Math Links Roundup

First, I’ll mention that I’ve updated my blogroll - there’s been a real burst in math blogs over the summer, at least in part instigated by my friends at the Secret Blogging Seminar, but also by the spurt of Fields Medalists with blogs. (Are we up to 10% of the total number now?) I’ve also added a few philosophy blogs that I’ve been reading for a while, and a couple that I should have been reading, but of course I’m sure I’m missing others.

Anyway, there’s new math job search gossip stuff going on on the web - I think the discussion on that post is interesting and relevant across disciplines for people trying to figure out whether this is generally a good thing or not.

Tim Gowers discusses the way logarithms and other abstract things should be taught. He advocates a way that’s a bit more formalist than some others suggest, but it sounds reasonable to me. There’s also interesting discussion of formalism there in the comments, though some of it sounds more like structuralism to me. See for example Terence Tao’s comment, “I guess there is a fundamental transition in mathematical learning when one realises that what mathematical objects are (and how they are constructed) may be less important than what mathematical objects do (e.g. what properties they obey).”

Also, a discussion about the Axiom of Choice at The Everything Seminar (I may add that one to my links later too), focusing on a puzzle I first heard from my friend Lukas Biewald. There’s interesting discussion in the comments that reveals implicit ideas about platonism and formalism among mathematicians. I think the anti-platonist majority there should be a bit more careful though, because similar issues apply in arithmetic, thanks to Gödel’s results. I think we should be much more hesitant to say that the natural numbers are just something we make up than they are with the universe of ZFC (or a topos, or whatever), as I mentioned before.

Job Search

As some of you probably already know, I’m going on the job market this year. I don’t know if that will affect my blogging, except possibly to make it lighter (especially since I won’t be at any conferences for a while, and they tend to give me blogging inspiration). I probably won’t blog about job-search-related stuff though. (You can find that sort of stuff here if you want it.)

But it looks like Aidan has put together some links on this (despite being only in his third year). One he missed is the academic job market wiki, including a philosophy section. I wasn’t too in touch with people on the market last year - does anyone have any comments as to how useful that wiki was then? I suspect it will only be useful if a lot of people use it.

Banff Proposals for 2009

I just got the following e-mail. People should definitely think about doing something like this - the workshop I went to organized by Richard Zach on “Mathematical Methods in Philosophy” was great, and I think there’s plenty of potential here for fruitful cross-disciplinary collaboration:

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is currently accepting proposals for its 2009 programme. The deadline for 5-day Workshop and Summer School
proposals is October 1, 2007.

Full information and guidelines are available at the website
http://www.birs.ca/

Proposal submissions should be made online at:
https://www.birs.ca/proposals/.

BIRS will be again hosting a 48-week scientific programme in 2009. The Station provides an environment for creative interaction and the exchange of ideas, knowledge, and methods within the mathematical, statistical, and computing sciences, and with related disciplines and industrial sectors. Each week, the station will be running either a full workshop (42 people for 5 days) or two half-workshops (20 people for 5 days). As usual, BIRS provides full accomodation, board, and research facilities at no cost to the invited participants, in a setting conducive to research and collaboration.

Nassif Ghoussoub,
Scientific Director, Banff International Research Station

Crazy Neo-Falsificationism

Karl Popper’s criterion of “falsifiability” for scientific theories (saying that a theory counts as scientific only if there is some hypothetical observation that would prove it to be false) is a very good heuristic for thinking about what science (or any sort of evidence-based procedure for finding out about the world) is like. However, regardless of what scientists say (whether they be physicists yelling about string theory, biologists yelling about intelligent design, or anyone railing at crackpots, or economists, or anyone they don’t like) it just isn’t right as even part of a criterion for what counts as science. But I think there is perhaps a way to use something like it as a criterion for what counts as a belief, though perhaps my suggestion is crazy.

First, a quick rundown of the problems with falsificationism as a criterion for science. As Popper was well aware, it can’t apply to statistical theories - in most cases, no evidence could actually rule out a statistical theory, rather than just making it extremely improbable, and you might think we shouldn’t rule something out just because it’s extremely improbable, because (in the long run) we’re bound to get unlucky and rule out the truth at some point. A bigger problem is the Quine-Duhem problem - basically no theory is falsifiable in a strict sense, because falsification of a theory by evidence always depends on auxiliary hypotheses, which can be let go of to save the theory. For instance, an observation of Uranus or Mercury in a place where you don’t expect it to be might look like a straightforward falsification of Newtonian mechanics, but there’s also room to postulate a so-far-unobserved planet (Neptune or Vulcan), or to argue that there was some optical artifact in the way the telescope was working, or even just that the astronomer misremembered or misrecorded the observation. Thus, there is no sharp line that can be drawn by a falsifiability criterion of this sort. In addition, theories that look straightforwardly unfalsifiable can still serve as useful heuristics for the further development of science - for instance, the theory that there actually are quarks (as opposed to the theory that protons and neutrons and cloud chambers and the like all behave “as if quarks existed”) can lead one to think of different modifications of the Standard Model in the face of recalcitrant data.

