[ExI] Gemini on Mathematical Realism

[Ben Zaiboc]

On 07/10/2026 18:50, Jason Resch wrote:
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> On Wed, Oct 7, 2026, 4:57 AM Ben Zaiboc via extropy-chat <extropy-chat at lists.extropy.org> wrote:
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>     On 06/10/2026 22:32, Jason Resch wrote:
>     > Godel used almost the same argument to reach the same conclusion:
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>     > "[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.
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>     This is not true. A creation often has properties that its creator didn't explicitly give it, and has no prior knowledge of.
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>     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.
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> 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? I don't really know what it means anyway. An object made of maths? Errr, never mind.

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.

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.



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> 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.
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> 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.
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>     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.
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>     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.
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> 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.
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> 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.
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> 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.
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> 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.

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.


>  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? 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. 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.

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'. 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'.
-- 
Ben