Which kind of mathematics for Quantum Mechanics?

The relevance of H. Weyl's a program of research.

Antonino Drago


Dept. of Physical Sciences, Univ. "Federico II", Naples


In 1918 Weyl's book Das Kontinuum planned to found anew mathematics upon more conservative bases than both rigorous mathematics and set theory. It gave birth to so-called Weyl's elementary mathematics, i.e. an intermediate mathematics between the mathematics rejecting at all actual infinity and the classical one including it almost freely.

Abstract

In 1918 Weyl's book Das Kontinuum planned to found anew mathematics upon more conservative bases than both rigorous mathematics and set theory. It gave birth to so-called Weyl's elementary mathematics, i.e. an intermediate mathematics between the mathematics rejecting at all actual infinity and the classical one including it almost freely.
The present paper scrutinises the subsequent Weyl's book Gruppentheorie und Quantenmechanik (1928) as a program for at the same time founding anew theoretical physics - through quantum theory - and developing his mathematics - through the improvement of group theory - a mathematical theory effacing - according to Weyl - the old distinction between discrete and continuous mathematics. Evidence from Weyl's writings is collected for supporting this interpretation.

Then Weyl's program is evaluated as unsuccessful, owing to some crucial difficulties of both physical and mathematical nature.The present clear-cut knowledge of Weyl elementary mathematics allows us to re-evaluate Weyl's program in order to look for more adequate formulations of quantum mechanics in this kind of mathematics as well as in any weaker kind of mathematics than classical one.


Info

Drago A.: Which kind of Mathematics for quantum mechanics? The relevance of H. Weyl's program of research, in A Garola & A Rossi (eds.): Proceedings Foundations of Quantum Mechanics. Historical Analysis and Open Questions World Scientific, Singapore, 2000, pp. 167-193