LESSON 14

Are There Separate Forms in Addition to the Objects of Mathematics and Sensible Things?

Chapter 6: 1002b 12-1002b 32

             284. But in general one will wonder why, in addition to sensible things and those which are intermediate, it is necessary to look for certain other things which we posit as the specific essences (or Forms) of sensible things.

             285. For if it is because the objects of mathematics differ in one respect from the things which are at hand, they do not differ in being many things that are specifically the same. Hence the principles of sensible things will not be limited in number but only in species; unless one were to consider the principles of this particular syllable or word, for these are limited in number. And this is likewise true of the intermediate entities; for in their case too there are an infinite number of things that are specifically the same. Hence, if in addition to sensible substances and the objects of mathematics there are not certain other things, such as some call the Forms, there will be no substance which is one both numerically and specifically. Nor will the principles of beings be limited in number, but only in species. Therefore, if this is necessary, it will also be necessary on this account that there should be Forms. And even if those who speak of the Forms do not express themselves clearly, although this is what they wanted to say, they must affirm that each of the Forms is a substance, and that nothing accidental pertains to them.

             286. But if we hold that the Forms exist, and that principles are one numerically but not specifically, we have stated the untenable conclusions that follow from this view.

COMMENTARY

             515. Having inquired whether the objects of mathematics are the principles of sensible substances, the Philosopher now inquires whether in addition to the objects of mathematics there are certain other principles, such as those which we call Forms, which are the substances and principles of sensible things. In regard to this he does three things. First (284:C 515), he presents the question. Second (285:C 516), he argues one side of the question ("For if it is because"). Third (286:C 518), he argues the other side ("But if we hold").

             Accordingly, he says, first (284), that if one assumes that the objects of mathematics are not the principles of sensible things and their substances, one will next have the problem why, in addition to both sensible things and the objects of mathematics (which are an intermediate class between sensible things and the Forms), it is necessary to posit a third class of entities, namely, the specific essences, i.e., the Ideas or separate Forms.

             516. For if it is because (285).

             Here he argues one side of the question. The reason why it is necessary to posit separate Forms over and above sensible substances and the objects of mathematics seems to be that the objects of mathematics differ in one respect "from the things at hand," i.e., from sensible things, which exist in the universe; for the objects of mathematics abstract from sensible matter. Yet they do not differ but rather agree in another respect. For just as we find many sensible things which are specifically the same but numerically different, as many men or many horses, in a similar way we find many objects of mathematics which are specifically the same but numerically different, such as many equilateral triangles and many equal lines. And if this is true, it follows that, just as the principles of sensible things are not limited in number but in species, the same thing is true "of the intermediate entities"--the objects of mathematics. For since in the case of sensible things there are many individuals of one sensible species, it is evident that the principles of sensible things are not limited in number but in species, unless of course we can consider the proper principles of a particular individual thing, which are also limited in number and are individual. He gives as an example words; for in the case of a word expressed in letters it is clear that the letters are its principles, yet there are not a limited number of individual letters taken numerically, but only a limited number taken specifically, some of which are vowels and some consonants. But this limitation is according to species and not according to number. For a is not only one but many, and the same applies to other letters. But if we take those letters which are the principles of a particular syllable, whether written or spoken, then they are limited in number. And for the same reason, since there are many objects of mathematics which are numerically different in one species, the mathematical principles of mathematical science could not be limited in number but only in species. We might say, for example, that the principles of triangles are three sides and three angles; but this limitation is according to species, for any of them can be multiplied to infinity. Therefore, if there were nothing besides sensible things and the objects of mathematics, it would follow that the substance of a Form would be numerically one, and that the principles of beings would not be limited in number but only in species. Therefore, if it is necessary that they be limited in number (otherwise it would happen that the principles of things are infinite in number), it follows that there must be Forms in addition to the objects of mathematics and sensible things.

             517. This is what the Platonists wanted to say, because it necessarily follows from the things which they held that in the case of the substance of sensible things there is a single Form to which nothing accidental belongs. For something accidental, such as whiteness or blackness, pertains to an individual man, but to this separate man, who is a Form, according to the Platonists, there pertains nothing accidental but only what belongs to the definition of the species. And although they wanted to say this, they did not "express themselves" clearly; i.e., they did not clearly distinguish things.

             518. But if we hold that (286).

             Then he counters with an argument for the other side of the question. He says that, if we hold that there are separate Forms and that the principles of things are limited not only in species but also in number, certain impossible consequences will follow, which are touched on above in one of the questions (249:C 464). But the Philosopher will deal with this problem in Book XII (1040:C 2450) and Book XIV of this work. And the truth of the matter is that, just as the objects of mathematics do not exist apart from sensible things, neither do Forms exist apart from the objects of mathematics and from sensible substances. And while the efficient and moving principles of things are limited in number, the formal principles of things, of which there are many individuals in one species, are not limited in number but only in species.