3 ms·

Actually, no, there are results from transfinite set theory which are foundational to so much mathematics that there would be extensive damage to the theoretica

Actually, no, there are results from transfinite set theory which are foundational to so much mathematics that there would be extensive damage to the theoretical foundations of very practical math if it were unavailable. For example, the distinction between countable and uncountable is important in analysis and measure theory. Countable subsets of the real line have Lebesgue measure zero, and this result is used in many theorems in probability and stochastic process theory with practical implications. Now that we've seen a statement which was rated as almost certain is incorrect, it seems to reinforce my original point: it is not so clear what kind of math has applications, so it is good to develop math broadly. I agree that it shouldn't be our only argument, but it does pack a significant punch.

cubefox 4y ago
Most mathematicians never use ZFC or any similar "foundational" system. That something can serve as such a foundation doesn't remotely mean that it is useful. Math is not like a house which breaks down without a foundation. The distinction between countable and unaccountable infinity is precisely something that finitists point out as being useless. For example, analysis existed well before Cantor, and it's notion of limits and convergence ironically provides a potential notion of infinity which doesn't treat it as a mathematical object, contrary to the set theoretical notion of actual infinity, where we have a zoo of transfinite numbers. And the fact that the countable/uncountable distinction can be integrated in some practical theories doesn't show that it has any practical implication. That would only be the case if those practical theories wouldn't exist otherwise. But analysis existed before Cantor, and other practical theories, like probability theory, could have existed before him. They are perfectly compatible with being a finitist.
- deleted 4y ago
  [deleted]
- practal 4y ago
  If we are talking about "most" mathematicians, then you should also admit that most mathematicians are not a member of the finitist church, and that most mathematicians appreciate the distinction between countable and uncountable.
  - civilized 4y ago
    Yeah, I'm not sure I follow the logic of "transfinite set theory isn't useful because any applications it has could be replaced with finitist math", when this is something that no one wants to do. Reminds me of that famous old HN comment about how Dropbox is pointless because an equivalent service could be set up with Linux utilities. Sure, but there are reasons people don't want to do that. Most people prefer the ease of use of mainstream foundations. To be fair to the original commenter, I would be willing to bet some money that, say, large cardinal theory won't inspire any applications in the next 20 years. But next 100 years or 1000 years, I wouldn't. Maybe there will be some weird cross-fertilization with other more applied fields that leads to something.
    - cubefox 4y ago
      "transfinite set theory isn't useful because any applications it has could be replaced with finitist math" would be a misleading way to phrase it. Like, analysis is not an "application" of transfinite set theory. It doesn't presuppose the existence of any transfinite numbers. No piece of applied math does. There is nothing which needs to be replaced with "finitist math".
      - civilized 4y ago
        To repeat my earlier comment: analysis on mainstream foundations uses infinite sets of multiple transfinite cardinalities, and there are results like "countable sets have Lebesgue measure zero", and the difference between finite, countable, and uncountable sums/unions/intersections, that are relevant to proving results that have practical significance, as well as (crucially) to avoiding erroneous calculations. If you want to prove the same results in a finitist framework, there is nontrivial work to do, and few mathematicians are interested in doing it. If your contention is that only calculations matter, not proofs, I would agree with you that transfinite set theory may not be relevant. You can do calculations without any rigor at all. But I think the position that proofs are of no practical value is untenable. It is historically simply not the case that engineer or physicist intuition is a sufficient guide to deriving results reliably.

cubefox 4y ago

Most mathematicians never use ZFC or any similar "foundational" system. That something can serve as such a foundation doesn't remotely mean that it is useful. Math is not like a house which breaks down without a foundation. The distinction between countable and unaccountable infinity is precisely something that finitists point out as being useless. For example, analysis existed well before Cantor, and it's notion of limits and convergence ironically provides a potential notion of infinity which doesn't treat it as a mathematical object, contrary to the set theoretical notion of actual infinity, where we have a zoo of transfinite numbers. And the fact that the countable/uncountable distinction can be integrated in some practical theories doesn't show that it has any practical implication. That would only be the case if those practical theories wouldn't exist otherwise. But analysis existed before Cantor, and other practical theories, like probability theory, could have existed before him. They are perfectly compatible with being a finitist.

deleted 4y ago
[deleted]
practal 4y ago
If we are talking about "most" mathematicians, then you should also admit that most mathematicians are not a member of the finitist church, and that most mathematicians appreciate the distinction between countable and uncountable.
- civilized 4y ago
  Yeah, I'm not sure I follow the logic of "transfinite set theory isn't useful because any applications it has could be replaced with finitist math", when this is something that no one wants to do. Reminds me of that famous old HN comment about how Dropbox is pointless because an equivalent service could be set up with Linux utilities. Sure, but there are reasons people don't want to do that. Most people prefer the ease of use of mainstream foundations. To be fair to the original commenter, I would be willing to bet some money that, say, large cardinal theory won't inspire any applications in the next 20 years. But next 100 years or 1000 years, I wouldn't. Maybe there will be some weird cross-fertilization with other more applied fields that leads to something.
  - cubefox 4y ago
    "transfinite set theory isn't useful because any applications it has could be replaced with finitist math" would be a misleading way to phrase it. Like, analysis is not an "application" of transfinite set theory. It doesn't presuppose the existence of any transfinite numbers. No piece of applied math does. There is nothing which needs to be replaced with "finitist math".
    - civilized 4y ago
      To repeat my earlier comment: analysis on mainstream foundations uses infinite sets of multiple transfinite cardinalities, and there are results like "countable sets have Lebesgue measure zero", and the difference between finite, countable, and uncountable sums/unions/intersections, that are relevant to proving results that have practical significance, as well as (crucially) to avoiding erroneous calculations. If you want to prove the same results in a finitist framework, there is nontrivial work to do, and few mathematicians are interested in doing it. If your contention is that only calculations matter, not proofs, I would agree with you that transfinite set theory may not be relevant. You can do calculations without any rigor at all. But I think the position that proofs are of no practical value is untenable. It is historically simply not the case that engineer or physicist intuition is a sufficient guide to deriving results reliably.