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Tình Yêu và Toán Học: Trái Tim của Thực Tại Ẩn Giấu
"Tình yêu" bất định và "toán học" minh xác, hai thứ tưởng chừng như không thể dung hòa đã được Edward Frenkel khéo léo hòa trộn, đan bện vào nhau, dẫn dắt người đọc bước vào thế giới toán học đẹp đẽ và thanh nhã, nơi cây cầu Langlands kết nối các lục địa bí ẩn của Đại số, Hình học, Lý thuyết số, Giải tích và Vật lý lượng tử. Những "vật thể kỳ quái" trong thế giới toán học như phiến đá Rosetta, nhóm Lie, đối xứng gương hay đa tạp cờ cũng được thuần hóa bởi tình yêu thuần khiết, trở nên gần gũi đến kinh ngạc.
Tình yêu và toán học còn là cuộc hành trình cam go tìm đến tự do với góc nhìn của một nhà toán học, bức tranh về xã hội Nga đầy biến động cuối những năm 1980 - đầu những năm 1990 và xã hội tự do kiểu Mỹ hiện lên qua những mảng màu hiện thực đối lập được pha trộn và khắc họa chân thực như một tác phẩm hội họa bậc thầy.
Tình yêu và toán học còn là cuộc hành trình cam go tìm đến tự do với góc nhìn của một nhà toán học, bức tranh về xã hội Nga đầy biến động cuối những năm 1980 - đầu những năm 1990 và xã hội tự do kiểu Mỹ hiện lên qua những mảng màu hiện thực đối lập được pha trộn và khắc họa chân thực như một tác phẩm hội họa bậc thầy.
390 pages
First published October 1, 2013
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About the author
Edward Frenkel
9 books87 followersEdward Frenkel (Russian: Эдвард Френкель, Edvard Frenkel'; born May 2, 1968) is a mathematician working in representation theory, algebraic geometry, and mathematical physics. He is a professor of mathematics at University of California, Berkeley.
Frenkel grew up in Kolomna, Russia to a family of Russian Jews. As a high school student he studied higher mathematics privately with Evgeny Evgenievich Petrov, although his initial interest was in quantum physics rather than mathematics.[1] He was not admitted to Moscow State University because of discrimination against Jews and enrolled instead in the applied mathematics program at the Gubkin University of Oil and Gas. While a student there, he attended the seminar of Israel Gelfand and worked with Boris Feigin and Dmitry Fuchs. After receiving his college degree in 1989, he was first invited to Harvard University as a visiting professor, and a year later he enrolled as a graduate student at Harvard. He received his Ph.D. at Harvard University in 1991, after one year of study, under the direction of Joseph Bernstein. He was a Junior Fellow at the Harvard Society of Fellows from 1991 to 1994, and served as an associate professor at Harvard from 1994 to 1997. He has been a professor of mathematics at University of California, Berkeley since 1997.
Jointly with Boris Feigin, Frenkel constructed the free field realizations of affine Kac–Moody algebras (these are also known as Wakimoto modules), defined the quantum Drinfeld-Sokolov reduction, and described the center of the universal enveloping algebra of an affine Kac–Moody algebra. The last result, often referred to as Feigin–Frenkel isomorphism, has been used by Alexander Beilinson and Vladimir Drinfeld in their work on the geometric Langlands correspondence. Together with Nicolai Reshetikhin, Frenkel introduced deformations of W-algebras and q-characters of representations of quantum affine algebras.
Frenkel's recent work has focused on the Langlands program and its connections to representation theory, integrable systems, geometry, and physics. Together with Dennis Gaitsgory and Kari Vilonen, he has proved the geometric Langlands conjecture for GL(n). His joint work with Robert Langlands and Ngô Bảo Châu suggested a new approach to the functoriality of automorphic representations and trace formulas. He has also been investigating (in particular, in a joint work with Edward Witten) connections between the geometric Langlands correspondence and dualities in quantum field theory.
Frenkel has co-produced, co-directed (with Reine Graves) and played the lead in a short film "Rites of Love and Math", a homage to the film "Rite of Love and Death" (also known as "Yûkoku") by the Japanese writer Yukio Mishima. The film premiered in Paris in April, 2010 and was in the official competition of the Sitges International Film Festival in October, 2010. The screening of "Rites of Love and Math" in Berkeley on December 1, 2010 caused some controversy.
Frenkel's book Love and Math The Heart of Hidden Reality was published in October 2013.
