empirical metamathematics

[Sam Sanders]

Dear José,

>   Motivated by this statement, I would like to ask the following questions:
> 
> 1) Are there references to empirical work in metamathematics? 

I would say the following paper:

https://arxiv.org/abs/1908.05676

provides empirical evidence for Platonism, where the latter is interpreted as 

“doing mathematics is the discovery of a world of ideal objects of which we can only know vague reflections” 

Goedel wrote something like that in an unpublished manuscript.  

In a nutshell, I show in the above paper that the “Big Five” of reverse math are only a reflection of a higher-order truth.  Both the reflection and the 
higher-order truth mostly consist of well-known mainstream mathematical concepts.  

> 2) Is there a theoretical way to explain the statistical distributions obtained by S. Wolfram in the networks of theorems?

The first thing that comes to mind is: human bias.  Misinterpretation of data is not uncommon:

http://www.stat.cmu.edu/~cshalizi/2010-10-18-Meetup.pdf

Best,

Sam