The Lumber Room

"Consign them to dust and damp by way of preserving them"

“Every good theorem must have a good counterexample”

with 4 comments

Abhyankar[1] attributes the quote to Severi.

[1]: Historical Ramblings in Algebraic Geometry and Related Algebra, Shreeram S. Abhyankar, The American Mathematical Monthly, Vol. 83, No. 6 (Jun. – Jul., 1976), pp. 409-448. Also available here, because it won a Lester R. Ford Award (“articles of expository excellence”) and also a Chauvenet Prize (“the highest award for mathematical expository writing”).

Abhyankar, after distinguishing between the flavours of “high-school algebra” (polynomials, power series), “college algebra” (rings, fields, ideals) and “university algebra” (functors, homological algebra) goes on to present his fundamental thesis (“obviously a partisan claim”):

The method of high-school algebra is powerful, beautiful and accessible. So let us not be overwhelmed by the groups-ring-fields or the functorial arrows of the other two algebras and thereby lose sight of the power of the explicit algorithmic processes given to us by Newton, Tschirnhausen, Kronecker, and Sylvester.

Perhaps for this reason, Dr. Z calls Abhyankar (“one of my great heroes”) “the modern prophet of high-school-algebra”.

Anyway, enough rambling. Back to “Every good theorem must have a good counterexample”. Discuss.

[Edited to add: The statement in its original context was referring to a phenomenon where a pleasing conjecture is found to have counterexamples, until it is resolved by realising that we must, say, count multiplicities the “right” way—the right way turning out to be whatever makes the conjecture true. Thus Bezout’s theorem, etc., and the quote just means, as he paraphrases, “don’t be deterred if your formula is presently invalid in some cases; it only means that you have not yet completely deciphered god’s mind”. On the other hand, what I (mis?)remembered was that one must know “counterexamples” to a theorem in the sense that one must know why the conclusion is not true if the hypotheses are weakened: thus one doesn’t really understand a theorem till one knows at least one “counterexample” (and at least two proofs).]

Written by S

Tue, 2009-03-10 at 06:34:18

4 Responses

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  1. Does that mean every good program must have a good bug?


    Tue, 2009-03-10 at 08:26:27

    • No, but we know all programs have bugs anyway :)

      Actually, the quote as I remembered it (before looking it up) was something like “a good mathematician knows a counterexample to every theorem”.


      Tue, 2009-03-10 at 22:29:17

    • Actually, I guess it does. The only programs that don’t have bugs are trivial “Hello, world”-like programs. So every good (interesting) program invariably has bugs. Lots of them. Except TeX, of course, where all the bugs are by design.


      Wed, 2009-07-22 at 16:37:37

  2. S

    Fri, 2014-11-28 at 13:05:40

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