But despite all these problems, I think there’s still something very useful about the idea of falsificationism. But rather than a logical criterion, as Popper considered it, I’d rather think of it as an epistemological, or perhaps even psychological one. Popper thought that a theory needed to be specific enough that certain observations would be logically inconsistent with it, in order to count as a scientific theory. I’d rather say that a belief needs to be flexible enough that certain observations would lead the agent to give it up, in order for the belief to count as a “rational” or “scientific” one. (Or perhaps even to count as a belief at all, rather than just an article of faith, or something like that.) That is, it doesn’t need to be inconsistent with any set of observations - it just needs to be held in a way that is not totally unshakable. Although this is a psychological criterion I’m suggesting, I don’t think that the observations that would lead an agent to give it up need to be known to the agent - they just need to actually have the relevant dispositions. This removes the worries about statistical theories and the Quine-Duhem problem - although it might be that any theory could logically be saved from the data by giving up enough auxiliaries, it seems plausible that any rational agent would have some limit to the lengths that they would go to to save the theory. (I don’t know if comparative amounts of evidence needed to shake one’s belief should say anything interesting about the comparison between two agents.) This also applies to the more “standardly unfalsifiable” theories that I’d like to defend - I say that they’re important because they give useful heuristics for how to modify theories that are different from their empirically identical peers. But if these heuristics never seem to lead one to good modifications, then eventually one would likely give up this theory. It can’t be falsified, but one can still be made to give it up by seeing how fruitless it is and how much more fruitful its competitor is (which is just as unfalsifiable in this respect).

One might have worries about mathematical truths, or other potential “analytic” truths. Popper explicitly set these aside and said that his criterion only applied to things that weren’t logical truths (or closely enough related to logical truths). However, I suspect that something like my criterion might still apply here - although there is no possible observation that is inconsistent with Cauchy’s theorem on path integrals in the complex plane, I suspect that there are possible observations that would make anyone give up their belief in this theorem. For instance, someone could uncover a very subtle logical flaw that appears in every published proof of the theorem, and then exhibit some strange function that is complex-differentiable everywhere but whose integral around a closed curve is non-zero. Or at least, someone could do something that looks very much like this and would convince everyone, even though I think they couldn’t actually discover such a function because there isn’t one. It’s tougher to imagine what sort of observation would make mathematicians give up their beliefs in much simpler propositions, like the claim that there are infinitely many primes, or that 2+2=4, but as I said, there’s no need for the agents to actually be able to imagine the relevant observations - the disposition to give up the belief in certain circumstances just has to exist.

I think this is a relatively low bar for a belief to reach - I suspect that just about all apparent beliefs that people have would actually be given up under certain observations. However, with logical beliefs and religious beliefs, people often claim that no possible observation would make them give it up (this is called “analyticity” for logical beliefs, and “faith” for religious beliefs). I don’t know if that should actually count as a defect for either of these types of belief, but I think it is good reason to worry about them, at least to some extent.

More than a Century Old

Joe Shipman recently posted an interesting e-mail on the Foundations of Math e-mail list:

I propose the thesis “any mathematics result more than a century old is suitable for undergraduate math majors”.

Note that the original proofs may be too difficult for undergraduates, I am only requiring that today a “boiled-down” proof (which may be embedded in a much larger theory than existed at the time of the original proof) could be taught.

So far I have only found one significant counterexample, Dirichlet’s theorem (which, in its logically simplest form, states that if a is prime to b, there exists a prime congruent to a mod b).

Can anyone think of better counterexamples? Does anyone know of a proof of Dirichlet’s theorem that does not require prerequisites beyond the standard undergraduate curriculum?

(Two other possible counterexamples, the Prime Number Theorem and the Transcendence of Pi, are proven sufficiently easily at the following links that they would, in my opinion, be appropriate for a senior seminar:

http://www.ma.utexas.edu/users/dafr/M375T/Newman.pdf

http://sixthform.info/maths/files/pitrans.pdf

).

