MATHEMATICS AS AN ADEQUATE LANGUAGE

(a few remarks)

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INTRODUCTION

This conference is called The Unity of Mathematics. I would like to make a few remarks on this wonderful theme.

I do not consider myself a prophet. I am simply a student. All my life I have been learning from great mathematicians such as Euler and Gauss, from my older and younger colleagues, from my friends and collaborators, and most importantly from my students. This is my way to continue working.

Many people consider mathematics to be a boring and formal science. However, any really good work in mathematics always has in it: beauty, simplicity, exactness and crazy ideas. This is a strange combination. I understood earlier that this combination is essential on the example of classical music and poetry. But it is also typical in mathematics. Perhaps it is not by chance that many mathematicians enjoy serious music.

This combination of beauty, simplicity, exactness and crazy ideas is, I think, common to both mathematics and music. When we think about music we do not divide it into specific areas as we often do in mathematics. If we ask a composer what is his profession, he will answer, "I am a composer." He is unlikely to answer, "I am a composer of quartets." Maybe this is the reason why when I am asked what kind of mathematics I do, I just answer, "I am a mathematician."

I was lucky to meet the great Paul Dirac, with whom I spent a few days in Hungary. I learned a lot from him.

In the 1930's, a young physicist, Pauli, wrote one of the best books on quantum mechanics. In the last chapter of this book, Pauli discusses the Dirac equations. He writes the Dirac equations have weak points because they yield improbable and even crazy conclusions:

1. These equations assume that, besides an electron, there exists a positively charged particle, the positron, which no one ever observed.
2. Moreover, the electron behaves strangely upon meeting the positron. The two annihilate each other and form two photons.
   And what is completely crazy:
3. Two photons can turn into an electron-positron pair.

Pauli writes that despite this, the Dirac equations are quite interesting and especially the Dirac matrices deserve attention.

I asked Dirac,
"Paul, why, in spite of these comments, did you not abandon your equations and continue to pursue your results?"

"Because they are beautiful."

Now there is a radical perestroika of the fundamental language of mathematics. I will talk about this later. During this time, it is especially important to remember the unity of mathematics, to remember its beauty, simplicity, exactness and crazy ideas. I want to remind you that when the style of music changed in the 20th century many people said that the modern music lacked harmony, did not follow standard rules, had dissonances, and so on. However, Shoenberg, Stravinsky, Shostakovich and Schnitke were as exact in their music as Bach, Mozart and Beethoven.

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DINNER TALK OF ISRAEL GELFAND

(given at the conference dinner at Royal East Restaurant on September 3, 2003)

It is a real pleasure to see all of you. I was asked many questions. I will try to answer some of them.

• The first question is: Why at my age I can work in mathematics?
• The second: What we must do in mathematics?
• And the third: What is the future of mathematics?

I think these questions are too specific. I will instead try to answer my own question:

• What is mathematics? (Laughter)

Let us begin with the last question: What is mathematics?

From my point of view, mathematics is a part of our culture like music, poetry and philosophy. I talked about this in my lecture at the conference.

There, I have mentioned the closeness between the style of mathematics and the style of classical music or poetry. I was happy to find the following four common features: first -- beauty, second -- simplicity, third -- exactness, fourth -- crazy ideas. The combination of these four things: beauty, exactness, simplicity and crazy ideas is just the heart of mathematics, the heart of classical music. Classical music is not only the music of Mozart, or Bach, or Beethoven. It is also the music of Shostakovich, Schnitke, Shoenberg (the last one I understand less). All this is classical music. And I think, that all these four features are always present in it. For this reason, as I explained in my talk, it is not by chance that mathematicians like classical music. They like it because it has the same style of psychological organization.

There is also another side of the similarity between mathematics and classical music, poetry, and so on. These are languages to understand many things. For example, in my lecture I discussed a question which I will not answer now, but I have the answer: Why did great Greek philosophers study geometry? They were philosophers. They learned geometry as philosophy. Great geometers followed and follow the same tradition -- to narrow the gap between vision and reasoning. For example, the works of Euclid summed up this direction in his time. But this is another topic.

An important side of mathematics is that it is an adequate language for different areas: physics, engineering, biology. Here, the most important word is adequate language. We have adequate and nonadequate languages. I can give you examples of adequate and nonadequate languages. For example, to use quantum mechanics in biology is not an adequate language, but to use mathematics in studying gene sequences is an adequate language. Mathematical language helps to organize a lot of things. But this is a serious issue, and I will not go into details.

Why this issue is important now? It is important because we have a "perestroika" in our time. We have computers which can do everything. We are not obliged to be bound by two operations -- addition and multiplication. We also have a lot of other tools. I am sure that in 10 to 15 years mathematics will be absolutely different from what it was before.

The next question was: How can I work at my age? The answer is very simple. I am not a great mathematician. I speak seriously. I am just a student all my life. From the very beginning of my life I was trying to learn. And for example now, when listening to the talks and reading notes of this conference, I discover how much I still do not know and have to learn. Therefore, I am always learning. In this sense I am a student. Never a "Fuhrer".

I would like to mention my teachers. I cannot explain who all my teachers were because there were too many of them. When I was young, approximately 15-16 years old, I began tutoring in mathematics. I did not have the formal education, I never finished any university, I "jumped" through this. At the age of 19, I became a graduate student, and I learned from my older colleagues.

At that time one of the most important teachers for me was Schnirelman, a genius mathematician, who died young. Then there were Kolmogorov, Lavrentiev, Plesner, Petrovsky, Pontriagin, Vinogradov, Lusternik. All of them were different. Some of them I liked, some of them -- I understood how good they were but I did not agree with their, let us say softly, point of view. (Laughter) But they were great mathematicians. I am very grateful to all of them, and I learned a lot from them.

At the end, I want to give you an example of a short statement, not in mathematics, which combines simplicity, exactness, and other features I mentioned. This is a statement of a Nobel Prize winner, Isaac Bashevis Singer: "There will be no justice as long as man will stand with a knife or with a gun and destroy those who are weaker than he is."

Text written by Tatiana V. Gelfand.

taken from the Unity of Mathematics website.

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All material is Copyright 2012 to T. V. Gelfand and T. I. Gelfand(unless otherwise noted)