Frenkel grew up in Kolomna, Russia to a family of Russian Jews. As a high school student he studied higher mathematics privately with Evgeny Evgenievich Petrov, although his initial interest was in quantum physics rather than mathematics.[1] He was not admitted to Moscow State University because of discrimination against Jews and enrolled instead in the applied mathematics program at the Gubkin University of Oil and Gas. While a student there, he attended the seminar of Israel Gelfand and worked with Boris Feigin and Dmitry Fuchs. After receiving his college degree in 1989, he was first invited to Harvard University as a visiting professor, and a year later he enrolled as a graduate student at Harvard. He received his Ph.D. at Harvard University in 1991, after one year of study, under the direction of Joseph Bernstein. He was a Junior Fellow at the Harvard Society of Fellows from 1991 to 1994, and served as an associate professor at Harvard from 1994 to 1997. He has been a professor of mathematics at University of California, Berkeley since 1997.
Jointly with Boris Feigin, Frenkel constructed the free field realizations of affine Kac–Moody algebras (these are also known as Wakimoto modules), defined the quantum Drinfeld-Sokolov reduction, and described the center of the universal enveloping algebra of an affine Kac–Moody algebra. The last result, often referred to as Feigin–Frenkel isomorphism, has been used by Alexander Beilinson and Vladimir Drinfeld in their work on the geometric Langlands correspondence. Together with Nicolai Reshetikhin, Frenkel introduced deformations of W-algebras and q-characters of representations of quantum affine algebras.
Frenkel's recent work has focused on the Langlands program and its connections to representation theory, integrable systems, geometry, and physics. Together with Dennis Gaitsgory and Kari Vilonen, he has proved the geometric Langlands conjecture for GL(n). His joint work with Robert Langlands and Ngô Bảo Châu suggested a new approach to the functoriality of automorphic representations and trace formulas. He has also been investigating (in particular, in a joint work with Edward Witten) connections between the geometric Langlands correspondence and dualities in quantum field theory.
Frenkel has co-produced, co-directed (with Reine Graves) and played the lead in a short film "Rites of Love and Math", a homage to the film "Rite of Love and Death" (also known as "Yûkoku") by the Japanese writer Yukio Mishima. The film premiered in Paris in April, 2010 and was in the official competition of the Sitges International Film Festival in October, 2010. The screening of "Rites of Love and Math" in Berkeley on December 1, 2010 caused some controversy.
Frenkel's book Love and Math The Heart of Hidden Reality was published in October 2013.
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April 12, 2023
Idealised Objects
Love makes us say and do silly things. But without love worse thing happen. So I can’t fault Frenkel for his loving devotion to his subject. Nevertheless what he says is often silly. And he needn’t say it in order to get his point across: math (or ‘maths’ for those in the Mother Country) is beautiful.
Here’s the love note from his introduction: “Math is a way to describe reality and figure out how the world works, a universal language that has become the gold standard of truth.” If math is a description of anything (and that is controversial), it is of numbers and their relationships, certainly not of reality. It tells us how numbers work with each other, often in surprising ways. But ‘the world’ is a big place and math doesn’t say much about how love works, among much else.
And while math does rely a great deal on the criterion of truth, what it means by truth is severely restricted. Truth in math is a definitional phenomenon which concerns numbers and their relationships and nothing else. Even then, math has to pretend that certain untruths about itself don’t exist (the name Gödel should not be mentioned in company). So the beloved has several imperfections that the lover doesn’t notice.
The language of mathematics is indeed unlike any other language.* This is because each element of it - its vocabulary and its grammar - is precisely and invariably defined. They are “idealised objects.” Everything about them is known because they are their definition tout court. In this sense at least math is entirely artificial. There is no need for a dictionary because all the words (numbers) have fixed relationships to one another. There is no ambiguity in the specification of prime numbers, for example. There are no ‘dialects’ in which a number is possibly prime or not. There is no variation in the relations among numbers despite the different notations used in 17th century France and 21st century Russia.
But like all languages, the language of mathematics is entirely ‘closed.’ That is, everyone of its components can only be described by other of its components. There is no connection, for example, between the number 1 and anything outside the mathematical language in which 1 is defined. There is no 1 in non-language. This becomes absolutely clear when dealing with numbers that are negative or irrational like the square root of 2 (the hypotenuse of the unit triangle), which simply cannot exist except in mathematical language.
