Comment by gjm11
1 month 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.)
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
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.
: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...?
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