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. But not even is any theorem true of
them, unless we want to change the objects of mathematics and invent
doctrines of our own. But it is not hard to assume any random
hypotheses and spin out a long string of conclusions. These
thinkers, then, are wrong in this way, in wanting to unite the objects
of mathematics with the Ideas. And those who first posited two kinds
of number, that of the Forms and that which is mathematical, neither
have said nor can say how mathematical number is to exist and of
what it is to consist. For they place it between ideal and sensible
number. If (i) it consists of the great and small, it will be the same
as the other-ideal-number (he makes spatial magnitudes out of some
other small and great). And if (ii) he names some other element, he
will be making his elements rather many. And if the principle of
each of the two kinds of number is a 1, unity will be something common
to these, and we must inquire how the one is these many things,
while at the same time number, according to him, cannot be generated
except from one and an indefinite dyad.
All this is absurd, and conflicts both with itself and with the
probabilities, and we seem to see in it Simonides 'long rigmarole' for
the long rigmarole comes into play, like those of slaves, when men
have nothing sound to say. And the very elements-the great and the
small-seem to cry out against the violence that is done to them; for
they cannot in any way generate numbers other than those got from 1 by
doubling
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