Brain Advance Access originally published online on March 4, 2008
Brain 2008 131(4):1156-1160; doi:10.1093/brain/awn038
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Book Review |
Thoughts of a Mathematician
How do we think? What is a thought? Are there many different kinds of thinking? How do our thoughts relate to the outside world? Such questions have been pondered over by philosophers for thousands of years. There are no real answers but the questions get clarified and we learn something in the process. The progress of modern neuroscience is now changing the picture and is beginning to shed light on this whole area. It is not unrealistic to expect that, by the end of the 21st century, our understanding of the human brain will mean that many of the old philosophical questions will simply have disappeared, rather in the way that no-one asks anymore ‘What is Life?’ Instead we have a new subject called molecular biology, encompassing DNA and the genetic code.But meanwhile, as we head towards this goal, it is useful to formulate more precise questions concerning the operations of the brain which lend themselves to experimental investigation. The variety of scanning techniques that are now available enables us to address certain questions relating psychology to neurophysiology. In this way we can begin to build up some understanding of the physiological basis of mental processes.
Already much has been discovered about the role of different regions of the brain, particularly in relation to vision. In fact vision is the great success story of evolution and its mechanisms occupy a substantial part of the human brain. So much so that it must surely have been exploited to underpin the thinking process. But the degree to which vision is involved in thinking is something that should lend itself to scientific analysis and the result is likely to be much more complex than just answering questions like: do you think in words or in pictures?
A basic scientific approach to a complex phenomenon is to study it in its simplest or purest form, where extraneous factors are removed. So what is ‘pure thought’? There is a strong case for arguing that mathematics is the purest form of thought, where the external world is almost banished from the scene. Most of the time our thoughts are intertwined with our sensory perceptions, so that it is difficult to disentangle thought from other activities in the brain. Mathematics appears to offer thought unencumbered in this way and this is particularly true of ‘pure mathematics’, as opposed to mathematics applied to the sciences.
So the question ‘how do mathematicians think?’ is one that is of broad scientific interest, although it has a particular appeal to us mathematicians. We can indulge in introspection under the no doubt mistaken impression that we are ideal guinea pigs. Certainly the three books under review indicate that the topic is one of consuming interest, and that mathematicians may have something special to contribute.
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It is important to recognize that mathematics exists at various levels, just as the English language ranges from that of the primary school to the higher flights of poetry and the plays of Shakespeare. The entire population learns elementary arithmetic but only a small number study the higher mathematics and an even smaller number create new mathematics, being the poets of the profession.
Any analysis of mathematical thought has to focus carefully on some particular level and the answers may not be uniform. Different parts of the brain may well be involved in the various levels of mathematics.
In fact mathematics is a particularly hierarchical subject where each layer is built on the ones before, which is why missing a year in the educational process can have such disastrous consequences. This hierarchical structure goes along with the development of abstraction, the process by which many instances of similar phenomena are grouped together into a single building block for the next level. The passage from arithmetic to algebra whereby definite numbers such as 1, 2, 3 are replaced by unknown or variable symbols like x, y, z is a familiar example of such abstraction.
So in examining ‘mathematical thought’ we have to examine the levels in different ways and the whole process of abstraction, moving from one level to the next, may itself correspond to some organizational aspect of the brain.
Much discussion of mathematical thinking tends to concentrate understandably on the elementary operations of arithmetic. These are familiar to everyone and, because of their simplicity, they can be easily tested experimentally. For example, multiplication tables, which tend to be learnt by heart (or at least used to be) appear to involve the same parts of the brain that are used to learn languages. On the other hand, understanding the significance of numbers is a different skill based elsewhere in the brain.
But to explore mathematical thinking more deeply we have to start by asking a more basic question, what is mathematics? Is it a creation of the human mind or is it ultimately derived from experience? Do we, like Plato, believe in an ideal world where perfect circles and exact straight lines have independent existence? ‘Platonists’ believe that the theorems of mathematics are ‘out there’, just waiting to be discovered, the mathematician being the explorer like Christopher Columbus who finds them. ‘Realists’ on the other hand believe that mathematical ideas emerge from our physical experience, that we search for patterns that lie behind and proceed to ‘invent’ theorems to help understand nature.
