Book 3
10
It is very much the same with the present moment in time. This too cannot be generated and destroyed; but nevertheless it seems always to be different, not being a substance. And obviously it is the same with points, lines and planes, for the argument is the same; they are all similarly either limits or divisions.For arguments against the substantiality of numbers and mathematical objects see Aristot. Met. 13.1-3, 6-9; Aristot. Met. 14.1-3, 5, 6.
1
In general one might wonder why we should seek for other entities apart from sensible things and the Intermediates:Cf. Aristot. Met. 3.2.20ff.. e.g., for the Forms which we Platonists assume.
2
If it is for the reason that the objects of mathematics, while differing from the things in our world in another respect, resemble them in being a plurality of objects similar in form, so that their principles cannot be numerically determined (just as the principles of all language in this world of ours are determinate not in number but in kind—unless one takes such and such a particular syllable or sound, for the principles of these are determinate in number too—
3
and similarly with the Intermediates, for in their case too there is an infinity of objects similar in form), then if there is not another set of objects apart from sensible and mathematical objects, such as the Forms are said to be, there will be no substance which is one both in kind and in number, nor will the principles of things be determinate in number, but in kind only.
4
Thus if this is necessarily so, it is necessary for this reason to posit the Forms also. For even if their exponents do not articulate their theory properly, still this is what they are trying to express, and it must be that they maintain the Forms on the ground that each of them is a substance, and none of them exists by accident.
