Sometimes it happens in physics again and again, that after the discovery of a new
phenomenon, a theory fitted out with all the criteria of beauty must be replaced by a
quite ugly one. Luckily, in most cases, the course of further development indeed
reveals that this ugly theory was only provisional. . . .
In mathematics this idea leads in many instances to the truth. One has an unsolved
problem, and, at first, has no insight at all how the solution should go, even less, how
one might find it. Then the thought comes to describe for oneself what the sought-for
truth must look like were it beautiful. And see, first examples show that it really seems
to look that way, and then one is successful in confirming the correctness of what was
envisaged by a general proof. . . . In general we find a [mathematical] formulation all
the more beautiful, the clearer, more lucid, and more precise it is. (*)

(*) Helmut Hasse, Mathematik als Wissenschaft, Kunst, und Macht (1952), 18-20.


Or, as Hasse says even more forcefully, "The true mathematician who has found
something beautiful, senses in it the irresistible pressure to communicate his
discovery to others. "

The Lebesgue's question: Can one demonstrate the existence of
a mathematical object without defining it? Rene Baire went even further than
Lebesgue and Borel in answering in the negative. (*)
For me progress in this range of ideas consists in delimiting the domain of what is
definable. And, in the final analysis, in describing its appearance, everything must lead
back to the finite. That is, provisional rules for constructions that admit infinitely
many steps are to be ruled out.


And Baire was quite explicit that given, say, the set of positive integers, "it is as
far as I am concerned false to consider the parts of that set as given. " It would
seem that even the set of positive integers itself is inconceivable as a whole from
this point of view.
Mathematics either is or is not independent of its human creators. Zermelo,
as well as Hadamard, recognized the distinction, as he wrote.....