Tuesday, May 20, 2014

On beauty and truth in science.

Philip Ball writes a thought-provoking article in Aeon with the thesis that the kind of beauty that scientists describe does not necessarily match the aesthetic notions of art, and is not even consistent among scientists.

It was hard for me to get beyond the casual conflating of beauty in mathematics (the four-color theorem, the proof of Fermat's theorem, and proofs in general) and beauty in scientific theories (relativity, evolution, and so on). But if one goes beyond the artificial duality constructed by the author, the idea of beauty as a driver in science (and mathematics) is a rich one to explore.
A particular example: for a long time (and even codified in books) it was taught that there were five natural classes of approximation hardness: PTAS, constant factor-hard, log-hard, label-cover (superlogarithmic) hard, and near-linear hard. There were even canonical members of each class. 

Of course, this nice classification no longer exists. There are even problems that are $\log^* n$-hard to approximate, and can also be approximated to that factor. And to be fair, I'm not sure how strong the belief was to begin with. 

But it was such a beautiful idea. 

At least in mathematics, the search for the beautiful result can be quite fruitful. It spurs us on to find better, simpler proofs, or even new ideas that connect many different proofs together. That notion of connection doesn't appear to be captured in the article above: that beauty can arise from the way a concept ties disparate areas together. 
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