Like many other mathematicians from the ancient Greeks onward, Frenkel tends toward the reification of numbers as things which exist independently of their definitions in language: “... mathematical concepts exist in a world separate from the physical and mental worlds – which is sometimes referred to as the Platonic world of mathematics” This is to some degree understandable. All language is mysterious regarding where it comes from and where it ‘lives’ when not being used. And for Frenkel, the language of mathematics has personally important associations: it provided a refuge in the oppressive intellectual regime of the Soviet Union, and eventually a way to escape the memory of that oppression.
Consequently, like many lovers, Frenkel considers his beloved as something effectively supernatural. Math is more real to him than is anything that is not math.** Math is an ideal world, indeed he calls it a “parallel world” of infinite possibility, beautiful symmetry, and perfect stability - in a word: heaven. He yearns for this world as much as any believer yearns for the next life, or nirvana, or the face of God. He all but asserts the divine character of mathematics: “Mathematics directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars.”
But Frenkel’s breathless praise of mathematics is both unnecessary and counter-productive. Unnecessary because the beauty of mathematics doesn’t need what amounts to idolatry to inspire admiration. And counter-productive because it makes mathematics into a quasi-religion which threatens to be as arrogantly smug and self-satisfied as any other religion. He is quite right to claim that “Mathematics is a way to break the barriers of the conventional, an expression of unbounded imagination in the search for truth.” And so is poetry, sculpture, or, for that matter, writing book reviews about hyperbolic math books.
* In fact there are a large number of mathematical languages. The Langlands Project, in which Frenkel is involved is committed to translating among them in order to consolidate mathematical advances across sub-disciplines.
** For example, Frenkel’s observation about the difficulty of conceiving of the square root of a negative number is telling: “The reason is purely psychological: whereas we can represent as the length of a side of a right triangle, we don’t have such an obvious geometric representation of the square root of minus 1. But we can manipulate the square root of minus 1 algebraically as effectively as the square root of 2.” He is wrong in both his analogy and his conclusion. But for him this is the ‘real world.’
Love makes us say and do silly things. But without love worse thing happen. So I can’t fault Frenkel for his loving devotion to his subject. Nevertheless what he says is often silly. And he needn’t say it in order to get his point across: math (or ‘maths’ for those in the Mother Country) is beautiful.
Here’s the love note from his introduction: “Math is a way to describe reality and figure out how the world works, a universal language that has become the gold standard of truth.” If math is a description of anything (and that is controversial), it is of numbers and their relationships, certainly not of reality. It tells us how numbers work with each other, often in surprising ways. But ‘the world’ is a big place and math doesn’t say much about how love works, among much else.
And while math does rely a great deal on the criterion of truth, what it means by truth is severely restricted. Truth in math is a definitional phenomenon which concerns numbers and their relationships and nothing else. Even then, math has to pretend that certain untruths about itself don’t exist (the name Gödel should not be mentioned in company). So the beloved has several imperfections that the lover doesn’t notice.
The language of mathematics is indeed unlike any other language.* This is because each element of it - its vocabulary and its grammar - is precisely and invariably defined. They are “idealised objects.” Everything about them is known because they are their definition tout court. In this sense at least math is entirely artificial. There is no need for a dictionary because all the words (numbers) have fixed relationships to one another. There is no ambiguity in the specification of prime numbers, for example. There are no ‘dialects’ in which a number is possibly prime or not. There is no variation in the relations among numbers despite the different notations used in 17th century France and 21st century Russia.
But like all languages, the language of mathematics is entirely ‘closed.’ That is, everyone of its components can only be described by other of its components. There is no connection, for example, between the number 1 and anything outside the mathematical language in which 1 is defined. There is no 1 in non-language. This becomes absolutely clear when dealing with numbers that are negative or irrational like the square root of 2 (the hypotenuse of the unit triangle), which simply cannot exist except in mathematical language.
Like many other mathematicians from the ancient Greeks onward, Frenkel tends toward the reification of numbers as things which exist independently of their definitions in language: “... mathematical concepts exist in a world separate from the physical and mental worlds – which is sometimes referred to as the Platonic world of mathematics” This is to some degree understandable. All language is mysterious regarding where it comes from and where it ‘lives’ when not being used. And for Frenkel, the language of mathematics has personally important associations: it provided a refuge in the oppressive intellectual regime of the Soviet Union, and eventually a way to escape the memory of that oppression.
