Mathematics of this century succeeded in harmonizing and unifying diverse fields on a scale probably never seen before. The most prominent role in this unification was played by set theory. Initially conceived by Cantor as a new chapter of mathematics, "the theory of infinity", set theory, gradually changed its status and developed into the
universal mathematical language. It was understood that starting with a rather short list of basic terms and operations, one could generate recursively the linguistic constructions which apparently conveyed equally well the intuition of the founding fathers of calculus, probability, number theory, topology, differential geometry and what not. Thus the whole mathematical community acquired a common idiom. Moreover, allowing the clear distinction
between the set-theoretic and geometric content of the mathematical constructions on the one hand, and their flexible linguistic expression (notations, formulas, calculation) on the other, set theory greatly simplified the interaction between the right and left brains of every working mathematician as an individual. This two-fold function of the set-theoretic
language became the basis for the development of new technical tools, for the solution of old problems as well as the formulation of research programs.

Document Body
Page Navigation Panel

Pages 1--6 from Untitled

Page 1
2

Good Proofs are Proofs that make us Wiser

Yuri I. Manin

Interview by Martin Aigner and Vasco A. Schmidt

This year's International Congress is the last ICM in this century. Do you think a Hilbert is

still possible? Are there any contemporary problems corresponding to Hilbert's Problems?

I don't actually believe that Hilbert's list had a great role in the mathematics of this

century. It certainly was psychologically important for many mathematicians. For example

Arnold told that while being a young graduate student he had copied the list of Hilbert

problems in his notebook and always kept it with him. But when Gelfand learnt about

that, he actually mocked Arnold on this. Arnold saw problem solving as an essential part of

great mathematical achievements. For me it's different. I see the process of mathematical

creations as a kind of recognizing a preexisting pattern. When you study something |

topology, probability, number theory, whatever | first you acquire a general vision of the

vast territory, then you focus on a part of it. Later you try to recognize "what is there?"

and "what has already been seen by other people?". So you can read other papers and

finally start discerning something nobody has seen before you.

Is the emphasis on problems solving a kind of romantic view: a great hero who conquers

the mountain?

Yes, somehow a kind of sportive view. I don't say it is irrelevant. It is quite important

for young persons, as a psychological device to lure young people to create some social

recognition for great achievements. A good problem is an embodiment of a vision of a

great mathematical mind, which could not see the ways leading to some height but which

recognized that there is a mountain. But it is no way to see mathematics, nor the way to

present mathematics to a general public. And it is not the essence. Especially when such

problems are put in the list, it is something like a list of capitals of great countries of the

world: it conveys the minimal possible information at all. I do not actually believe that

Hilbert thought this is the way organize mathematics.

Would you venture to predict some dominant patterns of mathematics in the next century?

This is very difficult. I think the mathematics of the 20th century is best presented

around programs, not problems. Sometimes they are explicitly formulated, sometimes

they are gradually emerging as a prevailing tendency. For example the development of

mathematical logic and the foundations of mathematics. That was certainly a development

of a program which was understood as such. After Cantor's discoveries it was clear that

we have to consider very deeply the ways we think about infinity. Or we have Langlands'

1
1

1
Page 2
3

program of understanding the Galois group. There is one program with which we enter

the next century. This program can be thought of as the quantization of mathematics.

When one looks at how many mathematical notions changed in the last twenty years in a

way that that the new notions are quantum versions of the old ones | it is amazing: Look

at quantum groups, quantum cohomology, quantum computing | and I think many more

are ahead of us. This is very strange because nobody actually conceived anything like that

as a program for developing mathematics in general. The desire was just to understand

the mathematical tools that physicists invented with fantastic intuition and which they

used in a very stimulating but somewhat careless way from the point of view of a pure

mathematician.

How do you think the 20th century will be looked at from an historical point of view?

Was it an important century?

