The question might also be raised as to what the good is which things derive from numbers because their mixture can be expressed by a number, either one which is easily calculable, Footnote 1i.e., a simple ratio. or an odd number. Footnote 2It is hard to see exactly what this means. If the terms of a ratio are rational, one of them must be odd. Alexander says a ratio like 1 : 3 is meant. Oddness was associated with goodness (cf. Aristot. Met. 1.5.6). For in point of fact honey-water is no more wholesome if it is mixed in the proportion three times three
Footnote 3Apparently the Pythagoreans meant by this three parts of water to three of honey.
Aristotle goes on to criticize this way of expressing ratios. ; it would be more beneficial mixed in no particular proportion, provided that it be diluted, than mixed in an arithmetical proportion, but strong.
Again, the ratios of mixtures are expressed by the relation of numbers, and not simply by numbers; e.g., it is 3 : 2, not 3 X 2 Footnote 4Cf. previous note. ; for in products of multiplication the units must belong to the same genus. Thus the product of 1 x 2 x 3 must be measurable by 1, and the product of 4 X 5 x 7 by 4. Therefore all products which contain the same factor must be measurable by that factor. Hence the number of fire cannot be 2 X 5 X 3 X 7 if the number of water is 2 x 3. Footnote 5sc. because if so, a particle of fire would simply equal 35 particles of water.
If all things must share in number, it must follow that many things are the same; i.e., that the same number belongs both to this thing and to something else. Is number, then, a cause; i.e., is it because of number that the object exists? Or is this not conclusive? E.g., there is a certain number of the sun’s motions, and again of the moon’s, Footnote 65 in each case, according to Aristotle; cf. Aristot. Met. 12.7.9, 11. and indeed of the life and maturity of every animate thing. What reason, then, is there why some of these numbers should not be squares and others cubes, some equal and others double?
There is no reason; all things must fall within this range of numbers if, as was assumed, all things share in number, and different things may fall under the same number. Hence if certain things happened to have the same number, on the Pythagorean view they would be the same as one another, because they would have the same form of number; e.g., sun and moon would be the same. Footnote 7Cf. previous note.
But why are these numbers causes? There are seven vowels, Footnote 8In the Greek alphabet. seven strings to the scale, Footnote 9In the old heptachord; cf. note on Aristot. Met. 5.11.4. seven Pleiads; most animals (though not all Footnote 10Cf. Aristot. Hist. An. 576a 6. ) lose their teeth in the seventh year; and there were seven heroes who attacked Thebes. Is it, then, because the number 7 is such as it is that there were seven heroes, or that the Pleiads consist of seven stars? Surely there were seven heroes because of the seven gates, or for some other reason, and the Pleiads are seven because we count them so; just as we count the Bear as 12, whereas others count more stars in both.
Indeed, they assert also that Ξ, Ψ and Ζ are concords, Footnote 11According to Alexander ζ was connected with the fourth, ξ with the fifth, and ψ with the octave. and that because there are three concords, there are three double consonants. They ignore the fact that there might be thousands of double consonants—because there might be one symbol for ΓΡ. But if they say that each of these letters is double any of the others, whereas no other is, Footnote 12θ, φ , and χ are aspirated, not double, consonants. and that the reason is that there are three regions Footnote 13Palate, lips, and teeth. of the mouth, and that one consonant is combined with σ in each region, it is for this reason that there are only three double consonants, and not because there are three concords—because there are really more than three; but there cannot be more than three double consonants.
Thus these thinkers are like the ancient Homeric scholars, who see minor similarities but overlook important ones.
Some say that there are many correspondences of this kind; e.g., the middle notes Footnote 14i.e., the μέση(fourth) and παραμέση(fifth), whose ratios can be expressed as 8 : 6, 9 : 6. of the octave are respectively 8 and 9, and the epic hexameter has seventeen syllables, which equals the sum of these two; and the line scans in the first half with nine syllables, and in the second with eight. Footnote 15i.e., a dactylic hexameter whose sixth foot is always a spondee or trochee has nine syllables in the first three feet and eight in the last three. For τὸ δεξιόν meaning the first part
of a metrical system see Bassett,Journal of Classical Philology 11.458-460.
And they point out that the interval from α to ω in the alphabet is equal to that from the lowest note of a flute to the highest, whose number is equal to that of the whole system of the universe. Footnote 16Alexander suggests that the number 24 may have been made up of the 12 signs of the zodiac, the 8 spheres (fixed stars, five planets, sun and moon) and 4 elements. We must realize that no one would find any difficulty either in discovering or in stating such correspondences as these in the realm of eternal things, since they occur even among perishable things.
As for the celebrated characteristics of number, and their contraries, and in general the mathematical properties, in the sense that some describe them and make them out to be causes of the natural world, it would seem that if we examine them along these lines, they disappear; for not one of them is a cause in any of the senses which we distinguished with until respect to the first Principles. Footnote 17Cf. Aristot. Met. 1.3.1, Aristot. Met. 5.1, 2.
There is a sense, however, in which these thinkers make it clear that goodness is predicable of numbers, and that the odd, the straight, the equal-by-equal, Footnote 18i.e., square. and the powers Footnote 19Probably their power
of being represented as regular figures; e.g. the triangularity of 3 or 6. of certain numbers, belong to the series of the Beautiful. Footnote 20Cf. Aristot. Met. 1.5.6. For the seasons are connected with a certain kind of number Footnote 21i.e., 4. ; and the other examples which they adduce from mathematical theorems all have the same force.
Hence they would seem to be mere coincidences, for they are accidental; but all the examples are appropriate to each other, and they are one by analogy. For there is analogy between all the categories of Being—as straight
is in length, so is level
in breadth, perhaps odd
in number, and white
in color.
Again, it is not the Ideal numbers that are the causes of harmonic relations, etc. (for Ideal numbers, even when they are equal, differ in kind, since their units also differ in kind) Footnote 22Aristotle has argued (Aristot. Met. 13.6-8.) that if the Ideal numbers differ in kind, their units must differ in kind. Hence even equal numbers, being composed of different units, must be different in kind. In point of fact, since each ideal number is unique, no two of them could be equal. ; so on this ground at least we need not posit Forms.
Such, then, are the consequences of the theory, and even more might be adduced. But the mere fact that the Platonists find so much trouble with regard to the generation of Ideal numbers, and can in no way build up a system, would seem to be a proof that the objects of mathematics are not separable from sensible things, as some maintain, and that the first principles are not those which these thinkers assume.