Opinion 136: Amir Alexander's "Infinitesimal" Should Be Read by ANYONE Interested in Math, Regardless of whether or not they are interested in the history of math, since it Says As Much about the Present, and more importantly, the FUTURE, of Math

By Doron Zeilberger

Written: April 8, 2014

I just finished reading, for the second time, Amir Alexander's long-awaited book, "Infinitesimal: How a dangerous mathematical theory shaped the modern world", and plan to read it at least one more time. Sure it tells you fascinating stories of "math wars" way back in the `dark ages' where mathematical `dissidents' were silenced, or else would be (if lucky) confined to house arrest, and the Jesuits tried to exploit the pedantic rigor of traditional, proof-based, Euclidean Geometry in order to further their religious agenda, and silence all dissidence.

As Alexander tells so well, this religious dogmatism cost Italy its scientific hegemony, that moved to England, where people (after Oliver Cromwell) were so afraid of dogmatism (not withstanding Thomas Hobbes, of Leviathan fame, who loved dictatorships), that they discouraged rigidity of any kind, scientific or otherwise, and encouraged people like John Wallis to let their mathematical intuition run wild, and do (17th-century-style) experimental math, that lead to Newton's (and Leibnitz', but especially Newton's more `applied' approach) calculus, that changed science for ever.

But every revolution turns sour and stale. As we all know, by the 19th century, the pendulum swung back towards (alleged) rigor, that would have pleased both Hobbes and the Jesuits. Cauchy and Weierstrass replaced the intuitive infinitesimals by a `rigorous' foundation of analysis based on the notion of limits, and made calculus (that acquired a gentrified name: `mathematical analysis') into the most boring subject, that I am sure turned away so many mathematically talented students (and it is a miracle that I survived!).

[ Ironically, one needs neither infinitesimals, nor Cauchy-Weierstrass style limits, to have both a rigorous and pleasant foundation for the calculus needed in science and engineering. Make everything discrete and finite! At the end of the day, all today's calculations are done, via modern number-crunching, by discretizing so-called differential equations, ultimately solving huge (but finite) systems of linear equations, part of discrete math! So a much more pleasant, and (as it turns out, much more rigorous, not that I care) approach, is to assume that the universe has a tiny, yet `strictly positive' "indivisible", and numerical analysis consists of approximating the true finite difference equations by ones of a much coarser grid. ]

So while the debate between Hobbes and Wallis, and Cavalieri and Guldin, seems to us, today, as ridiculous as the debate between different sects of a religion is to a non-believer, the struggle between mathematical dogmatism and free-style experimental math is as fierce as ever. Practitioners of experimental math (heavily using computers rather than infinitesimals), are the current underdogs. Luckily, we no longer have to fear house arrest, or (physical) torture, but our papers get frequently rejected by `good' journals (or `good journals wannabees', e.g., the Elect. J. of Combinatorics, whose current editor, Miklós Bóna, just rejected a paper written by one of my students [ironically that journal was founded by Experimental Math pioneer, the late Herb Wilf, I am sure that poor Herb is rolling in his grave], since it was

"Very high-level experimental math, but not of sufficient quality for the Elect. J. of Combinatorics, and the author should try a specialized journal in expermental math"

But enough whining! Reading Amir Alexander's masterpiece should endow us, experimental mathematicians, with new strengths, and energy, to defy the current `rigorous' dogma, and pursue a new kind of mathematics (to paraphrase another pioneering maverick, Stephen Wolfram). I strongly recommend that you read the insightful review, published today, by that icon of mathematical writing, John Allen Paulos, who in the concluding paragraph said:

"Now, however the opposition is between proof-driven `pure' mathematicians and computer-friendly, quasi-empirical mathematicians. The latter will dominate, I hope and expect- along with the kind of moderate pluralistic democracy that Dr. Alexander sees as a natural outgrowth of Wallis' open, experimental mathematics"

But even if you are a current, mainstream `rigorous-proof-dogmatist', you would still enjoy Amir Alexander's masterpiece, and very possibly draw opposite conclusions than mine, or just have many hours of enjoyable reading. Enjoy!

Opinions of Doron Zeilberger