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This sort of thing just strikes me as so pointless. How could one possibly decide? Obviously, if we're talking about new axioms that aren't provable from the

This sort of thing just strikes me as so pointless. How could one possibly decide? Obviously, if we're talking about new axioms that aren't provable from the old ones, you can't use those old ones to settle the matter. The axioms under discussion don't really affect normal math so you can't appeal to their usefulness there (unlike with, say, the axiom of choice, which famously was controversial for a while). And I don't think people can really have much of a claim to have any good intuition for such things, or else we'd have had such an axiom a long time ago. My bet is neither of these will become widely accepted. That said, if I had to bet on one of the two becoming accepted, I'd go with "V = ultimate L". Why? Because mathematics already has an axiom roughly analogous, that doesn't actually affect normal mathematics but prunes away sets we don't need so people don't waste their time with them. I'm talking, of course, about the axiom of foundation. It's like, why is that even in ZFC? Ordinary math doesn't actually care about the material content of sets; it really affects nothing. The only thing it does, is that, well, without it, non-well-founded sets are something of a problem, in that they could exist, or they could not, and you can't really prove much about them either way. There are non-well-founded set theories that replace the axiom of foundation with an axiom of antifoundation, allowing you to actually prove things about non-well-founded sets -- Peter Aczel wrote a good book on the subject -- but if you just take it out and don't put anything in to replace it, then, well, you just can't really say anything about the matter. The axiom of foundation prunes these irrelevant sets away and tells you to focus on actual math rather than this mystery that's not really much of a mystery at all. So I'd bet on another axiom along those lines becoming accepted, rather than one that does the opposite. After all, if we want to take the opposite perspective that anything that can exist ought to... well, then why not non-well-founded sets, too? (Note: Not a logician, my understanding of the two axioms being discussed is based pretty much entirely on this article, so the analogy I claim may not actually work as well as I think it does.)

KGIII 9y ago
They speak of the Philsophy of Mathematics. I typed a lot out and chose to delete it. In short, it is considered the highest order among Mathematicians. However, us Applied Mathematicians just get on with the real work and have pretty much figured out that they are insane. It's a bit like true random and infinity. It's best not to think about it to much and only useful for post doc bar conversations or impressing stoned Econ undergrads of your preferred gender. Even then, it's best not to think about it too much.
- eastWestMath 9y ago
  Eh, everyone but the set theorists ignores the set theorists at this point. I think some exciting things happened in the 60's with forcing, but a lot of the subsequent developments seem pretty contrived.
  - cgmg 9y ago
    To an outsider, practically any area of advanced, pure mathematics looks contrived. That doesn't mean it's not beautiful and interesting, particularly to those who study it. Also, the claim that 'everybody ignores set theorists' is patently false.
    - KGIII 9y ago
      They are a bunch of damned liars. 0.999... = 1 Not only can I prove it, I can tell you the excuses they use to justify it. I'm pretty sure that they just realized that it was 'proven' and had to then come up with a reason. I really do try to not think about it too much, though I may sometimes ponder it a bit while smoking. It's all horrible. I don't know what they hell they were thinking. Half of them probably belong in asylums.
      - Simon_says 9y ago
        People who get confused about that equality are in actuality confused about what a real number is. In particular they probably think that real numbers are digit expansions, and they have probably never seen a construction of the reals. You can show this person a proof, and you might get them to shut up about it, but they’ll never believe it on an intuitive level until you show them what real numbers actually are and what their relationship to their (non-unique!) digital representation is.
        enord 9y ago
        I believe the confusion is spread across several layers. For those in the know, infinite decimal expansion is syntactic sugar for a limit over a particular infinite series of sums, but this convention sidesteps how people are taught syntactic manipulation and how it's intetpteted in a particular algebra. 0.999... is presented as a term, that can be syntactically replaced with an actual infinite decimal expansion, upon which the normal syntactic opetations can be applied, specifically "=" as in "0.999...=1". This is ofcourse impossible, because even if the rules are valid you would be applying them forever. This is the apparent absudity of the equation. In defence of people who hold this intuition: it's a valid concern in many branches of math(discrete, constructive, infinitesimal etc.), just not in informal expositions of the properties of real numbers.
tbodt 9y ago
Higher mathematics is pointless and had always been pointless. People do it because it's interesting and fun.
- dwaltrip 9y ago
  Topology is a great example of higher mathematics with important connections to many applied endeavors: https://en.wikipedia.org/wiki/Topology#Applications https://en.wikipedia.org/wiki/Topology#Applications https://math.stackexchange.com/questions/73690/real-life-applications-of-topology https://math.stackexchange.com/questions/73690/real-life-app...
mikhailfranco 9y ago
Non-well-founded sets seem to be grammars from a CS point of view. As soon as you admit the Kleene-star '*' operator you have potential infinity. If you allow circular grammar definitions, then you are specifying non-well-founded-sets. When parsing finite structures, it just so happens that the parsing terminates, but it's easy to extend to open-ended (potentially infinite) streams. There are even type system constructs and language expressions for them, but only executable in lazy languages, or systems open to external input. CS has also made theoretical progress, perhaps the most surprising is differentiating regular expressions and other regular datatypes: http://en.wikipedia.org/wiki/Brzozowski_derivative http://en.wikipedia.org/wiki/Brzozowski_derivative http://blog.sigfpe.com/2005/05/derivatives-of-regular-expressions.html http://blog.sigfpe.com/2005/05/derivatives-of-regular-expres... http://strictlypositive.org/diff.pdf http://strictlypositive.org/diff.pdf I also recommend 'The Liar' by Barwise & Etchemendy.

