When I introduced this MISCELLANY to the Mathematical Part of the Inhabitants of this Kingdom, it was done with a view to turn their attention from that torrent of Politics and Metaphysics which was overwhelming all the countries of Europe, and carrying down its rapid stream, the Reason, and more particularly the Happiness of all degrees of society.—Instead of seriously contemplating on Mathematical and Philosophical Subjects, our Youth were contending, or more frequently quarreling with their neighbours, about the nature of Government; and puzzling themselves about matters they could not understand for want of experience, and sometimes of sagacity.

This being the case, I wished to turn the attention of as many as possible, from those idle and vain pursuits, to the more solid and convincing exercises of Mathematics and Natural Philosophy; it being the opinion of the greatest men, that these kind of Publications produce more emulation, and consequently more Mathematicians than any regular system of education whatever.

—— Thomas Whiting, London, August 10, 1798
Mathematical, Geometrical, and Philosophical Delights

About James Stevenson

Just as individuals have different personalities, mathematicians have different proclivities, interests, and mathematical biases. So I think it is appropriate to indicate what my mathematical background is, since I think it will relate to a number of current issues about training and professions in mathematics.

I received a doctorate in pure mathematics in the realm of locally convex topological vector spaces. A few years later I left the academic research world, with regret at leaving teaching, and tried my luck in non-academic pursuits. (Not all lovers and performers of music can or want to be composers.) I landed in the world of Defense contracting, in my case entailing the passive detection, localization and tracking of submarines for the Navy, which lasted through the 1970s, 80s, and 90s. This relied heavily on digital signal processing and stochastic estimation theory, neither of which I had studied in my academic life. I was involved in systems that performed computations on the spherical earth for the localization and tracking results, which then required map projections for the displays. These areas drew on my undergraduate math and physics training and my life-long interest in maps and mapping. As the analog-to-digital hardware advanced with higher sampling rates, my colleagues and I formed a small company in the 1980s and 90s that, besides continuing in the Navy work, developed a pioneering patent in the location of cellular phones for 911 purposes. And finally in the 2000s I migrated to working on a system for locating and rescuing downed pilots that relied on clandestine satellite telecommunications. Not only did this system use digital signal processing with even faster A2Ds, it exposed me to the ingenious realm of coding techniques that electrical engineers had come up with to cram more channels into restricted bandwidth and to still transmit the signals with minimal loss of information. A lot of this employed the theory of finite fields from abstract algebra.

I then retired after having more fun than I had ever imagined possible. To see mathematics and ingenious algorithms extract signals out of the physical environment and transform them into useful information was a heady experience. But the period of retirement has allowed me to go back and review my earlier academic experiences and renew my interest in basic mathematics and physics.

A word about my mathematical biases. Among the different categories of mathematicians, I have found a dominant division to be between analysts/topologists/geometers and abstract algebraists. The former conglomeration are highly visually oriented and think in terms of continuous spaces and shapes; the later are generally interested in discrete collections of abstract objects and operations on them that have various structures, such as collections with certain properties that are invariant under permutations, or symmetries. I belong to the former group. I am not an algebraist. I like tangible things I can visualize, and so rely heavily on diagrams and graphics to explain ideas. (Since linear algebra involves vectors, which you can see, I don’t consider it really an abstract algebra sort of thing.) Algebraists are probably more interested in number theory and combinatorics, which I dislike immensely. (Number theory is the diabolical field where trivial-sounding problems of no physical significance can be murderously difficult to prove. For example, Goldbach’s conjecture that every even integer greater than 2 is the sum of two prime numbers. No one has proved this yet.)

I should mention that a dominant theme in modern mathematics is the cross-fertilization of ideas and results from seemingly disparate fields of mathematics. And so the topologists and algebraists did join hands in such areas as algebraic topology, algebraic geometry, and continuous groups (Lie groups). This last is one of the foundations of quantum mechanics. But I have to admit that I personally find the combined structures jarring, though amazing and exciting.

Purpose of the Blog

Over the years since I retired I began to write up short essays on various aspects of mathematics (puzzles, problems, and public commentary) I had come across in daily life, as well as topics that had needled my mind for some time. I decided I would like to make them public, and so I created this blog as a way to publish them over time.

I was asked who I thought the audience for such writings might be. Since I often added explanations that someone untutored in math might understand, I was hoping for some sort of general audience, maybe with a high school exposure to plane geometry and algebra. However, some of the write-ups involve specific issues in math or physics and so may benefit from some exposure to elementary calculus and linear algebra (vectors and matrices).

Description of the Blog

I have organized the material into several sections flagged as categories which might make it more accessible.
Curiosities and Questions is a set of explorations into statements or topics that caught my interest in newspapers or other public forums, or they were just things that struck my mathematical curiosity, like the rotation of the dish in a microwave.
Puzzles and Problems are my solutions to various puzzles and problems that I came across, usually online, that involve simple math. In most cases, the problems I found did not include solutions, or I felt the solutions lacking or difficult to understand.
Math Inquiries is a section that is more heavily math oriented, involving issues that have nagged my mind over the years or that I wanted to describe more simply, like the equivalence of Kepler’s and Newton’s Laws. This is where calculus and linear algebra often show up.
Math Commentary is a set of essays that express my opinion about mathematical topics or controversies, usually related to teaching or explaining mathematics to novices, or that have a philosophical dimension. This collection got a bit polemical, so I will have to tone them down before making them public.
Other Voices is a collection of articles authored by others that I found especially interesting and supportive of themes I have raised or thought about.
Other Vistas consists of articles on subjects, often only tenuously related to mathematics, that grabbed my attention with their remarkable import.
Causality, Chance, and Connections is a somewhat amorphous and ambiguously defined group of essays that share a basic mathematical attitude towards finding chains of reasoning or causality to explain things, though the setting may not be strictly mathematical, but possibly more historical or cultural.
Administration consists of comments and remarks pertaining to the website itself.

There are, of course, now numerous very fine books of collections of short essays on similar mathematical topics, such as those by the prolific authors Ian Stewart and Paul J. Nahin, and the inimitable Martin Gardner. I do not intend to compete with such well-written efforts. But my idiosyncratic approach may offer some different slants to common problems that others might appreciate.

One final note.  Since I rely heavily on graphics and equations, I did not want to struggle with an entirely new and limited medium for expression on the web.  So I am reserving the bulk of my articles in PDF files from Word documents.  It is hard enough to create these articles in Word, but I generally like the results.  For the most part I do not like the way information is presented online and so chose the PDF route.  I feel I am in good company at least, since this also seems to be the way James Tanton constructed his blog Thinking Mathematics!

Logo. I built my Möbius “infinity” strip based on a graphic from the University of Michigan Virtual Reality Laboratory at the College of Engineering. (http://www.umich.edu/~vrl/project2/moebius/)