the physicalization of metamathematics (from thermodynamics to intuitionism)

Tue Mar 8 19:58:35 EST 2022

Just to share an idea about the physics-inspired framework

https://writings.stephenwolfram.com/2022/03/the-physicalization-of-metamathematics-and-its-implications-for-the-foundations-of-mathematics/

in which Wolfram proposes to formalize mathematics as it is done by real
mathematicians: the second law of thermodynamics in this model of
mathematics should be the opposite of that of ordinary physics, i.e.
entropy should decrease over time, rather than increase. We can imagine the
development of mathematics as a time-reversal of the development of the
universe. Because Wolfram's hypergraph for mathematics is finite at any
moment, the entropy is a finite number (if the system were infinite, the
entropy may not be defined).

Indeed, entropy is a measure of how much does the observer knows the
microstate of a system given the macrostate. The more developed a
mathematical theory is, the less its entropy should be. This is why
mathematicians like Grothendieck, instead of directly attacking a problem,
prefer to "clean" the theory so that the solution will be trivial. In
Grothendieck's own words, his methods can be described as follows:

I can illustrate the second approach with the same image of a nut to be
> opened.
> The first analogy that came to my mind is of immersing the nut in some
> softening liquid, and why not simply water? From time to time you rub so
> the liquid penetrates better,and otherwise you let time pass. The shell
> becomes more flexible through weeks and months – when the time is ripe,
> hand pressure is enough, the shell opens like a perfectly ripened avocado!
> A different image came to me a few weeks ago.
> The unknown thing to be known appeared to me as some stretch of earth or
> hard marl, resisting penetration… the sea advances insensibly in silence,
> nothing seems to happen, nothing moves, the water is so far off you hardly
> hear it.. yet it finally surrounds the resistant substance.

Reference: https://ncatlab.org/nlab/files/McLartyRisingSea.pdf

Gromov calls this approach, "proof by entropy", in contrast to the "proof
by energy", which he describes as follows:

Maybe, the simplicity of Kolmogorov's argument and an apparent
> inevitability with which it comes along with translation of
> "baby-Boltzmann" to "baby-Groethendieck" is illusory. An "entropy barrier"
> on the road toward a conceptual proof (unlike the "energy barrier"
> surrounding a "hard proof") may remain unnoticed by one who follows the
> marks left by a pathfinder that keep you on the track through the labyrinth
> under the ”mountain of entropy”.

>  All this is history. The tantalizing possibility suggested by entropy -
> this is the main reason for telling the story - is that there may be other
> "little somethings" around us the mathematical beauty of which we still
> fail to recognize because we see them in a curved mirror of our
> preconceptions.

Reference:
https://math.mit.edu/~dspivak/teaching/sp13/gromov--EntropyViaCT.pdf

According to some authors, e.g.,

Tanaka, Akinori, Akio Tomiya, and Kōji Hashimoto. *Deep Learning and
Physics*. Springer, 2021.
https://link.springer.com/book/10.1007/978-981-33-6108-9

the field of statistical learning evolved from statistical mechanics, and
the learning process is similar to how Maxwell's demon violates the second
law of thermodynamics. In physics, the possibility of Maxwell's demon is
prevented, because the updating of the memory of the demon will increase
the entropy of the system. In the same way, an entropy-oriented
mathematician, like Grothendieck, will play the role of Maxwell's demon to
decrease the entropy of a mathematical theory and there will be no
violation of the second law of thermodynamics since the act of thinking
will increase the entropy of the universe. Therefore, the development of
mathematics in Wolfram's framework could be interpreted as a freezing
process, where the "refrigerator" engine is the mathematician. If a theory
is undecidable, absolute zero should be unreachable.

It is also possible to heat up Wolfram's hypergraph for mathematics at any
time by adding new theorems, which are not an easy consequence of the
existing ones. Gödel's first incompleteness theorem can be interpreted as
the fact that Wolfram's mathematical hypergraph can always be heated,
provided that the theory described by this hypergraph is rich enough to
express Peano arithmetic. A decidable theory cannot be heated up
indefinitely.

In conclusion, when considering mathematics as a dynamic process (Wolfram's
approach) rather than a static system (Hilbert's traditional axiomatic
approach), the notion of entropy of mathematics at any given moment is
meaningful (but not unique, since there are several notions of entropy,
e.g., categorical cross-entropy, Shannon entropy, von Neumann entropy,
min-entropy, max-entropy, etc.). Analogies with thermodynamics and the
rather empirical field of statistical learning may provide insights for a
dynamical foundation of mathematics.

A question that I would like to ask the FOM member is: how does Wolfram's
dynamical conception of mathematics relate to intuitionism? Indeed, for
intuitionists, like for Wolfram, mathematics is a process:

Reference: https://plato.stanford.edu/entries/intuitionism/

Kind regards,
Jose M.
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