[ExI] Gemini on Mathematical Realism

[Jason Resch]

On Tue, Oct 6, 2026 at 9:02 PM Stuart LaForge via extropy-chat <
extropy-chat at lists.extropy.org> wrote:

> On 2026-10-06 14:31, Jason Resch via extropy-chat wrote:
>
>
>
[snip.. I just wanted to focus on this one statement, and a possible
contradiction in the hybrid view]


>
> >>> seems to imply] some form or other of Platonism or ‘realism’
> >> as to
> >>> the mathematical objects."
> >>> -- Kurt Gödel in “Some basic theorems on the foundations of
> >>> mathematics and their implications p. 311″ (1951)
>
> Gemini is saying that all axioms imply a universe of objective truth
> that is discoverable by any intelligence that accepts those axioms as
> true. Note that the objective truth, in this case, can apply equally to
> pure math or physical law. The objectivity of this truth stems from the
> fact that all beings that truly understood the question would agree upon
> the answer. In that instance, the answer is the objective truth for all
> intents and purposes.
>
>
>
To me, the Gödel realization implies truth ≠ proof. Formal systems deal
with proofs, but as Gödel showed, always leave some gap, there are truths
that lack proofs.
Where do such "truths without proofs" land in Gemini's view? To me, they
sit outside the given axiomatic system, and highlight the limit of purely
constructivist views.

Take any axiomatic system that can model the integers: There will always
remain infinitely many things that are true about the integers but which
that axiomatic system cannot prove. For example, the existence or
non-existence of solutions to particular Diophantine equations. This limit
emerges from just +, -, the equals sign, and the integers. In what sense
does mathematical truth concerning the integers exist in Gemini's view? If
Gemini gives an answer in terms of what a particular axiomatic system can
prove, then this only reaches some of that truth. As Gödel's reasoning
showed, this is a larger picture. Our finite axiomatic systems are finite
attempts to capture and define the infinite, and incompleteness proves all
such attempts are doomed to fail.

Jason


----------------
> Jason:
> >>>
> >>> I think there is also independent evidence of mathematicalism
> >> realism,
> >>> in the sense that many otherwise unexplainable features of our
> >>> physical universe, can be explained if we assume we exist in an
> >>> infinite ensemble (of the type that would exist under mathematical
> >>> realism). I summarize the set of explainable/derivable
> >> consequences
> >>> that others have discovered here:
> >>
> >
> https://zenodo.org/records/19474213/files/Deriving-Physics-from-the-Set-of-All-Computations.pdf?download=1
>
> I think that the completely different definitions of reality and realism
> between math and physics (most notably quantum) make any coherent
> discussion involving both physics and math using the same word "real"
> fraught with misunderstanding. I am working on some metaphysical
> formalism that addresses this that I am looking forward to sharing with
> you in the future.
>
>
I look foreward to it!

Jason
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