Monday, 26 October 2009

The Unreasonable Effectiveness of Mathematics in Science and Engineering

Why is mathematics - a product of the mind - so outstandingly successful at predicting the behavior of physical systems?

Einstein is said to have remarked that "The most incomprehensible thing about the universe is that it is comprehensible." This observation was elaborated by Eugene Wigner in his famous paper in Pure Mathematics (Volume 13, Number 1, February 1960) entitled 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences'

The theme was further developed in 'The Unreasonable Effectiveness of Mathematics' by R. W. Hamming in The American Mathematical Monthly (Volume 87, Number 2, 1980), which considered the predictive, as well as descriptive powers, of mathematics in relation to engineering.

Two surprising conclusions appear from these papers:

(1) Although it is a product of the human mind, mathematics is also involved in some strange metaphysical way at the deepest levels of physical existence. To quote Wigner:

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."

(2) There is no Darwinian explanation for the presence of mathematical abilities within the mind. The ability to understand physics could not have arisen by evolution. Although our bodies may well be the product of random mutation and selection al the way from amoeba to man, our minds have some 'unevolved' dimension. To quote Hamming:

"But it is hard for me to see how simple Darwinian survival of the fittest would select for the ability to do the long chains that mathematics and science seem to require".

"If you pick 4,000 years for the age of science, generally, then you get an upper bound of 200 generations. Considering the effects of evolution we are looking for via selection of small chance variations, it does not seem to me that evolution can explain more than a small part of the unreasonable effectiveness of mathematics."

Or Wigner again:

"Certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess."

(Note - this should not be taken as evidence for Genesis, Special Creation, Young Earth, Flat Earth and all the rest of the creationist mumbo-jumbo. It does, however point to aspects of the mind which have not arisen out of the purely physico-chemical processes of evolution. (The process of evolution can in fact be regarded as an algorithm. Evolution produces complex structures out of the endless algorithmic loop of replication, random error in copying, and resources which are insufficient to allow all replicated copies to survive to replicate themselves in turn. Evolution is the algorithm, and the DNA codes of living organisms are the corresponding datastructures which are modified)

So we are left with something of a mystery. According to the physicalist worldview, the mind (including mathematicians' minds) is an epiphenomenon of matter which has evolved solely to ensure the survival of the selfish genes which code for it. Why should this 'top-level' phenomenon have such intimate access to the 'bottom level' phenomena such as quantum physics? After all, the two levels are supposedly separated by less well-understood (in some cases) explanatory layers such as evolutionary psychology, neurology, cell biology, genetics, molecular biology, and chemistry.

- Sean Robsville


Rational Buddhism 

The Church-Turing-Deutsch Principle and Buddhist Philosophy

Buddhism versus Materialism



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