Book, Paragraph

 1 III, 1 |     believe both in Forms and in mathematical objects intermediate between
 2 III, 2 |       better, or worse," but the mathematical sciences take no account
 3 III, 2 |       attributes about which the mathematical sciences offer proofs),
 4 III, 2 |          with which they say the mathematical sciences deal?-The sense
 5 III, 2 |        astronomy is one of these mathematical sciences there will also
 6 III, 2 |       things of which optics and mathematical harmonics treat; for these
 7 III, 6 |      not-besides perceptible and mathematical objects-others such as some
 8  IV, 1 |      this part; this is what the mathematical sciences for instance do.
 9  VI, 1 |     still, it is clear that some mathematical theorems consider them qua
10  VI, 1 |          being; for not even the mathematical sciences are all alike in
11 VII, 10|         intelligible circles the mathematical, and by perceptible circles
12  XI, 1 |          sort of thing which the mathematical sciences demand.) Nor (b)
13  XI, 7 |  sciences, whether productive or mathematical. For each of these marks
14  XI, 7 |    universal or not. Each of the mathematical sciences deals with some
15 XII, 8 |    standpoint of that one of the mathematical sciences which is most akin
16 XII, 8 |       but eternal, but the other mathematical sciences, i.e. arithmetic
17 XII, 10|      them one. And those who say mathematical number is first and go on
18 XIII, 1|        classes-the Ideas and the mathematical numbers, and (2) some recognize
19 XIII, 1|         some others say that the mathematical substances are the only
20 XIII, 2|        That it is impossible for mathematical objects to exist in sensible
21 XIII, 2|          lines and points of the mathematical solid there must be others
22 XIII, 2|        that exist along with the mathematical solids to which these thinkers
23 XIII, 2|      latter exist along with the mathematical solids, while the others
24 XIII, 2|          others are prior to the mathematical solids.) Again, therefore,
25 XIII, 2|         planes, and those in the mathematical solids, and those which
26 XIII, 2|          apart from those in the mathematical solids; four sets of lines,
27 XIII, 2|         of these, then, will the mathematical sciences deal? Certainly
28 XIII, 2|       will be various classes of mathematical numbers.~Again, how is it
29 XIII, 2|         Again, there are certain mathematical theorems that are universal,
30 XIII, 2|          of what, and when, will mathematical magnitudes be one? For things
31 XIII, 3|     treat them qua sensible, the mathematical sciences will not for that
32 XIII, 3|        those who assert that the mathematical sciences say nothing of
33 XIII, 3|          definiteness, which the mathematical sciences demonstrate in
34 XIII, 6|        they say is the case with mathematical number; for in mathematical
35 XIII, 6|      mathematical number; for in mathematical number no one unit is in
36 XIII, 6| successive numbers. And so while mathematical number is counted thus-after
37 XIII, 6|    identical with the Ideas, and mathematical number being different from
38 XIII, 6|  sensible things; and others say mathematical number alone exists, as
39 XIII, 6|        in one kind of number-the mathematical; only they say it is not
40 XIII, 6|       alone exists, and some say mathematical number is identical with
41 XIII, 6|          of mathematics and in a mathematical way-viz. those who do not
42 XIII, 7|       without difference, we get mathematical number-only one kind of
43 XIII, 7|    number of this sort cannot be mathematical number; for mathematical
44 XIII, 7|         mathematical number; for mathematical number consists of undifferentiated
45 XIII, 8|         are so, and one supposes mathematical number alone to exist, the
46 XIII, 8|        worst,-the view ideal and mathematical number is the same. For
47 XIII, 8|          in the one opinion. (1) Mathematical number cannot be of this
48 XIII, 8|          and that this should be mathematical number, is impossible. For
49 XIII, 9|         ideal number and posited mathematical. But those who wished to
50 XIII, 9|    assumed these principles, how mathematical number was to exist apart
51 XIII, 9|       from ideal, made ideal and mathematical number the same-in words,
52 XIII, 9|     same-in words, since in fact mathematical number has been destroyed;
53 XIII, 9|          numbers only, and these mathematical, must be considered later;
54 XIV, 2 |   whether it is ideal number, or mathematical, that they construct out
55 XIV, 2 |         numbers), but who posits mathematical number, why must we believe
56 XIV, 3 |     Those, however, who say that mathematical number alone exists cannot
57 XIV, 3 |          Forms and that which is mathematical, neither have said nor can
58 XIV, 3 |        have said nor can say how mathematical number is to exist and of
59 XIV, 4 |         and element, but only of mathematical number). For on this view
60 XIV, 5 |          simultaneously with the mathematical solids (for place is peculiar
61 XIV, 5 |       are separate in place; but mathematical objects are nowhere), and
62 XIV, 6 |         these, and generally the mathematical relations, as some describe