Consequently, like many lovers, Frenkel considers his beloved as something effectively supernatural. Math is more real to him than is anything that is not math.** Math is an ideal world, indeed he calls it a “parallel world” of infinite possibility, beautiful symmetry, and perfect stability - in a word: heaven. He yearns for this world as much as any believer yearns for the next life, or nirvana, or the face of God. He all but asserts the divine character of mathematics: “Mathematics directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars.”
But Frenkel’s breathless praise of mathematics is both unnecessary and counter-productive. Unnecessary because the beauty of mathematics doesn’t need what amounts to idolatry to inspire admiration. And counter-productive because it makes mathematics into a quasi-religion which threatens to be as arrogantly smug and self-satisfied as any other religion. He is quite right to claim that “Mathematics is a way to break the barriers of the conventional, an expression of unbounded imagination in the search for truth.” And so is poetry, sculpture, or, for that matter, writing book reviews about hyperbolic math books.
* In fact there are a large number of mathematical languages. The Langlands Project, in which Frenkel is involved is committed to translating among them in order to consolidate mathematical advances across sub-disciplines.
** For example, Frenkel’s observation about the difficulty of conceiving of the square root of a negative number is telling: “The reason is purely psychological: whereas we can represent as the length of a side of a right triangle, we don’t have such an obvious geometric representation of the square root of minus 1. But we can manipulate the square root of minus 1 algebraically as effectively as the square root of 2.” He is wrong in both his analogy and his conclusion. But for him this is the ‘real world.’
November 29, 2013
I'd give this book a better rating if it advertised itself as the biography of a Jewish mathematician who faced discrimination in the Soviet Union, or as a pop-math book about the applications of pure math in theoretical physics. But no, Frenkel claims that he intends to dazzle you with so much mathematical beauty that it will make you - you, as in, you the layman reader - fall in love with math. I'm afraid this book will do no such thing.
September 27, 2016
I wanted so much to love this book, but it was difficult. About half of the book is about Frenkel's life; and it was fascinating. The other half, interleaved with his memoirs, are descriptions of Frenkel's mathematical work and discoveries. I had a great deal of trouble following the descriptions of the math. I am superficially familiar with many of the concepts, but it just gets more and more complex. Toward the end, especially, I became quite confused.
Frenkel grew up in a small town in the Soviet Union, a two-hour train ride from Moscow. Initially interested in physics, Frenkel's father introduced him to a mentor who showed him how modern physics, especially quantum mechanics, relies on some very modern concepts in mathematics. In that way, Frenkel got hooked on math.
When he was in high school, starting to think about college, Frenkel applied to the math department at Moscow State University. It was explained to him that he had no chance of being admitted. He had a Jewish last name, and anti-Semitism would prevent his admission. Frenkel applied for admission anyway, and he was grilled mercilessly during an oral entrance exam. The examiners found excuses to refuse him admission.
So, Frenkel went to undergraduate school at a different college, one that did not discriminate so much. Nevertheless, he attended lectures and seminars at Moscow State University. He didn't have a college ID, so he scaled the fence to get in! The Soviets put tight controls on photocopiers. While in undergraduate school, his research papers were secretly copied and smuggled out, and reached mathematicians around the world. One day, Frenkel received an invitation from the president of Harvard University to come to Harvard on a fellowship grant, and become an assistant professor. At that point in time, he didn't even have a PhD!
What I did get out of the math descriptions, is the inner beauty of math. Vastly different areas of math can be connected, through hidden connections, as if by magic. This attracted Frenkel to the Langlands Program, a grand unified theory of mathematics. Now a vast subject, the program tries to connect number theory, harmonic analysis, geometry, representation theory, and mathematical physics. Riemann geometry is the cornerstone of Einstein's Theory of General Relativity. It contains hidden connections with number theory. Frenkel's career goal is to establish connections between the dualities in physics and the dualities in mathematics.
In the last chapter, Frenkel spent a sabbatical in France, writing and producing a short film. He also co-stars in the film, titled Rites of Love and Math. The trailer is on a site on YouTube.com. In this allegorical film, a mathematician discovers a mathematical formula for love. He realizes the formula's importance, and that it could be used for good as well as for evil. He tattoos the formula on his lover's body. The film was screened at many film festivals to wide acclaim, but also received a lot of controversy.
You can read the book and skim or skip through the math. Frenkel does a good job of describing the hidden connections and beauty of math. But he leaves the average reader lost in the details.
