In his book Why Beliefs Matter: Reflections on the Nature of science, noted British mathematician E. Brian Davies surveys the sweeping landscape of modern philosophy of science and mathematics, with considerable skill and numerous thoughtful insights. Its closest analogue would be John Barrow’s 1992 book Pi in the Sky: Counting, Thinking and Being.
Davies is certainly qualified to write this book. He has published works in spectral theory, operator theory, quantum mechanics, and the philosophy of science. He served as the President of the London Mathematical Society from 2008-2009.
Some of Davies’ most intriguing comments relate to the nature of mathematics, which constitutes the whole of Chapter 3 and portions of several other chapters. After surveying the various philosophical schools of mathematics (constructivism, formalism, Godel’s result, etc), Davies focuses on Platonism, which he defines as the notion that “theorems are supposed to be true statements about timeless entities, and to be true whether or not they have ever been or will ever be formulated by human beings, and whether or not they have proofs.”
Davies points out that Platonism is implicitly assumed by many mathematicians. Famed mathematician Paul Erdos frequently referred to “God’s book” of the best possible proofs of all theorems. Roger Penrose argued that individual mathematicians can communicate because “each one [has] a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly.” Similarly, French mathematician Alain Connes declared
I maintain that mathematics has an object that is just as real as that of the sciences I mentioned above, but this object is not material, and it is located in neither space nor time. Nevertheless the object has an existence that is every bit as solid as external reality, and mathematicians bump up against it in somewhat the same way as one bumps into a material object in external reality.
But Davies is not so sure. He notes that “To assert that Platonism is obviously correct, and that to deny it is simply ridiculous, is to commit oneself to a quasi-religious world-view.” Davies adds that Platonism “diminishes the status of numerical analysis” and “has delayed the development of topics that focus on quantitative results rather than mere existence.” Producing efficient algorithms to discover mathematical objects might not be as glamorous as proving the existence of solutions of new problems, “but it is arguably just as important and certainly just as hard.” Further, Davies argues that Platonism “depersonalizes mathematics” and “diminishes the respect that we should have for the astonishing creativity of the most able mathematicians.”
Davies summarizes his discussion of Platonism in these terms:
Platonism is attractive to many pure mathematicians, but unsupported intuition is a bad basis for deciding about truth. … Platonism just replaces one mystery by another. Instead of wondering about how we are able to understand mathematics, one has to wonder how the Platonic realm can exert any influence on the physical world.
So what exactly is mathematics? After extended discussion, Davies in the end can offer no firm answers. Why, for example, do we believe in the consistency of mathematics (since we know from Godel’s theorem that this is an unattainable objective)? Davies answers in these pragmatic terms:
The answer is simply intuition. Over two thousand years of formal mathematics have revealed some important misconceptions, but these have been rectified and we feel confident that we can patch up any further problems that we might encounter in the future. There is no way of proving that this confidence is justified, but we have no choice but to rely provisionally on hard-won insights in every other sphere of investigation, and mathematics is no different.
He concludes:
The key to mathematical progress has been the possibility of recording our successes so that they may be transmitted to our descendants. After walking for over two thousand years down this one-way road, we have progressed a considerable distance. Every generation starts further down the road, and builds a new section using tools that are themselves becoming more efficient. We do not know where it will lead, but it is clear that we are not yet near the end.
Some additional observations on Davies’ book may be found in a thoughtful review by Gerald B. Folland in the April 2012 issue of the Notices of the American Mathematical Society.
A highly rewarding (and more technical) book to read in conjunction with Davies’ book is Jeremy Gray’s remarkable Plato’s Ghost: The Modernist Transformation of Mathematics. As described at Princeton University Press,
Plato’s Ghost evokes Yeats’s lament that any claim to worldly perfection inevitably is proven wrong by the philosopher’s ghost; Gray demonstrates how modernist mathematicians believed they had advanced further than anyone before them, only to make more profound mistakes. He tells for the first time the story of these ambitious and brilliant mathematicians, including Richard Dedekind, Henri Lebesgue, Henri Poincaré, and many others. He describes the lively debates surrounding novel objects, definitions, and proofs in mathematics arising from the use of naïve set theory and the revived axiomatic method–debates that spilled over into contemporary arguments in philosophy and the sciences and drove an upsurge of popular writing on mathematics. And he looks at mathematics after World War I, including the foundational crisis and mathematical Platonism.
Both books resonate with Math Drudge’s commitment to thoughtful experimental mathematics.