Wednesday, August 10, 2005

New Mathematical Principles

After consultation with a colleague who was here way too late (as I was), we came up with two useful new principles of mathematics.

Axiom of Generalization: If something works for three cases, it works for every case.
The Axiom of generalization is popular with math students everywhere; if we allowed for it, we could have all sorts of interesting new results, such as the following:
"No infinite series converges to any number."
Proof: It is fairly straightforward to see that the infinite series with nth term (1/2)^n cannot converge to 0, 1/4, or 1/2, since the partial sums are strictly increasing and the second partial sum is already bigger than 1/2. Since we have exhibited three numbers to which the series could not converge, it clearly does not converge to any number. Also, I can easily construct two more series for which this is true...
The Axiom of Chance I think speaks for itself:

Axiom of Chance: If a result looks plausible, publish it. (Alternative form: Retractions are easier than proofs.)

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