KGIII 9y ago

They speak of the Philsophy of Mathematics. I typed a lot out and chose to delete it. In short, it is considered the highest order among Mathematicians. However, us Applied Mathematicians just get on with the real work and have pretty much figured out that they are insane. It's a bit like true random and infinity. It's best not to think about it to much and only useful for post doc bar conversations or impressing stoned Econ undergrads of your preferred gender. Even then, it's best not to think about it too much.

eastWestMath 9y ago
Eh, everyone but the set theorists ignores the set theorists at this point. I think some exciting things happened in the 60's with forcing, but a lot of the subsequent developments seem pretty contrived.
- cgmg 9y ago
  To an outsider, practically any area of advanced, pure mathematics looks contrived. That doesn't mean it's not beautiful and interesting, particularly to those who study it. Also, the claim that 'everybody ignores set theorists' is patently false.
  - KGIII 9y ago
    They are a bunch of damned liars. 0.999... = 1 Not only can I prove it, I can tell you the excuses they use to justify it. I'm pretty sure that they just realized that it was 'proven' and had to then come up with a reason. I really do try to not think about it too much, though I may sometimes ponder it a bit while smoking. It's all horrible. I don't know what they hell they were thinking. Half of them probably belong in asylums.
    - Simon_says 9y ago
      People who get confused about that equality are in actuality confused about what a real number is. In particular they probably think that real numbers are digit expansions, and they have probably never seen a construction of the reals. You can show this person a proof, and you might get them to shut up about it, but they’ll never believe it on an intuitive level until you show them what real numbers actually are and what their relationship to their (non-unique!) digital representation is.
      - enord 9y ago
        I believe the confusion is spread across several layers. For those in the know, infinite decimal expansion is syntactic sugar for a limit over a particular infinite series of sums, but this convention sidesteps how people are taught syntactic manipulation and how it's intetpteted in a particular algebra. 0.999... is presented as a term, that can be syntactically replaced with an actual infinite decimal expansion, upon which the normal syntactic opetations can be applied, specifically "=" as in "0.999...=1". This is ofcourse impossible, because even if the rules are valid you would be applying them forever. This is the apparent absudity of the equation. In defence of people who hold this intuition: it's a valid concern in many branches of math(discrete, constructive, infinitesimal etc.), just not in informal expositions of the properties of real numbers.

tbodt 9y ago

Higher mathematics is pointless and had always been pointless. People do it because it's interesting and fun.

dwaltrip 9y ago
Topology is a great example of higher mathematics with important connections to many applied endeavors: https://en.wikipedia.org/wiki/Topology#Applications https://en.wikipedia.org/wiki/Topology#Applications https://math.stackexchange.com/questions/73690/real-life-applications-of-topology https://math.stackexchange.com/questions/73690/real-life-app...

mikhailfranco 9y ago

Non-well-founded sets seem to be grammars from a CS point of view. As soon as you admit the Kleene-star '*' operator you have potential infinity. If you allow circular grammar definitions, then you are specifying non-well-founded-sets. When parsing finite structures, it just so happens that the parsing terminates, but it's easy to extend to open-ended (potentially infinite) streams. There are even type system constructs and language expressions for them, but only executable in lazy languages, or systems open to external input. CS has also made theoretical progress, perhaps the most surprising is differentiating regular expressions and other regular datatypes: http://en.wikipedia.org/wiki/Brzozowski_derivative http://en.wikipedia.org/wiki/Brzozowski_derivative http://blog.sigfpe.com/2005/05/derivatives-of-regular-expressions.html http://blog.sigfpe.com/2005/05/derivatives-of-regular-expres... http://strictlypositive.org/diff.pdf http://strictlypositive.org/diff.pdf I also recommend 'The Liar' by Barwise & Etchemendy.