"Mathematics is an art for the initiated"
Mar. 30th, 2016 03:44 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
posicСтатья из книжки Mathematicians: An outer view of the inner world. Photographs/portraits by Mariana Cook. Princeton University Press, 2009.
John T. Tate
Algebraic number theory
Sid W. Richardson Foundation Regents Professor of Mathematics, University of Texas, Austin, and Professor Emeritus, Harvard University
I grew up in Minneapolis as an only child. My father was an experimental physicist at the University of Minnesota. My mother knew the classics and taught high school English until I was born. My father had some books of logic and math puzzles by H. E. Dudeney which fascinated me. Although there were very few I could solve when I was a child, I liked to think about the puzzles.
I would like to express my appreciation of my father. He never pushed me, but from time to time explained some simple fundamental idea, like the fact that the distance a body falls in x seconds is proportional to x2, or how one can describe points in the plane by coordinates and describe curves by equations. He gave me a very good general idea of what science was about at an early age. I liked math and science but wasn't particularly good at arithmetic and especially hated long division drills.
In high school I read E. T. Bell's book Men of Mathematics. Each chapter is a short account of the life and works of a great mathematician. From it I learned of such wonderful things as the quadratic reciprocity law and Dirichlet's theorem on primes in arithmetic progressions. From time to time I tried to imagine how the proof might go, in vain, of course. I have always preferred to think about something myself than read what others have done. Already as a child with the puzzle books, I didn't like to look at the answers in the back, though I could have learned a lot by doing so. This extreme desire to do things myself has been a strength, but I wish it were complemented by a greater interest and ability in reading works of others. One needs a balance.
Having read in Bell's book about such people as Archimedes, Fermat, Newton, Gauss, Galois, and others, I got the idea that there was no point in being a mathematician if one weren't a genius. I knew I wasn't. I felt that wasn't true of physics because my father was a physicist, so I started graduate school at Princeton in physics. In the first year though, it became clear that math was my true love and best talent, and I was allowed to switch to math.
Princeton would have been an excellent place to do graduate work in math in any case, but it was an especially lucky choice for me because Emil Artin was there. I had never heard of him, and was astonished to learn that he had proved the ultimate generalization of the theorem which interested me most, the quadratic reciprocity law, and that the math book I had most enjoyed reading, Bartel van der Waerden's Modern Algebra, was based on lectures by him and Emmy Noether. Artin was a great mathematician who also loved teaching. He became my mentor and PhD supervisor.
My research has been mainly in number theory and algebraic geometry. Although with the advent of modern computers these subjects have become of great importance as the mathematics behind public-key cryptography and the methods of encrypted electronic communication on which modern commerce is based, I did not dream of this as a student, or during most of my life. I loved these subjects for the same reasons they have been studied for centuries: for their own intrinsic interest, for the beauty of the deep relationships which have been discovered and the challenge of finding and proving new ones. It is like a magic book of interrelated puzzles in which the solution to one reveals new pages with several more, and there are no answers in the back. This book was discovered by ancient Greeks, and their solutions to the first puzzles in it were recorded by Euclid. For example, how to see that the sequence of prime numbers 2, 3, 5, 7, 11, … does not end, or that √2 is not a rational number. By now we have come far beyond Euclid, and it is next to impossible to describe to a non-mathematician in any but the vaguest terms the solutions we have found and the puzzles we are trying to solve. It is frustrating that mathematics is an art for the initiated. In contrast to music or painting, it is hard to appreciate or enjoy at a popular level without expert knowledge.
Mathematics in itself is a cold subject, completely impersonal, with no connection to people's everyday life and emotions. The warmth in a mathematician's career comes from interactions with colleagues and students, the sharing of ideas, the sense of world-wide community. I greatly appreciate my many mathematical friends for their comradeship and for all I have learned from them.
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Только что найдено поиском в Гугле на mathematics "art for the initiated". Откуда я когда-то почерпнул полюбившуюся со временем цитату, я, к сожалению, не запомнил. Вероятно, источник был английским. Может быть, это и впрямь была эта статья Тейта. Но мне почему-то кажется, что мысль эта старинная и давно вошла в фольклор. Тем не менее, Гугль других источников не находит, так что, может быть, я ошибаюсь и это, действительно, Тейт.
John T. Tate
Algebraic number theory
Sid W. Richardson Foundation Regents Professor of Mathematics, University of Texas, Austin, and Professor Emeritus, Harvard University
I grew up in Minneapolis as an only child. My father was an experimental physicist at the University of Minnesota. My mother knew the classics and taught high school English until I was born. My father had some books of logic and math puzzles by H. E. Dudeney which fascinated me. Although there were very few I could solve when I was a child, I liked to think about the puzzles.
I would like to express my appreciation of my father. He never pushed me, but from time to time explained some simple fundamental idea, like the fact that the distance a body falls in x seconds is proportional to x2, or how one can describe points in the plane by coordinates and describe curves by equations. He gave me a very good general idea of what science was about at an early age. I liked math and science but wasn't particularly good at arithmetic and especially hated long division drills.
In high school I read E. T. Bell's book Men of Mathematics. Each chapter is a short account of the life and works of a great mathematician. From it I learned of such wonderful things as the quadratic reciprocity law and Dirichlet's theorem on primes in arithmetic progressions. From time to time I tried to imagine how the proof might go, in vain, of course. I have always preferred to think about something myself than read what others have done. Already as a child with the puzzle books, I didn't like to look at the answers in the back, though I could have learned a lot by doing so. This extreme desire to do things myself has been a strength, but I wish it were complemented by a greater interest and ability in reading works of others. One needs a balance.
