1980 Balzan Prize for Mathematics
Interview with Enrico Bombieri 05.06.2014
How has mathematics changed? What is its current role and how is the discipline perceived? How should it be perceived?
These are questions that non-specialists in the field can ask themselves as they see today’s complex systems (technological, economic and in the natural world) being studied according to mathematical principles and models.
The Balzan Prize over the course of the last 50 years has been following the evolution in mathematics.
Seven mathematicians have been awarded with the prize during these years and an eighth will receive the prize in 2014. For this reason, we started a dialog on the role of mathematics with former prizewinners, starting with Enrico Bombieri, the first prizewinner in mathematics from 1980. A very interesting discussion ensued. Our next installment will be with Jacob Palis, the seventh winner of the Balzan Prize in mathematics. Born in Milan (Italy), currently Emeritus Professor at the Institute for Advanced Study at the University of Princeton (USA), Bombieri is considered one of the greatest contemporary mathematicians. In October 2008, he gave a Balzan Lecture in Washington at the Carnegie Institution for Science where he discussed the changes and evolution of the concept of truth in Mathematics.
We asked Bombieri if he thought that applied Mathematical models still made sense despite their inability to predict the financial crises of the last seven years.
Mathematical models while they make sense logically, cannot be expected to be adequate to understand everything. Models don‘t predict all of the crises that we have seen because modelling by standard deviations works well only on too short a time frame.
Exceptional events are costly and are more frequent than ordinary statistics might have imagined. The usual methods do not apply. External events that are very rare turn out to be not as exceptionally rare as one might expect by usual mathematical modeling. The nature of the events is also very varied. Standard modelling by computer can’t simulate these events; it would take too much time. A group of mathematicians in the 1960s found that they could rescale events and
accelerate time. They could then use computers to model rare events in a realistic way.
Applied mathematics does not take a phenomenon such as high-frequency trading into consideration of long-term economics predictions. High frequency traders play with high leverage analyzing in a fraction of a second overall trading patterns in stock prices and taking advantage of this to obtain very, very quickly small profits that in the end add up to very big profits. Standard statistics in models of economics can’t be applied in the same realistic way to understand the overall effect of these unusual ways of operating in the market.
There are limits to applied mathematics as Hardy suggested in his short essay “A Mathematicians Apology”, and Bombieri mentioned in his 2008 lecture: “Most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, by the brutal but sufficient reason that they do not fit the facts.“
What is your view of the relationship between computers and mathematics?
Computers won’t replace mathematics just as the printing press did not replace writers and poets. The role of computers is very positive for mathematicians, especially those that study complex things which require a space where the dimension goes to infinity. There have been big breakthroughs in prime numbers, for example.
Computers can also study aggregates of particles and the aid of the computer is needed because there are too many particles to study them one by one, for example studying grains of sand, piles of pebbles and molecules of water. The branch of mathematics called combinatorics is very important here. Working with computers has also expanded the possibilities of making predictions about the behavior of large systems.
Computers can also help to predict the behavior of exotic functions, say Riemann’s hypothesis. Mathematicians haven’t been able to solve this hypothesis but computers can verify the hypothesis on a very large numerical scale and prove that for all practical purposes it may be considered to be correct.
It would be impossible to calculate by hand more than a few thousand solutions of the Riemann equation.
On another point, working with networks of computers expands the possibilities enormously. Today, this is called cloud computing. The monstrous supercomputer of science fiction stories is not practical, it is being replaced by networks of small computers. Problems can be distributed and you can now study the logical part of the math. A new computer language is being developed that can be used to check consistent logic within a series of words. Some checking of proofs can also be done by computer, although it doesn’t work yet to the point of finding proofs. This was tested in an extremely important mathematical paper over 300 pages long to prove a basic theorem needed to understand symmetries. Checking the correctness of the proof was almost beyond what a single person could do. That paper was checked by computers and found it to be correct from beginning to end, only on a technical point an auxiliary result was used before it was proved, a fault in the logical presentation of the paper but not in its internal logic. This was a great
accomplishment because it was an extremely difficult paper that has been used again and again as a cornerstone of subsequent works.
Certain areas of mathematics don’t use the computer. Computers work in a discrete fashion and don’t work directly on continuum models of fluid mechanics, for example. Applied mathematics allows you to see if models are elegant, but computers help to see if the model you are using is realistic or not. They can help you to adapt your model to reality.
What are your thoughts about Quantum computing?
Quantum logic is a different system and information is transmitted in a very different way. For example, the act of reading information may destroy it. The theory is very complex. The quantum world is different than our normal world and our perception of the physical world does not fit easily in it. In quantum computing when you can do something you can do everything very fast, but there are things that do not fit in the quantum world. This has a great impact in complexity theory, that is the study of how efficient a computer is.
The logic of the quantum world is different. No one really knows how to make predictions in quantum theory or string theory.
