[FOM] On the Nature of Mathematical Objects
I. Natochdag
natochdag at elsitio.net.uy
Thu Aug 12 11:03:45 EDT 2004
“intuition without the "skeleton" of a formalism is amoeba like.
Analogously,
pure formalism without a (vague) intuition is dead.”
Roughly speaking, I agree with this: Thus, mathematics may be defined
(again, roughly speaking) as the sought of ideas through methods and of
methods through ideas: by “methods” I mean formal rules of deduction and
by ideas the faculty of the postulating imagination-intuition.
Accordingly, this seems the most precise way of interpreting
mathematical history. Let me quote some examples:
- Leibniz had the “idea” of logically and rigorously treating
infinitesimals: Abraham Robinson achieved it “methodologically” in the
sixties.
- Young Gauss had the intuitive idea of the prime number theorem:
it was proved elementary in the forties by Erdos and Selberg.
- Fermat had the idea of his theorem: Wiles rigorously proved it.
I quoted three well-known examples: there are uncountable more. It may
be interpreted from them that ideas and rigor are a key of working math.
Martin Davis quoted a few weeks ago E.T. Bells frase: “Sufficient onto
the day is the rigor there of”. Intuitively I agree with it. Yet,
formally it may be read as a cry for “self-evidence” in mathematics. A
somewhat archaic Euclidean approach.
I. Natohdag
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