the physicalization of metamathematics (predictions)

Sat Mar 12 13:50:54 EST 2022

Monroe Eskew asked:

> It sounds like this is a viewpoint that takes mathematics as a physical
> phenomenon and attempts to characterize what large-scale behavior
> mathematicians will exhibit. Does it make testable predictions?

 I will answer as an external fellow of the project:

https://www.wolframphysics.org/people/jose-manuel-rodriguez-caballero/

but my answer may be different from other project members' answers as there
are various viewpoints, all derived from the common framework of A New Kind
of Science. First, I would like to cite a paper that is in the direction of
the ideas that I will develop below:

Szilard, Leo. "On the decrease of entropy in a thermodynamic system by the
intervention of intelligent beings." *Behavioral Science* 9.4 (1964):
301-310.
https://indico.ictp.it/event/7644/session/9/contribution/18/material/1/0.pdf

Wolfram is developing a framework for expressing mathematics as it is done
by human mathematicians. This framework, just like calculus, can be
instantiated in several ways (for example, differential calculus can be
used to describe heat flux, string vibration, etc.). Any particular
instantiation can be empirically tested and this instantiation can be
either confirmed or refuted from real-world data. For example, measuring
graph-theoretical properties as Cezary Kaliszyk and Josef Urban did for the
HOL Light graph,

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

A dynamical picture of Mathematics is modeled as a graph obtained by the
application of rules. Humans do not apply mathematical rules at random.
Hence, a human mathematician may be a stochastic process concerning the
probabilities of applying the rules. Different stochastic processes may
produce graphs with different statistical properties. The only common
feature that I see among all the stochastic processes modeling human
mathematicians is that, given a set of theorems, they should reduce the
entropy, i.e., as time increases, the proofs should become shorter and new
connections should become explicit, e.g., connections between number theory
and analysis like the analytic proof that there are infinitely many prime
numbers of the form 4k+3. Of course, entropy can increase by artificially
adding new theorems with long proofs to the system.

A prediction from Wolfram's model for mathematics is that, given a fixed
set of theorems, the evolution of mathematics should decrease the entropy
of the system of theorems and proofs. Which precise notion of entropy
should be the most convenient to formalize this statement? This is a work
in progress.

A second prediction may be more speculative since the Kolmogorov complexity
is non-computable. Assuming an oracle that can compute the Kolmogorov
complexity, it would be possible to decide whether the human mind, in his
process of doing mathematics, is influenced by quantum randomness, e.g.,
generated as the result of radioactive disintegration in the brain. The
Kolmogorov complexity of an aperiodic deterministic system should increase
as a logarithmic function of time (for almost all values of time, ignoring
when time can be compressed). On the other hand, the Kolmogorov complexity
of an aperiodic truly random system should increase as a linear function of
time (for almost all values of time, ignoring when time can be compressed).
Therefore, a measurement of the Kolmogorov complexity by this hypothetical
oracle should determine whether the evolution of mathematics is
deterministic or if it involves truly randomness. The only known source of
true randomness is quantum mechanics, all other sources of randomness are
just pseudo-randomness, i.e., complicated deterministic systems.

If it happens that the evolution of mathematics, as done by human
mathematicians, involves true randomness, then the Wolfram model to
describe it should be grounded on quantum hardware. Its simulation should
be done using a quantum computer because a classical computer simulating
quantum phenomena cannot produce a linear increment of Kolmogorov
complexity as a function of time. Therefore, the only way for the universe
and mathematics to be described by the same model is that the evolution of
mathematics is triggered by truly random phenomena. For details about this
argument, I would recommend my preprint:

Caballero, José Manuel Rodríguez. "Incompatibility between 't Hooft's and
Wolfram's models of quantum mechanics." *arXiv preprint arXiv:2108.03751*
 (2021).
https://arxiv.org/pdf/2108.03751.pdf

Finally, there is a misconception that classical system can't simulate
quantum systems. This is based on a no-go theorem, where some assumptions
are made concerning time. But it is possible to avoid these assumptions and
obtain a classical simulation of quantum phenomena. The price to pay is to
assume the existence of multiple universes. In this preprint I show an
example of solution of the Schrodinger equation constructed from a 100%
classical system :

Caballero, José Manuel Rodríguez. "Renormalized Wolfram model exhibiting
non-relativistic quantum behavior." *arXiv preprint arXiv:2108.08300*
 (2021).
https://arxiv.org/pdf/2108.08300.pdf

Notice that the possibility of simulating quantum systems by classical
systems does not have engineering applications, since the simulation is
exponentially slow. This is why, if true randomness is involved in the
historical development of mathematics, a classical simulation of the
Wolfram model will be exponentially slow and only a simulation grounded on
quantum hardware can be accurate. Therefore, a prediction of the Wolfram
model of mathematics could be that an accurate simulation of the
development of mathematics can only be done using a quantum computer (at
least a quantum random number generator). This prediction (I take
responsibility for that) is grounded on the belief that the evolution of
mathematics and the evolution of the universe both share the fundamental
structures: quantum mechanics and relativity. If this prediction is
falsified, then mathematics will not be as quantum as the universe.

Kind regards,
Jose M.
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