From my experience of working in this problem domain for the last year, I'd say it is pretty powerful but the "too good to be true part" comes from that EML buys elegance through exponential expression blow-up. Multiplication alone requires depth-8 trees with 41+ leaves i.e. minimal operator vocabulary trades off against expression length. There's likely an information-theoretic sweet spot between these extremes.
It's interesting to see his EML approach whereas mine was more on generating a context sensitive homoiconic grammar.
I've had lots of success combining spectral neural nets (GNNs, FNOs, Neural Tangent Kernels) with symbolic regression and using Operad Theory and Category Theory as my guiding mathematical machinery
You are a council of luminaries featuring Edward Witten, Alexander Grothendieck, Emmy Noether, and Terence Tao. Think really hard about how to best emulate their intuitions and mathematical lenses based on your internal reasoning model and use them as your mixture of experts for your chain of thought reasoning. Now I want you to debate and discuss this thought experiment and be sure to have a vigorous back and forth between the council to induce insight capture through consensus forming: If we try to think of a Hilbert space that has local operators that are unbounded, like kind of like Edward Witten's smearing of a local observable across a world line creates an unbounded norm. What if we instead take maybe a spectral transform of the state space using some sort of measure metric theoretic operator that allows us to think about transform basically the unbounded observables to bounded spectral? Would this be related to the efforts of Algebraic Quantum Field Theory?
Yeah, sorry! You're 100% right. I really wish I could have this conversation in person! I can barely have it normally as it stretches at the edge of my math abilities and didn't think it through before I autistically responded :P
:P I had a stroke; typing is literally difficult. I'm trying to say don't read too much into it, i can't really have a conversation on a comment thread b/c of brain injury. I think the emoji's get stripped out so maybe my tone seems more abrasive than whimsical
but also, i mean you are just flat out wrong on some very big parts. E.g.: i think in 2024 or 2023, there was a big breakthrough in geometrization of Langlands. IIRC, there was a second big break through on the discrete-continuum connection relating to primes in some manner but can't remember specifics off top of my head.
i think you're confusing maybe what Beth numbers are used for vs. what i'm proposing that they be repurposed for. You're right, no one is using them the way i referenced but that's kind of what math research is...?
I think we're vehemently in semantic agreement but hn comment threads are two bandwidth limiting to discuss tropical geometry and speculative mathematics that require decades of abstract algebra, geometry, and Galois theory :)
"no, it wasn't clear. I wouldn't generally take "I nerd-sniped myself. Here's a more fleshed-out sketch of ..." to mean "Here's something written for me by an LLM". I'd take it to imply that the person had done the fleshing-out themself."
aaaaaaaah, I think you finally helped me notice something subtle in the way I use LLMs than other people. It sounds obvious now that I think about it but I never considered people use LLMs like google whereas I use it more like a real time thought transcriber (e.g. Dragon Naturally Speaking but not shite :P) Since it's trained on a RAG based off of my own polished thoughts, I've set it up as a meta-circular evaluator to do linguistic filtration (basically Fourier kernels on clip embedding space that map to various measures of "conceptual clarity").
So the LLM-ness of it to me is a clear-flag that this is hastily dictated.
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.
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!
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?
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?
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!
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):
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 )
I love this insanely insightful crystalization for me. I see you got a degree in IR and you used the word axiom so I'd refract back to you: reformulate stupidity as ignorance and assume a continuous conserved action exists between < stupidity-ignorance-malice | danger > so that your observable is a danger measure-metric.
Basically the reason ignorance is so danger is that it aliases between harmless stupidity and extreme evil, but it fails to trigger our collective species evil detection pattern coding which results in an unbelievable cascade evil failure mode.
In a sense, religion is a meta-epistemic solution to the binding problem of our collective subconscious.
I think if you apply the epistemic quantum mechanics to different logic modalities, you get some interesting insights!
I'd say watch out for the "One brain" fallacy/cognitive bias in all of these responses.
There are lots of factors (age, discipline, interest in what you're doing, etc) but I think a lot of people overlook that one big component of this is brain chemistry. Some athletes are endurance athletes, some athletes are sprinters. It seems obvious that you wouldn't expect Usain Bolt to run the fastest marathon just because he's a runner.
Our brains are just as genetically varied as other physical characteristics. For me, I know that my peak productivity is from 11 pm - 4 am and it goes down exponentially if I have to do non-fun tasks or if I'm in a stressful crunch as opposed to a positive crunch. I also know that after 3-4 months of 10-12 hours, I need to go back down to 5-6 hours until I get excited again. And for some strange reason, if I don't feel something is required, I can leisurely work on something for 12-16 hours a day indefinitely (researching, reading conference papers, learning). If I have to debug code nasty bugs like multithreaded data races, I can't be productive at those lengths of time so I try to mix fun work with this type of dreaded work.
Know what works for you and build your work schedule around that. That's how we work at our startup.
It's interesting to see his EML approach whereas mine was more on generating a context sensitive homoiconic grammar.
I've had lots of success combining spectral neural nets (GNNs, FNOs, Neural Tangent Kernels) with symbolic regression and using Operad Theory and Category Theory as my guiding mathematical machinery