On Arithmetic, Analytic Philosophy, and the Concept of Representation

Why did the first founders of analytic philosophy think that the function of words is to do what symbols do, namely represent things, rather than signify them? And why did this happen in conjunction with mathematical philosophy?

The concept of representation needs to be looked into more closely. “Representation,” which appears to be closely related to imitation and therefore to knowing, is evidently something that we do with our minds. But it may not be — it seems not to be — enough to try to understand it as a purely arbitrary act of the human mind, founded on nothing real.

If, as St. Thomas and others believed, Euclidean number is a kind of real quantity, the units of numbers must have a real relation to each other, and not only a relation which we conceive in our minds. (That is why, for St. Thomas, the arithmetic unit and the transcendental unit are not the same thing; and the former only exists in material things. An angel is not an arithmetic unit.)

But what is it in the very materiality of things that makes such a real relation possible? Perhaps it is that the act of matter is to represent, which act is completely fulfilled when it is formed. One could understand this as the foundation of the entire quantitative order.

Because recent philosophers of mathematics doubted that the relation between arithmetic units was anything real apart from the mind, they concluded that it was purely in the order that they call “logical.” This “logical” is not quite the same as what we call “logical,” nor the same as what we call “artificial.” It is between the two, because it is made by the mind, but yet conceived of by analytic philosophy (implicitly, at least) as the very principle of all intelligibility. And this is as it must be, if the fundamental ordering of matter to form is only in the mind.


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