Physical Mathematics or Mathematical Physics?

Earlier today I finished tutoring a precalc student in Barnes & Noble here in San Mateo and wandered over to check out the math books. One caught my eye: The Mathematical Mechanic: Using Physical Reasoning to Solve Problems by Penn State professor Mark Levi. What a brilliant idea: from the Pythagorean Theorem to the Cauchy Integral Formula, Dr. Levi uses strings, water, and mechanical devices to model mathematical formulas. Force, velocity and potential energy, among other ideas from physics class, are introduced to prove shortest distances, minimum areas and lines of best fit (linear regression) from math class. In this way the reader gets to "feel" what's going on and to sense when equilibrium has been reached, proving the mathematical theorem geometrically, visually and mechanically (and somatically?).

As a math teacher and tutor I don't hide my impatience with rigor. I think mathematical rigor can be traced to European youngsters (culturally speaking) pretending to Euclid's cred and medieval scientists challenging 1600 years of religious authority. Professional mathematicians may need or want rigor but math students in the 21st century need to understand topics and know how to use formulas, not how to prove the heck out of them.

Dr. Levi's book is a refreshing way to learn a lot of math and physics, and while it seems like an original way to link the two fields, Dr. Levi's scheme is grander:

Physical ideas can be real eye-openers and can suggest a strikingly simplified solution to a mathematical problem. The two subjects are so intimately intertwined that both suffer if separated. An occasional role reversal [physics serving math for a change] can be very fruitful, as this book illustrates. It may be argued that the separation of the two subjects is artificial.

With this book a reader can learn how to link math and physics intuitively and the experiments are free!