Mark Levi’s new book The Mathematical Mechanic reverses the usual interaction of mathematics and physics. Usually one analyzes a physical problem in order to reduce its solution to a mathematical problem. The Mathematical Mechanic takes well-known mathematical results and creates physical problems that establish the mathematical result. Levi thus creates physical “proofs” for mathematical theorems. He includes a 20-page appendix that reviews the physics necessary to follow his book.
The book’s subtitle is “Using Physical Reasoning to Solve Problems.” Strictly speaking, however, the book uses physical reasoning to create problems. For example, the book gives six physical proof for the Pythagorean theorem by appealing to such principles as the conservation of energy and Hooke’s law. The proofs are fascinating, but they use Newtonian physics to prove a theorem known centuries before Newton. (A mathematician reading The Mathematical Mechanic may find it disorienting to read less-than-rigorous proofs for results that can be proved rigorously without undue effort.) Most of the mathematical topics in the book are elementary, though some topics are much more advanced, such as the Gauss-Bonnet theorem from differential geometry.
Levi observes that the physical proofs are often shorter than their traditional mathematical counterparts. The physical arguments are not without mathematical manipulation, but the amount of calculation required to complete the physical proofs is usually less than would be required in a purely mathematical proof for the examples given in the book.
The physical proofs in The Mathematical Mechanic seem unmotivated in a sense. Given a physical problem statement, the solution is quite tangible and motivated. However, it is sometimes hard to imagine how one could have formulated the physical problem in the first place. Careful study of Levi’s book may train readers to think of physical companions to mathematical problems, though a mathematician is unlikely to have such insight when first picking up the book.
Mathematicians will find The Mathematical Mechanic provides exercise in new ways of thinking. Instructors will find it contains material to suplement mathematics courses, helping pysically-minded students approach mathematics and helping mathematically-minded students appreciate physics.
John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs daily at The Endeavour.