Frenkel grew up in a small town in the Soviet Union, a two-hour train ride from Moscow. Initially interested in physics, Frenkel's father introduced him to a mentor who showed him how modern physics, especially quantum mechanics, relies on some very modern concepts in mathematics. In that way, Frenkel got hooked on math.
When he was in high school, starting to think about college, Frenkel applied to the math department at Moscow State University. It was explained to him that he had no chance of being admitted. He had a Jewish last name, and anti-Semitism would prevent his admission. Frenkel applied for admission anyway, and he was grilled mercilessly during an oral entrance exam. The examiners found excuses to refuse him admission.
So, Frenkel went to undergraduate school at a different college, one that did not discriminate so much. Nevertheless, he attended lectures and seminars at Moscow State University. He didn't have a college ID, so he scaled the fence to get in! The Soviets put tight controls on photocopiers. While in undergraduate school, his research papers were secretly copied and smuggled out, and reached mathematicians around the world. One day, Frenkel received an invitation from the president of Harvard University to come to Harvard on a fellowship grant, and become an assistant professor. At that point in time, he didn't even have a PhD!
What I did get out of the math descriptions, is the inner beauty of math. Vastly different areas of math can be connected, through hidden connections, as if by magic. This attracted Frenkel to the Langlands Program, a grand unified theory of mathematics. Now a vast subject, the program tries to connect number theory, harmonic analysis, geometry, representation theory, and mathematical physics. Riemann geometry is the cornerstone of Einstein's Theory of General Relativity. It contains hidden connections with number theory. Frenkel's career goal is to establish connections between the dualities in physics and the dualities in mathematics.
In the last chapter, Frenkel spent a sabbatical in France, writing and producing a short film. He also co-stars in the film, titled Rites of Love and Math. The trailer is on a site on YouTube.com. In this allegorical film, a mathematician discovers a mathematical formula for love. He realizes the formula's importance, and that it could be used for good as well as for evil. He tattoos the formula on his lover's body. The film was screened at many film festivals to wide acclaim, but also received a lot of controversy.
You can read the book and skim or skip through the math. Frenkel does a good job of describing the hidden connections and beauty of math. But he leaves the average reader lost in the details.
October 28, 2016
I loved this book and found it riveting. My parents gave it to me last Christmas, and I avoided reading it because the cover and description didn't give many clues as to the content. (Sadly, I DO judge books by their covers.) I'd also never heard of Frenkel (yikes, that's embarrassing). Once I cracked it open, I was hooked. Equal parts autobiography and "mathematical-research-y," it was a fascinating true story of discrimination against Jews in Russia, Frenkel's journey to becoming a world-renowned mathematician, and an introduction to the exciting world of modern math (the Langlands program).
I would have given this 5 stars. However, I think Frenkel severely overestimates the math knowledge of the average American reader. I'm not sure why, as he's an experienced professor at Berkeley. As someone who is working on my master's in math, I caught on to the phrasing and style. I was with him through Galois groups and Riemann surfaces (mostly), but he lost me somewhere between Lie algebras and quantum physics.
As a college math instructor, I'd be willing to bet that 99% of my students would have put down this book in frustration after a few chapters. Someone who is not familiar with the language of formal proofs is going to be lost amidst phrases like "suppose, to the contrary" and "opening the brackets" and discussion of 2D analogues to problems of higher dimension. Frenkel tries to confine the math details to the notes in the back of the book, but even the main content is likely to turn away the average reader from a quite fantastic book.
And that's a tragedy. Because this is a timely book, written engagingly by a brilliant man. Last semester, while teaching an introductory unit on Number Theory, I referenced the recent proof of the "bounded gaps" conjecture. I actually had a student comment "Oh, that's so cool! I had no idea people were still working on math! I thought everything was finished - like, it was all proved and done!" I'd like to rewind time and hand her this book.
For all my students who constantly question, "Why do we need math?," I'd like to hand them this book and make them read Chapter 12. In fact, I think that's going to be required reading next semester. :) Frenkel explains his work analyzing data for a doctor. His patients' systems were not accepting transplanted kidneys, and he needed to find a way to objectively decide whether or not to try to save the kidney or remove it. Frenkel analyzed a large volume of data and used his mathematician's ability to see general patterns from specific cases, and he created a simple algorithm which gave a good diagnosis with only 4 questions. That's math, working to make a life-and-death decision process more effective. Tell me that's not fascinating!