I think so. Mathematics of this century succeeded in harmonizing and unifying diverse

fields on a scale probably never seen before. The most prominent role in this unification

was played by set theory. Initially conceived by Cantor as a new chapter of mathematics,

"the theory of infinity", set theory, gradually changed its status and developed into the

universal mathematical language. It was understood that starting with a rather short list

of basic terms and operations, one could generate recursively the linguistic constructions

which apparently conveyed equally well the intuition of the founding fathers of calculus,

probability, number theory, topology, differential geometry and what not. Thus the whole

mathematical community acquired a common idiom. Moreover, allowing the clear distinc-tion

between the set-theoretic and geometric content of the mathematical constructions on

the one hand, and their flexible linguistic expression (notations, formulas, calculation) on

the other, set theory greatly simplified the interaction between the right and left brains of

every working mathematician as an individual. This two-fold function of the set-theoretic

language became the basis for the development of new technical tools, for the solution

of old problems as well as the formulation of research programs. The diversification of

mathematics was connected first of all with external social phenomena: the rapid growth

of the scientific community in general and the ground-breaking discoveries in physics. In

my opinion, the mathematics of the last hundred years did not produce anything compa-rable

to quantum theory or general relativity in terms of the resulting change of our total

world perception. But I do believe that without of the mathematical language physicists

couldn't even say what they were seeing. This interrelation between physical discoveries

and mathematical way of thinking, the mathematical language, in which these discoveries

can only be expressed, is absolutely fantastic. In this sense the 20th century certainly will

be regarded as a century of great breakthroughs.

Are there certain specific topics that come to your mind, in which our century was really

at a top level?

In the 18th and 19th century mathematical language was much vaguer than we are accus-tomed

to. I think the 20th century started with rethinking the basics. When the basics

were clear enough there was a great search of technical methods of incredible strength

which led to the creation of powerful tools allowing us to develop and expand our geomet-ric

intuition to new domains. I have in mind topology, homological algebra and algebraic

geometry. As soon as the technical development was accomplished, the solution of several

2
2

2
Page 3
4

very difficult problems fell into the span of thirty years | Deligne's proof of the Weil

conjectures, Faltings' proof of the Mordell conjecture, Wiles' proof of Fermat. All of them

could not have been done in the last century just because mathematics was not developed

enough.

Some people | some of them mathematicians | proclaim the end of proof, partly in view

of the universal availability of computers. How would you comment on this?

If you are speaking of mathematics without proofs you are speaking of something intrin-sically

contradictory. The proof can not die | only together with mathematics. But

mathematics can die as an accepted part of the culture of humanity. I think, in our gen-eration,

mathematicians still keep doing mathematics as we understand it. Proofs are the

only way we know the truth of our thoughts; that is actually the only way of describing

what we have seen. Proof is not just an argument convincing an imaginary opponent. Not

at all. Proofs is the way we communicate mathematical truth. Everything else | leaps of

intuition, elation of sudden discovery, ungrounded but strong beliefs, remains our private

matter. And when we do some computer calculations we are only proving that in the case

we have checked things are as we have seen them.

Just recently, there was a notice in the newspaper that a computer has proved a conjecture

of Herbert Robbins by carrying out a full search of all possible strategies.

Of course this is possible. Why not? If you have invented a good strategy of proof which

includes however an extensive search or long formal calculations, and afterwards you have

written a program implementing this search, it's perfectly OK. But computer assisted

proofs, as well as computer unassisted ones, can be good or bad. A good proof is a proof

that makes us wiser. If the heart of the proof is a voluminious search or a long string

of identities, it is probably a bad proof. If something is so isolated that it is sufficient

to get the result popped up on the screen or a computer, then it is probably not worth

doing. Wisdom lives in connections. If I have to calculate the first 20 digits of ¼ by hand

I certainly become wiser afterwards because I see that that these formulas for ¼ that I

knew take too much time to produce 20 digits. I will probably devise some algorithms

which minimize my effort. But when I get two millions of digits of ¼ from the computer

using somebody else's library program I remain so stupid as I was before.

If you have a beautiful theorem with an equally beautiful proof but which needs the

calculation of one thousand cases, do you mind giving it away to the computer? Is this a

bona fide proof?

