Measurement (Q5891785)

Lockhart's book seeks to (re-)introduce the lay, educated reader to a classical view of mathematics as worth pursuing for its own sake, despite the difficulties. ``[T]his is an art book,'' he announces -- one that will take the reader into a jungle that ``will haunt your waking dreams.''NEWLINENEWLINETwo major divisions of the book give an exposition first of classical Euclidean geometry, then of differential and integral calculus. In both cases, measurement of conical sections motivates a wholly unexpected conclusion: in the first part, a foray into projective geometry; in the second, the well-known definition of the natural logarithm, using the integral of the reciprocal function (\(1/x\)). The text sometimes meanders with no apparent aim, much as a wanderer in admiration of the jungle's sights. On other occasions, the wanderer finds that several paths she has crossed converge in a fantastic discovery, explained masterfully in the text. Illustrations whose not-quite-straight lines look hand-drawn, along with unanswered questions that are centered and set in bold type, pepper the text and complement well the author's encouragement to explore independently. For an expository work, such touches radiate charm.NEWLINENEWLINEEarly on, Lockhart advises the reader that ``[a]ny measurement we make, whether real or imaginary, will necessarily depend on our choice of measuring unit''. A summary of the book ought perhaps use as its measuring unit the thoughts of an audience closer to its intended target, so I photocopied the sections relevant to the method of exhaustion, a major theme of the first part, and gave them to my Honors Calculus class, soliciting their opinions.NEWLINENEWLINEThe students enjoyed and and understood the material, with the caveat that their advanced academic level prepared them for the material better than, say, the average layman, who would likely have to reread some passages. One student expressed appreciation for finally learning ``where \(\pi\) comes from''. Several indicated that the illustrations were helpful, and enjoyed the creativity used to illustrate dilation. One wrote that it ``concisely taught me methods of measurement I had never heard of before''. (As their calculus teacher, I wonder if this should embarrass me.) One student engaged a bold-faced question enough to wonder what assumptions were in play for the angles; were they locked in place, or allowed to change? On the other hand, students felt the knowledge expected from the reader was inconsistent, ranging from a level that was ``almost insulting'' to ``taking a lot of things for granted''.NEWLINENEWLINELockhart should perhaps have exercised some restraint with his aesthetic judgments; while evaluations of elegance and wonder are certainly common in mathematical prose, some of the ones here are not merely trivial, but questionable. A remark that the name of ``the law of cosines'' is silly strikes the reader as unconvincing; no explanation is given, and the proffered alternative is the rather ungainly ``generalized Pythagorean theorem''. Likewise surprising are the repeated expressions of disappointment that a problem resolves to irrational or transcendental numbers that are ``inexpressible'' or ``unsolvable''. Given the earlier insistence that ``[t]he solution to a math problem is not a number; it's an argument, a proof'', one wishes he had taken a less Pythagorean attitude at the shattering of our rational preconceptions.