Tuesday, November 28, 2006

Unreasonable (in)effectiveness of ...

I am provoked by the post Is the world mathematical? in Philosophical bits by Phil Thrift.
Phil Thrift wrote:

Is the world mathematical?

One frequently encounters the question Why is the world (or nature) mathematical? (What is generally intended is Why does nature obey mathematical laws?, or Why does mathematics describe nature so well?) The jumping-off place for this type of question is the idea that mathematics has been shown to be effective in describing some important aspects of the world, particularly in the natural sciences. (The Unreasonable Effectiveness of Mathematics in the Natural Sciences, by Eugene Wigner)

But Why is the world mathematical? to me begs the question: IS the world mathematical? [...]

So the confusion about Why is mathematics effective? may lie in the confusion that follows when one makes a distinction or separation between the world and mathematics. The stuff of the world IS mathematics.
I cannot help but quote 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.
Besides being one of the most influential mathematicians (and mathematical physicists) of 20th century, Gelfand also has 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 cannibalised a fragment from my book]


Anonymous said...

I've always been troubled by Wigner's argument because we use mathematics to describe the phenomena being studied in physics. So, if mathematics was NOT effective in describing that phenomena, how would we ever know? What other way do we have to describe the phenomena? His statement is like saying that singing is unreasonably effective in communicating vocal music.

29/11/06 5:43 PM  
Anonymous said...

In so far as mathematics describes physical phemonena well, it is because some of the best mathematical minds for the last 350 years have been working on the description. If an equivalent amount of brain power were applied to describing biologicalphenomena with mathematics, I am sure we would see similar progress.

It is worthy of note that the sort of mathematics which could be best suited to biological phenomena -- ie, the mathematics of qualitative structures and relationships aka category theory -- was only invented in the last 60 years, some three centuries after the Calculus. Perhaps it would be fair to wait another 300 years before concluding that biology is not well represented mathematically when compared with physics.

3/12/06 6:07 PM  

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