Super-specialized mathematicians working in only one direction are disappearing and what we are seeing is a mathematics that requires at the same time a lot of different fields and techniques. This has been taking place over the last 20 years. The world is changing because of instant communications. This means more information is available in real time.
One person alone cannot do all that is required but research in certain fields can now be done by groups of persons working independently at the same but sharing all the findings in real time. This has been put in practice with the Polymath project of two recent Fields medalists, with good results. I cannot predict what will happen next, because mathematicians also need to consider innovative ideas that require long deep thinking in order to ask the right questions, but in our “publish or perish world” that can be a problem. For example there is a vision of a new theory called Langlands program that has taken 30 years to explore. You have to love the subject to be a mathematician of Langlands’s caliber and of course you need a lot of patience. Success will not come instantly.
During your lecture at Carnegie in 2008, you mentioned that mathematics is in someway Darwinian, in that there are theories that survive and others that do not. Is it possible today to understand which models will survive and which will not?
There are two types of mathematics: the mathematics that you can prove in a logical system and mathematics that you try to project as a meaning. You view the meaning not just as a collective of single statements but also by understanding the relations between them. A good example is the theory of Lie groups, continuous symmetries used in physics to describe the world of particles. You can study that as a theory by itself but when you look at the applications you discover that they are very important in physics as well as in number theory, analysis, and probability theory.
The role of the mathematician is to discover what is interesting and the reasons that would be apparent in the same problems. Some problems are no longer interesting to me today but in the future they may be the beginning of something important. You don’t want to be too categorical.
There is a fundamental unity in mathematics. What you are looking for is the fundamental role of logic that is becoming more and more important and helps to explain why some problems are so difficult to solve, especially once you enlarge the system. In a small enough logical system the truth or falsity of a statement can always be checked. However, in any logical system that is sufficiently large there will be statements that are true but can’t be proved. This makes you go to a higher level of thinking and even that may not be enough. There are statements in elementary mathematics whose truth or falsity cannot be decided. There is no universal method to determine whether a mathematical statement is decidedly true or not. The idea of mathematics as a collection of true statements is incomplete.
What do you think is the role of beauty in Mathematics? Is there any link between the beauty of a theorem and its applicability?
Mathematicians generally agree that beauty does exist in the structure of theorems and proofs, even if most of the time it is largely visible only to mathematicians themselves. I participated in an art experiment where ten mathematicians and physicists were asked to produce a formula of their own invention that they considered beautiful; the formula then would be set as a work of art, in this case as an etching in a limited edition. The work of art consisted in the totality of the 10 formulas, not just one formula at a time.
Our equations were paired with an explanation. I did mine on a problem about symmetries, the so-called Ree Groups. The problem was beautiful, the expected answer was also simple, the road to the solution at first impossibly difficult (no computer in the universe could do it directly!), but with a surprising solution, hence beautiful. The “impossibly difficult” Thompson equations had an inner secret beauty because they reflected the properties of the group and, in the end, their beauty could be unlocked.
Mathematics is in some ways similar to an art project. When you write a significant formula it can be a work of art with two pages of explanations. Other times there is the beauty at the initial starting point, a secret beauty that results in the answer. Beauty is important and I think it is something intrinsic in mathematics. Yes, beauty has a role to play but one should not look at beauty in a superficial way and lose the content. That happens too. Secret beauty can be discovered. Oversimplification can be a risk as well.
What do you see as the link between the Balzan prizewinners in Mathematics? From Andrej Kolmogorov (1962) to you (1980), to Jean-Pierre Serre (1985), Armand Borel (1992), Mikhail Gromov (1999), Pierre Deligne (2004) and Jacob Palis (2010)?
All of the prizewinners have been extraordinary. Borel, Gromov, Serre and Deligne in the development of algebra and geometry. Algebra used to be only a tool of geometry and now it is a very big field with applications everywhere. Algebra and geometry are no longer distinct areas of mathematics.
Kolmogorov and Palis are two great examples in analysis. They studied dynamical systems and have created new problems and new questions to be answered in depth. Their interest has been in how the world transforms.
Serre was one of my first mentors, who actually discovered my abilities in 1964, and enabled me to start my career at a truly international level. He brought a new way to look at algebraic geometry, transforming it completely from its foundations. I owe a lot to him in my studies in geometry.
The prizes all reflect new ways of thinking about mathematics, and all of the work has practical applications in physics and computer science. The theory of dynamical systems has found use in the prediction of weather as well. The Balzan Foundation has been very consistent in its policy of awarding its prizes.
It differs very much from other prizes that are given to individuals because of age (Field) or lifetime achievement (Abel, Balzan, etc).
The Balzan Foundation carefully chooses alternative precise fields to attract attention to. In some cases these fields or areas are very precisely detailed. This is a very interesting idea that l haven’t seen much in other prizes. I was very surprised to win it, but I had heard of the prize before.