Thanks to Ars Mathematica, I found an interesting article (.pdf) demonstrating some recent theorems in pure mathematics that emerged primarily from doing research arising from biology (construction and inference of phylogenetic trees in particular). Unfortunately, I couldn’t really work through the math enough to really understand any of the theorems.

But really, it doesn’t seem terribly surprising that interesting theorems arise when one considers sufficiently interesting applications of mathematics. These theorems don’t appear to be like the one I mentioned earlier, where the result could be very strongly supported by scientific argument – they’re the more standard results of applied mathematics that mathematicians came up with and turned out to be applicable. But the work was still motivated by the science.

I don’t see any particularly strong argument here that biology will be more productive in this way than physics has been, but it seems to me that any field that applies sufficiently interesting mathematics will lead to the development of new mathematics that eventually turns out to be interesting for purely mathematical reasons as well. Philosophy led to mathematical logic, and the Gödel results in particular, not to mention the results of social choice theory (which I suppose could equally be attributed to economics or political science or a variety of other social scientific enterprises). I don’t know about much mathematics that can be attributed to chemistry, but I did see a very interesting lecture at last year’s Canada/USA Mathcamp about levels of topological classification relevant for describing chirality of molecules (I don’t know whether this has spurred interesting new areas of topology).

I suppose the main problem is just that the paradigm of applied mathematics that many people have (or at least I do) is differential equations. Once we see that so much of mathematics is applied, or can be applied, and that the applications very often lead to interesting new methods of development that occasionally lead to substantial insights within mathematics itself, it should be clear that whatever the status of Hardy’s claims that the best mathematics is essentially pure, there’s no reason for mathematics to cut itself off from the other sciences, or even to seek to remain pure at all times.

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Robert(19:09:53) :George PĆ³lya wrote an enumerative combinatorics paper on chemical compounds where he generalized Burnside’s lemma (http://en.wikipedia.org/wiki/Polya_enumeration_theorem). This eventually led to Joyal’s theory of species of structures. I’m not sure it counts though, because Polya didn’t come up with it to be applied to chemistry, it was just eminently applicable there and it made a good example.