It covers both the limits of what can be proven in the Peano Axioms, and how one would begin bootstrapping Lisp in the Peano Axioms. All of the bad jokes are in the second section.
Corrections and follow-up questions are welcome.
In the "Why Lisp?" section there is a bit of basic logic defined. The first of those functions appears to have unbalanced parentheses.
>(defun not (x) > (if x > false > true)
I have a compulsion that I can't control when someone starts using parentheses. I have to scan to see who does what with who.
You later say in the same section
>But it is really easy to program a computer to look for balanced parentheses
Okay. This is pretty funny. Thanks for the laugh. I realize that you weren't doing that but it still is funny that you pointed out being able to do it.
This later comment in the "Basic Number Theory" section was also funny.
>; After a while, you stop noticing that stack of closing parens.
I really enjoyed reading this post. Great job. Though it has been a long time since I did anything with Lisp I was able to walk through and get the gist again.
Are you saying that parentheses introduce the problem of having to scan to see what goes with what?
As in, if we don't have parentheses, but still have recursive/nested structure, we don't have to scan?
I automatically count and match even today, 40 years later.
And I'm glad that someone liked some of my attempts at humor. As my wife sometimes says, "I know that you were trying to make a joke, because it wasn't funny."
Can we not ruin every technology we develop with ads?
Also is this somewhat not just lining Google's/ad networks pockets? They won't care about even a large-ish number of people using this add on.
The larger adtech companies will have some form of bot-activity-detection, _but_ plenty will let it all just slip on by. (I've personally written a Babashka script at $job that checks performance of our publishers, and flags any fraudulent-seeming activity)
Fewer people went to pay. That’s the crux. Companies wanting mass market penetration, rather than just accept the natural size of the market.
Very reminiscent of the New Math pedagogy of the 1960s. Built up arithmetic entirely from sets. Crashed and burned for various reasons but I always had a soft spot for it. It also served as my introduction to binary arithmetic and topology.
(I am not suggesting that that very specific sequence of events happened in this case. It's just an example of how "I was looking this up for other reasons and then I saw something about it on HN" could genuinely happen more often than by chance, without any need for weird tracking-and-advertising shenanigans.)
I also find it super weird that Peano axioms are enough to encode computation. Again, this might be trivial if you think about it, but that's one self-referential layer more than I've thought about before.
One question for you btilly - oddly enough, I just recently decided to learn more Set Theory, and actually worked on an Intro to Set Theory textbook up to Goodstein sequences just last week. I'm a bit past that.
Do you have a good recommendation for a second, advanced Set Theory textbook? Also, any recommendation for a textbook that digs into Peano arithmetic? (My mini goal since learning the proof for Goodstein sequences is to work up to understanding the proof that Peano isn't strong enough to prove Goodstein's theorem, though I'll happily take other paths instead if they're strongly recommended.)
I don't have good suggestions for a good set theory textbook. Grad school was 30 years ago, and I didn't specialize in logic.
The best set theory book that I read was https://www.amazon.com/Naive-Theory-Undergraduate-Texts-Math.... But that one is aimed at people who want to go into math but do not wish to specialize in set theory, and not at people who want to actually learn set theory.
https://justine.lol/sectorlisp2/
Also, lots of Lisp from https://t3x.org implement numerals (and the rest of stuff) from cons cells and apply/eval:
'John McCarthy discovered an elegant self-defining way to compute the above steps, more commonly known as the metacircular evaluator. Alan Kay once described this code as the "Maxwell's equations of software". Here are those equations as implemented by SectorLISP:
ASSOC EVAL EVCON APPLY EVLIS PAIRLIS '
Ditto with some Forths.
Back to T3X, he author has Zenlisp where the meta-circular evaluation it's basically how to define eval/apply and how to they ared called between themselves in a recursive way.
When it comes to bootstrapping a programming language from nothing, the two best options are Lisp and Forth. Of the two, I find Lisp easier to understand.
There are two places where you accidentally wrote “omega” instead of “\omega”.
I am away from my computer for the day, but I will fux it later.
I think the problem is that it's already fuxed.
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 )
There’s probably ten+ years of math education encoded in this single sentence?
Wish me luck!
https://github.com/ikrima/topos.noether/blob/master/discrete...
[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.)
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?
(In a different comment I make some actual criticisms of what you wrote. I see you replied to my comment there, and that's a more appropriate place to discuss actual ideas. I don't see much point in criticizing LLM output in a field LLMs are bad at.)
Anyway: (1) 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. And (2) no, the problem wasn't that you used words I don't understand. It's certainly possible that your ideas are excellent and I just don't understand them, but I'm a mathematician myself and none of the words scare me.
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.
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.
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!
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.)
For Beth numbers, the wikipedia article is plenty enough to get you started: https://en.wikipedia.org/wiki/Beth_number
I'm aware I'm being a bit dickish about this, which I regret, but I'm not sure how else to respond to what seem like repeated deliberate attempts to frame this as "ikrima, the expert, kindly condescends to provide some elementary mathematics education to gjm11, the novice" which doesn't appear to me to be an accurate characterization of the situation.
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'm not sure how progress on geometric Langlands means I'm "flat out wrong on some very big parts". I certainly didn't say "there has been no progress on the geometric Langlands conjecture recently" or anything like that, nor did I say anything like "there is never any connection between discrete and continuous things in mathematics". (Not least because that's obviously very false.) So I don't understand what it is I've said that you think progress on the geometric Langlands conjecture is a counterargument to.
I understand that you're describing things that (you hope) could be done, rather than things that are already standard practice. But I don't think you've given any actual indication of how beth numbers in particular are going to be relevant. They're a very specific thing, and nothing you've said about them seems to make any contact with what they are in a way that would distinguish them from (say) the alephs or even the ordinals.
What I'm not seeing is any sign that there's more here than (1) a very general idea that could be stated perfectly well without any very advanced mathematics (e.g., "wouldn't it be nice to have a unified theory of how continuous and discrete things relate to one another, and apply it to human perception and cognition?") and (2) a bunch of buzzwords from various fields of advanced mathematics that have some superficial connection to #1, and (3) a handwavy assertion that somehow these things are all deeply connected ... without any actual deeper connection than "these things all sound like they might relate to one another".
What makes all that advanced mathematics worth the effort is its precision and rigour. Maybe all the precision and rigour is actually there and you just haven't chosen to show it to us. Maybe you have a deep enough intuition for these things that we should just trust that the details can be filled in precisely and rigorously. If it were, say, Terry Tao saying these things then I'd be inclined to trust him (though only provisionally; very smart mathematicians do sometimes have grand visions that never work out). Maybe, maybe, maybe. But as yet I haven't seen the evidence. And, whereas very smart mathematicians do sometimes have grand visions that never work out, the grand visions of amateurs unfortunately almost never turn out to have much substance. (Not literally never. Ramanujan was the real deal, for instance. But for every Ramanujan there are thousands of people who are, well, Not Ramanujan.)
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.
https://github.com/ikrima/topos.noether/blob/aeb55d403213089...