[FOM] Foundational Challenge

[José Manuel Rodríguez Caballero]

Tim wrote:
>
> So again, I don't agree with M. Katz that ZFC as a universal
> foundation for all mathematics is not credible, if we understand that
> claim rightly.  The claim isn't that for every subfield X of mathematics,
> we must explicitly use raw set theory for the "foundations of X" and
> eschew any defined notions.  The claim, rather, is that set theory is
> still the most convincing candidate when it comes to Generous Arena,
> Shared Standard, and Metamathematical Corral for mathematics considered as
> a whole


There is a contemporary tendency in mathematics, physics and computer
science of substituting formulae by the so-called graphical calculus [1, 2,
3, 4], both in the statements of the theorems and in the proofs. In this
approach, the sets are not fundamental, neither from a formal point of view
nor from an intuitive point of view. The main focus is on the composition
of processes and the fundamental intuition comes from topology, especially
from knot theory. For this reason, this new tendency is known as
compositionality [5]. Compositionality cannot be reduced to the foundation
of X, because X is aimed to be everything, in mathematics and outside
mathematics, e.g., physicist, biology, social sciences, computer science,
etc. It is not unreasonable to predict that compositionality may become
someday a new Generous Arena, Shared Standard, and Metamathematical Corral
for mathematics as a whole.

As evidence that compositionality is already part of the current mainstream
scientific activity, I would like to share the following typical fragment
from an announce of Postdoctoral and Ph.D. positions in Edinburgh for a
project about Quantum Theory (notice that both Category Theory and
Causality, which are closely related to compositionality, are considered as
important for this field):

Applicants must have or be about to receive a degree in Computer
> Science, Mathematics, or Physics, with a background in one or more of
> the following areas:
> * Quantum computing
> * Category theory
> * Programming languages
> * Causality
> * Concurrency


Kind regards,
Jose M.

[1] Penrose, Roger. Applications of negative dimensional tensors.
Combinatorial mathematics and its applications 1 (1971): 221-244.
[2] Kauffman, Louis H. Introduction to quantum topology. *Quantum topology*.
1993. 1-77.
[3] Coecke, Bob, and Aleks Kissinger. Picturing quantum processes.
Cambridge University Press, 2017.
[4] Blinn, Jim, Using Tensor Diagrams to Represent and Solve Geometric
Problems. 2002
URL =
https://www.microsoft.com/en-us/research/wp-content/uploads/2002/01/UsingTensorDiagrams.pdf
[5] Compositionality
URL = https://compositionality-journal.org/about/
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