Comment by ikrima

Comment by ikrima 3 days ago

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Hey! This is fantastic and actually ties in some very high disparate parts of math. Basically, reorient & reformulate all of math/epistomology around discrete sampling the continuum. Invert our notions of Aleph/Beth/Betti numbers as some sort of triadic Grothendieck topoi that encode our human brain's sensory instruments that nucleate discrete samples of continuum of reality (ontology)

Then every modal logic becomes some mapping of 2^(N) to some set of statements. The only thing that matters is how predictive they are with some sort of objective function/metric/measure but you can always induce an "ultra metric" around notions of cognitive complexity classes i.e. your brain is finite and can compute finite thoughts/second. Thus for all cognition models that compute some meta-logic around some objective F, we can motivate that less complex models are "better". There comes the ultra measure to tie disparate logic systems. So I can take your Peano Axioms and induce a ternary logic (True, False, Maybe) or an indefinite-definite logic (True or something else entirely). I can even induce bayesian logics by doing power sets of T/F. So a 2x2 bayesian inference logic: (True Positive, True Negative, False Positive, False Negative)

Fun stuff!

Edit: The technical tldr that I left out is unification all math imho: algebraic topology + differential geometry + tropical geometry + algebraic analysis. D-modules and Microlocal Calculus from Kashiwara and the Yoneda lemma encode all of epistemology as relational: either between objects or the interaction between objects defined as collision less Planck hyper volumes.

basically encodes the particle-wave duality as discrete-continuum and all of epistemology is Grothendieck topoi + derived categories + functorial spaces between isometry of those dual spaces whether algebras/coalgebra (discrete modality) or homologies/cohomologies (continuous actions)

Edit 2: The thing that ties everything together is Noether's symmetry/conserved quantities which (my own wild ass hunch) are best encoded as "modular forms", arithmetic's final mystery. The continuous symmetry I think makes it easy to think about diffeomorphisms from different topoi by extracting homeomorphisms from gauge invariant symmetries (in the discrete case it's a lattice, but in the continuous we'd have to formalize some notion of liquid or fluid bases? I think Kashiwara's crystal bases has some utility there but this is so beyond my understanding )

ricardobeat 3 days ago

> Invert our notions of Aleph/Beth/Betti numbers as some sort of triadic Grothendieck topoi that encode our human brain's sensory instruments that nucleate discrete samples of continuum of reality (ontology)

There’s probably ten+ years of math education encoded in this single sentence?

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  • gjm11 3 days ago

    My apologies to ikrima for being critical, but I think anyone who thinks "aleph/beth/Betti numbers" is a coherent set of things to put together is just very confused.

    Aleph and beth numbers are related things, in the field of set theory. (Two sequences[1] of infinite cardinal numbers. The alephs are all the infinite cardinals, if the axiom of choice holds. The beth numbers are the specific ones you get by repeatedly taking powersets. They're only all the cardinals if the "generalized continuum hypothesis" holds, a much stronger condition.)

    [1] It's not clear that this is quite the right word, but no matter.

    Betti numbers are something totally different. (If you have a topological space, you can compute a sequence[2] of numbers called Betti numbers that describe some of its features. (They are the ranks of its homology groups. The usual handwavy thing to say is that they describe how many d-dimensional "holes" the space has, for each d.)

    [2] This time in exactly the usual sense.

    It's not quite true that there is no connection between these things, because there are connections between any two things in pure mathematics and that's one of its delights. But so far as I can see the only connections are very indirect. (Aleph and beth numbers have to do with set theory. Betti numbers have to do with topology. There is a thing called topos theory that connects set theory and topology in interesting ways. But so far as I know this relationship doesn't produce any particular connection between infinite cardinals and the homology groups of topological spaces.)

    I think ikrima's sentence is mathematically-flavoured word salad. (I think "Betti" comes after "beth" mostly because they sound similar.) You could probably take ten years to get familiar with all the individual ideas it alludes to, but having done so you wouldn't understand that sentence because there isn't anything there to understand.

    BUT I am not myself a topos theorist, nor an expert in "our human brain's sensory instruments". Maybe there's more "there" there than it looks like to me and I'm just too stupid to understand. My guess would be not, but you doubtless already worked that out.

    [EDITED to add:] On reflection, "word salad" is a bit much. E.g., it's reasonable to suggest that our senses are doing something like discrete sampling of a continuous world. (Or something like bandwidth-limited sampling, which is kinda only a Fourier transform away from being discrete.) But I continue to think the details look more like buzzword-slinging than like actual insight, and that "aleph/beth/Betti" thing really rings alarm bells.

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    • ikrima 3 days ago

      also you're onto the actual quantum mechanics paper I'm working on. QM/QFT is modern day epicycles: arbitrarily complex because it was the aliasing the natural deeper representation which was Fourier/Spectral analysis.

      Reformulating our entire ontology around relational mechanics is the answer imho. So Carlo Ravoli's RQM is right but I think it doesn't go far enough. Construct a grothendeik topos with a spacetime cohomology around different scales of both space and time with some sort of indefinite conservation and you get collision less Planck hyper volumes that map naturally to particle-wave duality interpretations of QM.

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    • ikrima 3 days ago

      lol, it's a sketch of a proof covering a large swath of unexplored math. the other poster wasn't wrong when he said I smashed 10y+ of graduate math in one sentence.

