Written: May 23, 2003
I just finished reading "Geometrical Landscapes: The Voyages of Discovery and the Transformations of Mathematical Practice", a fascinating book written by historian-of-science Amir R. Alexander and published by Stanford University Press.
This beautifully-written and thought-provoking book can be read on several levels, each of them very rewarding. On the "lowest" level, it is a great story of very brave and very smart people who discovered new lands and new mathematics. It makes us mathematicians aware how much we owe to the pre-Calculus giants on whose shoulders Newton and Leibniz stood (and there were quite a few besides Galileo: Cavalieri, Torricelli, Dee, and the indefatigable Thomas Hariot, the main hero of Alexander's narrative.)
On yet another level, but still neither philosophical nor historical, but merely that of day-to-day mathematical research, is a lesson that we, as practicing mathematicians, can learn from Hariot and his contemporaries. We should not worry too much about rigor, since the proof of the "proof" (pudding) is in the applications (eating), and it is often fruitful to develop heuristic and non-rigorous methods. Sooner or later they will be justified, and even if they don't, "so what?". Theoretical physicists and other scientists know it all too well, but we mathematicians seem to be hung-up on stifling logical rigor. Amir Alexander convincingly shows how Thomas Hariot used these "hand-waving" methods to great advantage, and got all correct results, that today can be easily justified with Calculus, but at the time were major breakthroughs. In this connection I urge everyone to read Pierre Cartier's beautiful article about "Mathemagicians" Euler and Feynman.
Finally, on the philosophical and especially meta-historical level this book presents a new and exciting paradigm to the history of mathematics. This view is summarized in an intriguing appendix, "The mathematical narrative". In this appendix, Alexander lays out, in very lucid and clear style, easily comprehensible to mere mathematicians, the foundations of his new and revolutionary approach to the history of mathematics. Mathematicians, and hence mathematics itself are influenced and even shaped by their zeitgeist and location, what Alexander calls the current narrative. It follows that mathematics's "party line" of platonic time-less absolute truths is just a fictitious make-believe.
Not surprisingly, most mathematicians will not buy this, because they are comfortable with their naive platonism, and think of themselves, or at least of their subject, as immortal. Hence it is not a shock that even such a sociologically-conscious and non-orthodox mathematician as Reuben Hersh, of Davis-Hersh and "What's Mathematics Really" fame, disagreed with Alexander's bottom line. In a largely positive book review in "American Scientist", Hersh states that Alexander has "no right to talk", since he is not a mathematician himself, and he did not even seem to have consulted with contemporary working research-mathematicians. Hersh claims that Alexander confused correlation with "cause" (like advocates of smoking who claim that the proven correlation between smoking and cancer does not prove that smoking causes cancer).
Myself, I was completely convinced by Alexander, and Hersh's claim that the motivation behind most mathematicians is "problem-solving" does not contradict Alexander's approach, but reinforces it. Isn't problem-solving like exploration challenges (especially the mountain-climbing metaphor, so dear to Hardy and many other mathematicians).
According to Hersh, only mathematicians are qualified to philosophize and historize about their subject. This is utter nonsense. Most practicing mathematicians are philosophically illiterate, even in their non-mathematical philosophy. It is clear that Amir Alexander knows mathematics very well, at least the past mathematics he was talking about. The history of mathematics is too important to be left to mathematicians.
Let me end with a personal note. I found out that I was scooped by Hariot! It turned out that my ulta-finitistic and discrete philosophy (also preached by Steven Wolfram and others) for the foundation of mathematics as elaborated in my article "Real" Analysis is a Degenerate Case of Discrete Analysis was already espoused by Hariot. In the above article, I assert that the discrete approach is far more rigorous than the continuous one, so I disagree with Alexander on one point. In the concluding paragraph (p. 202), Alexander says that the approaches of Newton and Leibniz (and we should add Cauchy and Weierstrass) were a step-up in rigor. I disagree, the philosophical foundation of Mathematical Analysis went all downhill after Hariot. I am sure that Hariot's discretism and mathematical atomism would be resurrected along the lines described in my article.
Even if you are a die-hard platonist, formalist, intuitionist, logicist, or whateverist, and do not want to change your mathematical worldview, I as sure that you will still enjoy Amir Alexander's masterpiece. Just pick the level(s) that you are comfortable with.