Another version of the thesis is “any mathematics result more than 200 years old is suitable for freshmen” (note that most high schools offer a full year of Calculus). Results that were merely conjectured more than 200 years ago but not really proved until later don’t count.

– JS

I’ve sometimes considered something like this. Can anyone else think of potential counterexamples? I wonder if there were some results known on solutions of differential equations in the 18th century that would be too advanced for first-years. And probably some particular calculations done in the 19th century that are just too large for an undergraduate to properly manage. I think it’s also possible that some of Cantor’s results on the possible Borel structures of the sets of discontinuities of real-valued functions might be too advanced, but it’s also possible that advanced seniors can manage them. Or perhaps the Riemann-Roch theorem? (I don’t actually even know enough to state that theorem myself.)

Another interesting corollary to this discussion - what’s the earliest result of the 20th century that is beyond the reach of an advanced undergraduate?

Betting Odds and Credences

I was just reading the interesting paper When Betting Odds and Credences Come Apart, by Darren Bradley and Hannes Leitgeb, at least in part because of some issues that are coming up in my dissertation about the relations between bets and credences. Their paper is a response to a paper by Chris Hitchcock arguing for the 1/3 answer in the Sleeping Beauty problem, where he shows that if Beauty bets as if her credences were anything other than 1/3, then she is susceptible to a Dutch book.

They end up agreeing that she should bet as if her credences were 1/3, but they argue that this doesn’t mean that her credences should actually be 1/3, because of some similarities this case has to other cases where betting odds and credences come apart. I know at least Darren supports (or has supported) the 1/2 answer in the Sleeping Beauty case, so he’s got a reason to argue for this position.

I think in the end though, their paper has convinced me of the opposite - the correct thing to do in this situation is to bet as if one’s credence is 1/2, even though one’s credence should actually be 1/3! I get the 1/3 credence argument from a bunch of sources (especially Mike Titelbaum’s work on the topic). But for the betting as if one’s credence is 1/2, I might be using the term “bet” in a somewhat non-standard way. However, I think my usage is inspired by my attempt to resist some of the claims of Bradley and Leitgeb.

They give some examples of other cases in which it might look as if one should bet at different odds than one’s credences. For instance, if one is offered a bet on a coin coming up heads, but knows that this bet will only be offered if the coin has actually come up tails, then it looks as if one should bet at odds different from one’s credences. However, they agree that in this case one’s credences change as soon as the bet is offered, and one should bet at odds equal to the new credences.

Their next example is very similar, but without the shift in credences. One is offered a bet on a coin coming up heads, but knows that if the coin actually came up heads then the bet is carried out with fake money (indistinguishably replacing the real money in your and the bookie’s pockets) and is real if the coin actually came up tails. In this case, it looks like one should bet at odds different from one’s credences, which should still be 1/2.

However, I think that in this case what’s going on is that one isn’t really being offered a proper bet on heads at odds of 1/2. Functionally speaking, the money transfer involved will be like a bet on heads at odds of 1. It might be described as a bet at different odds, but I think bets should be individuated in some sort of functionalist way here, rather than according to their description in this sense. Thus, since one’s credence in heads is less than 1, one shouldn’t accept this bet.

Bradley and Leitgeb then say that what goes on in Hitchcock’s set-up of the Sleeping Beauty bets is similar. The bet will be repeated twice if the coin comes up tails (because Beauty and the bookie both forget the Monday bet), and thus this is a situation like the one with the bet that might turn out to be with pretend money, but in the opposite direction. Thus, this bet ends up being one that costs the agent $20 if the coin comes up heads, and wins her $20 if it comes up tails, so it’s functionally a bet at odds of 1/2. I think this is the set of bets she should be willing to accept, but that her credence in heads should be 1/3, so her betting odds and credences should come apart.

Of course, there may be a slight difference between the situations. In this version of the Sleeping Beauty bets, the bet gets made twice if the coin comes up tails, rather than paying off double. Perhaps the fact that it’s agreed to multiple times doesn’t make the same difference that having money replaced by something twice as valuable would. If so, then this bet really was properly described as a bet at odds of 1/3, so that I would no longer think that this is an example where betting odds and credences should come apart.

So I think I don’t really accept the particular claims that Bradley and Leitgeb make in this paper, but it’s only because I’m trying to do something subtle about how to individuate bets in functional terms. I’m sure there are good examples out there on which betting odds and credences could rationally come apart, but I’m not convinced whether the Sleeping Beauty case is one of them.

