Re: mystification and categorification
While I'm airing my confusions, can anyone tell me what 'categorification' means? I don't know any such process; the simplest exanple, 'categorifying' natural numbers to get finite sets, seems to me rather 'remembering the finite sets and maps which gave rise to natural numbers by the abstraction of passing to isomorphism classes'.
A fair question. I attended John's Coimbra lectures on this stuff in 1999 but a lot of it leaked out afterwards. If I had to guess I'd say he was categorifying the free monoid on one generator to make it a monoidal category, but then how did the monoid end up as coproduct and the generator as the final object? One suspects some free association there---John, how *do* you make that connection? With regard to categorification in general, sets seem to play a central role in at least one development of category theory. The homobjects of "ordinary" categories are homsets. (In that sense I guess "ordinary" must entail "locally small.") 2-categories are what you get if instead you let them be homcats, suitably elaborated. Going in the other direction, if you take homsets to be vacuous, not in the sense that they are empty but rather that they are all the same, then you get sets. One more step in that direction makes everything look the same, which may have something to do with the Maharishi Yogi hiring category theorists for the math dept. of his university in Fairfield, Iowa. (When I spoke last with the MY's "Minister of World Health," an MD who like Ross Street was a classmate of mine but eight years earlier starting in 1957, the entire conversation seemed to be largely a skirting of the minefield of the sameness of everything, which may subconsciously have been behind my obscure reply to Peter Freyd's posting a while back about unique existence going back to Descartes, where I tried to one-up him by claiming it went *much* further back.) Categorification isn't the only way to get to 2-categories, which can be understood instead in terms of the interchange law as a two-dimensional associativity principle. However John has got a lot of mileage out of the categorification approach, which one can't begrudge in an era where mileage and minutes are as integral to a balanced life as one's checkbook. (Q: How many minutes in a month? A: Depends on your plan.)
Since in the category of sets, any nasty old infinite set satisfies the golden equation and many others, I have taken the liberty of interpreting 'nice' to mean at least 'satisfying no unexpected equations'.
Quite right. I would add to this "and satisfying the expected equations." The "nasty sets" of which Steve speaks fail to satisy such expected equations as 2^2^X ~ X. (The power set of a set is a Boolean algebra, for heaven's sake. Why on earth forget that structure prior to taking the second exponentiation? Set theorists seem to think that they can simply forget structure without paying for it, but in the real world it costs kT/2 joules per element of X to forget that structure. If set theorists aren't willing to pay real-world prices in their modeling, why should we taxpayers pay them real-world salaries? Large cardinals are a figment of their overactive imaginations, and the solution to consistency concerns is not to go there.) Vaughan Pratt
Vaughan Pratt wrote:
Quite right. I would add to this "and satisfying the expected equations." The "nasty sets" of which Steve speaks fail to satisy such expected equations as 2^2^X ~ X. (The power set of a set is a Boolean algebra, for heaven's sake. Why on earth forget that structure prior to taking the second exponentiation? Set theorists seem to think that they can simply forget structure without paying for it, but in the real world it costs kT/2 joules per element of X to forget that structure. If set theorists aren't willing to pay real-world prices in their modeling, why should we taxpayers pay them real-world salaries? Large cardinals are a figment of their overactive imaginations, and the solution to consistency concerns is not to go there.)
I will answer you in a Popperian key: Large cardinals are falsifiable, and are not yet falsified. They may in fact be figments of our imaginations, but then why do they keep on *working*? Could be just a coincidence -- but so could all other observation; that way lies the nullification of science in general. It's an illusion, by the way, to think that you can be rid of concerns about consistency by dumping large cardinals, that you can thus achieve a priori justification for apodictic certainty. That doesn't exist even for the natural numbers; Ed Nelson is quite right on this point. As to the question of taxpayer funding, I will not attempt to justify it (I'm a libertarian in politics), but will merely note that many taxpayers probably feel that way about *all* pure mathematics.
Vaughan Pratt wrote:
(The power set of a set is a Boolean algebra, for heaven's sake. Why on earth forget that structure prior to taking the second exponentiation? Set theorists seem to think that they can simply forget structure without paying for it, but in the real world it costs kT/2 joules per element of X to forget that structure. If set theorists aren't willing to pay real-world prices in their modeling, why should we taxpayers pay them real-world salaries? Large cardinals are a figment of their overactive imaginations, and the solution to consistency concerns is not to go there.)
Vaughan Pratt
Dear Vaughan, I like it! But there's still the question of just what structure the power set has. Constructively it's not a Boolean algebra in general, though it is a frame. And is it even a set? You can in fact only say that by removing the structure, which is exactly what you told the set-theorists not to do. And in this instance it's arguable. Topos theorists say it is a set, predicative type theorists say it isn't. Part of the structure of the power "set" is topological - the Scott topology, with the inclusion order as its specialization order. But to formalize it as topological space, point-set + topological structure, you again have to forget structure in order to get a point-set. Taking this seriously generally brings you to point-free topology in some form or other. Steve Vickers.
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Vaughan Pratt