Now and again, there arise certain trends in science and technology which prove deleterious. Take, for instance, the carbon nanotube. It is, as of 2024, 33 years old, and millions of man-hours have gone into practical nanotube development projects. To say that the reward has not been commensurate with the effort would be far too generous -- just about nothing has come of those millions of hours. In hindsight, this should perhaps have been more obvious; the theoretical benefits of nanotubes hinge on the production of pristine submicron fiber-like (giant-) molecules, and those have always been somewhere over the horizon.
I feel that Cantor's theories are much the same way. They have severe logical shortcomings, which were highlighted over 100 years ago by the superior logician Skolem; namely that you can construct an uncountable set out of any countable set, and that every so-called uncountable set has a perfectly isomorphic countable model. Further, the diagonalization argument only works in the limit, with very generous use of ". . .", and the finitists have put together a number of very compelling arguments against it. People claim that Cantor's set theory might be a good foundation for mathematics, but it is at best a foundation made of sand. As with the nanotube, I feel that many researchers have spent countless hours -- millions, perhaps -- following an intellectual/scientific trend, and nothing good has come of it.
Great series. It really humanizes Cantor for me to see him trying to walk back a claim in those letters. I did my senior project on the continuum hypothesis, up through Paul Cohen's proof of it's independence (he wrote a short book about it that is very clear and accessible). Thanks for sharing, was very interesting to learn a little more about Cantor's motivations and the contemporary reaction to his ideas outside mathematics.
Nah, infinities don't exist. There are just recursive procedures that emit ever-increasing numbers of digits with no bounds checking.
The diagonal proof is just arguing that two nested while(true) loops will run for longer than one. (And then we define this as "bigger" just to confuse undergraduates)
I don't understand what you mean by this. What would it mean for an infinity or any number to "exist", does the number 2 "exist" somewhere in a way that infinities do not?
In any case our best physical models right now are full of infinities. Space looks like it is infinitely big to an absurd degree of precision. The spectrum of the hydrogen atom contains two different infinities! A countable infinity of bound states and a continuum of free electron states at higher energies.
Mathematics is just an exercise in symbol manipulation according to a particular rule set. Mathematics is a _game_. Park Place exists in the context of a game of Monopoly and also exists in Atlantic City. The tokens in a game of Reversi don't correspond to anything in the real world. You might be playing it on a physical board of a particular size, but you could just as easily play it on an imaginary board, or on a piece of paper, or a computer, or in some imaginary board of infinite size. Whether or not either exists in "reality" doesn't matter in the context of the game, though.
If the rules of mathematics allow for infinities, then they exist in the context of mathematics. Whether they exist in reality doesn't matter. You can still say things that are true about these imaginary objects in the context of the game that everyone is playing with these symbols, just like you can say things that are true about the game of Reversi, even one played entirely in the minds of two players on an imaginary, infinite board.
Your critique of the use of ”bigger” in your strawman applies equally to your own use of ”longer” - neither loop will ever terminate. So in what sense is either one longer?
The diagonal proof is arguing that there is no bijection from N to {0, 1}∞ , despite the fact that both are infinite. The sense in which the latter is ”bigger” is that there are always elements left over that are covered by N.
Also, what has nested loops got to do with it? You can use one loop to generate the natural numbers, and a pair of nested loops to generate pairs of natural numbers (if you like, the rationals). But the diagonal proof doesn’t show that these will have ‘different’ run times — they’re in fact in bijective correspondence.
I learned about Cantor in university, and it blew my mind. I've always wondered if anyone's used his work in an applied field yet (like how non-euclidean geometry was just crazy maths for a while and now underpins most physics).
Was hoping this paper would tell me, but from what I've read its more if a (very nicely written) summary of his core work. Anyone know if there's any applications yet?
Of course. Theory of computation and formal logic are two examples of math built on top of Cantor. And theory of computation has one or two real-world applications or so I've heard :)
It's always fun to make fun of cranks. Thanks for linking that. The author really needs to find the right statement of what they call the Nested Interval Theorem. I cracked up at the complete misuse of it in the "Interval Argument for the Rationals" section
I learned about Cantor from this readable book:
Aczel, Amir D. "The Mystery of the Aleph Mathematics, the Kabbalah, and the Search for Infinity" Simon & Schuster (2001)
I don't understand most of Yuri Manin's mathematics, but I still find some of it interesting.
