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Mathematical Understanding of Nature: Essays on Amazing Physical Phenomena and Their Understanding by Mathematicians

V. I. Arnold
American Mathematical Society
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

This is a collection of 39 essays from the distinguished mathematician Vladimir Arnold. He was a man of very strong opinions. One of these is especially pertinent for this collection:

Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.

The book’s editor notes that from the age of eleven Arnold was a member of the “Children’s Learned Society”, which had been organized by A. A. Lyapunov and run at his home. The curriculum included mathematics and physics with a bit of chemistry and biology. The current book seems to have been motivated by these early experiences.

The essays are short — none longer than about six pages — and address topics mostly in mathematics and physics. The levels of sophistication and difficulty vary widely. Some topics would be accessible to high school students with a little bit of algebra and geometry; others might challenge even specialists. In the first category is an essay on the eccentricity of the orbit of Mars. It is implicitly a lesson on intelligent estimation. In the second category is a piece about adiabatic invariants.

Arnold’s introduction includes the comment: “Examples teach no less than rules, and errors, more than correct but abstruse proofs.” It is a wonderful teaching point that is amplified by at least a few unintentional errors in the text. One, noted by the editor, is about the forward motion of a bicycle. (If you attach a string to lower pedal of a bicycle at rest and pull straight back, does the bicycle move forwards or backwards?) Another occurs in an essay about the rainbow, and rather oddly attributes the blueness of the sky to a Moiré effect.

This is a wonderful book for browsing, for anyone drawn to physical applications of mathematics or to Arnold himself and the breadth of his interests. To sample some of his wizardly work, look at the essay on the maximum deviation of a light beam through a water drop and the following essay on the rainbow.

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Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.