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“The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”

Steve Russ

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 1930’s 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 particles, 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.

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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 that meant (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

unreasonable than he supposed”). This company includes the philosophers

Penelope Maddy (2009) and Steven French (2000), and (to some extent) the

historians Ivor Grattan-Guinness (2008) and two contributors to this issue,

Jeremy Gray and Jesper Lützen.

Re-visiting Wigner’s theme in the second decade of the 21st 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 has made claims for a new kind of science based on cellular

automata (2002). 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

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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 1960’s 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 Núñez, 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 topics in relation to his paper, and only

then in a later issue of ISR to address the numerous technical and disciplinary

issues, including topics such as computer modelling, use of statistics, automata,

game theory and wider applications to 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 remarkable 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

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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 20th 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

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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 draw

attention to the polarisation it produced and therefore was 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 in the

flowering in the 1960’s of interest in, and the intellectual 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 was

in modelling and particularly in the way computer technology enhances and

extends the modelling that humans inevitably do in trying to understanding 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.

REFERENCES

Djorgovski, George. 2005. Virtual Astronomy, Information Technology, and the New Scientific

Methodology. 7th Int. Workshop Computer Architecture for Machine Perception, 125 – 132.

French, Steven. 2000. The Reasonable Effectiveness of Mathematics: Partial Structures and the

Application of Group Theory to Physics. Synthese 125: 103-120.

Hamming, R.W. 1980. The Unreasonable Effectiveness of Mathematics. American Mathematical

Monthly 87(2) : 81-90.

Humphreys, Paul. 2004. Extending Ourselves: Computational Science, Empiricism, and Scientific

Method. Oxford University Press