Abstract

According to Plato the mathematical being has an existence of its own, apart from the world perceptible by the senses. The mathematical entities lie between the world of Ideas and the world of material and changeable realities. This point of view is essentially held by Boethius and Proclus. Aristotle rejects this standpoint of Plato and argues that the mathematical entities exist in the sensible things ; as such not actual, but only potential. They are only in an actual state in the human mind by means of the abstracting act of the mathematician. As such they are « abstracta » only, or, in other words : conceptions of the intellect with a foundation in reality. This opinion is also held by Averroes and St Albert. Avicenna agrees with Aristotle, that the mathematical beings are real in the things perceptible by the senses. But he does not accept, that they would be in these things in a potential way only, and that their potentiality could only be actualized through the abstracting activity of the mathematician. According to him the mathematical being is not a « thing-in-the-intellect » ; we have to think it as real as the object of physical science. The characteristic of the mathematical view is, that it has no relation to substances, but only to the accidental : quantity. Also St Thomas definitely opposes the standpoint of Pythagoras, Plato and the Neoplatonists. Fundamentally he agrees with Aristotle : the mathematical being is, in some way or other, real in the material things perceptible by the senses. Though many passages could be cited from which it could be deduced, that he like Aristotle, Averroes and St Albert considered the mathematical being as a « thing-in-the-intellect » with a foundation in reality, these passages prove to have little argumentative power, if we study them more thoroughly. It is, in any case, quite certain, that St Thomas doesn't explicitly say in any place, that the mathematical being is but a « thing-in-the-intellect », neither that it is only potential in things. On the contrary, in numerous passages he argues that the mathematical being is an accidental in the material things. This points to the opinion held by Avicenna. It may in fact be proved by several arguments that St Thomas in this question agreed with Avicenna. In our opinion two of those arguments may be held as conclusive. We may conclude that St Thomas thinks also the mathematical being equally as real as the object of science. The characteristic of the mathematical view is according to him only this : it has as its object not a substance, but the accidental : quantity