[T]he question of
what “really” exists pervades the sciences and human thought in general.
The belief that
the infinite does not really exist goes back at least to Aristotle. Parrnenides even questioned the reality of plurality and
change. (Einstein’s vision has much in common with Parmenides). Towards the end of the nineteenth century an acrimonious exchange took place between Kronecker and
Cantor regarding the reality of the actual (as opposed to potential) infinite. Kronecker claimed that only the finite integers
really exist and all else is merely the work of man.
Cantor countered that the essence of mathematics was its freedom and that he had attained a larger vision than Kronecker had who could not see the infinite. Most mathematicians have followed Cantor and found his paradise a more beautiful and alluring universe.
…. But this seeing is not explained by
modus ponens. In his beautiful book
Proofs and Refutations, Lakatos (1976) has shown that
the mathematical process itself is dialectical and not Euclidean. At all times our ideas are formally inconsistent. But inconsistency, while still recognized as a pathology, is no longer seen to be a fatal disease. If we come across a contradiction, we localize it, isolate it, and try to cure it. But
we have to get over our neurotic phobias concerning this disease and recognize it as inseparable from life itself.