by Derek Smith

I would like to share a collection of mathematical articles I read every so often for inspiration and motivation. The only real criteria I used for inclusion in this list is that they be available online. The first in the list is G. H. Hardy’s “A Mathematician’s Apology.”

As somebody interested in “applications” of mathematics and the inspiration mathematics draws from the natural world, I must include a counterpoint to Hardy’s viewpoint. In fact, I provide links to two articles. In the first article, Eugene Wigner demonstrates the improbability of Hardy’s hope that mathematics should lie separate from the physical world. In the second article, Richard Hamming explores these concepts further, though by his own admission incompletely.

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

The Unreasonable Effectiveness of Mathematics

A final article by P. R. Halmos in this vein investigates the “elements” of mathematics.

Does mathematics have elements?

I would like to include a second article from Prof. Hamming. In comparison to the “why” questions posed in the previous links, this article focus on the mechanics of asking questions. As a graduate student, I find this perspective immensely insightful.

I discovered the next article only recently, forgive me that all may not have access to a full text version. Again the topic is similar to Hamming’s article, delving into perils of a career in research. In particular, this article by Donald Weidman ties in with the recent blog posts by Vishal.

Emotional Perils of Mathematics

A more positive note on the social nature of problem solving is to be found in the article by William Thurston, recently referenced in an AMS post. Like Kareem, I too have been deeply influenced by the numerous ideas made explicit by Prof Thurston. If ever anybody claimed to know the “secret” to how mathematics operates, surely the kernel of those secrets could be found in this article.

On Proof and Progress in Mathematics

I began earlier this month composing another blog post about teaching, which will wait until next time, but I’d like to include a take on the topic by Paul Lockhart. I didn’t take away from this article any practical, or “actionable”, piece of advice. But as somebody who spent hundreds of hours in high school and college creating and recording music with friends simply for our own enjoyment, I can’t help but connect with his comparison of mathematics to music: the difference between the notes on a page and the joy of performing.

The final piece I share with you is chock full of tidbits to improve your technical writing. I can’t suggest that you sit down and read this as you would the other pieces linked, but offer it as a companion to the TeX system also produced by Donald Knuth. Similar to Prof Thurston’s article, it contains a neat exposition of the social processes which go into publishing (in addition to the writing tips).

I hope as the semester winds down this post provides some useful distractions!