Comment by 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.)