Here is a mathematics book with attitude. Right from the opening sentences of the foreword by Carver Meade (a well-known electrical engineer and computer science professor at Caltech):
Most of us took mathematics courses from mathematicians – Bad Idea!
Mathematicians see mathematics as an area of study in its own right. The rest of us use mathematics as a precise language for expressing relationships among quantities in the real world, and as a tool for deriving quantitative conclusions from these relationships. For that purpose, mathematics courses, as they are taught today, are seldom helpful and are often downright destructive.
More from the author’s preface:
Too much mathematical rigor teaches rigor mortis: the fear of making an unjustified leap even when it lands on the correct result. Instead of paralysis, have courage – shoot first and ask questions later. Although unwise as public policy, it is a valuable problem-solving philosophy, and it is the theme of this book: how to guess answers without a proof or an exact calculation.
The author goes on to say that his book complements works such as Polya’s How to Solve It and Mathematics and Plausible Reasoning. Where Polya teaches how to solve precisely stated problems exactly, the author wants to teach us how to find moderately accurate solutions to the imperfectly defined problems that “life often hands us.” He does this by offering six broad approaches. These are: using dimensional analysis, starting from easy cases, lumping (essentially estimating integrals and derivatives with finite sums and differences), thinking pictorially, working with the big part (estimating via fractional changes), and using analogy. None of these is new, and indeed most are part of Polya’s repertoire. Yet the book offers much of value: good questions, lots of encouragement for students to use their wits, and especially the number and variety of examples worked out in detail.
Here are a few examples: guessing the value of a Gaussian integral that depends on a parameter, estimating drag in fluid dynamics using dimensional analysis, computing the period of a pendulum over a range of amplitudes, guessing the bond angle for the methane molecule, summing series, and estimating the depth of a well. There are many more! Nowhere is there an invitation to wild and carefree guessing. The author is always testing his estimations, approximations and guesses to see if they make sense. When they don’t, he asks why and pushes to understand what failed.
This is a book I would recommend strongly for supplemental use in undergraduate mathematics. My experience with mathematics majors who work in industry suggests that they arrive in their new jobs almost completely lacking in the (highly valued) skills of estimation and rough approximation. It’s almost as if they are afraid of producing a solution that’s not – in some sense – absolutely, precisely correct. It takes some time, development and maturity to understand when a very accurate solution is needed, and when a rough estimate will do.
Aside from the anti-mathematician bias in the early parts, this is a useful and valuable resource.
Bill Satzer (firstname.lastname@example.org) 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.
The author and MIT Press have make this book available online under the Creative Commons Noncommercial ShareAlike license.