Monday, December 03, 2007

6.36111 The Kantian problem of the right and left hand, that one cannot make cover each other, exists already in a plane, indeed in one-dimensional space, where the two congruent figures a and b also cannot be made to cover each other without moving them outside this space:

-----o——x-----x——o

a b

Right and left hand are actually completely congruent. And the fact that one cannot make one cover the other has nothing to do with it.

A right-hand glove could be put on the left hand if one could turn it around in four-dimensional space.

This problem is discussed by Kant in the Prolegomena. Schopenhauer refers to it twice in the Fourfold Root, once on p. 40 and on p. 194. His claim is that the difference between a left glove and a right glove can only be seen, not explained from concepts alone. That is, it cannot “be made intelligible except by means of intuition” (p. 194). The idea is that space is thereby shown to be an a priori form of intuition, since such differences in space must be intuited.

Is Wittgenstein’s last sentence a joke, or a real solution? My version of the figure is based on Letters to Ogden p. 61, and is different from the one in the other translations.

Henk Visser in “Wittgenstein’s Debt to Mach’s Popular Scientific Lectures” Mind (1982) Vol. VCI, pp. 102-105, says that what Wittgenstein says about Kant is conspicuously similar to what Mach says on the same issue, and Mach in turn credited M√∂bius as the source of the idea.

Black (p. 363) says that according to Kant things such as left and right hands or gloves can be “exactly alike in all spatial respects” and yet do not fit the same space, i.e. they are different. In intuition we get the difference, but the Understanding cannot (according to Kant) grasp it. Black (pp. 363-4):

"Kant used the argument several times—and to prove opposite conclusions. It was omitted from the second edition of the Critique—because, according to Kemp Smith, Kant had realized it was based ‘upon a false view of the understanding’ (op. cit. p. 165).

"W. says that the impossibility of making counterparts fill the same space (at least without entry into a higher dimension) leaves their congruence unchallenged. But Kant would readily have agreed: W. does nothing to explain how the congruent counterparts can be numerically distinct, which was Kant’s puzzle. On the face of it, the possibility of non-identical counterparts does not square with 6.3611 (3)—unless we take W. to be suggesting that the counterparts must have different causal antecedents by which alone they can be distinguished? (And this is now close to Kant’s conclusion.)"

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