Wednesday, September 05, 2018

What Do Philosophers Do?

For myself, I just notice such facts as:

(A) The overwhelming majority of professional mathematicians are not going to be wrong about what numbers are.

(B) The overwhelming majority of mathematicians assume, in their professional work, that numbers are axiomatic sets.

(C) Numbers are not axiomatic sets.

The conjunction of (A), (B) and (C) would be a contradiction, were the mathematicians of (B) not just assuming that numbers are axiomatic sets for the purposes of proving theorems from axioms, as I suspect they do. But many analytic philosophers deny (C), because of that contradiction. Such philosophers also ask questions like: “Do numbers (or sets) exist? If they do, where are they? If they don’t, then what does ‘2’ refer to?”

Since numbers (or sets) are abstract objects, hence if they exist then they exist in some Platonic realm of abstract objects, which raises the question: “How is it that we can access that realm, in order to know such properties of numbers as arithmetic?” To see how stupid such questions are, one only has to ask such questions as: “Does value exist; and if so, where is it?” Clearly some things have value, but it makes no sense to ask where it is (or what colour it is); such questions hardly further the analytical task of describing accurately what value is.

A question similar to the one about numbers might be: “Do shapes exist?” Shapes are instantiated in, and abstracted from, shaped things, clearly; and similarly, numbers are instantiated in, and abstracted from, numbers of things. That is basically what John Stuart Mill said (in passing); it is only common sense, although his observation was jumped on by a founder of analytic philosopher, Gottlob Frege (incorrectly). Numbers are not just whatever we say they are.

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