empirical metamathematics

[José Manuel Rodríguez Caballero]

Dear Sam,
  Thank you for the references. Concerning the statistical law pointed out
by Wolfram (Wolfram's law), according to which

in a system consisting of N theorems (where N is huge), the number of
> theorems that depend (directly or indirectly) on k theorems behaves as an
> exponential decay in k.


I think that it deserves more research before being dismissed as a
misinterpretation of data. Some patters have been observed in human
languages, e.g., the Zipf law [1] and it is likely that a mathematical
theory should exhibit similar patterns (mathematics is a language after
all). An interesting application of Wolfram's law could be to identify a
deductive system from its statistical properties, e.g., the empirical decay
constant (assuming Wolfram's law) may be different in ZFC, Simple Type
Theory, Calculus of Constructions, Category Theory, etc. In
cryptography, it is well-known how to identify a human language from the
frequency of the letters used in a long text [2].

Kind regards,
José M.

References:
[1] Dahui, Wang, Li Menghui, and Di Zengru. True reason for Zipf's law in
language. Physica A: Statistical Mechanics and its Applications 358.2-4
(2005): 545-550.
https://www.sciencedirect.com/science/article/pii/S0378437105004085

[2] Solso, Robert L., and Joseph F. King. Frequency and versatility of
letters in the English language. *Behavior Research Methods &
Instrumentation* 8.3 (1976): 283-286.
https://link.springer.com/article/10.3758/BF03201714

El mar., 29 sept. 2020 a las 10:20, Sam Sanders (<sasander at me.com>)
escribió:

> 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 Goedel wrote something like that in an unpublished manuscript.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
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