links all these together: the ratio of either factor to the product is the same as the ratio of one to the other factor; and the ratio of the product to either factor is the same as the ratio of the other factor to one.
These matters, and others like them discussed in arithmetic, are intelligible matters shared by all rational people. Everyone knows something of them, for such knowledge is self-evident. Hence the example they give: “One is half of two.” There is no doubt that the propositions of arithmetic are universal, must be accepted, and admit no exceptions whatsoever.
This was the starting point of their philosophy, founded by Pythagoras, whose followers were called the adherents of number. They supposed that abstract numbers existed outside the mind. Then Plato and his followers recognized the error of this view. They supposed instead that abstract essences, such as man and horse considered without qualification, existed outside the mind and had neither beginning nor end. Then Aristotle and his followers recognized the error of that view and said: “Rather, these unqualified essences exist in external reality together with individual things.” Some of Aristotle’s later followers adopted this position, but it too is mistaken. For nothing in external reality is universal at all; everything that exists externally is a particular, determinate thing.
When it is said that the natural universal exists in external reality, this means that what is universal in the mind exists externally. But when it exists externally, it can only be a determinate particular, not a universal. Its being universal is therefore conditional upon its being in the mind. And whoever posits an essence that exists neither in the mind