Thursday, November 28, 2013

Opposing views

The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel in 2013 was awarded to Eugene Fama, Lars Peter Hansen and Robert Schiller.  In an article in the New York Times, Professor Schiller commented on how he disagrees with the other 2 winners of the Prize even though they are all cited for their work on "empirical analysis of asset prices".  This reminds me that in virtually all sciences, even in hard sciences such as physics, there are disagreements on the validity of various theories.  One notable exception is in mathematics, where a formal proof of a theorem from antiquity is still accepted today.  Of course there are disagreements in mathematics, such as whether the proof of the four color theorem is valid, since it was only checkable using a computer program, but that is a debate whether there is a mistake in the proof, not whether the proof methodology is invalid.  Mathematical logic is a branch of mathematics that contemplates questions about consistency and completeness of formal mathematical proofs.  For instance, is it possible to provide a formal proof of a statement and its negation?  Consistent theories means that you cannot prove both a statement and its negation.  A complete theory means that for every statement you can prove the statement or its negation.  Most practicing mathematicians are probably not worried much about consistency and completeness of the mathematical framework they are working in, they just assume that it is consistent and complete.  It was quite a shock when Kurt Gรถdel proved in 1931 that various models of arithmetic are incomplete, i.e. there are true statements about natural numbers that cannot be proven with these axiomatic theories.

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