Pure mathematics is the subject "in which we do not know what we're talking about and don't care". Thus Einstein began a talk at Cambridge University on relativity theory in May 1932, soon after Kurt Gödel had shaken the very foundations of mathematical logic. Despite his apparent scepticism, Einstein's approach became increasingly mathematical and distant from experiment.
In common with Einstein, almost all physicists believe the laws of their subject are best expressed in mathematical form. Many mathematicians welcome this philosophy, but the leading mathematician E. Brian Davies of King's College London is a refreshingly dissident voice. In Science in the Looking Glass , he takes aim at those who worship mathematics and believe unthinkingly that it is necessarily the primary language of science.
For Davies, the great success of the mathematical theories of physics - such as quantum theory and general relativity - should lead us to believe that they are true or that nature is governed by mathematics. He is contemptuous of what he describes as the "feebleness" of mathematical Platonism as a guiding philosophy for scientists and especially of notions such as that favourite quotation of many a theoretical physicist: "God is a mathematician."
Davies also argues that pure mathematics is not quite as pure as it's cracked up to be. For example, he argues on psychological and historical grounds that mathematics is a human creation and that extremely large numbers - say 101000 - cannot be counted "in principle" in any meaningful sense. Moreover, he states that "it is not necessary to believe this in order to be interested in pure mathematics".
After Davies has displayed his mistrust of much of the mathematics most scientists take for granted, he moves on to the sciences.
I was uneasy about his approach from the outset, when he remarks that the superseding of Newton's well-established theory of gravity by Einstein's "poses a severe challenge to any theory of scientific knowledge". Why is this a challenge? In science, all theories are provisional and are at the mercy of experiment. Certainly, physical scientists are well used to theories being succeeded by ones that explain more in terms of fewer principles, even if one does not need to use the more general theories to solve most problems.
Every physics student learns that simple quantum theory and relativity theory are incompatible, that both work extremely well in their respective domains and that ultimately we need an overarching theory to encompass them (for example, string theory or M-theory), even though we shall probably not need to use the superior theory for most applications. That is hardly a serious problem for epistemologists and, if it were, it would be of little concern to scientists.
One of the most impressive aspects of Davies' treatment is its breadth - he covers both the physical and life sciences and touches on philosophy. He is at pains to underline the success of the essentially non-mathematical theory of plate tectonics (although he is not so keen to stress that its non-mathematical nature makes it much less falsifiable than, say, quantum theory).
In a subsequent chapter, he asks: "Is evolution a theory?". As far as I can see, he does not fully engage with this question by examining how researchers use Darwin's ideas, but instead chooses to review the opinions of a few leading popularisers. For this reason, his account has a derivative feel to me.
Likewise, his opposition to what he sees as reductionist science is scarcely an original perspective. He makes all the usual arguments against there being some sort of hierarchy of theories, from the subnuclear to the macroscopic, with explanations in terms of the smallest being the most fundamental. Yet he is unable to take on the big guns of the physicist Steven Weinberg who has argued powerfully that, when all is said and done, the most fundamental description of matter is in terms of the smallest entities and their interactions, even if their laws do not immediately enable an understanding of larger-scale phenomena.
In view of the amount of good popular science writing that Davies has read, it is surprising that Science in the Looking Glass appears so confused about its audience. It is too technical to appeal to lay readers, but not technical enough to appeal to the experts. However, those who read the book will find much to set them thinking, especially about the blind worship of mathematics that is often taken for granted in popular science books. The arguments he makes here are well worth reading as an antidote to lazy thinking.
Davies reminds us that Einstein memorably remarked in 1921 that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality". In spite of Davies' eloquence, I remain unconvinced that he has anything significant to add to that.
Graham Farmelo is senior research fellow, Science Museum.
Science in the Looking Glass: What Do Scientists Really Know?
Author - E. Brian Davies
Publisher - Oxford University Press
Pages - 295
Price - £25.00
ISBN - 0 19 852543 5