[ExI] Gemini on Mathematical Realism

[Jason Resch]

On Wed, Oct 7, 2026, 3:50 PM Ben Zaiboc via extropy-chat <
extropy-chat at lists.extropy.org> wrote:

> On 07/10/2026 18:50, Jason Resch wrote:
> >
> > On Wed, Oct 7, 2026, 4:57 AM Ben Zaiboc via extropy-chat <
> extropy-chat at lists.extropy.org> wrote:
> >
> >     On 06/10/2026 22:32, Jason Resch wrote:
> >     > Godel used almost the same argument to reach the same conclusion:
> >     >
> >     > "[The existence of] absolutely undecidable mathematical
> propositions, seems to disprove the view that mathematics is only our own
> creation; for the creator necessarily knows all properties of his
> creatures, because they can’t have any others except those he has given to
> them.
> >
> >
> >     This is not true. A creation often has properties that its creator
> didn't explicitly give it, and has no prior knowledge of.
> >
> >     John Conway didn't know all the properties of his Game of Life, that
> has become a famous example of not knowing what a system will do before it
> does it. Mandelbrot didn't know all the properties of fractals. Those are
> just two immediately obvious examples. There are many more.
> >
> >
> >
> > But note the two examples you gave are mathematical objects--the very
> kind that Godel would say were discovered rather than created.
>
>
> Are they? I thought they were computer algorithms. They can probably be
> represented mathematically, but that doesn't make them 'mathematical
> objects'. Does it?


The Game of Life is a particular instance of a cellular automata (
https://en.wikipedia.org/wiki/Cellular_automaton ) and automata theory is a
field that branches across computer science and mathematics. John von
Neumann famously discovered a self replicating machine (
https://en.wikipedia.org/wiki/Von_Neumann_universal_constructor ) that
existed within a cellular automaton, and he did this without using a
computer and more remarkably, before the discovery of DNA.

Turing machines themselves were first defined mathematically, and in a
sense they only exist abstractly in mathematics given their stipulated
infinite memory, so I would also class computer science as a subset of
mathematics. Turing's teacher Alonzo Church beat Turing to the punch with
his "lambda calculus" a way of defining programs via recursive mathematical
functions.

The Mabdlebrot set is itself a recursive function applied in a complex
plane ( https://en.wikipedia.org/wiki/Complex_plane ) . It is where the
function: f(z) = z^2 + c
For the complex point c, is applied to itself over and over again to
itself, setting z = 0 for the first iteration, and checking whether this
process diverges towards infinity or not.

So it too is mathematical, but without the aid of computers it is quite
onerous to draw at any reasonable resolution. I remember a friend wrote a
program to draw it on a TI-83 graphing calculator in middle school.

There remain many unknown properties of the Mabdlebrot set. There's still
quite a bit of uncertainty concerning its area, for example.


I don't really know what it means anyway. An object made of maths? Errr,
> never mind.
>

It is a difficult question. Especially when one considers they our whole
universe might itself exist as a mathematical object.

https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis


> But leaving that aside, there are things in the systems that can be
> discovered, yes, but the point is that they can only be discovered /after
> the system has been invented/. No glider gun existed to be discovered
> before Conway created the Game of Life. It's a thing that can be
> discovered, but only as a consequence of (and only within) a system that is
> invented. A system that didn't exist before.
>

Would intelligent aliens likely come upon numbers like e and Pi? Would they
have invented such numbers or would they have discovered their unavoidable
applicability across a diverse range of problems, just as they might make
similar discoveries of gravity and radio waves?

Is it possible or perhaps even likely that aliens would also happen upon
automata theory, and find the GoL particularly interesting among possible
rule sets, given it's capacity for emergent complexity?

You are right that as we move from more fundamental mathematical things
like Pi and e, to cellular automata, and on to things like the game of
chess, then the word "invented" does seem to become more appropriate than
"discovered." And finally when you get to things like fictional stories
then discovered seems a very strange word to use.

I think it all comes down to our sense of the probability of someone else
on earth happening upon something. The more inevitable it is the more apt
we are to use the word discovered.

But across a spatially infinite universe, countless authors on countless
planets "discover" the works of Shakespeare, and countless composers
"discover" the works of Beethoven. If nothing is ever truly unique or novel
across the whole of reality, then invention only has meaning locally.
Across the whole of reality, everything is merely (re)discovered.



> Yes, you can claim that it 'existed' in some hypothetical sense, in that
> it's one of an infinite number of things that could possibly exist, but
> that means nothing, really, and leads nowhere.
>

It's fun to think about. :-)


