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XiXiDu comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong
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The following is a comment by John Baez, posted on Google+ where I linked to this thread:
— Jerry Bona
This makes it sound like believing in an uncountable ordinal is equivalent to AC, which would make things easier - lots of mathematicians reject AC. But you might not need AC to assert the existence of a well-ordering of the reals as opposed to any set, and others have claimed that weaker systems than ZF assert a first uncountable ordinal. My own skepticism wasn't so much the existence of any well-ordering of the reals (though I'm willing to believe that no such exists), my skepticism was about the perfect, canonical well-ordering implied by there being an uncountable ordinal onto whose elements all the countable ordinals are mapped and ordered. Of course that could easily be equivalent to the existence of any well-ordering of the reals.
Since I found the other replies insufficiently stark here, let me just say that it is not. The details are in this subthread.
No they don't (*). Your saying this explicitly somewhat confirms my brain's natural, automatic assumption that your error here (and in similar comments in the past -- "infinite set atheism" and all that business) is as much sociological as philosophical: all along, I instinctively thought, "he doesn't seem to realize that that's a low-status position".
ZFC is considered the standard axiom system of modern mathematics. I have no doubt that if an international body (say, the IMU) were to take a vote and choose a set of "official rules of mathematics", the way (say) FIDE decides on the official rules of chess, they would pick ZFC (or something equivalent).
Now it's true, there are some mathematicians who are contrarians and think that AC is somehow "wrong". They are philosophically confused, of course; but, more to the point here in this comment, they are a marginal group. (In fact, even worrying about foundational issues too much -- whatever your "position" -- is kind of a low-status marker itself: the sociological reality of the mathematical profession is that members are expected to get on with the business of proving impressive-looking new theorems in mainstream, high-status fields, and not to spend time fussing about foundations except at dinner parties.)
See also this comment of mine.
(*) I don't know the numbers, or how you define "lots", and there are a large number of mathematicians in the world, so technically I don't know if it's literally false that "lots" of mathematicians would say that they "reject AC" . But the clear implication of the statement -- that constructivism is a mainstream stance -- most definitely is false.
This seems problematic. Many mathematicians work on foundations and are treated with respect. It isn't that they are low status so much that a) most of the really big foundational issues are essentially done b) foundational work rarely impact other areas of math, so people don't have a need to pay attention to foundations. There also seems to be an incredible degree of confidence in claiming that those skeptical of AC are " philosophically confused, of course".
It's somewhat pertinent to point out that the highest rated contributor at MathOverflow is none other than Joel David Hamkins of 'foundations of set theory' fame.
More than that, I daresay that they'd pick something much stronger than ZFC, probably ZFC with a large cardinal axiom. (And the main debate would be how large that cardinal should be.)
I think you are stating these things too confidently.
Most mathematicians could not state the axioms of ZFC from memory. My suspicion is that AC skepticism is highest among mathematicians who can.
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.
You could make a stronger case that skepticism about infinity is regarded as low-status.
But what do status considerations have to do with whether Yudkowsky's beliefs and hunches are justified?
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.
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.
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...
And anecdotally it seems that the AC skepticism that does exist seems to largely come from constructivism, so if we rule out that (since it doesn't seem that Eliezer wants to go all constructivist on us :) ), it's even less so.
I'm not sure what you mean by "constructivism" here; I usually hear that term referring to doubting the law of excluded middle (when applied to statements quantified over infinite sets), but I know several mathematicians who doubt the axiom of choice without doubting excluded middle.
I should also clarify the difference between doubting AC and denying AC. If you deny AC, then you believe that it is false, and hence any theorem whose only known proofs use AC is no theorem at all; it might be true, but it has not been proved. (And if AC follows from it, then it must in fact be false.) If you only doubt AC, however, then you simply believe that a theorem with a proof that uses AC is a weaker result than the same theorem with a proof that doesn't, and so the former theorem is still worth publishing but the latter is naturally preferred.
This seems such an obvious position to me that I doubt everything in mathematics (although there is a core which I generally assume since mathematics without it seems uninteresting (although I'm open to being proved wrong about this)).
Both AC and its negation can be made sense of in set theory. One or the other can be considered more interesting, or more relevant in the context of a particular problem, but given the extensive experience with mathematics of foundations we can safely study the properties of either. The question of which way "lies the truth" seems confused, since the alternatives coexist. Ultimately, some axiomatic options might turn out to be morally irrelevant, but that's not a question that human philosophers can hope to settle, and all simple things are likely relevant at least to some extent.
On the contrary, you need almost the full strength of AC to establish that a well-ordering of the reals exists. Like you say, you don't need it to construct uncountable ordinals, or to show that there is a smallest such. Cantor's argument constructively shows that there are uncountable sets, and you can get from there to uncountable ordinals by following your nose.
Is this because you can't prove aleph-one = beta-one? I'm Platonic enough that to me, "well-order an uncountable set" and "well-order the reals" sound pretty similar.
No something sillier. You can prove the axiom of choice from the assumption that every set can be well-ordered. (Proof: use the well-ordering to construct a choice function by taking the least element in every part of your partition.)
If one doesn't wish to assume that every set has a well-ordering, but only a single set such as the real numbers, then one gets a choice-style consequence that's limited in the same way: you can construct choice functions from partitions of the real numbers.
I'd hardly call a well-ordering on one particular cardinality "almost the full strength of AC"! I guess it probably is enough for a lot of practical cases, but there must be ones where one on 2^c is necessary, and even so that's still a long way from the full strength...
I just have a hard time imagining someone who was happy with "c is well-ordered" but for whom "2^c is well-ordered" is a bridge too far.
Hm, agreed. I guess not so much "the full strength" but "the full counterintuitiveness"? Where DC uses hardly any of the counterintuitiveness, and ultrafilter lemma uses nearly all of it?
Uh, that's a lot more than "Platonism"... how was anyone supposed to guess you've been assuming CH?
Edit: To clarify -- apparently you've been thinking of this as "I can accept R, just not a well-ordering on it." Whereas I've been thinking of this as "Somehow Eliezer can accept R, but not a cardinal that's much smaller?!"
Edit again: Though I guess if we don't have choice and R isn't well-orderable than I guess omega_1 could be just incomparable to it for all I know. In any case I feel like the problem is stemming from this CH assumption rather than omega_1! I don't think you can easily get rid of a smallest uncountable ordinal (see other post on this topic -- throwing out replacement will alllow you to get rid of the von Neumann ordinal but not, I don't think, the ordinal in the general sense), but if all you want is for there to be no well-order on the continuum, you don't have to.
That's how I remember it, although I don't know a reference (much less a proof). All we know is that omega_1 is not larger than R.