I propose the thesis "any mathematics result more than a century old is suitable for undergraduate math majors".This is something that I've heard before, in the weaker form "a good undergraduate math education gets you to around 1900". (Which raises the question: in 2100, will a good undergraduate math education still get you to 100 years from the present? Or will it only get you to, well, 1900 -- which will be two centuries in the past by that point? Will the words "undergraduate math education" even mean anything in a hundred years?) The only counterexample he saw was Dirichlet's theorem (which says that if a is prime to b, then there exists a prime congruent to a mod b).

Wayne Aitken, on the same mailing list, argues otherwise; so does Timothy Chow.

Shipman then suggests that this is actually a 100-year rule, to anticipate my earlier question. The reason for this is that although the current best-known proof of Fermat's last theorem, for example, isn't accessible to undergrads (or even most grad students!), in a hundred years we will have significantly shortened and simplified the proof, to the point where it could be explained to a seminar of intelligent, mathematically trained 21-year-olds.

Of course, we are creating more and more mathematics as time goes on, and the rate of growth is often stated to be exponential. It's been claimed that as much mathematics has been created in the last ten years as in all time before that, and even if it's not quite that fast it seems quite likely that mathematics is growing faster than world population. (Note that this trend can't continue forever.) I am inclined to attribute the claim of the exponential rate of growth to Andrew Odlyzko but I cannot find it.

The boiling down has some unintended consequences, though. Carl Sagan, in the chapter of The Demon-Haunted World: Science as a Candle in the Dark entitled "No Such Thing as a Dumb Question", wrote about the lure of psuedoscience and the fact that science is not emphasized in the schools. He quotes the philosopher John Passmore, on how science is generally presented in schools:

[science is often presented as a matter of learning principles and applying them by routine procedures. It is learned from textbooks, not by reading the works of great scientists or even the day-to-day contributions to the scientific literature... The beginning scientist, unlike the beginning humanist, does not have an immediate contact with genius. Indeed... school courses can attract quite the wrong sort of person into science -- unimaginative boys and girls whoI suspect that this is because in mathematics and science we are constantly doing this "boiling down", so the simplest way known to currently illustrate a result is very rarely the way it was originally shown. Furthermore, the standards of exposition are constantly changing, as (for example) new notations evolve, so it can be astonishingly difficult to read mathematical texts from a couple centuries ago, whereas the same evolution doesn't seem to have happened to literary texts, or at least not to the same degree. And even when a line of argument does stand the test of time, it's constantly reworded, tweaked, renotated... the argument is seen as something to be passed on, but the words which were used are not. The argument becomes a sort of communal property. Perhaps there is only one mathematician, working in many brains.likeroutine. (p. 335)

## 3 comments:

will a good undergraduate math education still get you to 100 years from the present?Which is why I claim that undergraduates will need category theory in the future.

"Furthermore, the standards of exposition are constantly changing, as (for example) new notations evolve, so it can be astonishingly difficult to read mathematical texts from a couple centuries ago, whereas the same evolution doesn't seem to have happened to literary texts, or at least not to the same degree."

Read Beowolf, Chaucer or even Shakespeare [remember all those notes even in the easy plays and sonnets].

"Indeed... school courses can attract quite the wrong sort of person into science -- unimaginative boys and girls who like routine."

I agree that I do not think many high school science and math courses let you see how messy things are.

And the mess can be half the fun. Also, for most kids the only real math course they take is Plain Geometry.

Is there enough time in a year to teach high school physics from experiments?

I took an Artificial Intelligence course which revolved around learning LISP. Usually after your first programing course and programing language you'd get told what language the code had to written in and then it was your responsibility to learn it. This course was so boring, which was difficult for me because I worked at night and I would literally fall asleep in boring courses. He spoon fed us LISP and some programing principals: blah.

Later in a used book store I stumbled on a book on LISP that started with dotted pairs moved from there. That was interesting.

If he wanted to teach LISP he could taught LISP from dotted pairs up showing how it was built stone by stone - a course on operating systems did that for UNIX and C. If he wished to show us how to build an AI system he could have told us what LISP book to buy for reference and then started building a small system.

Too many bright students are spoon fed.

I think art students are better educated in high school. They are given paint or clay and told to make a mess.

A good undergraduate education should include some thoery of computing, say GĂ¶dels theorems, the undecidabiluity of the halting problem, some algorithms and some basic complexity theory. In analysis it should include some distribution theory and functional analysis.

All topics invented after 1900.

On the other hand the thery of elliptic functions/integrals isn't neccessary so it wouldn't quite include all math invented before 1900.

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