Philosophers can be found on both sides of the debate with David Hume arguing strongly for realism. Many mathematicians are intuitively Platonists, they believe in the independent existence of the world of ideas. Immanuel Kant thought deeply about these questions and his ideas evolved over time, but his conclusion was that certain truths like the basic facts of Euclidean geometry are innate to the human mind while others are acquired. Some historians of mathematics argue that the development (or discovery?) of non-Euclidean geometry showed that Kant was wrong and that no truths were innate. But this may be too shallow an interpretation and one which modern neurophysiology may not support. Simple experiments seem to indicate that the abstract notion of magnitude (distinguishing more from less) is hard-wired in the brain and so is innate. Further experiments may show that much more mathematics, both factual and structural, is hard-wired.
Of course this brings us back to an evolutionary standpoint. The human brain has evolved in order for homo sapiens to succeed in the evolutionary struggle. A brain which has built into it mathematical principles extracted by experience from the physical world is clearly at an advantage and this may help to explain how we are born with an innate capacity for mathematics. The arguments between Hume, Kant and other philosophers, originating in pre-Darwinian times, tend to overlook the fact that the human brain is itself part of the natural world and has been shaped by the forces of nature. The discovery of non-Euclidean geometry does not alter the fact that Euclidean geometry is a very good approximation. Primitive man would not have been helped by the knowledge of non-Euclidean geometry. This evolutionary perspective may blur the lines between Platonism and Realism but mysteries still remain.
Galileo famously said that the book of nature is written in the language of mathematics. By the 20th century it had become impossible to describe the laws of fundamental physics, such as quantum mechanics, without extremely sophisticated mathematics. This fact so impressed the Hungarian physicist Eugene Wigner that he referred to ‘the unreasonable effectiveness of mathematics in physics’. Evolution goes some way to explaining this but only if we deal with physics that is apparent at human scales. Remarkably it appears that our mathematical theories of physics work superbly well at both microscopic and macroscopic scales, from the level of subatomic particles to that of galaxies. We did not need to handle black holes in order to evade the tigers of the jungle, so evolution cannot explain everything.
If the ultimate origin and nature of mathematics remains a deep problem we can try the alternative route of trying to describe mathematics. What is its subject matter? Physics, Chemistry, Biology are not hard to describe, though boundaries alter and subjects fuse. But the content of mathematics is more elusive. The ‘study of space and number’, the ‘study of patterns’, the ‘study of order and disorder’ have all been put forward as descriptions. But mathematics is hard to confine, it is constantly extending itself. In desperation formalists try to identify mathematics as a branch of logic. This is the axiomatic approach, exemplified by the Euclidean style of proof, and developed extensively in the first part of the 20th century, in which one starts with undefined objects and data (e.g. points and straight lines) and uses the strict rules of logic to deduce interesting consequences.
But most mathematicians now recognize this as of limited validity. It describes the scaffolding and the foundations but omits all other architectural aspects, notably the aesthetic one. There is no poetry in axioms and rules.
Non-mathematicians, with unhappy memories of classroom struggles with meaningless formulae tend to identify mathematics with the sterile formalist approach.
This is why they assume that computers have made mathematicians obsolete, or reduced them to the level of technicians. We mathematicians view ourselves as creative artists, guided by considerations of elegance and beauty. New insights are not produced by formal manipulation, they depend on ideas and original ideas cannot be manufactured to order.
If there is one over-riding theme common to all three books under review it is this view of mathematics as a creative enterprise, akin to creation in the arts. Mathematics is a human activity and it has to be understood in that light. It has strict rules, but so do painting, music and poetry. Its soul does not lie in the rules but outside them. Moreover, on occasion, rules have to be broken or at least re-interpreted. In fact the greatest steps forward, either in mathematics or in the arts, occur when genius inspires a break of the rules. For example, the familiar rule of algebra that xy = yx (the commutative law) was broken in the 19th century with profound and productive consequences.
