[FOM] Foundational Challenge

[José Manuel Rodríguez Caballero]

L. Paulson wrote:

> It might help if more mathematicians were familiar with some basic
> principles of computer science. To a computer scientist, it's natural that
> a say a graphics application might be built on top of certain data
> structures or libraries for computer graphics, which in turn are built on
> lower-level numerical libraries, and so on down until raw binary is
> reached. For mathematics, ZFC typically serves as the equivalent of binary,
> while category theory, et cetera, are the libraries.
>

 In this precise framework, let us imagine a software having a positive
integer as input and a list of positive integers as output, containing the
prime factors of the input. If the hardware does not exploit quantum
effects (classical computation) then this software can be analyzed until
the level of interactions among bits as explained in the quote above. On
the other hand, if the hardware does exploit quantum effects (quantum
computation) and the algorithm used for factorization is Shor's algorithm
or any improved version of it [1], then there will be a level of analysis
when there will be a whole which cannot be reduced to the interaction among
its parts (quantum entanglement). I quote Schroedinger [2, page 555]:

Another way of expressing the peculiar situation is: the best possible
> knowledge of a whole does not necessarily include the best possible
> knowledge of all its parts, even though they may be entirely separate and
> therefore virtually capable of being ‘best possibly known,’ i.e., of
> possessing, each of them, a representative of its own. The lack of
> knowledge is by no means due to the interaction being insufficiently known
> — at least not in the way that it could possibly be known more completely —
> it is due to the interaction itself.


According to the Oxford School of Categorical Quantum Mechanics, even if
quantum mechanics can be developed using ZFC, it is more natural to do it
using category theory (and category theory can be developed in an axiomatic
way, independent of ZFC). I quote from the introduction of a recent course
on this topic [3]:

The judicious language for this story is that of categories. Category theory
> teaches that a lot can be learned about a given type of mathematical
> species by
> studying how specimens of the species interact with each other, and it
> provides
> a potent instrument to discover patterns in these interactions. No
> knowledge of
> specimens’ insides is needed; in fact, this often only leads to tunnel
> vision.
> The methods of categories might look nothing like what you would expect
> from a treatise on quantum theory. But a crucial theme of quantum theory
> naturally fits with our guiding principle of compositionality:
> entanglement says
> that complete knowledge of the parts is not enough to determine the whole.
> In providing an understanding of the way physical systems interact,
> category
> theory draws closely on mathematics and computer science as well as
> physics.
> The unifying language of categories accentuates connections between its
> subjects. In particular, all physical systems are really quantum systems,
> including those in computer science.


There was an attempt to develop a rather set-theoretic approach to quantum
theory, which is mostly forgotten today: von Neumann's continuous
geometries [4]. Indeed, von Neumann recognized that he was inspired by
Cantor's set theory in order to develop his notion of spaces having as
dimension a real number between zero and one. Furthermore, von Neumann
proved that for each real number between zero and one, there is a space
having this real number as dimension. Some of the ideas originated in von
Neumann's continuous geometries have been successfully applied to quantum
field theory [5]. It may be interesting to consider von Neumann's
continuous geometries as primitive objects in a set-theoretic foundation of
mathematics, oriented towards quantum mechanics.

Kind Regards,
Jose M.

References:
[1] Bernstein, Daniel J.; Heninger, Nadia; Lou, Paul; Valenta, Luke (2017).
Post-quantum RSA. International Workshop on Post-Quantum Cryptography:
311–329.

[2] Schroedinger, E., 1935. Discussion of Probability Relations Between
Separated Systems, Proceedings of the Cambridge Philosophical Society, 31:
555-563; 32 (1936): 446-451.

[3] Chris Heunen and Jamie Vicary, Categorical Quantum Mechanics: An
introduction, University of Oxford, 2019
URL = https://www.cs.ox.ac.uk/files/10510/notes.pdf

[4]  John von Neumann, Continuous Geometry, Princeton Landmarks in
Mathematics and Physics (Book 46)
URL =
https://www.amazon.com/Continuous-Geometry-John-von-Neumann/dp/0691058938

[5] Yngvason, Jakob. The role of type III factors in quantum field theory.
Reports on Mathematical Physics 55.1 (2005): 135-147.
URL = https://www.sciencedirect.com/science/article/pii/S0034487705800096
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