Thursday, October 21, 2004

Counter-examples

Mathematics has a long tradition of using counter-examples as a way of illuminating structure in theory. Especially in more abstract areas like topology, canonical counterexamples provide a quick way of teasing out fine structure in sets of axioms and assumptions.

A brief foray through Amazon.com revealed catalogues of well known counter examples in topology, analysis, and graph theory. On the web, there are pages on counterexamples in functional analysis, Clifford algebras, and mathematical programming.

What would be good candidate areas for a list of counter-examples in theory ? Complexity theory springs to mind: simple constructions (diagonalization, what have you) that break certain claims.

In combinatorial geometry, one might be able to come up with a list of useful structures. Personally, I find the projective plane to be a useful example to demonstrate the limits of combinatorial arguments when reasoning about geometric objects.


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