Why is progress in mathematics possible? A computational approach to answer this question.

[José Manuel Rodríguez Caballero]

Dear FOM members,
The difficulty of this problem of progress in mathematics is based on the
fact that almost all strings of n symbols are incompressible in the sense
of Kolmogorov complexity, provided that n is large enough. Roughly
speaking, this means that most mathematical statements cannot be proved.
Hence, from a naive point of view, progress in mathematics should be
expected to be rarer than it has been in history.

In a recent essay [1], Stephen Wolfram proposed the following solution to
the problem of progress in mathematics.

One of the mysteries of mathematics is that it’s possible to make progress
> without continually getting mired in computational irreducibility and
> phenomena like undecidability. But in some sense this is just the
> mathematical analog of the fact that we as observers in the physical
> universe can identify slices of computational reducibility-and not be mired
> in computational irreducibility and utter unpredictability.


This explanation is based on the interaction between computational
irreducibility [2] and computational reducibility in a way that is similar
to what happens in S. Wolfram's theory of consciousness [3]. In this
approach, it is fundamental to consider the role of the "mathematical
observer", which is analogous to the physical "observer of the universe":

But to get further I think we have to consider "perception by mathematical
> observers" just as we consider "perception by physical observers". And
> what’s crucial about physical observers like us is that they manage to
> sample computationally reducible slices of the rulial universe—which they
> perceive as containing "definite things" consistent for example between
> different observers.


Here, the concept of rulial space [4, 5] is fundamental:

Rulial Space: The space defined by allowing all possible rules of a given
> class to be followed between states of a system. Different foliations of
> rulial space can be thought of as corresponding to different languages for
> describing behavior. The Principle of Computational Equivalence implies a
> fixed maximum speed ρ in rulial space. Rulial space for Turing machines is
> obtained by allowing all possible non-deterministic transitions.
>

My question is: How S. Wolfram's approach to explaining the possibility of
progress to mathematics is related to existing approaches to solve the same
problem?

Kind regards,
Jose M.

References
[1] Stephen Wolfram, "Why Does the Universe Exist? Some Perspectives from
Our Physics Project", Stephen Wolfram Writings, April 2021. URL =
https://writings.stephenwolfram.com/2021/04/why-does-the-universe-exist-some-perspectives-from-our-physics-project/

[2] Stephen Wolfram. A new kind of science. Vol. 5. Champaign, IL: Wolfram
Media, 2002. Page 737, URL =
https://www.wolframscience.com/nks/p737--computational-irreducibility/

[3] Stephen Wolfram, "What Is Consciousness? Some New Perspectives from Our
Physics Project", Stephen Wolfram Writings, March 2021. URL =
https://writings.stephenwolfram.com/2021/03/what-is-consciousness-some-new-perspectives-from-our-physics-project/

[4] Stephen Wolfram, "Exploring Rulial Space: The Case of Turing Machines",
Stephen Wolfram Writings, June 2020. URL =
https://writings.stephenwolfram.com/2020/06/exploring-rulial-space-the-case-of-turing-machines/

[5] Stephen Wolfram, "Glossary of the Wolfram Physics Project", URL =
https://www.wolframphysics.org/glossary/
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