Amir Alexander is a historian at UCLA and has written some books that have drawn attention. I haven’t read them but only summaries and some reviews. Alexander’s modus operandi apparently is: find a good narrative and weave history around it. The story sells, and history hitch-hikes along.
For example Alexander’s book Geometrical Landscapes: The Voyages of Discovery and the Transformation of Mathematical Practice uses the narrative of exploration and imposes this on mathematics too. Reuben Hersh debunks it. Be sure to first read Davis & Hersh, The Mathematical Experience, and then check out Hersh’s review. Hersh quotes Alexander:
“I find a narrative approach to be a most promising avenue for historicizing mathematics. (…) Mathematical work does, I argue, contain a narrative. Once this narrative is identified, it can be related to other, nonmathematical cultural tales that are prevalent within the mathematicians’ social circles.”
Hersh finds this okay but indicates that Alexander goes too far in suggesting that such narratives are also driving forces within mathematics. Reuben Hersh:
“If there existed a prevalent social or cultural story that was analogous or parallel to the mathematical story, it by no means follows that such a story “shaped” or “guided” the mathematics. Such a social or cultural story may have simply served as a model for how one talked about or advertised the mathematics.” (American Scientist)
Alexander recycles narratives and weaves history around it, and for some reason he has chosen mathematics to be his victim.
We see this even clearer in his earlier book Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics. Check these reviews:
(1) Reviel Netz: “Everyone is unhappy after his or her own fashion, but the font of possible stories is limited. Hence stories tend to flatten the many particular sorrows into a few single types of narrative misery. Amir Alexander is interested in one such type: the misery of the Outsider Mathematical Genius.” (Common Knowledge)
(2) Daniel S. Silver: “Mathematicians might have been the only scientists who acquired the popular label of heroic misfit, but Duel at Dawn can leave readers with the impression that most mathematicians wore the unhappy brand. Besides Galois, we find Abel and Bolyai offered as examples, and, arguably, Cauchy, too. Yet there were many well-known mathematicians in the nineteenth century who were seen as anything but misfits: Cayley, Dirichlet, Gauss, and Hamilton are just a few names that come to mind. And there are plenty today. So why are sad tales of unstable mathematicians from Galois to Perelman so popular? Duel at Dawn suggests a reason. The public sees mathematicians as being like artists, preferring to live in a virtual world that bears little resemblance to what G. H. Hardy called “this stupidly constructed ‘real’ one”. Such people cannot possibly be happy or sane. Another reason, one not offered by Alexander, can be found in our public high schools.” (AMS Notices)
Translate this as: A so-called “historical” tale relies on current (bad) education to sell well.
(3) Michael Patrick Brady: “(…), author Amir Alexander argues that the popular image of mathematicians as strange, reclusive figures springs from the early 19th century. It was then that mathematics began to evolve from a science based in the empirical realities of the Enlightenment to an art form informed by the ideals of Romanticism, concerned only with its own internal truths.” (Forbes)
Translate this: Brady fell into Alexander’s trap. Brady never heard about the distinction between Greek math and engineering ? Brady neither has the criticism on the math as given by Silver.
Now in 2014 Alexander published this new book Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World.
From Dirk Struik’s history of mathematics, I already knew that around 1600 most universities were quite conservative and that over the next century new academies were created for the new demands from society and trade and industry. Now Alexander reports that five leaders of the Society of Jesus convened on August 10 1632 and actually forbade the use of individibles (later giving rise to infinitesimals). Alexander claims that the Jesuits succeeded in keeping Italy in the Euclidean mold. What I regarded as common human conservatism thus would actually be organised. The countries that were relatively free from Rome also allowed for more freedom in mathematics, whence they discovered new methods, like the infinitesimal. Henri VIII (1491-1547) and his six marriages thus indirectly contributed to the creation of calculus.
(b) Judith V. Grabiner has a review at MAA. An important point:
“Aristotle, arguing for the potentially infinite divisibility of the continuum, had explicitly ruled out both indivisibles and the actual infinite.”