>
>
> >
> > Compare this against the properties a fiction author creates for the
> characters in his book, all of which are assigned explicitly or implicitly
> through the text put on paper. There is no more to those characters than
> what the author happens to write down.
> >
> > But this is never true for objects in math. The number 3 has more
> properties than could ever be written down in any book, even if you used
> all the atoms in the observable universe as pages.
> >
> >
> >     Given this, it would seem that Godel was wrong, and the existence of
> undecidable propositions does not disprove the view that maths is our
> creation.
> >
> >     In fact, the Game of Life is a good illustration of the whole thing:
> Glider guns had to be discovered, they weren't known until someone chanced
> upon them, but neither did they exist prior to Conway creating the rules of
> the game. So there was no pre-existing 'platonic' glider gun, until someone
> set up the rules that brought it into existence.
> >
> >
> > Yes and note you used the word "discovered" which I think is the
> appropriate word for finding things like glider guns or other possible
> structures or properties in GoL.
> >
> > GoL was defined (I won't use created or discovered here to not beg the
> question) in 1970, but it wasn't until 1982 that it was proven that one
> could build a Turing machine in the game of life. And it wasn't until 2018
> that someone found  a way to build a universal Turing machine with a
> working tape.
> >
> > This possibility was always there in the basic rules. But none of this
> was immediately known to Conway when he out the rules down on paper.
> >
> > So as Godel asked: could such a thing like GoL be "only our own
> creation" when it has so many rich properties beyond what Conway originally
> knew? And if these later-discovered properties were not known by Conway,
> and therefore not specified by him, then where do they come from?
>
>
> They are a consequence of the rules of the system. You could call them
> emergent properties, probably. As soon as any system crosses a certain
> threshold (which must lie somewhere between a dynamic system like GoL and a
> fixed one like a printed story), then unexpected and unknown properties can
> emerge.
>

I consider something as objectively real through the test of
intersubjective agreement, i.e. if two or more independent researchers
investigating the same thing reliably converge on discovering the same set
of previously unknown properties, then that thing is real.

Admittedly this test has its flaws, but I don't know of a better one, and
if you know of a better definition or test, then I hope you will share it.
(By better I mean having a lower false negative rate, as well as a lower
false positive rate).

One consequence of this definition is it makes anything that can be
consistently defined real, and don't leads to platonism. This may be a
weakness or a strength, depending on your inclinations. But one advantage
is it explains many features of our universe and its laws


> GoL was definitely created, not discovered in any but the most trivial and
> meaningless sense. If you can claim that it was discovered, then absolutely
> everything, without exception, is 'discovered', and the word loses its
> meaning.
>

I agree the difference between invented and discovered can lose its
distinction.



>
> >  How can two isolated observers both discover the same properties of GoL
> or the Mandelbrot set, unless there is some sort of objective truth exists
> concerning these objects? If the properties are already there, waiting to
> be found, then mathematical objects are not *only* our creations.
>
>
> It would be very strange if two different observers discovered conflicting
> properties of the same system, don't you think?


If the thing is real then it shouldn't happen. If it is fictional, however,
then different observers might reach different opinions deriving previously
unknown properties, for example in deciding how many hairs Ebenezer Scrooge
has on his head, or what Santa Claus's thumbprint looks like.

I'd go as far as to say that it's probably impossible, unless the system
> somehow changes depending on who is investigating it.
>
> As I keep saying (and as I think that Gemini is saying, if I understand
> correctly), the properties come into being as a consequence of the system
> being created (and used.


I don't buy it. How can one finite act, by finite beings, in a finite
universe, create infinite properties?

As I see it, either finite acts only create finite things, and there aren't
infinite primes or digits of Pi, or there are infinite primes and digits of
Pi, and our finite acts can only finitely grope at them (but they can't
create them).

Gemini's hybrid view (which I think is best described as contingent
platonism) seems to combine the least intuitive aspects of both platonism
and constructivism. It has the infinitely incomprehensible objects of
platonism, but it conditions their existence on finite and temporal human
activity.

If Conway created GoL but nobody ever played it, I don't think it would
> really make sense to claim that glider guns existed. After all, nobody
> would have heard of them! The term, and the concept, would never exist).
> There may be an infinity of such properties, but before the system existed,
> they didn't exist either. And if nobody ever reveals them by playing the
> game or running the system, whatever it might be, then it probably makes
> sense to say that they still don't exist yet, even though the potential for
> them does.
>

What would you say if someone in another far away galaxy (outside our light
cone) discovered them, would they exist? If you say it would not exist in
that case, the must the question of existence be answered separately for
every observer? If you meet a guy on the street who's never heard of Conway
or his GoL, does it not exist for him?


> Otherwise, you end up saying that an infinity of things that we have
> absolutely no knowledge of, can't even imagine, and will never ever see,
> definitely 'exist'.


That is a view I have learned to make peace with. Setting aside the
hundreds of billions of galaxies and the 10^22 other star systems in the
observable universe, there are at least 10^120 bubble universes with
different vacuum energies, and at least 10^500 possible string theory
universes. There are also very many integers and digits of Pi and primes we
will never see (most of them in fact).

This is what it means to be finite beings with finite brains in a finite
observable universe which sits in a plausibly infinite space, in a
plausibly infinite multiverse, which may fall out of an infinite Ruliad or
the set of all mathematical objects or all possible computations.

I can't see such a viewpoint being any use at all, and of course you can
> never know if it's true. It becomes something like another kind of 'halting
> problem'.
>

It is easy to think that such an ontology has no consequences, but quite
surprisingly, that is a false assumption. If all computations exist, then
there are testable consequences that follow. I documented 98 examples of
these consequences from a survey of the literature here:

https://zenodo.org/records/19474213

Jason
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