Self-organized criticality and the development of mathematics.

[José Manuel Rodríguez Caballero]

Dear FOM members,
  There have been some attempts [1, 2] to find structural analogies between
the development of mathematics and physical processes. The statistical
study of computer-verified mathematics libraries [3] may be a laboratory
for experiments in this field.

  In the physics of complex systems, there is a remarkable phenomenon known
as self-organized criticality, which, according to Per Bak [4, page 2]:

 is so far the only known general mechanism to generate complexity.


Self-organized criticality manifests itself when a complex system evolves
towards an attractor, which turns out to be a critical point. According to
the Oxford Languages dictionary, a critical point is

noun
> 1.
> CHEMISTRY
> a point on a phase diagram at which both the liquid and gas phases of a
> substance have the same density, and are therefore indistinguishable.
> 2.
> US-MATHEMATICS
> a point on a curve where the gradient is zero.


In Bak's context, the word “critical” should be understood in the sense of
“chemistry” (thermodynamics, to be more precise). If the development of
mathematics is due to self-organized criticality, like many other complex
systems like the stock market [5], the immunological system [6], the brain
[7], etc., it should satisfy a version of the Gutenberg-Richter law. The
original law is formulated as the linear regression,

log N = a - b*M + small error,

where M is the magnitude of an earthquake and N is the number of
earthquakes of magnitude at least M that will happen during a year. To
adapt this law to the evolution of mathematics, we should introduce a
notion of the magnitude of a theorem, which should reflect its importance
for mathematics. The function N could be the number of theorems of
magnitude at least M proved during a given period of intrinsic time. By
intrinsic time, I don't mean the extrinsic physical time measured in years,
but a notion of time, defined in terms of the deductive system, using tools
from the theory of proof complexity.

My questions would be:

1) Do you agree with Bak that self-organizing criticality is so far the
only known general mechanism for generating complexity?

2) Does the theory of self-organized criticality apply to the development
of mathematics?

Kind regards,
Jose M.

References

[1] Baez, John, and Mike Stay. "Physics, topology, logic and computation: a
Rosetta Stone." *New structures for physics*. Springer, Berlin, Heidelberg,
2010. 95-172.
https://arxiv.org/pdf/0903.0340.pdf

[2] Wolfram, Stephen. "The Physicalization of Metamathematics and Its
Implications for the Foundations of Mathematics." arXiv preprint
arXiv:2204.05123 (2022).
https://arxiv.org/pdf/2204.05123.pdf

[3] Kaliszyk, Cezary, and Josef Urban. "Learning-assisted theorem proving
with millions of lemmas." Journal of symbolic computation 69 (2015):
109-128.
https://www.sciencedirect.com/science/article/pii/S074771711400100X?via%3Dihub

[4] Bak, Per. *How nature works: the science of self-organized criticality*.
Springer Science & Business Media, 2013.
https://jasss.org/4/4/reviews/bak
outline of this book (slides):
https://people.scs.carleton.ca/~arpwhite/courses/5002/notes/SI%20Lecture%2010.pdf

[5] Bartolozzi, Marco, Derek B. Leinweber, and Arthur W. Thomas.
"Self-organized criticality and stock market dynamics: an empirical
study." *Physica
A: Statistical Mechanics and its Applications* 350.2-4 (2005): 451-465.

[6] Tsumiyama, Ken, Yumi Miyazaki, and Shunichi Shiozawa. "Self-organized
criticality theory of autoimmunity." PLoS One 4.12 (2009): e8382.

[7] Cranstoun, S. D., et al. "Self-organized criticality in the epileptic
brain." Proceedings of the Second Joint 24th Annual Conference and the
Annual Fall Meeting of the Biomedical Engineering Society][Engineering in
Medicine and Biology. Vol. 1. IEEE, 2002.
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