I would expect those who know the axioms from memory to be more philosophically sophisticated (i.e. non-Platonist), and to be more likely to be familiar with technical results such as Gödel’s theorem that ZFC is as consistent as ZF.
They’re also more likely to know Cohen’s theorem that ZF + not(AC) is also just as consistent. And of course, being philosophically sophisticated, it’s clear to me that they would be more likely to realise that the axioms of ZFC are fairly arbitrary and no better than many others. They’re also more likely to know, and to appreciate the philosophical significance of, that there are many axiom systems that are strong enough to do most mathematics (including all concretely applied mathematics) and yet much weaker (hence more surely consistent) than ZFC (although this has little to do with AC as such).
However, when arguing about what philosophically sophisticated people are going to think, we’re both naturally inclined to think that they’ll agree with ourselves, so our impressions about that prove nothing.
If AC skepticism were not low-status, you would expect to find papers and textbooks actively rejecting AC results
You do find such things (but they are mostly published in certain journals, which we can tell are low-status, since such things are published in them).
However, when arguing about what philosophically sophisticated people are going to think, we’re both naturally inclined to think that they’ll agree with ourselves, so our impressions about that prove nothing.
I’m not sure about that. You and komponisto seem to be using ‘philosophically sophisticated’ to contrast with Platonism. This use strikes me as similar to how arguing that ‘death is good’ is sophisticated, i.e., showing of your intelligence by providing convincing arguments for a position that violates common sense. In this case arguing that mathematical statements don’t have inherent truth value.
Remember just because you can make a sophisticated sounding argument for a preposition doesn’t mean its true.
Mathematica statements do have inherent truth value, but that value is relative to the axioms. And as far as the axioms go, the most you can say is that a system of axioms is consistent, and beyond that you get into non-mathematical statements. What exactly is sophisticated about this?
However, when arguing about what philosophically sophisticated people are going to think, we’re both naturally inclined to think that they’ll agree with ourselves, so our impressions about that prove nothing.
I’m not sure about that. You and komponisto seem to be using ‘philosophically sophisticated’ to contrast with Platonism.
Yes, which agrees with my complaint quoted above. Neither of us is a Platonist, so we both assume that philosophically sophisticated people won’t be Platonists, although we derive different things thereafter.
showing of your intelligence by providing convincing arguments for a position that violates common sense. In this case arguing that mathematical statements don’t have inherent truth value.
I’m certainly not trying to show off my intelligence. I just think that the idea of inherent truth value for abstract statements about completed infinities violates common sense!
what accounts for your intuition that ZF and other systems for reasoning about completed infinities are consistent?
To the extent that I have this intuition, this is mostly because people have used these systems without running into inconsistencies so far. (At least, not in the systems, such as ZF, that people still use!)
But strictly speaking, ‘ZF is consistent.’ is not a statement with an absolute meaning, because it is itself a statement about a completed infinity. I have high confidence that no inconsistency in ZF has a formal proof of feasible length, but I really have no opinion about whether it has an inconsistency of length 3^^^3; we haven’t come close to exploring such things.
(Come to think of it, I believe that my Bayesian probability as to whether ZF is consistent to such a degree ought to be quite low, for essentially the same reason that a random formal system is likely to be inconsistent, although I’m not really sure that I’ve done this calculation correctly; I can think of at least one potential flaw.)
I cannot speak for komponisto about any of this, of course.
But strictly speaking, ‘ZF is consistent.’ is not a statement with an absolute meaning, because it is itself a statement about a completed infinity. I have high confidence that no inconsistency in ZF has a formal proof of feasible length, but I really have no opinion about whether it has an inconsistency of length 3^^^3; we haven’t come close to exploring such things.
These feasibility issues are definitely interesting. Another possibility is that there is a formal proof of feasible length, but no feasible search will ever turn it up. (Well, unless P = NP). Yet another possibility is that a feasible search will turn it up, I certainly regard it as more likely than most people do.
To the extent that I have this intuition, this is mostly because people have used these systems without running into inconsistencies so far. (At least, not in the systems, such as ZF, that people still use!)