The Principal Principle

A very plausible normative principle relating subjective degree of belief to objective chance is David Lewis’ “Principal Principle”. In a simplified version, this principle says that if you know the objective chance of some inherently chancy outcome, then your degree of belief in that outcome should equal the chance. Thus, if you know that the coin is fair, then you should have degree of belief 1/2 that it will come up heads.

This has some added bite because the chance information overrules a lot of other information - if you know the coin is fair, then it doesn’t matter how it happened to come up on the last 1000 flips, you should still believe in heads to degree 1/2. Even if the last 1000 flips were all tails - this is one idea of what’s fallacious about the gambler’s fallacy (or inverse gambler’s fallacy).

Of course, some sorts of information can overrule the chance information - if a very accurate fortuneteller has told you that the coin will come up heads, then maybe you should believe to a degree higher than 1/2, even though you still believe the coin is fair. This sort of information is what Lewis called “inadmissible” information. The question for the Principal Principle then is just what counts as inadmissible information?

To answer this, I think we need to consider just what chance really is. On one notion of chance, it requires that the world be objectively indeterministic, so that there is no fact of the matter about future chancy events. On this account, the idea of an accurate fortuneteller for chancy events doesn’t even make sense. This might be a natural view of chance that arises from the many-worlds interpretation of quantum mechanics. On this view, the chance of an event could potentially depend on anything for which there is a fact of the matter - but this only includes facts about the past and present. But since you’d need to know all this information (or the relevant parts anyway) to know the chances, there will trivially be no possibility of inadmissible evidence, so the Principal Principle stands (if at all) in a very simple form!

But there are other notions of chance I’ve heard people talk about. One is supposed to be compatible with strict determinism. I don’t know too many of the details, but I suspect that the idea is that there’s some natural class of “nearby worlds”, and chance is just some sort of probability measure on those worlds. This can definitely give rise to non-extreme values for chances, even though there is no possibility other than necessity. However, on this interpretation of chance, I don’t see why anything like the Principal Principle would have any normative force at all. I suppose if you can somehow narrow things down enough to know what the chances are, but can’t eliminate any of the worlds in the class that defines the chances, then it would make sense. But it’s far from clear to me why this situation would be at all common.

Then of course there’s Lewis’ own characterization of chance. I believe his idea is that one can read off the natural laws of a world by seeing what best systematizes the entire history of it. If there are certain types of events that have no interesting pattern to them at all except for a certain limiting frequency, then the best way to systematize these will be with chancy laws. In this setting it’s not clear how one would justify the Principal Principle, or how one would claim to have knowledge about the chances.

At any rate, the Principal Principle seems to say different things on these different interpretations of chance, and it gives rise to either different justifications or different accounts of what should count as “inadmissible evidence”.

Coding Truth and Satisfaction

I just got back from a week visiting the Canada/USA Mathcamp, where I spent a few days teaching a class on Gödel’s theorems. I only got to the incompleteness theorems on the last day, when I introduced Gödel numbering, showed that some syntactic relations were definable, suggested that provability was therefore definable, and showed that truth (in the standard interpretation) is not definable. Thus, I didn’t get to anything like the full incompleteness theorems, but just showed that there is no recursive set of axioms such that truth in the standard interpretation corresponds to provability from those axioms.

The thing about showing that truth is not definable is that you normally go through satisfaction instead. Given the fact that syntactic relations are definable, we can define functions (which returns the code number for the numeral expressing ) and (which takes as the code of a formula with one free variable, and substitutes in the term that is the code of, and returns the code of the resulting sentence). Then (which says that is the code of a formula satisfied by number ) can be defined in terms of by .

Then, it’s fairly straightforward to show that is not definable. If it were, then we could define , which would have some code number . But then would be true iff didn’t satisfy , which is a contradiction. Therefore, is not definable.

Afterwards, someone pointed out that this proof of the undefinability of doesn’t make any assumption about how the coding works, which seems somewhat surprising. After all, I could take highly non-effective codings, and the undefinability of would still hold. This is surprising, because can be defined under certain encodings. One such encoding is just to separate the sentences into true and false ones, and code them respectively by even and odd numbers, using some sort of lexicographic ordering. Then, since being even and being odd are both definable, truth would be definable as well. But it turns out that no trick like this is going to let satisfaction be definable, because of the diagonalization argument I gave above.