I'll finish my lengthy review with perhaps my favorite thought from the book:
I would have given this 5 stars. However, I think Frenkel severely overestimates the math knowledge of the average American reader. I'm not sure why, as he's an experienced professor at Berkeley. As someone who is working on my master's in math, I caught on to the phrasing and style. I was with him through Galois groups and Riemann surfaces (mostly), but he lost me somewhere between Lie algebras and quantum physics.
As a college math instructor, I'd be willing to bet that 99% of my students would have put down this book in frustration after a few chapters. Someone who is not familiar with the language of formal proofs is going to be lost amidst phrases like "suppose, to the contrary" and "opening the brackets" and discussion of 2D analogues to problems of higher dimension. Frenkel tries to confine the math details to the notes in the back of the book, but even the main content is likely to turn away the average reader from a quite fantastic book.
And that's a tragedy. Because this is a timely book, written engagingly by a brilliant man. Last semester, while teaching an introductory unit on Number Theory, I referenced the recent proof of the "bounded gaps" conjecture. I actually had a student comment "Oh, that's so cool! I had no idea people were still working on math! I thought everything was finished - like, it was all proved and done!" I'd like to rewind time and hand her this book.
For all my students who constantly question, "Why do we need math?," I'd like to hand them this book and make them read Chapter 12. In fact, I think that's going to be required reading next semester. :) Frenkel explains his work analyzing data for a doctor. His patients' systems were not accepting transplanted kidneys, and he needed to find a way to objectively decide whether or not to try to save the kidney or remove it. Frenkel analyzed a large volume of data and used his mathematician's ability to see general patterns from specific cases, and he created a simple algorithm which gave a good diagnosis with only 4 questions. That's math, working to make a life-and-death decision process more effective. Tell me that's not fascinating!
I'll finish my lengthy review with perhaps my favorite thought from the book:
(Pg. 188, bold emphasis mine.)This area appeared to be pure and abstract, without immediate applications. But we have to realize that fundamental scientific research forms the basis of all technological progress. Often, what looked like the most abstract and abstruse discoveries in math and physics subsequently led to innovations that we now use in our everyday life. Think of the arithmetic modulo primes, for example. When we see it for the first time it looks so abstract that it seems impossible something like this could have any real world applications....But...many apparently esoteric results in number theory...are now ubiquitous in, say, online banking....We should never try to prejudge the potential of a mathematical formula or idea for practical applications.
History shows that all spectacular technological breakthroughs were preceded, often decades earlier, by advances in pure research. Therefore, if we limit support for basic science, we limit our progress and power.
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November 28, 2013If you're a math geek, Love and Math will be up your alley. If you're a math geek who is Jewish and whose parents (or you) come from Eastern Europe, it's probably a must read. Love and Math is a hybrid book, kind of a mix between Surely You're Joking, Mr. Feynman and George Polya's How To Solve It. The Feynman type bits - Frenkel dealing with anti-Semitism in Russia and making an erotic math film - are a mix of amusement and pathos. But mostly this book is an autobiography of math problems examined in the career of a top flight mathematician. The problems are explained succinctly and lucidly. If you aren't a math type, they'll be over your head. If you are a math type, this book is an intellectual treasure trove for casual reading. I note (not advertising, mind you) that when I was working on my novel about a Polish mathematician and her family (out next fall, but really I'm not advertising, that would be in bad taste), I read a fair number of math autobiographies while looking for a proper tone for the narrator. I settled on using Polya's book as a guide. Had this book been published a couple of years ago, it would have influenced my writing.
July 21, 2018
Uma discussão legal sobre a matemática por trás da física moderna, que gira em torno da auto-biografia do autor. Muito legal ouvir o que era alguém judeu se formar em matemática na época da União Soviética. Acabei apreciando mais pela história do que pelos conceitos passados. Já que perdi bastante por ouvir o audiolivro, que não dá espaço nem para as fórmulas e as demonstrações, nem para me concentrar o suficiente para entender bem o conteúdo (sem contar a minha limitação com matemática em geral).
November 6, 2015
The general tone of negative and middling reviews for this book suggest why it is (especially in countries like the United States) that while popular science seems to be embraced more and more by the masses, actual scientific and mathematical literary seems to be on the decline or finds itself at least consistently below average.
