Musings

Here I will be posting thoughts on education and teaching of mathematics, the university, faculty and students, and on applied mathematics. (Some are better taken with a grain of salt.)

While mathematics is not an empirical science, it has connections to the natural sciences, draws from them, and its development is very closely linked with the natural sciences. Below is an excerpt from an article written by John von Neumann (reading the entire article is highly recommended).

“I think that it is a relatively good approximation to truth—which is much too complicated to allow anything but approximations—that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science. There is, however, a further point which, I believe, needs stressing. As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from “reality,” it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l’art pour l’art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical. In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas. I am convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future.”

John von Neumann, The Mathematician, in The Works of the Mind,
edited by Robert B. Heywood, The University of Chicago Press,
Chicago, 180–196, 1947, 180–196.

In all we do, we need to look forward, plan for the future, have a vision and know where we are going.

“I skate to where the puck is going to be, not where it has been.”

Wayne Gretzky

“Teachers should prepare the student for the student’s future, not for the teachers past.”

“It seems to me the difficulty of knowing the future does not absolve the teacher from seriously trying to help the student to be prepared for it when it comes.”

Richard W. Hamming, The Art of Doing Science and Engineering:
Learning to Learn
, Gordon and Breach Science Publishers, 1997.

We must look to the future even though it is hard to predict

“It’s tough to make predictions, especially about the future.”

Yogi Berra

since

“If you don’t know where you’re going, you might not get there.”

Yogi Berra

This might as well be called you and your studies or you and your work. Students, educators, and researchers can find much valuable information on doing research, good high quality work, and teaching, in these writings of Richard Hamming.

”Just hard work is not enough − it must be applied sensibly”

”If you do not work on an important problem, it’s unlikely you’ll do important work.”

”If you are to do important work then you must work on the right problem at the right time and in the right way.”

“You need a vision of who you are and where your field is going. A suitable parable is that of the drunken sailor. He staggers one way and then the other with independent, random steps. In n steps he will be, on the average, about √n steps away from where he started. but if there is a pretty girl in one direction he will get a distance proportional to n. The difference, over a life time of choices, between √n and n is very large and represents the difference between having no vision and having a vision. The particular vision you have is less important than just having one – there are many paths to success. Therefore, it is wise to have a vision of what you may become, of where you want to go, as well as how to get there. No vision, not much chance of doing great work; with a vision you have a good chance.”

Richard W. Hamming, You and Your Research.

Apparently Richard Hamming gave a talk titled You and Your Research many times and on many occasions. I am aware of the following two versions. A long version which is a transcript by J. F. Kaiser of a talk given by Richard Hamming at Bell Labs (on March 7, 1986).

This version has been recently published in the New School Economic Review 3 (1), 2008, pp. 5–26; as well as in the book Simula Research Laboratory – by thinking constantly about it, A. Teveito, A. M. Bruaset, and O. Lysne editors, Springer, Berlin, 2010, pp. 37–60.

The short version, Richard Hamming, You and Your Research. A stroke of genius: striving for greatness in all you do, appeared in IEEE Potentials, October 1993, pp. 37–40.

There was also an interview of Richard Hamming by David Gilbert which appeared in IEEE Computer Futures, Spring 1991, pp. 10–17 (unfortunately I have been unable to obtain a copy of this interview).

Finally, there is the book: Richard W. Hamming, The Art of Doing Science and Engineering: Learning to Learn, Gordon and Breach Science Publishers, 1997.

Students, researchers, and educators would do well to read these and take Richard Hamming’s advice to heart.

Education is what, when, and why to do things, Training is how to do it.”

“Either one without the other is not of much use. You need to know both what to do and how to do it.”

Richard W. Hamming, The Art of Doing Science and Engineering:
Learning to Learn
, Gordon and Breach Science Publishers, 1997.

Many universities try to economize by relegating a large portion of the teaching to instructors and graduate students, on the one hand, while on the other hand, encouraging their research faculty to obtain grants (and spend their time working on funded research rather than teaching) thus generating additional revenue for the university.

This seems antithetical to the idea of a university.

Charles Vest a former president of MIT and now the president of the National Academy of Engineering observes:

“For nearly twenty years I was active in both classroom teaching and research, teaching both undergraduate and graduate subjects every term. As a teacher I have seen the value to students of learning from–and working with–men and women who are discovering the future through their research, not just teaching the history of their fields.”

“One may start out as an effective and even brilliant teacher, but without the kind of continual renewal that research and scholarship provide, one may not grow in wisdom and breadth, and over time may lose rather than gain in effectiveness as a teacher.”

“That is why I believe that the very best learning environment is one in which undergraduate and graduate education are blended with the conduct of research and scholarship. The issue should not be teaching versus research, it should be the proper interweaving of the two.”

Charles Vest, Learning in a Research University, (President Vest’s letter to the parents of MIT undergraduates) MIT Parents News, Spring 1994.

Those interested in rankings of universities (perhaps ones that are a little more serious and objective) should take a look at the Academic Ranking of World Universities (the so called “Shanghi Ranking”).

While it ranks the top 500 universities from around the world, you can also easily view university rankings of specific countries (and how they relate to the worldwide ranking), in particular: USA, UK, Germany, Canada, France, Australia, Japan, Switzerland, Sweden, Netherlands, Denmark, Belgium, Israel, Norway, Finland, Russia, and China. You can also view the top 200 universities in broad areas: Natural Sciences and Mathematics, Engineering / Technology and Computer Sciences, Life and Agriculture Sciences, Clinical Medicine and Pharmacy, and Social Sciences; and top 200 universities in specific disciplines: Mathematics, Physics, Chemistry, Computer Science, and Economics / Business.

As with all rankings, this one too, should be taken with a grain of salt (and those interested should look what the rankings are based on and how they were created).

Peter D. Lax (one of the greatest mathematicians of our time) in his acceptance speech of the Abel Prize:

“Traditionally mathematics is divided into two kinds: pure and applied. The relation of the two is delicate. The great applied mathematician Joe Keller’s definition is: pure mathematics is a branch of applied mathematics. He meant that mathematics, beginning with Newton, was originally concerned with answering question[s] in physics, it is only later that the tools and concepts used were elaborated into theories that took on lives of their own. It was remarked by von Neumann that after a while abstract mathematics needs to be invigorated by the injections of new empirical material, like a new scientific theory, new experimental facts, or numerical studies.”

Peter D. Lax, Abel Prize acceptance speech, 2005 Abel Prize Multimedia

See also, an Interview with Peter D. Lax, Notices of the AMS, 53 (2), February 2006, 223–229 and his SIAM Review article The Flowering of Applied Mathematics in America, SIAM Rev. 31 (4), 1989, 533–541.

Goodhart’s Law

“When a measure becomes a target, it ceases to be a good measure”

M. Strathern, “Improving ratings”: audit in the British University system,
European Review 5, 1997, 305–321.

We see, all too often: universities manipulating students’ data, class sizes, etc. in order to improve their rankings; publishers and editors manipulating Impact Factors.

Relying on superficial measures virtually guarantees that they will be manipulated and abused.

Douglas N. Arnold and Kristine K. Fowler wrote an interesting article on the Impact Factor: Nefarious Numbers, Notices of the AMS, 58 (3), March 2011, 434–437.

The following is attributed to Richard W. Hamming (though I suspect it was actually written by someone else; it is clearly based on Hamming’s books, writings, and lectures). Students (faculty, and university administrators) would do well to read and ponder….

As best I can tell this first appeared in the Faculty Association of the University of Waterloo Forum, Number 106, March 2001, see FAUW Forum, p. 19.