...a companion blog to "Math-Frolic," specifically for interviews, book reviews, weekly-linkfests, and longer posts or commentary than usually found at the Math-Frolic site.

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." ---Bertrand Russell (1907) Rob Gluck

"I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal." ---Bertrand Russell (1957)

******************************************************************** Rob Gluck

Wednesday, September 10, 2014

So You Want Some Math History...

Recently finished a quickread of history professor Amir Alexander's "Infinitesimal," which has received plenty of rave reviews, a few of which I'll reference below, (as I won't fully review it here):


For whatever reason, I've never been much fascinated with math history (at least not pre-19th century history). This book recounts the sometimes surprisingly controversial disputes over the early history of infinitesimals in mathematics. It ends right before the part that I would've found interesting… the Newton - Leibniz rivalry and history of infinitesimals and limits since then (David Berlinski previously wrote a popular book, "A Tour of the Calculus," covering that history which I find more intriguing). Thus, though Alexander's writing is good, and the narrative sometimes dramatic, it wasn't subject matter that particularly grabbed me. But if history is among your foci this volume comes very highly recommended and you should definitely give it a look. It certainly captures events and conflicts that most people will be unacquainted with.

A few days back I briefly mentioned "Mathematics and the Real World" by Zvi Artstein… a read, which once again, because of its older historical context started off slowly for me. However, the last half of the book, moving into more modern math history, improved considerably. The writing is still somewhat stilted, but I did enjoy more of the material being covered from chapter 5 (an introduction to randomness) on, and especially the last few chapters. If you're looking for a more lively narrative and interesting story-line than "Infinitesimal" will better suit your taste, but if you want a broader sweep of math history I do recommend the Artstein book, even though parts of it are a bit of a slog. I'm not sure it's all that helpful, but the most extensive review of the book I've seen is here:

Recently on social media, Steven Strogatz and currently Evelyn Lamb have talked about teaching math history courses… don't know if this is a new trend, or such courses have long been around. In any event, both these works could be useful to an instructor of such a course -- the Alexander book, while perhaps more engaging, only deals with a sliver of history, while Artstein chronologically covers a wider swath and range of history, and is, I think, more useful. An older, more concise, historical overview of mathematics I've previously enjoyed, again comes from Berlinski, "Infinite Ascent."

By the way, I was not previously familiar with the name "Zvi Artstein," but he certainly appears to be a well-established, well-credentialed professional Israeli mathematician -- I mention this only to indicate he is not some crackpot opinionated writer, because of what follows:

Artstein finishes his volume with a 30-page, almost polemical, chapter on math education that is quite an assault on Dr. Keith Devlin's well-distributed viewpoint regarding "mathematical thinking." (He never mentions Devlin by name, but seems to clearly be directing much of his commentary toward Devlin's writings.) I've seen others sometimes voice arguments similar to Artstein's, but not with the same tenacity and self-assuredness of Artstein. I think Artstein makes some valid points along the way… but then too, I don't believe in any 'one-size-fits-all' solution to teaching youngsters math. He seems to favor a form of teaching first proposed by Louis Bénézet in the earlier 1900's that never caught on widely.
Anyway, Artstein asserts, "The truth is, there is no such thing as mathematical thinking." He argues there are only two types of thinking ('comparative' and 'creative,') which apply to most all disciplines, not just math, and that they can't be taught, but can only be learned through experience and practice!
He goes so far as to say that, "Presenting 'mathematical thinking' as a necessity, or as beneficial to life beyond mathematics and its applications, is likely to cause harm." I won't try to fully explain Artstein's viewpoint (because I'm afraid I would misstate it in some manner), but he does offer many examples, and it makes for an interesting, if oddly contentious, final chapter (in fact, I suspect he could write a whole 'nuther controversial book based on the ideas expressed in this chapter -- the chapter sticks out oddly from the rest of the book, though I think in his mind it was more of a natural conclusion that the book was always building towards).
So, I do recommend the Artstein offering (despite sometimes pedantic writing style, and the harsh ending), based on the second half historical discussion he delivers, and issues he will force you to think about.

On short notice, I asked Keith Devlin (via email) if he'd care to respond with any comment about Artstein's views -- I haven't heard anything back, but will add it here later if I do.

[ ==> housekeeping note: the usual Friday potpourri links here will, this week, begin appearing either late Friday or early Saturday (instead of early Friday) -- this is simply a practical change to better accommodate capture of Friday math posts that were showing up hours after my linkfest was being posted.]

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