It will be a bona fide proof with the same reservations as I would have for any proof written

on paper. There can be possible mistakes in the programming, there can be possible

mistakes in implementing the calculations and finally there can be possible mistakes in

our understanding of how to classify all the cases and so one. We have examples of those

proofs. We have the Four-Color-Problem and the classification of finite simple groups.

In both cases a huge amount of combinatorial material was partly treated by computer

calculation. So there is still room for doubts and the need to recheck the calculations, but

most important, to devise way for seeing things in a new light.

Let me ask you a question about mathematics internally. In recent years, the mathemati-3
3

3
Page 4
5

cal community seems to emphasize applications. Do you think that pure mathematics will

have problems, as compared to applied mathematics? Do you have the impression that

the money will go in the future only to those fields?

Applications ask for and get much more money that pure mathematics. But I don't think

it's actually the problem of money in terms of allocating limited resources. Mathemati-cians

don't need and don't spend much money. It's a problem of the public attention

and the public scale of values. I see the growing estrangement of our society from the

traditional Enlightenment values, and the public just does not want to spend on mathe-matics,

probably on universities in general. Mathematics | if it will be a victim | will

be a victim of of this general process, not of the fact of that money goes to applications.

But, surely, I do think that there will be a continuous shift to applications in terms of

the quantitative resources allocated to applications, and the attractiveness of this kind of

occupation for young persons. Applied mathematics is connected with computer simula-tion

| computer as large, database programs and things like that. I have once translated

a talk by Donald Knuth into Russian. In Usbekistan there was a meeting dedicated to

Al'khorezmi. Knuth started his talk with a funny statement. In his opinion the primary

importance of computers for the mathematical community is that those people finally took

to mathematics who were interested in mathematics but had an algorithmic sort of mind.

Now they were able to do what they wanted. Before that, this subculture didn't exists. I

take this argument quite seriously and I do believe that among the community of future

potential mathematicians there is subcommunity whose minds are better to writing com-puter

programs than for proving theorems. In the last century they probably would have

proved theorems but nowadays they do not. I have a great suspicion that for example

Euler today would spend much more of his time on writing software because he spent so

much of his time, e. g., in effort of calculating tables of moon positions. And I believe that

Gauß as well would spend much more time sitting in front of the screen.

Let us go back to the question of applied mathematics, Isn't true that mathematics is often

successful but that the computer science people receive most of the credit? A standard

example is computer tomography. No one I ever talked to had ever heard of the Radon

transform, the core of computer tomography. Even educated people think that this is the

work of computer scientists.

The point is that there is an inherent weakness in trying to justify one's concerns by

saying that they are useful. Useful is a world of engineering. Whatever you understand of

quantum mechanics (or chips or whatever), it is only understanding of formulas on piece

of paper. There is nothing useful about it. It becomes useful if it is implemented in things,

and if it becomes engineered.

Should the mathematicians go on the offensive? Should they step out into the world and

say "here we are"? Are we too reluctant to advertise our achievements?

I am a rather reclusive person and I hate imposing my views on the public. I think

whatever is good will come out anyway, although there is a general problem of selling

culture | assuming that we are producing something of cultural value. It is up to the

public to pay for it or not to pay for it. Of course, some of us probably must try to prove

that they are important, but I think it is difficult. How could Rembrandt have argued

against the fact that he was dying in total misery as a poor man? How could he argue?

4
4

4
Page 5
6

I don't really know what mathematics is all about. But this is so with culture, because

in the same way, we don't really know what Rembrandt's pictures are about, why he

portrayed persons | as he did | an old man and background. Why is it important? We

don't know. That's the problem of culture: You can not say "why".

What do you think is the cultural role of mathematics?

In my opinion, the basis of all human culture is language, and mathematics is a special

kind of linguistic activity. Natural language is an extremely flexible tool of communicat-ing

essentials required for survival, of expressing one's emotion and enforcing one's will ,

of creating virtual worlds of poetry and religion, of seduction and conviction. However,

natural language is not very well for acquiring, organizing and keeping our growing un-derstanding

of nature, which is the most characteristic trait of the modern civilization.

