Mathematics and the Real World (59 page)

Beyond the range of research topics that covers a vast number of possibilities, mathematical research also offers totally different models of research itself, models that require various types of expertise. For example, some mathematicians are known for their excellent problem-solving abilities. Give them a specific problem in mathematics, and they will solve it. Others excel in opening up new paths and constructing new mathematical theories. Some have the knack, in addition to their other talents, of asking the right questions. Open problems in mathematics played a central role in research in the past and still do so today. It is not easy to ask the right question, one that will be interesting and that has a reasonable chance of being solvable.

We have mentioned Hilbert's address to the Second International Congress of Mathematicians in 1900, in which he presented a list of mathematical problems that he considered would leave their mark on the mathematics of the twentieth century. The list appeared in the proceedings of the congress and included twenty-three problems. The first problem on the list included Hilbert's question about the foundations of mathematics—the program that Gödel showed could not be carried out—and the continuum hypothesis, solved by Paul Cohen in 1964. Another problem was Fermat's last theorem, which was solved by Andrew Wiles in 1995. Hilbert was right with regard to the position these problems would occupy in mathematics in the twentieth century, but it may well be that the fact that they were presented in such an illustrious forum contributed to their great
impact. Most of Hilbert's problems have been solved by now, but some still remain unsolved.

Around the turn of the millennium several similar lists of problems were published. The best known is the list of seven problems published by the Clay Mathematics Institute (CMI), which also offered a prize of one million dollars to anyone who solved one of them. We described one of the problems in section 53, namely, whether the P class of problems is the same as the NP class. Another problem on the CMI list was Poincaré's conjecture, which was first introduced by Henri Poincaré as early as in 1904, and which was solved only in 2002. We will describe later the problem and the controversy that its solution sparked. Paul Erdős was known as a serial presenter of open problems in his area of research, number theory and combinatorics. To motivate potential problem solvers, Erdős would offer monetary prizes for the first person to solve any of his problems, with the amount of money, between five dollars and thousands of dollars, reflecting the difficulty of the problem. His purpose was to promote mathematics, and indeed, the challenge and the financial rewards led to the solution of many interesting problems, more than any single person could have solved. I was present at a lecture at which Erdős was asked how he could shoulder the financial burden imposed by the distribution of many payments. He answered that he paid by checks, signed by him, and he hoped that the recipient would not deposit the check but would prefer to keep it as a memento. He was surprised, he said, when one of the mathematicians who won a prize did deposit the check and then asked Erdős if he could have the paid check back as a memento!

Offering and solving open problems with different levels of difficulty is part of the practice of research in the mathematics community. At a conference I attended in Warsaw, the well-known Polish mathematician Czesław Olech presented an open problem and promised a bottle of vodka to anyone who solved it. I managed to solve it before the end of the conference and showed the solution to Olech on the way to the airport. A bottle of vodka duly arrived with a messenger some two months later. The solution appeared in an article I coauthored with another participant in the conference, who had also solved the problem. I won another bottle of vodka, also in Poland, after
Ron Stern, a mathematician from Montreal, promised it as a prize to anyone solving a problem he presented. The problem was solved in parallel by Piermarco Cannarsa, and each of us received our own bottle. I gave a tasty bottle of dessert wine to Felipe Pait, who produced an interesting example as a solution to a question I had posed at a conference held at Rutgers University in New Jersey. The example completed the results in an article I was writing at the time, and I included it in the article, giving due credit to Felipe for his contribution. The reader should not conclude from these instances that research in mathematics consists of drinking liquor that we win as prizes for solving problems at conferences. These are just a piquant, but very small, part of the work we do. In most cases no prize is offered for finding a solution to a problem, and whoever finds a solution must settle for professional esteem, and even more, the satisfaction derived from the solution itself.

Much research in mathematics, as in other disciplines, is the product of cooperation between several mathematicians. The special nature of mathematical research resulted in the development of types of cooperation that are different than those among our natural-scientist colleagues. Even when the research is in cooperation with other mathematicians, most of the thinking is carried out alone, either separately or with the coresearchers sitting together, staring at the blackboard in silence. That is also why, when trying to identify the contribution of each of the coauthors of a joint paper, the sum of the parts, whether in mathematics or in other fields, turns out to be greater than the whole. On more than one occasion my collaborators in a research project and I have come back to it the next day, or even just after a lunch break, and we have the same solution in mind. That is also why in mathematics it is the practice, though not an unbroken rule, to list the coauthors of a paper in alphabetical order, unlike in other disciplines that require experiments, where generally the first author listed will be the one who made the major contribution, and the last-mentioned will be the head of the laboratory. The list of authors of a mathematical paper is usually shorter than that of papers in biology or physics, subjects in which large-scale experiments are performed. That said, if we were to draw a chart of joint authors of papers in mathematics, we would obtain a denser network than we would have expected.