Having read in Bell's book about such people as Archimedes, Fermat, Newton, Gauss, Galois, and others, I got the idea that there was no point in being a mathematician if one weren't a genius. I knew I wasn't. I felt that wasn't true of physics because my father was a physicist, so I started graduate school at Princeton in physics. In the first year though, it became clear that math was my true love and best talent, and I was allowed to switch to math.
Princeton would have been an excellent place to do graduate work in math in any case, but it was an especially lucky choice for me because Emil Artin was there. I had never heard of him, and was astonished to learn that he had proved the ultimate generalization of the theorem which interested me most, the quadratic reciprocity law, and that the math book I had most enjoyed reading, Bartel van der Waerden's Modern Algebra, was based on lectures by him and Emmy Noether. Artin was a great mathematician who also loved teaching. He became my mentor and PhD supervisor.
My research has been mainly in number theory and algebraic geometry. Although with the advent of modern computers these subjects have become of great importance as the mathematics behind public-key cryptography and the methods of encrypted electronic communication on which modern commerce is based, I did not dream of this as a student, or during most of my life. I loved these subjects for the same reasons they have been studied for centuries: for their own intrinsic interest, for the beauty of the deep relationships which have been discovered and the challenge of finding and proving new ones. It is like a magic book of interrelated puzzles in which the solution to one reveals new pages with several more, and there are no answers in the back. This book was discovered by ancient Greeks, and their solutions to the first puzzles in it were recorded by Euclid. For example, how to see that the sequence of prime numbers 2, 3, 5, 7, 11, … does not end, or that √2 is not a rational number. By now we have come far beyond Euclid, and it is next to impossible to describe to a non-mathematician in any but the vaguest terms the solutions we have found and the puzzles we are trying to solve. It is frustrating that mathematics is an art for the initiated. In contrast to music or painting, it is hard to appreciate or enjoy at a popular level without expert knowledge.
Mathematics in itself is a cold subject, completely impersonal, with no connection to people's everyday life and emotions. The warmth in a mathematician's career comes from interactions with colleagues and students, the sharing of ideas, the sense of world-wide community. I greatly appreciate my many mathematical friends for their comradeship and for all I have learned from them.
-------
Только что найдено поиском в Гугле на mathematics "art for the initiated". Откуда я когда-то почерпнул полюбившуюся со временем цитату, я, к сожалению, не запомнил. Вероятно, источник был английским. Может быть, это и впрямь была эта статья Тейта. Но мне почему-то кажется, что мысль эта старинная и давно вошла в фольклор. Тем не менее, Гугль других источников не находит, так что, может быть, я ошибаюсь и это, действительно, Тейт.
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no subject
Date: 2016-03-30 03:46 pm (UTC)no subject
Date: 2016-03-30 04:13 pm (UTC)Even in this journal, I am often publishing my math diary notes in the open, everyone can look and see. One can also follow an occasional link to the arXiv given in my or some other mathematician's journal. This is a bit better as formulas are typeset more attractively in LaTeX than in HTML. Maybe what I blog about or even publish on the arXiv lately is more boring than the average, though. Or, say, written presentation may be more boring than oral one. Listening to a conversation about etale cohomology, wild ramification, curves on algebraic surfaces etc. might be more entertaining than reading about contraderived categories of contramodules.
Are these things really comparable? Is the perception of modern art or music by the uninitiated as superficial as their perception of etale cohomology and contraderived categories? Maybe one needs to find somebody versed in both to get an intelligent answer to such a question...
no subject
Date: 2016-03-30 04:38 pm (UTC)1. to belong to something like the most mathematically talented 0.1% (if not 0.01%) of the general population;
2. something like 5 years of diligent study on top of a typical undergraduate math. major education (or, say, 4 years on top of most math Ph.D. degrees)
How many years does one need to study to just begin to appreciate some of the most complicated pieces of modern music? and how many people are just plain incapable of it, no matter how much effort they invest?
no subject
Date: 2016-03-30 07:14 pm (UTC)A lot of people study music for some time, but only very few go on to acquire a professional level in music. This is similar to mathematics: lots of people study some mathematics, but almost no one is capable of becoming a mathematician. So that's why we say "you need to be very talented to do this".
However, I don't know how to estimate the numbers. Is it 0.1% or 0.01% of people who are "talented"? For example, I think about 1% of people have absolute musical pitch, which is not necessary to become professionally successful but helps a lot. It can be that mathematics is more selective than music, in a quantitative sense (maybe 1% of people are "capable" of being musicians and only 0.1% of people are "capable" of being mathematicians). But I would say it is similar in a qualitative sense.
no subject
Date: 2016-03-30 07:32 pm (UTC)I do not know how to estimate the numbers, either, but what I think is special about mathematics is that the talents form a long scale with thick tails. Everyone has his own top abstraction level that he can climb to as a student, and then another, perhaps higher one that he gradually masters over the decades of his research life. For most research mathematicians, both levels are not very high.
E.g., most professional research mathematicians never even learn the basics of homological algebra or algebraic geometry. In fact, it seems that most algebraists don't. And among those who do, only a tiny fraction proceeds to such heights as the etale cohomology. Most of us live our lives with the uncomfortable feeling that learning this or that subject would make us considerably better mathematicians, still we cannot do it.
Still, there must be at least many tens, perhaps several hundred people in the world nowadays who know etale cohomology. Math majors who learn homological algebra and algebraic geometry in their undergraduate years are even more common. Even people with the very top achievements in original research in mathematics, by any standards, are counted at least by the tens in every generation.
no subject
Date: 2016-04-02 03:27 am (UTC)