Book 13
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SomePlato. distinguish mathematical objects from those which come after the Ideasi.e., the (semi-)Ideal lines, planes, etc. Cf. Aristot. Met. 1.9.30.; and of those who treat the subject in a different manner someSpeusippus; cf. sect. 7 above. speak of the mathematical objects and in a mathematical way—viz. those who do not regard the Ideas as numbers, nor indeed hold that the Ideas exist—and othersXenocrates. For his belief in indivisible lines see Ritter and Preller 362. Aristotle ascribes the doctrine to Plato in Aristot. Met. 1.9.25. speak of the mathematical objects, but not in a mathematical way; for they deny that every spatial magnitude is divisible into extended magnitudes, or that any two given units make 2.
11
But all who hold that Unity is an element and principle of existing things regard numbers as consisting of abstract units, except the Pythagoreans; and they regard number as having spatial magnitude, as has been previously stated.sect. 8.
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It is clear from the foregoing account (1.) in how many ways it is possible to speak of numbers, and that all the ways have been described. They are all impossible, but doubtless somesc. the view of Xenocrates (cf. Aristot. Met. 13.8.8). are more so than others.
1
First, then, we must inquire whether the limits are addible or inaddible; and if inaddible, in which of the two ways which we have distinguished.Aristot. Met. 13.6.2, 3. For it is possible either (a) that any one unit is inaddible to any other, or (b) that the units in the Ideal 2 are inaddible to those in the Ideal 3, and thus that the units in each Ideal number are inaddible to those in the other Ideal numbers.
2
Now if all units are addible and do not differ in kind, we get one type of number only, the mathematical, and the Ideas cannot be the numbers thus produced;
