Aristotle - index for METAPHYSICA Book 13 Part 2

The objects of mathematics

Paragraph 1 That it is impossible for mathematical objects to exist in sensible things, and at the same time that the doctrine in question is an artificial one, has been said already in our discussion of difficulties we have pointed out that it is impossible for two solids to be in the same place, and also that according to the same argument the other powers and characteristics also should exist in sensible things and none of them separately.

Paragraph 2 But, again, it is not possible that such entities should exist separately.

Paragraph 3 Again, how is it possible to solve the questions which we have already enumerated in our discussion of difficulties?

Paragraph 4 Again, there are certain mathematical theorems that are universal, extending beyond these substances.

Paragraph 5 And, in general, conclusion contrary alike to the truth and to the usual views follow, if one is to suppose the objects of mathematics to exist thus as separate entities.

Paragraph 6 Again, by virtue of what, and when, will mathematical magnitudes be one?

Paragraph 7 Again, the modes of generation of the objects of mathematics show that we are right.

Paragraph 8 Again, the solid is a sort of substance;

Paragraph 9 Grant, then, that they are prior in definition.

Paragraph 10 It has, then, been sufficiently pointed out that the objects of mathematics are not substances in a higher degree than bodies are, and that they are not prior to sensibles in being, but only in definition, and that they cannot exist somewhere apart.

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Paragraph 1	That it is impossible for mathematical objects to exist in sensible things, and at the same time that the doctrine in question is an artificial one, has been said already in our discussion of difficulties we have pointed out that it is impossible for two solids to be in the same place, and also that according to the same argument the other powers and characteristics also should exist in sensible things and none of them separately.
Paragraph 2	But, again, it is not possible that such entities should exist separately.
Paragraph 3	Again, how is it possible to solve the questions which we have already enumerated in our discussion of difficulties?
Paragraph 4	Again, there are certain mathematical theorems that are universal, extending beyond these substances.
Paragraph 5	And, in general, conclusion contrary alike to the truth and to the usual views follow, if one is to suppose the objects of mathematics to exist thus as separate entities.
Paragraph 6	Again, by virtue of what, and when, will mathematical magnitudes be one?
Paragraph 7	Again, the modes of generation of the objects of mathematics show that we are right.
Paragraph 8	Again, the solid is a sort of substance;
Paragraph 9	Grant, then, that they are prior in definition.
Paragraph 10	It has, then, been sufficiently pointed out that the objects of mathematics are not substances in a higher degree than bodies are, and that they are not prior to sensibles in being, but only in definition, and that they cannot exist somewhere apart.