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Alan,
I apologize for misunderstanding your position. Although, let me explain why I don't think your position has much merit.
I suggest that the very definition of numbers themselves involves axioms and inferences that already require correspondence with "the facts"; that math might be pre-defined as true in Tarski perspective.
Let's look at the Peano axioms:
1. Zero is a number.
2. If a is a number, the successor of a is a number.
3. zero is not the successor of a number.
4. Two numbers of which the successors are equal are themselves equal.
5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.
(Source: mathworld.wolfram.com)
Now, what in any of these axioms and rules of inference requires correspondence with the 'facts'? 'True' in convention T might mean (I don't profess to have gotten it right) that there is a way to express "2+2=4" in a mathematical abstract reality and that "2+2=4" actually obtains in a mathematical abstract reality.
It might be the case that everything we discuss or can imagine requires a correspondence with the world ('facts') and therefore any and all formal systems we invent are circular, but why do most formal systems fail to be consistent? That is, it is far easier to create an inconsistent formal system than one like mathematics that appears consistent. If your contention is correct, then why is it that Peano arithmetic can be made inconsistent by changing ever so slightly the axioms stated above? Also, by eliminating axioms we greatly reduce the scope of Peano arithmetic, and in which case the formal system created would hardly garner as much attention as it has. I just don't see how you can make this kind of suggestion with any compelling argument. If you take Euclid's fifth postulate, for example, it was once held self-evident and corresponding with the 'facts', only to be found vunerable to Gauss' review. What's worse, this is an instance where the axiom both works (Euclid) and doesn't work (non-Euclidean). Does this mean that reality is schizophrenic? Rather, I think it means that there's a wide range of discretion in creating formal systems and that your suggestion is not very compelling. |