The autobiographical part of the book weaves through the mathematics well enough and is very interesting in its own right, from its reflections on Russian mathematics to the antisemitism implicit in Soviet university administration that the author was directly confronted with on his academic journey. As for the maths, you'd think reading such an earnest and painstakingly clarified account of the forefront of mathematical research (producing at least a couple of Fields Medals in the last few years) merits taking out paper and pencil to try and reproduce some of the author's arguments (mostly verifying basic properties), or at least a careful read of the chapter notes. Apparently not!- it's bite-sized memes and pretty pictures or nothing (as with popular science on television and certain intelligent-seeming books by Gladwell and the like; on the other hand some sources try not to compromise too much; for instance the Numberphile Youtube channel on which Frenkel has had appearances).
This response is disappointing, but I still think a keen high school student (and not necessarily as keen as Frenkel appears to have been) or even a not so keen one might find something beautiful about mathematics in this book and be persuaded to follow it (likely with the odds considerably more in their favour than for Frenkel in Soviet Russia). Certainly Frenkel's greatest achievement here is communicating as simply as possible (but making it no simpler) the essential motivation behind Galois groups, representations, Lie theory and Langlands duality. There's also a compelling bird's eye view from his experience working on the programme, but the returns for the average reader are greatest when it comes to the motivations and basic definitions and arguments. Such persuasion would be a considerable enough ambition fulfilled, whatever the book's success with a lay audience not willing to deviate the slightest from the expectation of a novel-like experience whatever the gains that might be in store.
Edited to add:
Having written this review, I was further disheartened to read this blog post-
http://www.dam.brown.edu/people/mumfo...
Nature apparently found this article unnatural (for scientists!) and rejected it. Honestly, I didn't get a nuanced picture of what Grothendieck's work was all about from this, but that's not the point- the basic ideas, in spite of the almost unavoidable jargon (given the length), do get through to a non-trivial degree and what's most valuable about a piece like this is having experts in a notoriously difficult field deign to simplify their eagle's eye view for the public- the least the public could do, especially the scientific public, is make an effort!
The autobiographical part of the book weaves through the mathematics well enough and is very interesting in its own right, from its reflections on Russian mathematics to the antisemitism implicit in Soviet university administration that the author was directly confronted with on his academic journey. As for the maths, you'd think reading such an earnest and painstakingly clarified account of the forefront of mathematical research (producing at least a couple of Fields Medals in the last few years) merits taking out paper and pencil to try and reproduce some of the author's arguments (mostly verifying basic properties), or at least a careful read of the chapter notes. Apparently not!- it's bite-sized memes and pretty pictures or nothing (as with popular science on television and certain intelligent-seeming books by Gladwell and the like; on the other hand some sources try not to compromise too much; for instance the Numberphile Youtube channel on which Frenkel has had appearances).
This response is disappointing, but I still think a keen high school student (and not necessarily as keen as Frenkel appears to have been) or even a not so keen one might find something beautiful about mathematics in this book and be persuaded to follow it (likely with the odds considerably more in their favour than for Frenkel in Soviet Russia). Certainly Frenkel's greatest achievement here is communicating as simply as possible (but making it no simpler) the essential motivation behind Galois groups, representations, Lie theory and Langlands duality. There's also a compelling bird's eye view from his experience working on the programme, but the returns for the average reader are greatest when it comes to the motivations and basic definitions and arguments. Such persuasion would be a considerable enough ambition fulfilled, whatever the book's success with a lay audience not willing to deviate the slightest from the expectation of a novel-like experience whatever the gains that might be in store.
Edited to add:
Having written this review, I was further disheartened to read this blog post-
http://www.dam.brown.edu/people/mumfo...
Nature apparently found this article unnatural (for scientists!) and rejected it. Honestly, I didn't get a nuanced picture of what Grothendieck's work was all about from this, but that's not the point- the basic ideas, in spite of the almost unavoidable jargon (given the length), do get through to a non-trivial degree and what's most valuable about a piece like this is having experts in a notoriously difficult field deign to simplify their eagle's eye view for the public- the least the public could do, especially the scientific public, is make an effort!
February 2, 2024
There I was, after a discussion with a "Arzt" pondering Frenkels' "Love and Math: The Heart of Hidden Reality." Some known methods can and may be used to redirect those in need of relief from non life threatening concerns. This book “Love and Math” is an ode to illuminating Frenkel’s work, via the Langlands program, a “grand unified theory of mathematics.”Such states that hard questions are solved using harmonic analysis (number theory). Frankel helps us to comprehend via simple polynomial equations as X²=2. There are two solutions that are not surprisingly irrational numbers.
