Most mathematicians could not state the axioms of ZFC from memory. My suspicion is that AC skepticism is highest among mathematicians who can.
I don’t see why this is even relevant, but for what it’s worth, I don’t particularly share this suspicion: 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.
My own impression is that professed “AC skepticism” (scarequotes because I think it’s a not-even-wrong confusion) is most correlated not with interest in logic and foundations, but with working in finitary, discrete, or algebraic areas of mathematics where AC isn’t much used.
One piece of evidence that AC skepticism is not low-status is that papers and textbooks will often emphasize when a proof uses AC, or when a result is equivalent to AC. People find such things interesting.
The fact that people find such things interesting is at best extremely weak evidence for the proposition that constructivism and related positions are mainstream. (After all, I find such things interesting!)
As I pointed out in the comment linked to above, there is a difference between dinner-party acknowledgement of constructivism (which is widespread) and actually taking it seriously enough to worry about whether one’s results are correct (which would be considered eccentric).
If AC skepticism were not low-status, you would expect to find papers and textbooks actively rejecting AC results, rather than merely mentioning in a remark or footnote that AC is involved. (Such footnotes are for use at dinner parties.)
And also, texts just as frequently do not bother to make apologies of the sort you allude to. A fairly random example I recently noticed was on p.98 of Algebraic Geometry by Hartshorne, where Zorn’s Lemma is used without any more apology than an exclamation point at the end of the (parenthetical) sentence.
But what do status considerations have to do with whether Yudkowsky’s beliefs and hunches are justified?
It tends to irritate me when people get something wrong which they could easily have gotten right by using a standard human heuristic (such as the “status heuristic”, noticing what the prestigious position is).
My own impression is that professed “AC skepticism” (scarequotes because I think it’s a not-even-wrong confusion) is most correlated not with interest in logic and foundations, but with working in finitary, discrete, or algebraic areas of mathematics where AC isn’t much used.
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.)
If AC skepticism were not low-status, you would expect to find papers and textbooks actively rejecting AC results, rather than merely mentioning in a remark or footnote that AC is involved. (Such footnotes are for use at dinner parties.)
Not entirely. If the only known proof for a result assumes choice, then a proof that doesn’t use choice will almost certainly be publishable.
And also, texts just as frequently do not bother to make apologies of the sort you allude to. A fairly random example I recently noticed was on p.98 of Algebraic Geometry by Hartshorne, where Zorn’s Lemma is used without any more apology than an exclamation point at the end of the (parenthetical) sentence.
Using an exclamation mark like that is a pretty rare thing to do. You wouldn’t for example see this if one used the axiom of replacement. The only other axiom that would be in a comparable position is foundation but foundation almost never comes up in conventional mathematics. Hartshorne is writing for a very advanced audience so I think putting an exclamation mark like that is sufficient to get the point across especially when one is using choice in the form of Zorn’s lemma.
is most correlated not with interest in logic and foundations, but with working in finitary, discrete, or algebraic areas of mathematics where AC isn’t much used.
This seems to fit my impression as well.
Incidentally, for what it is worth, your claim that rejection of AC is low status seems to be possibly justified. I know of two prominent mathematicians who explicitly reject AC in some form. One of them does so verbally but seems to be fine teaching theorems which use AC with minimal comment. The other keeps his rejection of AC essentially private.
Using an exclamation mark like that is a pretty rare thing to do. You wouldn’t for example see this if one used the axiom of replacement. The only other axiom that would be in a comparable position is foundation but foundation almost never comes up in conventional mathematics.
Of course it’s worth noting that axiom of replacement doesn’t come up much either, though obviously the case there isn’t quite as extreme as with foundation.
We appear to have misunderstood each other, having something different in mind by words like “skepticism” and “reject.” I agree Con(ZF) entails Con(ZFC), and that every educated mathematician knows it. Beyond that I don’t have a good handle on what you’re saying, or even whether you disagree with Yudkowsky, or me. Are you saying that mathematicians pay lip service to constructivism, but ignore it in their work? Are you additionally saying that there is something false about constructivist ideas?
It tends to irritate me when people get something wrong which they could easily have gotten right by using a standard human heuristic (such as the “status heuristic”, noticing what the prestigious position is).
That doesn’t sound like such a great heuristic to me...
I don’t see why this is even relevant, but for what it’s worth, I don’t particularly share this suspicion: 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.
My own impression is that professed “AC skepticism” (scarequotes because I think it’s a not-even-wrong confusion) is most correlated not with interest in logic and foundations, but with working in finitary, discrete, or algebraic areas of mathematics where AC isn’t much used.
The fact that people find such things interesting is at best extremely weak evidence for the proposition that constructivism and related positions are mainstream. (After all, I find such things interesting!)
As I pointed out in the comment linked to above, there is a difference between dinner-party acknowledgement of constructivism (which is widespread) and actually taking it seriously enough to worry about whether one’s results are correct (which would be considered eccentric).
If AC skepticism were not low-status, you would expect to find papers and textbooks actively rejecting AC results, rather than merely mentioning in a remark or footnote that AC is involved. (Such footnotes are for use at dinner parties.)
And also, texts just as frequently do not bother to make apologies of the sort you allude to. A fairly random example I recently noticed was on p.98 of Algebraic Geometry by Hartshorne, where Zorn’s Lemma is used without any more apology than an exclamation point at the end of the (parenthetical) sentence.
It tends to irritate me when people get something wrong which they could easily have gotten right by using a standard human heuristic (such as the “status heuristic”, noticing what the prestigious position is).
This is also my experience.
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.)
Not entirely. If the only known proof for a result assumes choice, then a proof that doesn’t use choice will almost certainly be publishable.
Using an exclamation mark like that is a pretty rare thing to do. You wouldn’t for example see this if one used the axiom of replacement. The only other axiom that would be in a comparable position is foundation but foundation almost never comes up in conventional mathematics. Hartshorne is writing for a very advanced audience so I think putting an exclamation mark like that is sufficient to get the point across especially when one is using choice in the form of Zorn’s lemma.
This seems to fit my impression as well.
Incidentally, for what it is worth, your claim that rejection of AC is low status seems to be possibly justified. I know of two prominent mathematicians who explicitly reject AC in some form. One of them does so verbally but seems to be fine teaching theorems which use AC with minimal comment. The other keeps his rejection of AC essentially private.
Of course it’s worth noting that axiom of replacement doesn’t come up much either, though obviously the case there isn’t quite as extreme as with foundation.
We appear to have misunderstood each other, having something different in mind by words like “skepticism” and “reject.” I agree Con(ZF) entails Con(ZFC), and that every educated mathematician knows it. Beyond that I don’t have a good handle on what you’re saying, or even whether you disagree with Yudkowsky, or me. Are you saying that mathematicians pay lip service to constructivism, but ignore it in their work? Are you additionally saying that there is something false about constructivist ideas?
That doesn’t sound like such a great heuristic to me...