I agree that this counts as evidence, but it’s possible to overestimate it. Foundational issues hardly ever come up in everyday mathematics, so the fact that people are able to prove astonishing things about 3-manifolds without running into contradictions I regard as very weak evidence in favor of ZF. There have been a lot of man-hours put into set theory, but I think quite a bit less than have been put into other parts of math.
(Come to think of it, I believe that my Bayesian probability as to whether ZF is consistent to such a degree ought to be quite low, for essentially the same reason that a random formal system is likely to be inconsistent, although I’m not really sure that I’ve done this calculation correctly; I can think of at least one potential flaw.)
JoshuaZ and I had a discussion about this a while ago, starting here.
Another possibility is that there is a formal proof of feasible length, but no feasible search will ever turn it up. (Well, unless P = NP).
This reminds me of people who argue that, because P != NP, we will never prove this. (The key to the argument, IIRC, is that any proof of this fact will have very high algorithmic complexity.) I’m not sure how to find this argument now. (There is something like it one of Doron Zeilberger’s April Fools opinions.)
the fact that people are able to prove astonishing things about 3-manifolds without running into contradictions I regard as very weak evidence in favor of ZF
Yes, these results should be formalisable in higher-order arithmetic (indeed _n_th order for n a single-digit number). It is the set theorists’ work with large cardinals and the like that provides the only real evidence for the consistency of such a high-powered system as ZF.
I would expect those who know the axioms from memory… to be more likely to be familiar with technical results such as Gödel’s theorem that ZFC is as consistent as ZF.
They’re also more likely to know Cohen’s theorem that ZF + not(AC) is also just as consistent.
Yes; that’s definitely within the scope of my “such as”!
However, when arguing about what philosophically sophisticated people are going to think, we’re both naturally inclined to think that they’ll agree with ourselves, so our impressions about that prove nothing.
Not quite. Remember that I gave a specific meaning for “philosophically sophisticated”: I said it meant “non-Platonist”. And what I meant by that, here, is not believing that AC (or any other formal axiom) represents some kind of empirical claim about “the territory” that could be “falsified” by “evidence”, despite being part of a consistent axiom system.
I claim the situation with AC is like that of the parallel postulate: it makes no sense to discuss whether it is “true”; only whether it is “true within” some theory.
You do find such things (but they are mostly published in certain journals, which we can tell are low-status, since such things are published in them).
What I meant was more like: you would find some substantial proportion (say 20% or more) of textbooks being used to teach analysis (say) to graduate students in mathematics omitting all theorems which depend on AC.
Remember that I gave a specific meaning for “philosophically sophisticated”: I said it meant “non-Platonist”.
Yes, and I was happy to take it this way, as I am certainly no Platonist. Surely only a Platonist could believe that AC is true; we philosophically sophisticated people know that you can make whatever assumptions you want! And so naturally a theorem with a proof using AC is a weaker result than the same theorem with a proof that doesn’t, since it holds under fewer sets of assumptions, and thus the latter is preferred. Meanwhile, a theorem with a proof using not(AC) is just as valid as the same theorem with a proof using AC; it’s less useful only because it has fewer connections with the published corpus of mathematics, but that’s merely a sociological contingency.
Is it often the case that you need to assume the negation of AC for a proof to hold? AC comes up in seemingly-unrelated areas when you need some infinitely-hard-to-construct object to exist; I can’t imagine a similar case where you’d assume not(AC) in, e.g., ring theory.
As usual, the negation of a useful statement ends up not being a useful statement. I don’t think anyone works with not(AC), they work with various stronger things that imply not(AC) but actually have interesting consequences.
Sniffnoy may have more examples, but here are some that I know:
Every subset of the real line is Lebesgue-measurable.
Every subset of the real line has the Baire property (in much the same vein as the preceding one).
The axiom of determinacy (a statement in infinitary game theory).
Adding the first two to ZF + DC (dependent choice) is consistent (assuming that ZFC + Con(ZFC) is consistent, as just about everybody believes), and this gives a “dream universe” for analysis in which, for example, any everywhere-defined linear operator between Hilbert spaces is bounded.