Of all the arts, perhaps architecture is the one that most closely resembles mathematics. Strict engineering principles prevent a building falling down, there is geometry in its design, intricate detail in the finish but it is the overall architectural vision that is paramount. Oddly enough, mathematicians tend to compare themselves to poets who are superficially the least mathematical. The great German mathematician Hermann Weyl showed his poetic instincts when he said that ‘My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful.’
Karl Weierstress, an earlier German mathematician responsible for the rigorous establishment of analysis was the surprising author of the statement that ‘A real mathematician has to have the soul of a poet.’
While the three books have much in common and agree on its poetic aspect, they cover different approaches to the subject. Very roughly (and with overlaps) they deal with the following three topics:
- What mathematics is
- How mathematicians think
- The mathematical personality.
- How mathematicians think
David Ruelle is a mathematical physicist who tries to explain to the general reader what mathematics is and how mathematicians go about their work. Any mathematician reading it will recognize his description of our enterprise, both the formal day to day aspect and the more fundamental and creative process that gives the subject its life. The book is well organized, clearly written and gives a fair impression of the working mathematician.
William Byers has written much more discursively and covers a lot more ground. He is also more ambitious in that he wants to identify the essence of mathematical creativity, what makes it tick. His subtitle ‘using ambiguity, contradiction and paradox to create mathematics’ gives a clear indication of his main theme that mathematics, at its core, is not nearly as dependent on logic and order as most people think.
Because he is writing a large book Byers can afford to spend much time on the history of mathematics and the major problems that it has faced on the way. He enjoys pointing out, with many examples, that contradiction and paradox have played a vital part in a subject where logic is supposed to be supreme. In fact contradictions are like the no-entry sign that faces the motorist who is making an incorrect turn, while a paradox is like one of those junctions where all roads have no-entry signs. The highway engineer clearly has something to learn from such junctions and perhaps comes up with a new solution such as an underpass or flyover. So, on reflection, it should not be a surprise that a subject like mathematics that operates by a ‘highway code’ has to handle paradoxical situations. Moreover the lessons it learns in the process may play a fundamental part in its future.
But it is ‘ambiguity’ that really appeals to Byers. In a subject where clarity and precision are given the highest priority he is attracted to the situations where such clarity breaks down, where things are ambiguous. Rather than this being a weakness, a failure of the system, Byers sees such ambiguity as the catalyst for the creative process. As long as several different outcomes are possible there is flexibility and openness. Studying ambiguities carefully can show hidden meanings, and open the door to new adventures.
In all this Byers is quite persuasive and makes a good case, but I am not totally convinced. Trying to put one's finger on the nature of the creative process is over ambitious and doomed to failure. Byers gives many examples to support his view but he stretches the meaning of ambiguity to suit his purpose.
Though God may, in the Big Bang, have created the universe from a vacuum, human creation has antecedents. Almost by definition this pre-creation stage has to be incomplete, or ambiguous. What is created cannot have been clear in advance. But this use of ambiguity to explain creativity is almost a tautology.
The famous French mathematician Henri Poincaré analysed his own experiences and concluded that creativity emerged by chance. The key breakthrough would appear to pop up spontaneously after a dormant period where the conscious brain had signed off.
The book by Fitzgerald and James represents a collaboration between a psychologist and a mathematician. What interests them is the personality of the mathematician and how this is related to his work. Do mathematicians have certain personality traits in common? If so, does the mathematics produce the personality—too much work makes Jack a dull boy—or are those with such personalities just automatically attracted to mathematics? Certainly, in the public eye, mathematicians must be prime candidates for the caricature of the absent-minded professor, who cannot remember if he has had lunch.
Fitzgerald and James both start with an interest in autism and specifically in that form of autism that is known as Asperger's Syndrome. This is characterized by a withdrawal from the external world, often accompanied by an intense concentration on mathematics or music. Moreover, such concentration is either a reflection of, or produced by, unusual ability in the specialized field. An excellent fictional work which illustrates all this is ‘The curious incident of the dog in the night-time’ by Mark Haddon.