      Aleph numbers = these are cardinals sizes of infinity; depending on your choice of axioms, ZFC or not, you have the continuum hypothesis of aleph0 = naturals, aleph1= 2^N = Continuum

      Beth numbers are transfinite ordinals => they generalize infinitesimals like the 1st, 2nd, 3rd. so you can think of them as a dual or co-algebra (I'm hand waving here, it's been twenty years since real analysis).

      Betti numbers are for persistent cohomology; they track holes similar to genus

      I mean there's a lot to cover between tropical geometry, differential geometry, and algebraic analysis. So sometimes alarm bells are false alarms and your random internet commenter knows what he's talking about but is admittedly too sloppy but it's 5 pm on a Saturday and I wrote that in the morning while making breakfast eggs, not for submission to the annals of Mathematics!

      Thank you for coming to my TED Stand Up Talk.

      More math at the GitHub: http://github.com/ikrima/topos.noether

      Also, if you're really that uptight, most of this is actually to teach algebraic topology to my autistic nonverbal nephew because I'm gonna gamify it as a magic spell system

      So it'll be open source and that begs the question, if you use it to learn something, did that mean I just zero-proof zero-knowledge something out of you that I didn't even need to know by making a self referential statement across both space & time?

      peace out my ninja!

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      • gjm11 2 days ago

        The comment you're replying to already explained what aleph, beth and Betti numbers are. (But a few nitpicks: 1. Beth numbers are not ordinals, they're cardinals. They're indexed by ordinals, just as the alephs are, but if that's what you care about why not use the ordinals themselves? 2. I'm not seeing how you get from "Beth numbers are indexed by ordinals" to "they generalize infinitesimals" to "you can think of them as a dual". Not saying there isn't something there, but I think you could stand to unpack it a bit if so. 3. Betti numbers are not only for persistent (co)homology; they were around long before anyone had thought of persistent (co)homology.)

        It's certainly possible (as I explicitly said before) that my bad-math-alarms have hit a false positive here. You haven't convinced me yet, for what it's worth. (You need not, of course, care whether you convince me or not. It's not as if my opinion is likely to have any effect on you beyond whatever you might feel about it.)

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    • ikrima 3 days ago

      I mean you wouldn't be wrong to assume so but how can you expect anyone to saliently condense the entirety of a 10 year long proof of Grothendieck topos to 3 or 4 sentences my guy!

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  • ikrima 3 days ago

    you know what, I nerd sniped myself, here's a more fleshed out sketch of the [Discrete Continuum Bridge

    https://github.com/ikrima/topos.noether/blob/master/discrete...

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    • gjm11 3 days ago

      It seems to be entirely written by an LLM.

      [EDITED to add:] This is worth noting because today's LLMs really don't seem to understand mathematics very well. (This may be becoming less so with e.g. o3-pro and o4, but I'm pretty sure that document was not written by either of those.) They're not bad at pushing mathematical words around in plausible-looking ways; they can often solve fairly routine mathematical problems, even ones that aren't easy for humans who unlike the LLMs haven't read every bit of mathematical writing produced to date by the human race; but they don't really understand what they're doing, and the nature of the mistakes they make shows that.

      (For the avoidance of doubt, I am not making the tired argument that of course LLMs don't understand anything, they're just pattern-matching, something something stochastic parrots something. So far as I can tell it's perfectly possible that better LLMs, or other near-future AI systems that have a lot in common with LLMs or are mostly built out of LLMs, will be as good at mathematics as the best humans are. I'm just pretty sure that they're still some way off.)

      (In particular, if you want to say "humans also don't really understand mathematics, they just push words and symbols around, and some have got very good at it", I don't think that's 100% wrong. Cf. the quotation attributed to John von Neumann: "Young man, in mathematics you don't understand things, you just get used to them." I don't think it's 100% right either, and some of the ways in which some humans are good at mathematics -- e.g., geometric intuition, visualization -- match up with things LLMs aren't currently good at. Anyway, I know of no reason why AI systems couldn't be much better at mathematics than the likes of Terry Tao, never mind e.g. me, but they aren't close enough to that yet for "hey, ChatGPT, please evaluate my speculation that we should be unifying continuous and discrete mathematics via topoi in a way that links aleph, beth and Betti numbers and shows how our brains nucleate discrete samples of continuum reality" to produce output that has value for anything other than inspiration.)

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      • ikrima 3 days ago

        Yup, it's 100% generated by an LLM. I thought that was intentionally clear? (I'm recovering from a TBI so I'm still adjusting to figuring out how to relearn typing; I use the LLMs as my voice mediated interface to typing out thoughts).

        I'm not sure there's an argument I'm hearing here other than you seem to have triggered some internal heuristic of "this was written by an LLM" x "It contains math words I don't understand" => "this is bullshit"

        which you wouldn't be wrong but I am making a specific constructionist modal logic here using infinity-groupoids from category theory. infinite dimensional categories are a thing and that's what these transfinite numbers represent

        you have hyperreal constructionists of the reals as well which follows nonstandard analysis. you can also use the Weil cohomology which IIRC gets us most of calculus without the axiom of choice but someone check me on that.

        so....again, not sure what your specific critique is?

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ikrima 3 days ago

You know what, since you put in all that work, here's my version using p-adic geometry to generalize the concept of time as a local relativistic "motive" (from category theory) notion of ordering (i.e. analogous to Grothendieck's generalization of functions as being point samples along curves of a basis of distributions to generalize notions of derivatives):

https://github.com/ikrima/topos.noether/blob/aeb55d403213089...

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