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?

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.

Frank P. Ramsey Appreciation Society

I recently stumbled upon the FPRAS webpage through a fortuitously placed ad in Gmail. It’s good to know that there’s a society devoted to this important intellectual figure, but it’s a bit distressing to know that they have such poor web design sensibilities. Also, the only description it has of the society suggests that it’s all about Ramsey Theory, ignoring his philosophical and economic contributions. Ramsey Theory is definitely very interesting stuff - on one level it basically says that if you’re looking at a big enough collection, then there’s bound to be some ordered substructure. (More precisely, for any positive integers n and k, there is an N such that any coloring of the edges of a complete graph on N vertices with at most k colors has some set of n vertices where all edges between them are the same color. For 3 and 2, the value is 6, so that if you have 6 people at a party, there are bound to be either 3 mutual acquaintances, or 3 mutual strangers.)

Random Links

Metaphysicians and ontologists may want to consult this list in doing their work. Or perhaps make contributions. I added “numbers”.

Math And Science Song Information, Viewable Everywhere - unfortunately, there are no songs about set theory it seems.

1997

1997 seems like it must have been quite a year in the philosophy of mathematics. Mike Resnik published Mathematics as a Science of Patterns, and Stewart Shapiro did Philosophy of Mathematics: Structure and Ontology, which are two strong arguments in favor of different versions (I think) of structuralism, which had been a popular idea over the previous few decades, but I think not terribly well-developed before those books. At the same time, John Burgess and Gideon Rosen outlined and attacked fictionalism in their A Subject with no Object, and Penelope Maddy advocated an end to all this investigation into the ontology and epistemology of mathematics in Naturalism in Mathematics. In addition, Synthese did a special issue on proof in mathematics, and in 1998 a couple other important books came out - Mark Balaguer’s Platonism and Anti-Platonism in Mathematics and the collection Truth in Mathematics edited by Dales and Oliveri.

I don’t know of any other particular year that had such a proliferation of interesting books coming out nearly simultaneously in one relatively small area of philosophy like this. And I feel like there’s another book from 1997 that I’m leaving out as well. Does anyone know any other examples of years like this?

The Unexpected Examination

Timothy Chow published an article about seven years ago about the Unexpected Examination paradox. Along with the article, he has a bibliography of just about every philosophy paper that discusses the paradox, which seems quite interesting to me. I just found this article because the bibliography was recently updated on the arXiv (which is something that we in philosophy could do well to adapt, I’m sure).

Anyway, someone should do something like this for the Sleeping Beauty problem, which seems to have been first discussed around the time that article was written. (The link there is a good start, but definitely needs some updating. Especially with the good new work being done on the problem right now.)

Horgan on Common Sense

There’s an interesting discussion between John Horgan (of The End of Science fame - a very interesting set of interviews with scientists and philosophers of scientist, and a bit amateurish in terms of the content, as one might expect - I recall it as being worth reading though) and Leonard Susskind (a relatively important physicist and string theorist, from what I see) and several others (whose points are less interesting) about string theories and the ability of common sense to lead us to reject them. I’m not (yet) a philosopher of science, but it seems that Horgan is onto something - scientific theories can’t be totally unconstrained by some sort of reasonableness. But I think he’s definitely wrong on this particular point. I’m not a believer in Penelope Maddy’s position that “philosophy cannot criticize [nor] can it defend” science. But this is because I think philosophy and science are continuous with one another, and each should be able to shed light on the other. However, it has to be done far more carefully than Horgan is doing it here. Susskind seems to make this point quite eloquently by talking about “uncommon sense”.

At the end, Horgan repeats the common complaint about string theory that it’s not experimentally falsifiable or confirmable. According to Susskind (and I certainly can’t say myself) this just isn’t the case. But even if Horgan is right about it’s non-testability, I assume he means it in the sense that there is no experiment that could confirm or verify string theory as against the contemporary mix of quantum mechanics and relativity. I would be surprised if the theory was totally unable to be falsified or confirmed at all. Even if different settings of parameters make the theory consistent with different numbers of fundamental particles, different sizes of the universe, and so on, I assume that these parameters will have to be set to account for some amount of the observations one has, and once they’re set, they would make further predictions approximately in line with current scientific theory, and not just allow for all the seeming laws to suddenly change at any moment. I certainly hope string theory is verifiable or falsifiable in at least this weak sense - this seems like an important criterion for scientific theories (though the Quine-Duhem problem shows that we need to be somewhat more careful in phrasing this weak requirement).

But there is no important need for this stronger sense of falsifiability and confirmability (that is, the kind that requires there to be some experiment to differentiate it from current theory). Sure, it would be nice to be able to have an experiment to settle which one of the two theories was better, but even if there’s not, that doesn’t mean the new theory isn’t scientific. That would make science too much a matter of historical accident - the Copenhagen Interpretation of quantum mechanics would be science, but the many-worlds theories and Bohm’s theories would be dismissed as unscientific. If they had come in a different order, a different one would have been scientific. And even apart from this problem, it seems there are often benefits to debating two theories that are empirically identical. Oftentimes, one theory will suggest different modifications in the face of recalcitrant evidence. Or one theory will make all the calculations far easier. Or one theory postulates fewer invisible elves pushing electrons around, and otherwise fits together more aesthetically.

This is the sort of debate in which philosophy and common sense (or perhaps better, uncommon sense) are important in science. Horgan has staked an extreme position that seems indefensible on these grounds, but makes approximately the right broad claim. However, this broad claim then undermines his more specific argument against string theory, that it’s untestable. (For an example of a theory that philosophical concerns should drive us against, despite its empirical adequacy, see here.)

In other news, I’m leaving for Sydney airport in a few hours, and then will be on a roadtrip through Arizona and New Mexico for about a week or so before returning to Berkeley, so I’m unlikely to post until the end of the month.