Wednesday, January 05, 2011

Why math is unique

The brilliant mathematician Timothy Gowers wrote an elementary book on Math,
and makes this point:
For example, as I was re-reading an interesting little maths primer this week by Timoth Gowers (A Fields medal winner and maths prof at Cambridge) I found him briefly discussing axiomatics and the role of Russell and Whitehads' "Principia Mathematica". The importance of this text, for Gowers, is that it establishes the axiomatic hermeneutic (my phrase not his btw) which "means that any dispute about the validity of a mathematical proof can always be resolved".
That is correct. Furthermore, as Gowers goes on to explain:
Nevertheless, the fact that disputes can in principle be resolved does make mathematics unique. There is no mathematicd equivalent of astronomers who still believe in the steady-state theory of the universe, or of biologists who hold, with great conviction, very different views about how much is explained by natural selection, or of philosophers who dis- agree fundamentally about the relationship between consciousness and the physical world, or of economists who follow opposing schools of thought such as monetarism and neo-Keynesianism. [p.49]
That is correct, and it is a consequence of logicism. Math is the only subject that resolves all of its disputes.

You might think that the The hard sciences would be able to resolve disputes, but they are not. Physics has a dispute over the merits if string theory, and there is no hope of any resolution.

Update: I see that the Wikipedia article on Mathematics has some nonsense about philosphers and mathematicians deciding that math must be like a science because Goedel proved that it was not reducible to logic. As explained below, they are wrong. You can find a more accurate description of the logical nature of math in the article on Axiomatic set theory. I'd like to see those philosophers and mathematicians named.

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