[FOM] On the Nature of Mathematical Objects

[I. Natochdag]

“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