Adding the first two to ZF + DC (dependent choice) is consistent (assuming that ZFC + Con(ZFC) is consistent, as just about everybody believes)
This isn’t quite right. The consistency of ZF + DC + “every subset of R is Lebesgue measurable” is equivalent to the consistency of an inaccessible cardinal, which is a much stronger assumption then the consistency of ZFC + Con(ZFC).
Surely only a Platonist could believe that AC is true; we philosophically sophisticated people know that you can make whatever assumptions you want!
Yes, indeed!
And so naturally a theorem with a proof using AC is a weaker result than the same theorem with a proof that doesn’t, since it holds under fewer sets of assumptions, and thus the latter is preferred.
Yes—but it needs to be stressed that this doesn’t distinguish AC from anything else! (Also, depending on the context, there may other criteria for selecting proofs besides the strength or weakness of their assumptions.)
If only people would talk about whether they prefer working in ZFC or ZF+not(C) (or plain ZF), or better yet what they like and don’t like about each, rather than whether AC is “true” or how “skeptical” they are.
If only people would talk about whether they prefer working in ZFC or ZF+not(C) (or plain ZF), or better yet what they like and don’t like about each, rather than whether AC is “true” or how “skeptical” they are.
Yes, indeed, that would be much more sophisticated! But scepticism of the orthodoxy can be the first step to such sophistication. (It was for me, although in my case there were also some parallel first steps that did not initially seem connected.)
They’re also more likely to know Cohen’s theorem that ZF + not(AC) is also just as consistent. And of course, being philosophically sophisticated, it’s clear to me that they would be more likely to realise that the axioms of ZFC are fairly arbitrary and no better than many others. They’re also more likely to know, and to appreciate the philosophical significance of, that there are many axiom systems that are strong enough to do most mathematics (including all concretely applied mathematics) and yet much weaker (hence more surely consistent) than ZFC (although this has little to do with AC as such).
However, when arguing about what philosophically sophisticated people are going to think, we’re both naturally inclined to think that they’ll agree with ourselves, so our impressions about that prove nothing.
You do find such things (but they are mostly published in certain journals, which we can tell are low-status, since such things are published in them).
I’m not sure about that. You and komponisto seem to be using ‘philosophically sophisticated’ to contrast with Platonism. This use strikes me as similar to how arguing that ‘death is good’ is sophisticated, i.e., showing of your intelligence by providing convincing arguments for a position that violates common sense. In this case arguing that mathematical statements don’t have inherent truth value.
Remember just because you can make a sophisticated sounding argument for a preposition doesn’t mean its true.
Mathematica statements do have inherent truth value, but that value is relative to the axioms. And as far as the axioms go, the most you can say is that a system of axioms is consistent, and beyond that you get into non-mathematical statements. What exactly is sophisticated about this?
Yes, which agrees with my complaint quoted above. Neither of us is a Platonist, so we both assume that philosophically sophisticated people won’t be Platonists, although we derive different things thereafter.
I’m certainly not trying to show off my intelligence. I just think that the idea of inherent truth value for abstract statements about completed infinities violates common sense!
If that’s so, what accounts for your intuition that ZF and other systems for reasoning about completed infinities are consistent?
To the extent that I have this intuition, this is mostly because people have used these systems without running into inconsistencies so far. (At least, not in the systems, such as ZF, that people still use!)
But strictly speaking, ‘ZF is consistent.’ is not a statement with an absolute meaning, because it is itself a statement about a completed infinity. I have high confidence that no inconsistency in ZF has a formal proof of feasible length, but I really have no opinion about whether it has an inconsistency of length 3^^^3; we haven’t come close to exploring such things.
(Come to think of it, I believe that my Bayesian probability as to whether ZF is consistent to such a degree ought to be quite low, for essentially the same reason that a random formal system is likely to be inconsistent, although I’m not really sure that I’ve done this calculation correctly; I can think of at least one potential flaw.)
I cannot speak for komponisto about any of this, of course.
I’m mostly with you.
These feasibility issues are definitely interesting. Another possibility is that there is a formal proof of feasible length, but no feasible search will ever turn it up. (Well, unless P = NP). Yet another possibility is that a feasible search will turn it up, I certainly regard it as more likely than most people do.
