves pure a priori intuitions, they must be mere forms of our sensibility. They precede all our empirical intuition, i.e. all our perceptions of real objects; through them we can know objects a priori, though indeed only as they appear to us ·and not as they are in themselves·. Section 11 That solves the problem about how mathematics is possible. Pure mathematics is possible only because it bears on mere objects of the senses. The empirical intuition of such objects is grounded a priori in a pure intuition of space and time, and this pure intuition is merely the mere form of our sensibility. It precedes the actual appearance of objects, since it makes it possible for them to appear to us. ·Objects can appear to us only through our sensibility; and anything we get through our sensibility bears the marks of the form of sensibility·. Our a priori intuitions don’t involve the content of the appearance, the element of sensation in it, for that belongs to the empirical realm; they involve the form of the appearance, namely space and time. If you suspect that space and time are features of things in themselves rather than mere features of how things relate to sensibility, then tell me: How in that case could we know a priori - in advance of any experience of things what the intuitions of space and time must be like? Yet we do know this. There is no problem about this knowledge so long as space and time are taken to be nothing more than formal conditions of our sensibility, and the objects are taken to be merely appearances. For then the pure intuition that embodies the form of sensibility can be understood as coming from us - from our side of the transaction with objects - which means that it can be had a priori rather than empirically. Section 12 To clarify and confirm all this, we need only to look at how geometers do (and absolutely must) go about proving that two figures are completely congruent, meaning that one can be replaced at all points by the other. All such proofs ultimately come down to this: The two figures coincide with each other; which is obviously a synthetic proposition resting on immediate intuition. This intuition must be given pure and a p riori, otherwise the proposition couldn’t hold as absolutely certain and necessary. If it rested on an empirical 21 intuition, it would only have empirical certainty, and would mean: So far as our experience has shown us, this proposition has held up till now. That space has three dimensions, and that no space could have more, is built on the proposition that not more than three lines can intersect at right angles in a point. This proposition can’t be shown from concepts, but rests immediately on intuition, and indeed (because it is necessary and certain), on pure a priori intuition. That a line can be drawn to infinity, or a series of changes continued to infinity, presupposes a representation of space and time...
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This note was uploaded on 03/12/2013 for the course PHIL 105 taught by Professor Mendetta during the Spring '13 term at SUNY Stony Brook.

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