My guess for why this is is because satisfaction requires something like a coding of some of the syntactic structure as well as some of the semantic structure. Standard codings make the syntax definable, but then the semantics fails. Certain bizarre codings let the semantics be definable, but they make the syntax fail. But since the syntax and semantics aren’t recursive in terms of each other, there’s no way to make them both recursive, the way satisfaction seems to require. (Of course, you could probably choose some awful coding that makes both sets of relations undefinable, but why would you want to do that?)

Anyway, going back to camp is always great for precisely this sort of reason - I get to try to teach something new and interesting that I haven’t taught before, and people ask questions that also help me understand the material in a new way.

Back from Australia

I’m back from spending three weeks in Australia again - as usual, it was a very productive trip. It was also nice to get to attend the workshops on Norms and Analysis and Probability that went on last week. There were a lot of interesting talks there, so I won’t go through very many of them. Overall, I think the most interesting was Peter Railton’s talk in the first workshop, where he seemed to be supporting a framework for metaethics and reasons that is broadly compatible with the framework of decision theory. However, he brought in lots of empirical work in psychology to show that for both degree of belief and degree of desire, there seem to be two distinct systems at work - one more immediately regulating behavior, while the other being more responsive to feedback and generally regulating the first. It reminded me somewhat of what Daniel Kahneman was talking about in a lecture here at Berkeley a few months ago. But not being an expert in any of this stuff, I can’t say too much more than that.

Another particularly thought-provoking talk was Roy Sorensen’s in the Norms and Analysis workshop. He presented a situation in which you are the detective in a library. You just saw Tom steal a book, so you know that he’s guilty. However, before you punish him, the defense presents an envelope that may either contain nothing, or may contain exculpatory evidence (something like, “Tom has an identical twin brother in town”, or “The librarians have done a count and it seems that no books are missing”, which would make you give up your belief that Tom was guilty). Given that you know Tom is guilty, should you open the envelope or not? On the one hand, it seems you should, because you should make maximally informed decisions. On the other hand, it seems you shouldn’t, because either the envelope contains nothing, or it contains information you know is misleading, and in either case it’s no good.

Sorensen was arguing that you shouldn’t open the envelope, but I don’t think he succeeded in convincing any of the audience. But I think the puzzle sheds interesting light on what it takes to know that evidence is misleading, and how apparent evidence or the lack thereof really plays out when you know other background facts about where the evidence is coming from.

Philosophical Sophistication as Mathematical Sophistication

How much philosophical sophistication does someone need to count as being “mathematically sophisticated” enough to follow a graduate course in algebra?

When reading through a draft paper by Colin McLarty (addressing different issues entirely), I came upon the following passage from Serge Lang’s canonical text:

I think it illustrates a lot of issues that often arise in understanding mathematical writing.

In actuality, mathematicians almost never write the statement that Lang wrote, except in the same sort of definitional statement. In particular, in place of “f(x)” they would write some expression in terms of “x” that one might use, such as “x³+2x-1″ or the like. Because this expression is just a placeholder, we might expect some neutral term, like “t“. But instead he uses a term that gives the reader the idea of what the overall expression is supposed to mean, at the expense of some abuse of notation.

Another issue of use and mention at work here is what the term “x” to the left of the arrow is doing. He doesn’t say whether “x” is a placeholder for a term denoting a specific element of A, or whether it is a sort of meta-placeholder, representing a variable that itself takes values in A.

In practice, I believe that both options are allowed. By a minor abuse of notation, one can write either “Under function f, 3 \mapsto 9″, or “Under function f, x \mapsto x²“. (I’m using “\mapsto” to stand for the arrow used in Lang’s statement.) In particular, the latter type of statement derives from the former by the standard practice of ignoring certain types of use-mention distinction, and allowing variables to stand either for elements of A or the names of elements of A. This abuse is allowed just about everywhere except in some parts of model theory, where it’s important to distinguish objects and their names.

So getting back to my original point, I think that an ability to know when a term is being used or mentioned, and whether it’s standing for itself, an expression that is partly composed in the way that it’s composed (this might relate to Lang’s famous statement that “notation should be functorial over the language”), or something totally different is important. I suspect that a non-sophisticated math student (or a sophisticated philosopher) would read the statement Lang wrote and suspect that the arrow would never be useful, because we’d always have to specify in some other place what f(x) was (that is, what expression “f(x)” refers to).

One aspect of mathematical sophistication seems to rely being aware of these different levels on some subconscious level, so that you can always jump to the right one, even through multiple abuses of notation.