21 September 2009

Counterexamples in X

Counterexamples in Probability And Statistics (Joseph P. Romano and A. F. Siegel) and Counterexamples in Probability and Real Analysis (Gary L. Wise and Eric B. Hall) both seem to be books in the tradition of Counterexamples in Analysis (Bernard Gelbaum and John Olmsted) and Counterexamples in Topology (Lynn Arthur Steen and J. Arthur Seebach. These are books that collect the examples just "outside" the boundaries of the various standard theorems, the point being to explain why one needs the seemingly strange collections of hypotheses that seem to begin every analytic theorem. (Hence the tags "education" and "teaching"; I've often seen these counterexample books described as "anti-textbooks", and as being complementary to standard textbooks which often spend most of their time telling you what's true.)

It seems that these books are concentrated on the analytic end of mathematics; I couldn't find, for example, books of counterexamples in algebra, combinatorics, or number theory. There is, however, Theorems and Counterexamples in Mathematics. My sense is that the nonexistence of these books is connected to the fact that those fields don't seem quite as rife with theorems where all the work is hidden in the definitions.

10 comments:

unapologetic said...

I think it also has something to do with the fact that results in algebra are relatively robust. They have a lot to do with the general contours of mathematics rather than niggling about the borders to see the fine structure there. In a very loose analogy, algebra is the snowman in the Mandelbrot set while analysis is the seahorse valley.

Tom said...

The subject matter is the same vein but unfortunately the price is not. Oh how I love Dover books.

Mitch said...

- "snowman"? "seahorse"? That's some metaphor. It took me a while but I see what you're saying now.

- One explanation (of the lack of algebra/combbinatorics examples) could be because (from what I see) those fields tend to have all the counterexamples right there as motivators for structure. For example, the chain of inclusion of integral domain, UFD, PID, Euclidean domain, and field. These are usual presented almost immediately after their definitions with the counterexample (a PID that's not a UFD, etc).

Or it could be just no one's got around to it.

Michael Lugo said...

Mitch,

that's a good explanation. The need for these books might have to do with the way that textbooks are written, then; for some reason authors of algebra textbooks are more likely to include the counterexamples than authors of analysis textbooks.

equatorialmaths said...

It has something with layman intuition one can try to apply in topology, analysis, and probability, but not so in algebra and NT. In the latter one tends to operate in a more rigorous fashion, just as there is little else to do...

unapologetic said...

Laymen have "intuition" in analysis? Sounds like someone's never taught a calculus class.

Rani said...

Let me help you write Counterexamples in Combinatorics.

Chapter 1: Graphs
Section 1.1: Petersen's graph
Section 1.2: The random G(n,p) graph.

letterman said...

Um, Mitch, every PID is a UFD. You probably meant the other way, like Z[x].

I've been collecting counterexamples for properties of integral domains like these for a while. Not likely ever to make them public, though--I'm no mathematician.

peter-arndt said...

More for the collection:

Fornaess J., Stensones B. Lectures on Counterexamples in Several Complex Variables (Princeton, 1987)(252p)

Khaleelulla S. Counterexamples in Topological Vector Spaces (LNM0936, Springer, 1982)(ISBN 354011565X)(199p)

Two (not so convincing) lists of counterexamples in Algebra:

http://mathoverflow.net/questions/29006/counterexamples-in-algebra

http://ncatlab.org/nlab/show/counterexamples+in+algebra

peter-arndt said...

And more:

Michael Capobianco, John C. Molluzzo: Examples and Counterexamples in Graph Theory

P. Lounesto: Counterexamples in Clifford algebras. Advances in Applied Clifford Algebras 6 (1996), 69-104.
(see also the link)
http://users.tkk.fi/ppuska/mirror/Lounesto/counterexamples.htm

A.B. Kharazishvili: Strange Functions in Real Analysis, Second Edition, 410p.

J. Stoyanov: Counterexamples in Probability. John Wiley, Chichester, 1987, 1997.