In the establishment of such a network too Paul Erdős has fundamental rights. He was one of the most prolific cooperators ever. He coauthored papers with more than five hundred mathematicians. We can obtain an interesting mapping if we draw a chart in which we denote everyone who coauthored an article with Erdős as “Erdős 1.” We denote by “Erdős 2” everyone who coauthored a paper with someone denoted by Erdős 1 but who did not personally coauthor a paper with Erdős. Anyone who coauthored a paper with someone denoted by Erdős 2 but who is not himself Erdős 1 or Erdős 2, we denote by “Erdős 3,” and so on (I currently am denoted Erdős 3). This network has some surprising properties, particularly in light of the way in which mathematical research advances. The number of Erdős 2 mathematicians at the end of 2010 was close to ten thousand, and of course the number of Erdős 3 mathematicians was far greater. There are some mathematicians who are not connected to Erdős via this chain of coauthored papers (we usually refer to these as having an infinite Erdős number), but of those who are linked with him, the average Erdős number is 4.5. With the tools available on the Internet today, it is easy to determine the chain of joint authorship that links any mathematician with Paul Erdős, and even the link between any two mathematicians. These chains are short, a surprising fact considering the range of topics covered by such an individualistic discipline.

Before mathematical papers are published in the professional literature, they are reviewed, in general anonymously, by professional colleagues (peer review). The referee decides whether to accept the paper for publication, basing the decision mainly on the degree of innovation in the paper and on being convinced that the results are correct. The term
convinced
is open to interpretation. The referee is not meant to perform a detailed a check to see whether there is an error in the paper. That is just too hard to do. The results are written in a logical, well-ordered manner, and as we have stated previously, following logical claims without the attendant intuition is an extremely difficult task for the human brain to perform. The cure might have been to attempt to present the underlying intuition in writing. Good authors try to do that, but it is a formidable undertaking. The difficulty in identifying errors is the reason for many errors coming
to light later, when others, in addition to the authors and referees, become interested enough in the results to check the proof.

In contrast to the public image of mathematics, mathematical errors are commonplace, and although attempts are made to avoid them, success is only partial, at best. One of the best-known errors in the history of mathematics relates to the four-color problem, which we referred to in section 56. The problem was presented in the middle of the nineteenth century, and a solution was published in 1879 by Alfred Kempe (1849–1922). His solution was highly regarded by many in the mathematics community. It was not until eleven years later that Percy Heawood (1861–1955) found a flaw in the proof. As mentioned above, the four-color problem was fully solved only in 1976, almost one hundred years after the publication of the flawed proof that was initially accepted as correct.

Mathematicians are not very dismayed that mistakes are made. On more than one occasion I have seen comments by referees and reviewers that although an article was flawed, it was still very important, as it contained novel and fundamental ideas. Generally, in research in mathematics, the proof of a hypothesis is no less important, and may be even more important, than the correctness of theorem that it verifies. Many mathematical results have been confirmed by different proofs over the years, either to simplify the existing proof or by means of a new proof that gives a different and sometimes even deeper understanding of the issue.

Mathematical results are presented in an ordered, logical, and precise form, and that is also how mathematics is learned in school and university, creating the impression that with the exception of possible occasional errors, a mathematical proof is irrefutable. In the previous section we showed that no foundations have been discovered yet that raise mathematics in general to the level of complete certainty. However, even if we agree that the foundations accepted today, that is, the axioms of set theory and the use of logic, ensure contradiction-free mathematics, it would not be practical to base the proofs themselves directly on those foundations. Therefore, even in ordered, logical writing there is no choice but to rely on the reader's and the author's prior knowledge, knowledge that cannot be checked fundamentally. We have already emphasized that belief and trust in what you are told
is a characteristic promoted by evolution, and the same applies in the practice of mathematics. As a result, there is no answer accepted by the whole mathematics community to the questions of what is permitted and what is forbidden in a mathematical proof, and what constitutes a complete proof. Every subgroup of mathematicians develops its own standards of what it considers to be a complete proof. This practice sometimes leads to serious disputes, an example of which we now describe.

Poincaré's conjecture relates to geometry and is simple enough for us to describe here in full and in intuitive terms. As mentioned previously, the problem was presented in 1904 in the context of attempts to understand the geometry of the world more fully, including the properties of geometric bodies. The following is a graphic description of the conjecture.

Consider the boundary (i.e., the surface) of an ordinary ball in three dimensions. We will compare this boundary with the boundary of an egg or of a cube. Even if these three objects look different, they are very similar. For example, we can match points on the boundary of the ball continuously, one by one, with those on the boundary of the cube. That is, points close to each other on the ball are matched with points close to each other on the cube, and vice versa. Mathematicians call this relation homeomorphism and say that two bodies are homeomorphic or topologically equivalent. Thus, the boundary of the ball is homeomorphic to that of the cube and to that of the egg, and also to many other bodies that may not be round. For example, if we knead and squeeze the boundary of the ball at will without tearing it or sticking parts of it together, we will still have a body homeomorphic to the boundary of the ball. There are other bodies, however, that are not homeomorphic to the boundary of the ball, for example, a ring or a pretzel shape. The boundary of a ball has another property that can be checked easily: if we think of a loop on the ball, we can shrink it to a single point without removing it from the surface. The same applies to the surface of the cube or egg, or to any other body that is homeomorphic to them. The boundary of a ring, however, does not have this property. A loop around a section of the ring facing the center (see the diagram below) will remain like that even if it is moved along and if we try to shrink it, unless we cut it. An interesting question arises: Is there a
body such that any loop on its boundary can be shrunk to a point without breaking contact with the boundary and that is not homeomorphic to (i.e., topologically equivalent) the boundary of the ball? The answer is that there is no such body in our physical, three-dimensional space. In other words, the property of shrinking the loop to a point is characteristic of bodies that are homeomorphic to the boundary of the ball.