George Green and mathematics
by Rev. Dr. John Polkinghorne FRS
Mathematical talent has a way of asserting itself even in circumstances which are unfavourable. The life of George Green is a striking example of this phenomenon. Anyone with any pretensions to being a mathematician is well aware of the importance of Greens functions and many of us have made a living out of their exploitation. Greens theorem, which relates volume effects to surface effects, is a beautiful result, fundamental to gravitational and electromagnetic theories. This astonishing fertility came from someone who had fifteen months formal schooling and whose adolescent years were spent working in the bakery and the mill. It is a most astounding story of genius and achievement under difficulty.
But the work of George Green illustrates an even more remarkable principle than that of the irresistible assertion of mathematical talent. In his celebrated essay of 1828 Green wrote that
In the somewhat oratund style of the day he was pointing to the phenomenon which a present day theoretical physicist of great distinction, Eugene Wigner, would describe in crisper terms as "the unreasonable effectiveness of mathematics". In other and plainer words, there is some deep relationship between the structures of mathematics and the structures of the physical world, so that mathematics and science enjoy a fruitful interplay between themselves, of mutual enhancement and understanding. Mathematics proves to be the key to unlock the secrets of the physical world. Successful fundamental physical theories are always expressed in terms of beautiful mathematics.
I suppose the greatest theoretical physicist I have known was Paul Dirac. He spent his life in the search for beautiful equations, in the expectation - fulfilled in his own researches with sensational success - that they would be the clue to understanding the pattern and structure of nature. In fact, Dirac once said that it was more important to have beauty in your equations than to have them fit experiment! By that he did not mean, of course, that empirical adequacy was unimportant. But if your equations didnt seem to fit the facts, maybe that was because you hadnt solved them properly, or maybe the experiments were wrong (it has been known to happen). At least there was a possibility of putting it right. But if your equations were ugly then there was no hope. Nature never led to ugly mathematics. This is a view instinctively held, I think, by all who work in fundamental physics.
There is an interesting example of the power of mathematical beauty in the work of that "apostolic succession" from Green through Kelvin to Clerk Maxwell, which resulted in elucidating the structure of electromagnetism. Maxwells great discovery of the displacement current arose from the completion of the equations of electromagnetic theory in a way which was compact and elegant - in a word, mathematically satisfying and beautiful.
We are so used to using mathematics to help physics in this way that most of the time we simply take it for granted. Yet when you stop to think about it, something very remarkable is going on. After all, what is mathematics? It arises from the free explorations of the human mind. Yet some of the most beautiful patterns that our mathematical friends dream up in their studies are found actually to occur in the structure of the physical world around us. Why do the reason within (mathematics) and the reason without (the physical world) fit together so perfectly? What gives the human mind this strange power to penetrate the mysteries of the universe? It surely cant just be evolutionary biology. Of course, our everyday thinking must match the everyday world in which we have to survive, but the fact that our ancestors had to dodge sabre-toothed tigers does not seem at all to require George Greens ability to discover Greens functions, or Paul Diracs ability to discover the relativistic equation of the electron. Intellectually fascinating as these great discoveries are, they seem to have no obvious survival value. Scientists are motivated by the thirst for understanding, and surely we should seek an explanation of this unreasonable effectiveness of mathematics, and not just take it for granted.
The rational beauty of the universe (which makes physics such a satisfying endeavour) and the rational transparency of the universe to our inquiry (which makes physics a possible endeavour) speak to me of a world shot through with signs of Mind. I believe that the universe is mathematically beautiful and accessible to us in its structure, precisely because it is a creation. Writers of semi-popular books on science seem to like talking about the Mind of God. Sometimes, as in the celebrated passage at the end of Stephen Hawkings Brief History of Time, such writing seems to be a kind of rhetorical flourish. But I believe that these writers speak truer than perhaps they know. Science is possible, George Greens great achievements were possible, because the universe is a creation, endowed by its Creator with a wonderful order. Our minds have access to that order through mathematics because we ourselves - as that deep theological book Genesis tells us - are made "in the image of God".
I did not start with a text, but I will permit myself to end with one. It is taken from the Psalms, and Maxwell had it inscribed over the archway of the original Cavendish Laboratory. I am glad to say that when that famous laboratory moved to modern buildings on the outskirts of Cambridge the text moved with it, and it adorns the entrance of the new Cavendish as it did that of the old.
Further information about Rev. Dr. John Polkinghorne and his contributions to the debate on science and theology can be found in the essay on God and Science.
George Green web pages