Michael Spivak's tiny but dense Calculus on Manifolds (1965) is another of those classic books that I've tried to read at intervals over the years, alas without much success in my eternal quest for enlightenment. Today a friend's copy came into my hands. I flipped through it, looking for examples of a mysterious "Yellow Pig" in-joke among mathematicians that Spivak is associated with. (He is also responsible for a system of gender-neutral pronouns – e, em, eir, eirs, eirself – a cute but rather unæsthetic attempt at social language engineering.)

Instead of a Yellow Pig, however, this time in Calculus on Manifolds I discovered something far more interesting and important: superb wisdom on how understanding can evolve over time toward ever-greater generality and power and simplicity. In the Preface (pps. viii-ix) Spivak writes:

The reader probably suspects that the modern Stokes' Theorem is at least as difficult as the classical theorems derived from it. On the contrary, it is a very simple consequence of yet another version of Stokes' Theorem; this very abstract version is the final and main result of Chapter 4. It is entirely reasonable to suppose that the difficulties so far avoided must be hidden here. Yet the proof of this theorem is, in the mathematician's sense, an utter triviality — a straightforward computation. On the other hand, even the statement of this triviality cannot be understood without a horde of difficult definitions from Chapter 4. There are good reasons why the theorems should all be easy and the definitions hard. As the evolution of Stokes' Theorem revealed, a single simple principle can masquerade as several difficult results; the proofs of many theorems involve merely stripping away the disguise. The definitions, on the other hand, serve a twofold purpose: they are rigorous replacements for vague notions, and machinery for elegant proofs. ...

This conclusion appears even more strikingly at the end of Chapter 4 itself, when Michael Spivak concludes:

Stokes' theorem shares three important attributes with many fully evolved major theorems:

1. It is trivial.

2. It is trivial because the terms appearing in it have been properly defined.

3. It has significant consequences.

Since this entire chapter was little more than a series of definitions which made the statement and proof of Stokes' theorem possible, the reader should be willing to grant the first two of these attributes to Stokes' theorem. The rest of the book is devoted to justifying the third.

So also in many areas of life, where to get to the trivial often requires a struggle through dense thickets of complexity ...

(cf. NotByAddingFeatures (24 May 1999), ComplexityFromSimplicity (5 Aug 1999), ComplexSimplicity (12 Feb 2000), AwesomelySimple (26 Jan 2001), No Concepts At All (22 Feb 2001), ProofsAndRefutations (24 Jun 2004), ApprovedMethods (12 Nov 2005), ...)

TopicScience - TopicLiterature - TopicPhilosophy - 2006-01-27

Comment 29 Jan 2006^ at 15:30 UTC

a correspondent writes:

Oliver Wendell Holmes said, " I would not give a fig for simplicity this side of complexity. I would give my life for simplicity the other side of complexity."

Comment 29 Jan 2006^ at 15:32 UTC

a correspondent writes:

Beresford Parlett has characterized a number of properties of symmetric eigenvalue problems by saying there is "an ounce of insight from a pound of preparation." Sometimes he says "notation" rather than "preparation," but I think the moral is the same -- and certainly that moral applies to Stokes theorem.

The fundamental theorem of calculus, the divergence theorem, and Stokes theorem are little flowers of analysis. I sometimes think the generalized Stokes theorem is the mathematical version of "a rose by any other name would smell as sweet."

(correlates: HardCoreBelievers, NothingnessShowsThrough, EssenceOfEducation, ...)