Yes, in the continuum there are no indivisibles. Yes, for a line going to infinity there is no reason to hold that it is an actual infinite (since what would that be ?). However, it would be strange to say that the continuum, say the interval [0, 1], cannot be divided infinitely (or indefinitely). In terms of division, the continuum is an actual infinite. If “actual infinite” is to have any meaning, it is the continuum. It would be strange to think that Aristotle thought otherwise. I haven’t studied Aristotle here though. I still allow for the possibility that the Jesuits were right. Calculus may have been discovered by Newton’s use of infinitesimals, but Cauchy developed it with limits, in order to solve the mathematical problems that he and others saw w.r.t. infinitesimals. Grabiner’s critique: “Alexander’s book contains occasional imprecise statements. Notably, although the book’s main title is “Infinitesimal,” the Jesuit condemnations he quotes denounce indivisibles, not infinitesimals.””
Thus I fully agree with Grabiner:
“But I think Alexander overestimates the importance of the disputes he describes. He doesn’t show that the Jesuit condemnation of indivisibles was anywhere near as influential as the Church’s condemnation of Galileo’s Copernicanism. Nor does he show that the ideas of Cavalieri and Wallis were widely seen as dangerous and disruptive. And, since a key point of Alexander’s book is the importance of the invention of the calculus, I’d argue that the calculus would still have been invented in the seventeenth century even had the Jesuits convinced everyone in Europe that indivisibles were not rigorous mathematics.” (MAA)
But also Grabiner falls in Alexander’s trap: “Nonetheless, the stories Alexander tells about these disputes are fascinating, and they deserve to be better known.” I would rather hold that the truth be known, and not the stories that Alexander tells.
Alexander thus surprises me on the organised conservatism in Italy. But it seems that he puts too much emphasis on the power of organisation in other countries. Given such freedom, there still was room for good old common human conservatism that doesn’t need organisation. While Newton discovered calculus and derived his laws of gravity using calculus, he still presented his Principia with Euclidean methods. To see Alexander’s evidence I would need to read his book, but his modus operandi makes me wary. My fear is that I will just read a common narrative of freedom against tyranny, adorned with elements from the history of calculus to justify his job contract at UCLA.
(c) Check Alexander’s own text in the Scientific American followed by critical comments by readers. His final paragraph reads:
“By transforming the calculus into a rigorous mathematical system, Cauchy ended a conflict that had lasted more than two millennia. In the 5th century B.C. Hippasus had shown that mathematics could never fully describe the world. In the 19th century A.D. Cauchy showed that it didn’t have to: Mathematics would survive, and thrive, on its own, freed from the shackles of material reality. Modern mathematics was born.”
This is a crooked paragraph. Mathematics has always been free from the shackles of material reality. Mathematics namely is abstraction. Hippasus didn’t do what Alexander claims he did. A statement like “Modern … was born” is story-telling and an empty shell.
(d) The review by Alan Hirschfeld, professor of physics, is informative, but: “That a mathematical theory can be characterized as dangerous, much less world-shaping, pings my skeptic’s radar. To earn this dual distinction for what appears to be merely a centuries-old quibble over the nature of points, lines and planes sets a formidable test for any author. But Mr. Alexander succeeds, weaving the strands of a colossal mathematical dispute into the fabric of Western cultural history. The result is an interpretive tapestry whose richness justifies his exclamatory subtitle. ” (Wall St. Journal)
Translate this as: Another victim fell into the trap.
Note that infinitesimals were still a problem at the time of Cauchy. It is somewhat strange to have infinitesimals in the title when the story ends with limits. The true umbrella is the birth of calculus, but Alexander cannot tell that story, since he wants to focus on infinitesimals since the Jesuits forbade indivisibles. The general readership will be interested in the struggle of science against tyranny. To relive that same old narrative, they now are treated on somewhat crooked mathematics and somewhat crooked history of mathematics.
People should be able to ask their money back if a book does not deliver what is promised.
Of course, my book Conquest of the Plane shows that calculus can be developed with algebra, without limits and without infinitesimals. But I don’t give a money-back-guarantee, because traditionally minded mathematicians have shown that they don’t read well, see the earlier entry.
NB 1. In a way I feel a bit vindicated that the narrative “democracy and mathematics” gets such attention. I have been advocating that his combination gets more attention. However, the better narrative is:
- mathematics is liberating itself, since no authority can force you to accept a theorem except your own understanding of the proof
- democracy deteriorates when people get bad education in mathematics
- mathematicians have been destroying democracy since Kenneth Arrow presented his “impossibility theorem” in 1950 / 1951, see my book Voting theory for democracy.
NB 2. Everything hangs together. For math education, see these suggestions for improvement:
- the book Elegance with Substance (2009)
- the book Conquest of the Plane (2011)
- this note on infinity
- all leading to Neoclassical mathematics for the schools.