Abstract
Why does mathematics work so well in describing some parts of the natural world?This question is profound, ancient, far-reaching and compelling. It seems to become more so in each respect as time goes by, at least for some people. For them it is an intellectual catalyst, serving as stimulus for further thought and questions at many levels without ever being significantly resolved itself. It was put in a particularly evocative form by the physicist Eugene Wigner as the title of a lecture in 1959 in New York: ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’. He was well-qualified for the task having discovered in the 1930s that the well-established mathematical theory of groups was just what he needed to make important progress in atomic physics. He received a share of the Nobel Prize in Physics in 1963 ‘for his contributions to the theory of the atomic nucleus and the elementary par- ticles, particularly through the discovery and application of fundamental symmetry principles’. His 1959 lecture was published in 1960 (presumably with a minimum of editorial attention, hence its rather informal style). His paper and the themes around it, being re-visited fifty years on, form the main subject of this issue of ISR.
Wigner's central thesis was that mathematical concepts are often defined and developed in one context and then, perhaps much later, turn out to have a completely unanticipated but highly effective application in another context. Instead of citing the example of group theory and particle physics he mentions the way in which complex Hilbert space (developed as a natural part of functional analysis around 1900) turned out to be invaluable in the formulation of quantum mechanics a few decades later. In reference to such unexpected application he says, ‘It is difficult to avoid the impression that a miracle confronts us here’. Under the term ‘effectiveness’ he includes the fact that a mathematical formulation ‘leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena’, with accuracy ‘beyond all reasonable expectations’. He describes the usefulness of mathematics in the sciences as ‘bordering on the mysterious’ and declares that ‘there is no rational explanation for it’.
Perhaps it is the juxtaposition of ‘unreasonable’ and ‘effectiveness’ that stirs the imagination, but it has been a memorable and provocative title. It has proved to be not only a much-cited paper but one whose theme has stimulated sustained commentary from scientists, philosophers and historians. The core question, however, remains a puzzle that refuses to yield, or even be modified, in spite of these attentions.
There was a rejoinder from the computer scientist Richard Hamming in 1980, with almost the same title, in which he attempted to give some solution. He concluded, however, that ‘… all of the explanations I have given when added together simply are not enough to explain what I set out to account for ’. The language Wigner used — the language of ‘mystery’, ‘miracle’ and ‘gift’ — had an almost religious tenor, which seems to divide readers into two groups: those who are sympathetic to mystery (let's call them the ‘transcendentalists’), and those who prefer to reduce any sense of mystery in their lives to a minimum (let's call them the ‘positivists’). According to Aristotle, the Pythagoreans believed that ‘all is number’, and he refers to their association of number with music and astronomy. Whatever they meant by that (and we know very little) we can perhaps make them the first of the transcendentalists. Another must be Galileo, with his oft-quoted statement that ‘the universe … [thought of as a book] is written in the language of mathematics’. Note that he does not refer to a platonic realm separate from nature but instead writes simply of mathematics as the language necessary for understanding nature. But the idea of the creator of the universe as a mathematician definitely has its appeal, including a neat though perhaps too easy solution to Wigner's problem. Mario Livio's Is God a Mathematician? (2009) is a lively and highly accessible book-length commentary on Wigner's paper clearly on the side of the transcendentalists. There are many other distinguished scientists among them, including James Jeans, Steven Weinberg, Freeman Dyson and Roger Penrose.
Among the positivists is an equally distinguished company (classified here, perhaps unfairly, from their desire to show Wigner's claim was, ‘less unrea- sonable than he supposed’). This company includes the philosophers Penelo- pe Maddy (2009) and Steven French (2000), and (to some extent) the histori- ans Ivor Grattan–Guinness (2008) and two contributors to thi
Re-visiting Wigner's theme in the second decade of the twenty-first century needs little justification. Science has changed dramatically since he wrote — not least through computer technology. We have seen the emergence of computational science, in which modelling and simulation have ‘introduced a distinctively new, even revolutionary, set of methods into science’ (Humphreys 2004, 57). Stephen Wolfram (2002) has made claims for a new kind of science based on cellular automata. Complexity science has come into being. There are new applications of mathematics in areas such as biology, psychology, finance and economics. The very means of application to many subjects through statistics, computer modelling, visualisation and game theory have transformed many problem areas. Mathematics itself has changed with experimental and empirical ‘turns’ in practice and in philosophy, and the wide influence of computer-based methods, algorithmic thinking and probabilistic methods. Non-linear dynamical systems, chaos theory, fractal geometry are all new subjects since 1960 but now rather well-established. Powerful computer packages — Mathematica and MatLab for example — are commonplace tools in the environment of science students and research teams. The use of simulation packages and virtual worlds, data mining techniques and even collaborative and distributed working are ubiquitous. Such changes are obvious and they invite scrutiny from Wigner's perspective for examples of both effectiveness and unreasonableness.
Less obvious, but surely also significant, are the changes since Wigner's lecture in the way we think about knowledge. The 1960s saw the beginning of an extraordinary flowering of fruitful study and interest in the cultural context of disciplines — especially in the sciences and mathematics, which were not accustomed to this treatment. Kuhn and Lakatos are obvious examples of highly influential authors from that period, but there were dozens of others. In the succeeding decades national and international societies, journals, university departments and degrees all sprang up in areas such as the history and philosophy of science. Science studies grew to emphasise the social context of scientific disciplines. Lakoff and Nüñez, Grosholz, Byers and others have taken account of the cognitive and psychological aspects of the emergence and development of mathematics. The importance of examining the entire context and practices of the sciences and of mathematics in order to understand properly their development is now widely respected even if it is far from being universally welcomed. In short, compared with fifty years ago, there is a broad sympathy in the academic world with an interdisciplinary approach to knowledge. This journal itself testifies to the interest and success of such an outlook.
