The unreasonable effectiveness of modular forms

[José Manuel Rodríguez Caballero]

In the lecture/discussion

5.8 Discussion "The unreasonable effectiveness of modular forms" introduced
by P. Sarnak
https://youtu.be/jFtu4asyolk

several number theoreticians try to explain why modular forms are so useful
in mathematics. Is there any insight from the foundations of mathematics
why this should be the case?

For example, in Wolfram's physical interpretation of metamathematics, an
unreasonably effective mathematical object is modeled as a massive body
that exerts a gravitational attraction on other bodies (mathematical
objects), in a graph-theoretical space-time given by the rewriting rules of
mathematics (in the case of modular forms, the ZFC rewrite rules). The
problem with Wolfram's interpretation is that, at this stage, it is quite
empirical

https://writings.stephenwolfram.com/2020/09/the-empirical-metamathematics-of-euclid-and-beyond/

This essay is one of the attempts to formalize physical metamathematics
within the framework of homotopy type theory:

https://www.wolframphysics.org/bulletins/2020/08/a-candidate-geometrical-formalism-for-the-foundations-of-mathematics-and-physics/

and this is a more elaborate homotopy type theoretical formalization of the
same framework:
https://arxiv.org/pdf/2111.03460.pdf

So, in the physical metamathematical framework, stating that modular forms
are unreasonably effective is a statement about the density of connections
in regions of infinite graphs.

Kind regards,
Jose M.
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