"Manin: I think that people engaged in research
in mathematics today are doing so the same way
it was done 200 years ago. This is partly because
we don’t choose mathematics as our profession,
but rather it chooses us. And it chooses a certain
type of person, of which there are no more than
several thousand in each generation, worldwide.
And they all carry the stamp of those sorts of
people mathematics has chosen."
[+] [-] az00123|2 years ago|reply
[+] [-] A_D_E_P_T|2 years ago|reply
I feel that Cantor's theories are much the same way. They have severe logical shortcomings, which were highlighted over 100 years ago by the superior logician Skolem; namely that you can construct an uncountable set out of any countable set, and that every so-called uncountable set has a perfectly isomorphic countable model. Further, the diagonalization argument only works in the limit, with very generous use of ". . .", and the finitists have put together a number of very compelling arguments against it. People claim that Cantor's set theory might be a good foundation for mathematics, but it is at best a foundation made of sand. As with the nanotube, I feel that many researchers have spent countless hours -- millions, perhaps -- following an intellectual/scientific trend, and nothing good has come of it.
[+] [-] superb-owl|2 years ago|reply
[+] [-] woopsn|2 years ago|reply
[+] [-] unknown|2 years ago|reply
[deleted]
[+] [-] nairboon|2 years ago|reply
[+] [-] ludston|2 years ago|reply
The diagonal proof is just arguing that two nested while(true) loops will run for longer than one. (And then we define this as "bigger" just to confuse undergraduates)
[+] [-] eigenket|2 years ago|reply
In any case our best physical models right now are full of infinities. Space looks like it is infinitely big to an absurd degree of precision. The spectrum of the hydrogen atom contains two different infinities! A countable infinity of bound states and a continuum of free electron states at higher energies.
[+] [-] empath-nirvana|2 years ago|reply
If the rules of mathematics allow for infinities, then they exist in the context of mathematics. Whether they exist in reality doesn't matter. You can still say things that are true about these imaginary objects in the context of the game that everyone is playing with these symbols, just like you can say things that are true about the game of Reversi, even one played entirely in the minds of two players on an imaginary, infinite board.
[+] [-] dlkf|2 years ago|reply
The diagonal proof is arguing that there is no bijection from N to {0, 1}∞ , despite the fact that both are infinite. The sense in which the latter is ”bigger” is that there are always elements left over that are covered by N.
[+] [-] xanderlewis|2 years ago|reply
Also, what has nested loops got to do with it? You can use one loop to generate the natural numbers, and a pair of nested loops to generate pairs of natural numbers (if you like, the rationals). But the diagonal proof doesn’t show that these will have ‘different’ run times — they’re in fact in bijective correspondence.
[+] [-] snthpy|2 years ago|reply
[+] [-] benrutter|2 years ago|reply
Was hoping this paper would tell me, but from what I've read its more if a (very nicely written) summary of his core work. Anyone know if there's any applications yet?
[+] [-] andrepd|2 years ago|reply
[+] [-] jabowery|2 years ago|reply
https://www.academia.edu/93528167/Interval_Arguments_Two_Ref...
[+] [-] scapp|2 years ago|reply
[+] [-] xtiansimon|2 years ago|reply
[+] [-] johnthescott|2 years ago|reply
[+] [-] unknown|2 years ago|reply
[deleted]
[+] [-] prvc|2 years ago|reply
[+] [-] hackandthink|2 years ago|reply
I don't understand most of Yuri Manin's mathematics, but I still find some of it interesting.
"Manin: I think that people engaged in research in mathematics today are doing so the same way it was done 200 years ago. This is partly because we don’t choose mathematics as our profession, but rather it chooses us. And it chooses a certain type of person, of which there are no more than several thousand in each generation, worldwide. And they all carry the stamp of those sorts of people mathematics has chosen."
https://www.ams.org/notices/200910/rtx091001268p.pdf