Aristotle was arguably the last great mind that that stretched this capability of language

to its limits. With the advent of Galileo, Kepler and Newton, the natural language in

sciences was relegated to the role of a high level mediator between the actual scientific

knowledge encoded in astronomical tables, chemical formulas, equations of quantum field

theory, databases of human genome on the one hand, and our brains on the other hand.

Using the natural language in studying and teaching sciences, we bring with it our values

and prejudices, poetical imagery, passion for power and trickster's skill, but nothing really

essential for the content of the scientific discourse. Everything that is essential, is carried

out either by long list of more or less well structured data, or by mathematics. For this

reason I believe that mathematics is is one of the most remarkable achievements of culture,

and my life-long preoccupation with mathematics in the capacity of researcher and teacher

still leaves me with awe and admiration by the end of every working day. However, I do

not believe that I can convincingly defend this conviction in the context of contemporary

public debate on science and human values.

Why are you so pessimistic?

I will start explaining my pessimism by reminding that in the current usage "culture" be-came

a profoundly self-referential word. Namely, it is taken for granted that any definition

of culture is determined by the pre-existing cultural background, even if the latter is not

made explicit. This means that no objective account and evaluation of culture is possible.

Furthermore, any statement about culture that becomes authoritative changes the public

image of culture and thus changes the culture itself. Most importantly, the modern dis-course

on culture is largely subordinate to the political discourse. We were less aware of

all this when four decades ago C. P. Snow launched the discussion of the "two cultures".

Basically, Snow was worried by the fact that in his milieu the scientific knowledge was not

considered as an organic part of the education of a cultured person, as opposite to the

Greeks and Shakespeare. Moreover, one could openly and even boastfully acknowledge

his or her image as a cultured person. Snow saw this is a result of the distorted public

perception of what constituted the actual content of culture and hoped that public debate

and reformed education could help to restore the balance.

Is the thesis of two cultures still relevant?

The relevance of this observation for us depends on our ability to identify ourself with

respect to his idealized Culture with capital C, embracing Homer and Bach, Galileo and

5
5

5
Page 6

Shakespeare, Tolstoy and Einstein. I am afraid that this ability is largely lost. In fact, the

popular idea of multiculturalism creates the image of many equally valid cultures. Grand

culture of European origin and/ or cultivation is put on a par with other regional cultures

and is diminished in stature by such pejorative connotations as cultural imperialism and

eurocentrism. Enviromentalists blames sciences and technology for the destructive uses

we made of them, thus further diminishing their cultural appeal. Ironically, the same

arguments that scientists employed in order to justify their occupation, are now turned

against them. Deconstructionist and postmodern trends of discourse put in doubt the

basic criteria of recognizing the scientific truth going back at least to Galileo and Bacon,

and try to replace them by wildly arbitrary intellectual constructions. In this way many of

the influential thinkers do not just ignore but aggressively dismiss the scientific counterpart

of the contemporary culture. I may (as I do) find this situation deplorable, but I can not

realistically count on an improvement in the foreseeable future.

Coming back to the future of mathematics, do you personally have a theory for which you

say: "If I live long enough, this is what I would like to see."?

This I do not know for the following reason: During my scientific career I have changed

my subjects several times and not so much because I found something more interesting

than something else. Basically I find everything very interesting, but there is no possibility

to do everything at the same time. The second best strategy is to try mastering several

fields in turn. Two main things I was always interested in were number theory of the

one hand and physics on the other. So I think in both domains I always tried to use the

intuition developed in both domains. Understanding problems in number theory helped

me to understand problems in physics and vice versa. On my private list of values a place

of honor is held by the Renaissance term "variet` a" | richness of life and world matched

with variety of experience and thought, achieved by great minds which we try to emulate.

Yuri I. Manin is professor at the Max-Plank-Institut f¨ ur Mathematik, Bonn.

6
6

Page Navigation Panel

1
2
3
4
5
6

(no subject)

Pages 1--6 from Untitled

Page Navigation Panel

no subject

no subject

(no subject)

(no subject)

(no subject)

(no subject)

(no subject)

(no subject)

(no subject)

(no subject)

(no subject)

(no subject)