I agree that this counts as evidence, but it’s possible to overestimate it. Foundational issues hardly ever come up in everyday mathematics, so the fact that people are able to prove astonishing things about 3-manifolds without running into contradictions I regard as very weak evidence in favor of ZF. There have been a lot of man-hours put into set theory, but I think quite a bit less than have been put into other parts of math.
JoshuaZ and I had a discussion about this a while ago, starting here.
This reminds me of people who argue that, because P != NP, we will never prove this. (The key to the argument, IIRC, is that any proof of this fact will have very high algorithmic complexity.) I’m not sure how to find this argument now. (There is something like it one of Doron Zeilberger’s April Fools opinions.)
Yes, these results should be formalisable in higher-order arithmetic (indeed _n_th order for n a single-digit number). It is the set theorists’ work with large cardinals and the like that provides the only real evidence for the consistency of such a high-powered system as ZF.
Yes; that’s definitely within the scope of my “such as”!
Not quite. Remember that I gave a specific meaning for “philosophically sophisticated”: I said it meant “non-Platonist”. And what I meant by that, here, is not believing that AC (or any other formal axiom) represents some kind of empirical claim about “the territory” that could be “falsified” by “evidence”, despite being part of a consistent axiom system.
I claim the situation with AC is like that of the parallel postulate: it makes no sense to discuss whether it is “true”; only whether it is “true within” some theory.
What I meant was more like: you would find some substantial proportion (say 20% or more) of textbooks being used to teach analysis (say) to graduate students in mathematics omitting all theorems which depend on AC.
Then you would have a controversy on your hands.
Yes, and I was happy to take it this way, as I am certainly no Platonist. Surely only a Platonist could believe that AC is true; we philosophically sophisticated people know that you can make whatever assumptions you want! And so naturally a theorem with a proof using AC is a weaker result than the same theorem with a proof that doesn’t, since it holds under fewer sets of assumptions, and thus the latter is preferred. Meanwhile, a theorem with a proof using not(AC) is just as valid as the same theorem with a proof using AC; it’s less useful only because it has fewer connections with the published corpus of mathematics, but that’s merely a sociological contingency.
Is it often the case that you need to assume the negation of AC for a proof to hold? AC comes up in seemingly-unrelated areas when you need some infinitely-hard-to-construct object to exist; I can’t imagine a similar case where you’d assume not(AC) in, e.g., ring theory.
As usual, the negation of a useful statement ends up not being a useful statement. I don’t think anyone works with not(AC), they work with various stronger things that imply not(AC) but actually have interesting consequences.
That’s intriguing. Do you have any examples of what people actually work with?
Sniffnoy may have more examples, but here are some that I know:
Every subset of the real line is Lebesgue-measurable.
Every subset of the real line has the Baire property (in much the same vein as the preceding one).
The axiom of determinacy (a statement in infinitary game theory).
Adding the first two to ZF + DC (dependent choice) is consistent (assuming that ZFC + Con(ZFC) is consistent, as just about everybody believes), and this gives a “dream universe” for analysis in which, for example, any everywhere-defined linear operator between Hilbert spaces is bounded.
This isn’t quite right. The consistency of ZF + DC + “every subset of R is Lebesgue measurable” is equivalent to the consistency of an inaccessible cardinal, which is a much stronger assumption then the consistency of ZFC + Con(ZFC).
Sorry, my mistake. Still, set theorists usually believe this.
Yes, indeed!
Yes—but it needs to be stressed that this doesn’t distinguish AC from anything else! (Also, depending on the context, there may other criteria for selecting proofs besides the strength or weakness of their assumptions.)
If only people would talk about whether they prefer working in ZFC or ZF+not(C) (or plain ZF), or better yet what they like and don’t like about each, rather than whether AC is “true” or how “skeptical” they are.
Yes, indeed, that would be much more sophisticated! But scepticism of the orthodoxy can be the first step to such sophistication. (It was for me, although in my case there were also some parallel first steps that did not initially seem connected.)