Thus it is that we read today with some disappointment the sparse and seemingly superficial references Wigner makes to the history and nature (or philosophy) of mathematics, and the almost complete absence of account of the psychological aspects of mathematics and its applications. It has therefore seemed a first priority to address these subjects in relation to his paper, and only then in a later issue of ISR to address some of the developments in scientific and technical topics, such as computer modelling and simulation, uses of statistics, cellular automata and game theory, as well as the wider applications of such methods in the social sciences. Our contributors in this issue are distinguished historians and philosophers of mathematics or science who, in good interdisciplinary style, have been prepared to ‘transgress’ into each others’ domains with impunity.
Jeremy Gray's contribution is a hard, thoughtful and critical assessment sharply focussed on Wigner's text. His concise opening summary concludes with a statement of the five core problems posed by Wigner (somewhat buried from sight in his 1960 exposition). Why are there laws of physics? And why are they knowable by us? Why are they expressible in mathematics?
And especially in mathematics not created with this in mind? Why, finally, does mathematics ‘improve’ the laws in terms, for example, of such remark- able degrees of accuracy? After distinguishing questions he calls deep from those he calls unintelligible, Gray proceeds to analyse Wigner's problems in some detail, drawing on his expertise in both the history of mathematics and the role of mathematics in twentieth century physics. He does us the excellent historical service — for the sake of this current issue — of arguing that Wigner's own treatment of these questions, especially in the case of the first two, does little to rescue them from being unintelligible. At the same time Gray betrays in his own remarks a sympathy for the view that all these problems are in fact deep questions and worthy of further investigation.
In a nicely constructed paper Jesper Lützen gives us a detailed and richly multi-disciplinary account of why Wigner might have been less surprised at the effectiveness of mathematics if he had not held a ‘dogmatic adherence to the formalist philosophy’ and if he had paid more serious attention to the historical influence of physics for the development of mathematics. Both charges are perhaps unexpected for an eminent physicist but perhaps less so for one (as in Wigner's case) who had worked closely with Hilbert. Lützen's discussion of formalism and related philosophies of mathematics as well as his interesting examples of the strong, sustained physical context of the development of mathematics through the eighteenth and nineteenth centuries make for a persuasive case while not removing all elements of surprise at the success of mathematics.
At first glance we might wonder if the paper by Patrick Suppes had strayed from its proper home in a journal of Greek philosophy, for it contains a substantial amount of quotation from Aristotle's work On the Soul. But closer scrutiny will soon show how relevant in fact this work is, given that we are prepared, at least provisionally, to buy into his series of transformations summarised in the initial abstract. First, he suggests, there has been an evolutionary development of the wiring of the brain so as to produce, out of sensory nerve signals, images isomorphic (in some sense) to whatever it is in the world that triggered those signals. Then that isomorphism can be related to what Aristotle called ‘form’, and in particular the way form operates in the context of perception and thinking. Next, visual perception is closely involved in the emergence of intuitive geometry, and finally, when the more systematic Euclidean geometry was applied (by Ptolemy) to astronomy this was a major early piece of mathematical physics and an illustration of the effectiveness of mathematics. Of course, each of these transformations, it could be argued, is highly speculative: but they are also each fascinating, have considerable plausibility and could become a series of research topics in themselves. On the Soul is a work of psychology (or philosophy of mind depending upon your perspective) and draws parallels between perception and thinking while maintaining form as the key concept in both. So Suppes offers us here a very broad historical perspective, a philosophical challenge in seeking to relate Aristotle's notion of form to modern structural isomorphism, and the diagnosis that at least some of Wigner's surprise was due to a lack of attention to the psychological aspects of mathematical thought.
Alan Baker draws attention to the fact that none of the philosophies of mathematics familiar from the first half of the twentieth century (logicism, intuitionism and formalism) paid significant attention to the applicability of mathematics to the world. The decades from 1960 have seen a reversal of that trend in work on the philosophy of mathematics. Baker brings us right up to date with reference to the work of several of the major players, such as van Fraassen, Steiner, Colyvan, Field and Leng. His paper is both a useful commentary on some of these contributions and a brief statement of his own views in combining the important indispensability argument with the explanatory role of mathematics in science.
Each of the contributions here offer an important position in the context of the issues concerned, but they also each offer a programme for future work in developing and resolving the questions that Wigner raised.
The discussion above of the reactions to Wigner's paper deliberately draws attention to the polarisation it produced and therefore is crudely binary. The literature over the last fifty years is substantial and the many reactions, including those of our present contributors, are nuanced, range across the disciplines and cannot be neatly categorised. One reason for the complexity and depth of the issues around the usefulness of mathematics is what I referred to above as the flowering in the 1960s of interest in, and the intellec- tual shaping of, the cultural context of technical disciplines — especially in the history and the philosophy of science. Another important strand of work on the usefulness of mathematics is in modelling and particularly in the way computer technology enhances and extends the modelling that humans inevitably do in trying to understand the world around them. Influential works on the role of modelling in science began to appear at this time such as those by Mary Hesse and Max Black. But it was the widespread use of powerful computers from the 1980s that made a step-change in the way mathematics could be applied in science. Thus Djorgovski (2005) could say, ‘applied computer science is now playing the role which mathematics did from the seventeenth through the twentieth centuries …’. The significance of modelling, and the multitude of uses of computer technology are large subjects calling out for consideration in relation to Wigner's theme. They must be taken up, along with accounts of the current state of the ‘unreasonable effectiveness of mathematics’ in other technical and scientific fields, at a later date.