While the first part of the book discusses mathematics in general terms, overlapping with the other two books under review, the last part is a collection of short biographies of many famous mathematicians from history. These biographies, which include some less well-known figures, are of interest in themselves, but they have been selected because of the light they may throw on the personality of mathematicians and the possible presence of Asperger's Syndrome.
Any attempt to draw conclusions on the psychology of mathematicians from a small number of selected cases from famous figures of the past is fraught with difficulty. The kind of careful statistical analysis that is usually applied to such questions is hardly possible with small numbers, not randomly selected but, on the contrary, picked out as supporting evidence.
This would be true even if all the biographies clearly showed the prevalence of Asperger's Syndrome, but in fact this is very far from the case and, to be fair to the authors, they make no such claim. A few clearly had psychological problems, but many were quite well-balanced and normal.
It is sometimes even suggested that unusual psychological states (e.g. mania) can enhance creativity, but this is disputed. On the other hand it is quite plausible that intense concentration, whether natural or the result of some syndrome, may be productive.
Instead of examining some famous figures from the past, I prefer to use my own experience of the very many mathematicians I have known over 50 years. This includes most of the outstanding mathematicians of our time. My conclusion, for what it is worth, is that there are indeed a small number of extremely eccentric individuals, some of whom may be diagnosed as having Asperger's Syndrome. Of some it is said, only half in jest, that ‘they cannot tie their shoelaces’. Fortunately the mathematical world tolerates such eccentrics and even allows them to flourish. However, we are talking about very small numbers, as a proportion of the whole. Outside this small group mathematical personalities show as much variety as the rest of the academic population. So Asperger's Syndrome cannot, in my opinion, play a large or typical part in mathematical creativity.
Another class of psychological cases that have attracted attention in mathematical circles are the calculating prodigies, including the ‘idiots savants’. Many of these are well-documented and exhibit a remarkable capacity for spectacularly large and rapid calculations. Understanding the physiological basis of such prodigies is a serious challenge and it might shed light on the subconscious processes that underlie mathematical calculations. It would appear, in particular, that the conscious and the subconscious (in so far as this distinction makes sense) can operate at vastly different speeds. Again this is easy to understand in evolutionary terms: escaping the proverbial tiger in the jungle requires speed. Conscious thought is no help at the time, though reflection at leisure later may suggest that dense jungles should be avoided.
Since all the three books agree that the creative mathematician is really a poet, not a logical computer, where does this leave the neurophysiologist who wants to understand mathematical thinking in its higher reaches? Formal procedures such as those of elementary arithmetic or algebra are clearly not adequate. One has to dig more deeply.
An attractive idea is to focus on ‘beauty’ as perceived by mathematicians. All agree that it plays a vital role even though it is hard to pin down. As in the arts, we can list many of the desirable characteristics that make up beauty: elegance, proportion, subtlety, depth, significance, but in the end we know it when we see it. Importantly, ‘beauty is in the eye of the beholder’ and so subjective. We do not all agree, and much of art or mathematics is an acquired taste.
Despite all the difficulties associated with understanding or defining beauty we can still ask what mechanisms in the brain are involved in its appreciation. This can be asked about beauty in the various arts or in mathematics. It is a fascinating question whether there is any commonality across all areas. Are we just misled by the inadequacies of language and the misleading power of metaphor?
As I argued at the beginning, mathematics is a pure form of thought and so it may provide an easier field for physiological study. It is still a daunting task. We have to identify many instances of what a mathematician finds beautiful and see, by experiment, if there is any region of the brain that is common to them. We can for example compare beauty, as illustrated in geometric form, with the more formal beauty of an elegant algebraic formula or of a subtle abstract argument.
Here is a challenging programme, already begun, that will require much refinement and experimentation. It will also have to remain in contact with ‘real mathematics’. It will not be easy and may take a long time, but the pay-off could be substantial. Not only might we learn something about mathematical thought at the higher level but this might provide a useful first step for understanding other areas, such as the arts, where more complex issues are involved.
Since mathematics and beauty were central to Greek thought, understanding the link between them, in the human mind, would surely have appealed to Plato.
School of Mathematics
University of Edinburgh
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