foundational implications for the future of mathematics

[José Manuel Rodríguez Caballero]

Harvey Friedman wrote:

[quote from Wolfram] " Among the implications of this view is that only
> certain collections
> of axioms may be consistent with inevitable features of human
> mathematical observers. A discussion is included of historical and
> philosophical connections, as well as of foundational implications for
> the future of mathematics."


I have had longstanding projects aimed at showing the usual systems of
> f.o.m. are inevitably unique in certain senses, but have only
> scratched the surface of this. What I have in mind is couched entirely
> in the traditional f.o.m. But I think you have something very
> different in mind which I would like to see a simplified hint of. Also
> something about the "foundational implications for the future of
> mathematics".


I would like to share an example of the "foundational implications for the
future of mathematics" that I developed at the beginning of the Wolfram
Physics Project in collaboration with a student of High Energy Physics,
Chris Pratt. Our starting point was a toy model of cosmic inflation known
as the Eternal Symmetree (word constructed from symmetry and tree):

[1] Harlow, D., Shenker, S., Stanford, D. and Susskind, L., 2011. Eternal
symmetree. *arXiv preprint arXiv:1110.0496*.
https://arxiv.org/pdf/1110.0496.pdf

[2] Marcolli, Matilde, and Nicolas Tedeschi. "Multifractals, Mumford curves
and eternal inflation." *P-Adic Numbers, Ultrametric Analysis, and
Applications* 6.2 (2014): 135-154.
https://arxiv.org/pdf/1311.5458.pdf

The mathematical version of the Eternal Symmetree could be as follows. Let
s_0, s_1, s_2, ... be a sequence of statements in ZFC such that each
statement of ZFC or its negation belongs to this sequence, but not both.
Assume that s_0 is a true statement. Consider the following infinite tree
labeled by statements of ZFC defined by the (non-computable) rules:

(i) the root (at level 0) is the statement s_0;

(ii) given a vertex at level n, if s_{n+1} is consistent with the
statements in the path from the root to the current vertex, then s_{n+1} is
a descendant of the current vertex.

(iii) given a vertex at level n, if the negation of s_{n+1} is consistent
with the statements in the path from the root to the current vertex, then
NOT( s_{n+1} ) is a descendant of the current vertex.

Every infinite path in the mathematical Eternal Symmetree is a "future of
mathematics" (physicists use the terminology "future boundary" for the set
of these paths). The mathematical observer will perceive the acceptance or
rejection of an undecidable statement as a truly random event, just like
the result of a measurement in quantum mechanics (at least according to
Hugh Everett III). Notice that pruning corresponds to avoidance of
contradictions.

Harvey Friedman wrote:

[quote from Wolfram] " A physicalized analysis is given of the bulk limit
> of traditional
> axiomatic approaches to the foundations of mathematics, together with
> explicit empirical metamathematics of some examples of formalized
> mathematics."


What does "bulk limit" mean here? Is there a suggestion here that
> there is a legitimate approach to the foundations of mathematics that
> is other than the traditional axiomatic approach through classical
> fom.? Or is the idea that the usual f.o.m. is either wrong headed or
> is uniquely inevitable? Can you give a simple condensed empirical
> finding that is significant?


In the framework of the Eternal Symmetree, it is natural to define an arrow
of time for the evolution of mathematics as physicists do with the arrow of
time of the universe, i.e., using the so-called fractal flow. According to
[1], page 31,

The fractal flow depends on the existence of an initial condition but not
> on its details. The only important feature of the initial condition is that
> it allows eternal inflation to take place. There are initial conditions
> that would preclude eternal inflation; for example an initial condition in
> which all space is occupied by a terminal vacuum. The theory of fractal
> flows cannot tell us why the initial condition is not of this type. But
> what it does imply, is that if eternal inflation occurs, the final
> attractor will have an arrow of time.


A candidate for what Wolfram calls the "bulk limit" could be this final
attractor, which will have an arrow of time and will be independent of the
details of the axioms and undecidable propositions accepted or rejected at
the beginning. We can see the seed of this idea in the chapter on
computational equivalence from A New Kind of Science

https://www.wolframscience.com/nks/p715--basic-framework/

where the independence from initial conditions in thermodynamics is one of
the inspirations for this principle, and there is a section about the
implications for foundations of mathematics. The passage from the
traditional axiomatic approach to the bulk limit of mathematics is similar
to the passage from considering water as a set of interacting molecules
(particle mechanics) to a liquid (continuum mechanics). The advance is the
same: to reduce the complexity of the object of study, e.g., engineers
prefer continuous mechanics rather than particle mechanics (a metaphor for
bulk mathematics) to do simulations of fluids (a metaphor for traditional
axiomatic mathematics). Perhaps we still don't see the bulk of the
mathematics because we are in the infancy of mathematics by the standards
of the Eternal Symmetree model, I quote from [1], page 33:

The cellular model is really about phenomena that take place on scales so
> large that they involve points which are out of causal contact. It is also
> about a sector of the theory, that from the boundary point of view,
> involves fields of extremely small dimension Delta << 1. From the bulk
> point of view these fields only vary between different Hubble patches. One
> of the lessons of the cellular model is that the number of such fields is
> enormous.


In my opinion, the main problem concerning the bulk of mathematics is to
define the appropriate stochastic process representing the work of
mathematicians. This may be done from empirical data about the existing
mathematics. I also believe that this research has applications in the
field of big data, e.g., astronomical data:

Zhou, Lixiao, and Maohai Huang. "Challenges of software testing for
astronomical big data." *2017 IEEE International Congress on Big Data
(BigData Congress)*. IEEE, 2017.

Kind regards,
Jose M.
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