Posted by: Alexandre Borovik | December 31, 2020

Unreasonable ineffectiveness of mathematics in biology

This post appeared first 2006 in a now-defunct blog, reposted in 2011 and in 2018.  I repost it again as a source of a quote from Israel Gelfand which appeared in Wikipedia.

Israel Gelfand:

Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. He meant physics, of course. There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.

I heard that from Israel Gelfand in a private conversations (and more than once. Because of Gelfand’s peculiar style of work (see  Conversations with A. S. Golubitski — it is about him), I was often present during his conversations with his biologist co-authors about structure of proteins. The quote was included by me in an earlier version of the book ‘Mathematics Under the Microscope‘, but is not present in the published version (originally I planned to insert it in Section 11.6).

Besides being one of the most influential mathematicians (and mathematical physicists) of 20th century, Gelfand also had 50 years of experience of research in molecular biology and biomathematics, and his remark deserves some attention.

Indeed biology, and especially molecular biology, is not a natural science in the same sense as physics. Indeed, it does not study the relatively simple laws of the world. Instead, it has to deal with molecular algorithms (such as, say, the transcription of RNA and synthesis of proteins which ensures the correct spatial shape of the protein molecule) which were developed in the course of evolution as a way of adapting living organisms to the changing world. If they solve a particular problem in an optimal way, they should allow some external description in terms of the structure of the problem. Indeed, this is the principal paradigm of physics; it is an experimental fact that the behavior of physical systems is governed by various minimality / maximality principles, and the optimal points have, as a rule, especially nice mathematical properties.

But why should a biological system to be globally optimal? Evolution is blind, and there is no reason to assume that the optimal solution is reached. The implemented solution could be one of myriads of local optima, sufficiently good to ensure survival. Lucky strikes could be so rare that the huge search space and billions of years of evolution produced just one survivable algorithm, which, as a result, dominates the living world, and is perceived by us as something special. But it might happen that there is absolutely no external characterization which allows us to distinguish it from other possible solutions, and that its evolutionary phylogeny is its only explanation.

However, I am not a philosopher and cannot claim that my solution of Gelfand’s paradox is correct. What I claim is that philosophers ask wrong questions. The classical conundrum of relations between mathematics and physical world starts to look very different — and much more exciting — as soon as we include biology into consideration. I will try to continue this discussion.

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