[ikrima]

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!

[gjm11]

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.)

[ikrima]

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 :)

For Beth numbers, the wikipedia article is plenty enough to get you started: https://en.wikipedia.org/wiki/Beth_number

[gjm11]

It would be plenty enough if I needed to get started. But you don't seem to be paying sufficient attention to what I wrote to notice that I already know what the beth numbers are and that unlike you I haven't written anything flatly false about them in this discussion.

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.

[ikrima]

: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...?

[gjm11]

(Conversational difficulties acknowledged. And, if trying to conduct this conversation here is just too inconvenient or annoying, I promise I will not take dropping out as admission of defeat or anything like that.)

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.)

[ikrima]

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