Circular Arguments

Recently, Paul Krugman reminded us of circular reasoning in his blog on Brexit. The same was said earlier by Mervyn King.

Economists use circular reasoning all the time but I’ll digress from economics in this post on something I came across recently involving Fermat’s Last Theorem.

There’s a proof of the irrationality of 21/n , where n is an integer for n > 2 (proof doesn’t work for n = 2) which goes something like this:

Suppose 21/n = p/q, where and are integers and n > 2 . Then:

pn = qn + qn

violating Fermat’s last theorem. So 21/n  is irrational by contradiction.

Sounds cool. But not so. It’s circular argument. A comment at mathoverflow by a person named JS Milne points that that Andrew Wiles’ proof of Fermat’s Last Theorem itself uses the irrationality of 21/.


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