[FOM] Quantum Mathematics

[José Manuel Rodríguez Caballero]

Dear FOM members,
  In Helemskii's book on Functional Analysis [1], Quantum Mathematics is
defined as follows:

quantum mathematics emerges from classical one after replacing functions by
> operators. [...] The outstanding role that is played in classical
> mathematics by functions with their commutative (pointwise) multiplication,
> in quantum mathematics passes to operators with their non-commutative
> multiplication (composition). It turns out (and this is what we
> should come to believe in) that fundamental notions and results of
> classical mathematics do have substantial quantum analogs or versions. We
> can say that these classical
> notions represent a small and hardly visible (classical) part of a huge
> quantum iceberg. To comprehend all of this iceberg we should realize
> (guess?) how to reasonably
> replace functions lying in the foundation of these notions (results,
> methods, problems)
> with operators.


A typical example of quantum mathematics is von Neumann's Continuous
Geometry [2]. Another example is Connes' Noncommutative Geometry [3].

For historical reasons, Quantum Mathematics is developed from Classical
Mathematics. Could be possible/desirable to invert this logical hierarchy,
i.e., to develop classical mathematics from quantum mathematics? In
particular, does quantum mathematics deserves its own foundation?

Sincerely yours,
José M.

References:
[1] Helemskii, Aleksandr Yakovlevich. *Lectures and exercises on functional
analysis*. Vol. 233. Providence, RI: American Mathematical Society, 2006.
[2]  Von Neumann, John. *Continuous geometry*. Vol. 25. Princeton
University Press, 1998.
[3] Connes, Alain. Noncommutative geometry. *San Diego *(1994).


--

José Manuel Rodríguez Caballero

arvutiteaduse instituut / Institute of Computer Science
Tartu Ülikool / University of Tartu

Personal Research Page: https://josephcmac.github.io/
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