Calculus and infinity and how to explain them (Part 1)
I have been reading three books more or less simultaneously at the moment, and thinking about others. I have become interested, not for the first time, in calculus, and I have been a keen reader of popular works on mathematics, most of them far over my head, for many years.

The three books are:

Everything and More: a Compact History of Infinity by David Foster Wallace. (Wallace uses the infinity sign for ‘Infinity’)

Calculus Made Easy by Silvanus P Thompson, revised by Martin Gardner

The Mathematical Universe by William Dunham

Calculus depends on infinity. Either you have to find the sum of an infinite series, or deal with infinitesimal quantities, which means treating one divided by infinity as greater than zero. These three books all explain calculus in various ways and with varying degrees of success. I am going to review them all informally and look at some other books by the same authors, and books by other authors in the same fields.

Let’s look at the most irritating book first: Everything and More: a Compact History of Infinity by David Foster Wallace. I got this book – actually as a very much solicited birthday present – because I had recently read Infinite Jest, a weird and amazing and brilliant novel by Wallace, and wondered if his brilliance extended to popular science writing. Infinite Jest is set in the near future and combines science fiction elements like a north-eastern USA that has been devastated and ceded to Canada (Americans call it The Great Concavity; Canadians call it The Great Convexity) and the well-known SF device of a work of art that is such a powerful meme that it kills people (in this case a film that people watch over and over again till they die of dehydration and starvation) with a passionate depiction of life at a tennis academy, and what it feels like to be a hopeless addict and then a recovering addict, and other things I’ve half forgotten, like the group of wheel-chair-bound Canadian terrorists who keep popping up. Infinite Jest is both irritating and hypnotic because of the relentlessly modern ‘fast-food’ language – ‘with’ is always, I think, abbreviated to w/ - a pointless abbreviation if ever I saw one, but I think it is used commercially in America – similarly, ‘without’ becomes w/o, and the word ‘morning’ is always replaced by ‘a.m.’. This is all right in a novel. In fact I like it. It adds to the feeling of dislocation and not being here or now but somewhere familiar but definitely elsewhere.

But I must look at Everything and More. Wallace has used his strange tropes and eccentricities – which I thought were part of his novelistic technique, to be used as needed and only as needed – in this non-fiction book. Why? It adds nothing and takes a lot away. It wilfully distorts and spoils our common heritage. Sir Isaac Newton is not called I. Newton. Some examples, with comments;

[p.10] “Here are two more quotations from towering figures. M. Kline: […] F.d.l. Saussure: […]”. No, these people are actually Morris Kline and Ferdinand de Saussure. Many people have heard of them under those names, and it annoys those who know, and misleads those who don’t, to give them bizarre abbreviated forms. I assume that David Foster Wallace would object if he was called D. F. Wallace by reviewers, just as J. R. R. Tolkien chose not to be known authorially as John Ronald Reuel.

[p. 108] “So then once the Story of [infinity] hits the late 1600s we’re now barreling at high and irreversible speed toward Cantor et al., […]”. This is very bad writing. Does it mean anything? No, I think Wallace is just drawing breath and figuring out what to say while his fingers patter over the keyboard.

[p. 127] “1635ish—G. P. de Roberval’s discovery that a curve’s tangent could be expressed as a function of the velocity of a moving point whose path composed the curve” [footnote:] “Sorry about the hideous syntax here; there’s no nice way to compress Roberval.” Actually it is the job of the author to find good ways to express meanings. Compression is made easier by dropping the footnote and thinking for longer before writing the sentence. The Britannica does it this way, which is more elegant and tells us almost as much – it fails to mention velocity, which could easily be worked in: “He discovered a general method of drawing tangents, by treating a curve as the result of the motion of a moving point and by resolving the motion of the point into two simpler components.” And it would be better to write Gilles Personne de Roberval.



- posted by Graham @ 1:26 pm