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.
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.
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:. 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. 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. Moreover the lessons it learns in the process may play a fundamental part in its future.
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 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.
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.
But it is his experience as a teacher that gives the book some of its extraordinary salience and authority Good mathematics teaching should not banish ambiguity, but enable students to master it Everyone should read Byers His lively and important book establishes a framework and vocabulary to discuss doing, learning, and teaching mathematics, and why it matters.
Perfectly formalized ideas are dead, while ambiguous, paradoxical ideas are pregnant with possibilities and lead us in new directions: they guide us to new viewpoints, new truths Integrating his experience as a mathematician and as a Buddhist, Byers examines the validity of this assumption. Much of mathematical thought is based on intuition and is in fact outside the realm of black-and-white logic, he asserts. Byers introduces and defines terms such as mathematical ambiguity, contradiction, and paradox and demonstrates how creative ideas emerge out of them.
He gives as examples some of the seminal ideas that arose in this manner, such as the resolution of the most famous mathematical problem of all time, the Fermat conjecture. Next, he takes a philosophical look at mathematics, pondering the ambiguity that he believes lies at its heart. Finally, he asks whether the computer accurately models how math is performed. The author provides a concept-laden look at the human face of mathematics. What Byers's book reveals is that ambiguity is always present You can't quite say that nobody has said this before.
But nobody has said it before in this all-encompassing, coherent way, and in this readable, crystal clear style Or you assign to each boy his weight—that's a function, okay? You see, there's a psychological element to mathematics that most people aren't aware of. Have you ever heard of 'Occam's Razor'? It's named for the 13th century English philosopher William of Occam. In plain language, Occam's Razor means that you try to reduce your thinking to a few principles, the idea being that if you have your principles laid out clearly and you're logical, then if you reach a conclusion you don't want to reach, the only possible explanation is that there was something wrong with your original principles.
Mathematics supposedly follows Occam's Razor more closely than almost any other subject. There's nothing ad hoc in mathematics. The problem is figuring out where to start. You have to start with ideas that you can't really define, because everything is defined in terms of something else. One of the things you start with is function.
I've defined it—and you were satisfied with the definition I gave you, but you had only two minutes to think about it. The trouble is, I used the word 'rule,' and rule is not very precise. For instance, I could create the following function: It assigns to all women the number '1,' provided that there is life as we know it on Mars.
And it assigns them the number '0' if there is not life on Mars. That is a well-defined rule, but it's not very satisfying because we don't know if there is life on Mars. So when you start in mathematics, you either have to take 'function' or 'set' as the undefinable. The point I am trying to make is that for your undefinables you want something that is so logical, so intuitively appealing, that not many people are going to argue with it. Function and set seem to be surviving.
Everything else is based logically on these. I guess that's a sign that they system is healthy. He makes his living worrying about what mathematicians should be assuming, what our axioms ought to be. Many of his papers deal with questions like the following: Suppose we change our axioms to this, how will it change mathematics? It was an amazing piece of work, because if you believed it, it wiped out whole areas of philosophy. The response of the working philosophers at the time was, "This is terrible.
It puts us out of business. We have to ignore it. In mathematics, everybody has accepted the axiom system—with a few notable exceptions—everybody has accepted the way we do mathematics. That's not to say there aren't other psychological problems in mathematics. There is a famous list of questions posed in by Hilbert—there's Hilbert. In , he gave a speech to an international body of mathematicians in which he charted what he felt were the 20 or 25 most important problems that mathematics ought to consider in the 20th century.
People get promotions and tenure just for solving Hilbert problems. However, it turns out that some of the most intuitively appealing ones, the ones you most feel you would like to have answered, have no answer.
They can't be answered within the framework of mathematics as we know it. The axiom system that we work in—the Zermelo-Frankel set theory—people are not inclined to change. Your average working mathematician is not inclined to change it. Well, suppose you have a polynomial equation with integer coefficients. If you stare at the equation, is there some way to tell if it has integer solutions? That's a simple-minded version of a Hilbert question.
Ideally, what you would like to do is to take a polynomial—here's a polynomial. You set it equal to zero and solve for x, or more generally, it looks like this. The answer is, there is no way to do this. That answer would have been inconceivable in the 19th century. I don't mean, we can't do it or it's too hard or we don't know the right theory yet, I mean it can't be done. That's one of the psychological features of mathematics. Because the first one was a specific example of a polynomial and the second one represents any old polynomial.
I was just falling into the trap of being a mathematician then. I've mentioned some of them. Another one is that some proofs now are so complicated that no one can understand them. Everybody loves this problem.
Suppose you are Rand McNally and your job is to make a map, you don't want two adjacent countries to have the same color. Now, you are going to be printing maps from now to the end of time, and you don't know what kind of maps are going to come up, but you want to have the right number of colors in supply.
How many colors do you need? The professor couldn't answer it. People fiddled and fiddled with the question and after a while they proved that five colors would always do. They thought that four would do, but they couldn't prove it. One solution was published and believed for 20 years until somebody found a mistake. Finally about 10 years ago, two mathematicians at the University of Illinois, Appel and Haken, came up with a solution.
They proved that four colors will suffice—the University of Illinois is so proud of this that when you get a letter from their math department, the cancellation mark says 'Four colors suffice. Nobody can check their proof in the usual way. Previous research has found that these nonlinguistic areas are active when performing rudimentary arithmetic calculations and even simply seeing numbers on a page, suggesting a link between advanced and basic mathematical thinking.
In fact, co-author Stanislas Dehaene, director of the Cognitive Neuroimaging Unit and experimental psychologist, has studied how humans and even some animal species are born with an intuitive sense of numbers—of quantity and arithmetic manipulation—closely related to spatial representation.
This work raises the intriguing question of whether an innate capability to recognize different quantities—that two pieces of fruit are greater than one—is the biological foundation on which can be built the capacity to master group theory.
Ansari suggests that a training study, in which nonmathematicians are taught advanced mathematical concepts, could provide a better understanding of these connections and how they form. Moreover, achieving expertise in mathematics may affect neuronal circuitry in other ways. Although additional studies are needed to determine whether mathematicians actually do process faces differently, the researchers hope to gain further insight into the effects that expertise has on how the brain is organized.
Jordana Cepelewicz is a science writer based in New York City.