L. E. J. Brouwer and the Sub-Axiomatic Foundations of Mathematics

[José Manuel Rodríguez Caballero]

Dear FOM members,
Traditionally, mathematics has developed from intuition, formalized in a
set of axioms. The statements of a theory are proven or disproven by means
of symbolic manipulations that are driven by algebraic, geometric,
analytical, combinatorial, and other kinds of intuition. If I understood
correctly, subaxiomatic foundations are the attempt to circumvent the
intelligibility of mathematical practice and develop mathematics directly
from the computational evolution of a complex system. From a technological
point of view, subaxiomatic foundations are the presentation of mathematics
that would be most compatible with the most powerful artificial
intelligence algorithms, for example, deep learning. Following the analogy
between symbolic machine learning for the conscious brain and statistical
machine learning for the unconscious brain, we could say that subaxiomatic
foundations are the mathematics of the subconscious part of the mind. That
is, this type of mathematics tries to operate directly on the process that
produces intuition rather than on its symbolic representation. Concerning
this possibility, S. Wolfram wrote [1]:

But if we actually want to make full use of the computational capabilities
> that our universe makes possible, then it’s inevitable that the systems
> we’re dealing with will be equivalent in their computational capabilities
> to our brains. And this means that—as computational irreducibility
> implies—we’ll never be able to systematically “outthink” or “understand”
> those systems.


But then how can we use them? Well, pretty much like people have always
> used systems from the natural world. Yes, we don’t know everything about
> how they work or what they might do. But at some level of abstraction we
> know enough to be able to see how to get purposes we care about achieved
> with them.


In statistics, there are two goals: explainability and prediction. The best
methods for one of these objectives are not the same as for the other.
Traditional mathematics is like methods for the explanation of a data set,
while subaxiomatic mathematics is like methods for prediction. It would be
nice to have a translator from subaxiomatic mathematics to a human-readable
mathematical system, such as Isabelle/HOL [2], but I am not sure that it
would be possible. The question of the meaning of a mathematical proof is
far from trivial. We can imagine proofs translated from this mathematical
source code into Isabelle/HOL language and still feel unintelligible.

There are many candidates as starting points of the subaxiomatic
foundations. One of them are Kauffman's calling and crossing symbols [3].
Other candidates are combinators [4]. Here is a series of videos developing
this idea, following a rather empirical approach [5] and here is a
computational notebook on this subject [6].

According to L. E. J. Brouwer [7]:

Completely separating mathematics from mathematical language and hence from
> the phenomena of language described by theoretical logic, recognizing that
> intuitionistic mathematics is an essentially languageless activity of the
> mind having its origin in the perception of a move of time. This perception
> of a move of time may be described as the falling apart of a life moment
> into two distinct things, one of which gives way to the other, but is
> retained by memory. If the twoity thus born is divested of all quality, it
> passes into the empty form of the common substratum of all twoities. And it
> is this common substratum, this empty form, which is the basic intuition of
> mathematics.


If we interpret the “substratum” mentioned by Brouwer as the evolution of a
complex system and the lack of mathematical language as unintelligibility,
I think we get the two main properties of the subaxiomatic foundations. Of
course, there is a lot of work associated with the word “intuitionism”
which Brouwer may reject as a betrayal of his original point of view. My
question would be: To what extent does L. E. J. Brouwer's intuitionism
resonates with the ideas of the subaxiomatic foundations?

Kind regards,
Jose M.

References
[1] Stephen Wolfram, Logic, Explainability and the Future of Understanding,
Stephen Wolfram Writings, URL:
https://writings.stephenwolfram.com/2018/11/logic-explainability-and-the-future-of-understanding/

[2] Nipkow, Tobias, Markus Wenzel, and Lawrence C. Paulson, eds.
Isabelle/HOL: a proof assistant for higher-order logic. Berlin, Heidelberg:
Springer Berlin Heidelberg, 2002.

[3] Kauffman, Louis H. "Iterants, Fermions and the Dirac Equation." *arXiv
preprint arXiv:1406.1929* (2014). https://arxiv.org/pdf/1406.1929.pdf

[4] Schönfinkel, Moses. "Über die Bausteine der mathematischen Logik."
Mathematische annalen 92.3-4 (1924): 305-316.

[5] Metamathematics: Sub-Axiomatic Foundations:
https://youtube.com/playlist?list=PLM_MgbAF1uYK4vSUEZtjnRVMfvcy-PshC

[6] [WELP22] Sub-axiomatic foundations of group theory in SK combinators
https://community.wolfram.com/groups/-/m/t/2818259

[7] Brouwer, L.E.J., 1981, Brouwer’s Cambridge lectures on intuitionism, D.
van Dalen (ed.), Cambridge: Cambridge University Press, Cambridge.
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