"Tools can be used for good and ill...forcing us to reckon with math's real world effects... The essence in mathematics lies in freedom."
--Edward Frenkel
With this nouveau understanding we now have a way to solve problems in number theory (that did not previously exist). By this act we ascertain the apparent deep and fundamental connections living between the varied fields in mathematics. Finding patterns may be key to something as “the same spooky patterns” hiding in areas of mathematics like: quantum physics and geometry.
"Tools can be used for good and ill...forcing us to reckon with math's real world effects... The essence in mathematics lies in freedom."
--Edward Frenkel
With this nouveau understanding we now have a way to solve problems in number theory (that did not previously exist). By this act we ascertain the apparent deep and fundamental connections living between the varied fields in mathematics. Finding patterns may be key to something as “the same spooky patterns” hiding in areas of mathematics like: quantum physics and geometry.
This entire review has been hidden because of spoilers.
November 11, 2016
This is, comfortably, the best popular math book I’ve ever had the fortune to lay my eyes on.
I have a couple degrees in the subject. One in applied math, that I studied in college, and one in pure math that I got twelve years later. Indeed, I coincided with the author at Harvard, and his description of the Math department in the Science Center, the ping pong table and that hidden gem of a library brought back memories of my first semester in college, which was largely spent poring over impossible math assignments. (It also reminded me of the rather poor personal hygiene of some of the guys there, which goes unmentioned in the book!) This is now my favorite popular math book.
I don’t know math anymore, but I regardless LOVED LOVED LOVED reading this. The man has an unbelievable gift for explaining stuff and flattering you into thinking you understand it too. I can probably now blag about SO(3) and SU(2) and SU(3) and fibres and sheaves and fundamental groups with the best of them and I feel like I own the material (which of course I don’t.)
For three days of my life I did math with Grothendieck and Gelfand, Drinfeld and Kac, basically, I sat there at the table with them. It was a massive high!
A couple months ago, to resolve a dispute with an almost equally spent former mathematician, I wasted hours scouring the Internet to remind myself what the real meaning of the cross product is, to no avail. Frenkel slips the answer en passant on page 121: the tangent space to the unity element of a Lie group is a simpler construct called a Lie algebra. The Lie algebra of an n-dimensional Lie group is a vector space of dimension n, basically. Well, the operation under which 3D space is such an algebra is the cross product. Cool, eh?
And so on. You close your eyes and you follow this boy, who grew up as a little dreamer in a city two hours out of Moscow and just wanted to know how the universe works. And you meet his teachers and take part in his search as he tries to solve the problem that consumed Einstein in the last 20 years of his life, that of coming up with a single theory that can reach beyond the “standard model” and also cover gravitation. You start with his first ever big triumph (something to do with the “knot groups,” which are explained beautifully) and you move on from there to the most intuitive ever explanation of Galois’ insight regarding the solution of polynomials (basically drawing an analogy between irrational numbers and imaginary numbers I wish somebody had told me about 30 years ago) and eventually you make it to the “Langlands problem” which I must admit went 101% over my head.
But I had enough there to cheer the author on, all the way.
The bit about the “equation of love” was beyond weird, of course, but I’m alright with it. Nobody forced me to read it. It was a bit as if somebody cooks you the best meal in history and then offers you the option of flamingo feather pie as desert: you don’t have to eat it. It did not undo the amazing first 228 pages, or that phenomenal set of notes in the back.
Also, the guy loves math. It’s so evident, it’s infectious. And he gives you a very long warning that you cannot judge math from the abject crap that passes for math in the world’s high school curricula. He hated that too, despite the fact that he could do it. Math is the amazing construct that, whether we like it or not, surrounds us, underpins the physical world and is waiting for us to go, discover it and get the answers we seek.
Chris Isham aside (who does not know me from Adam, but I once took his class), this is the most a man has ever inspired me to read math, and he did it from a book. What can I say?
I’m not sure I’ve had more fun reading a book, frankly. Did not really learn anything, it would be beyond presumptuous to say so, but It flattered me so hard, I got a proper high.
Thank you, Bernard, for suggesting this!
I have a couple degrees in the subject. One in applied math, that I studied in college, and one in pure math that I got twelve years later. Indeed, I coincided with the author at Harvard, and his description of the Math department in the Science Center, the ping pong table and that hidden gem of a library brought back memories of my first semester in college, which was largely spent poring over impossible math assignments. (It also reminded me of the rather poor personal hygiene of some of the guys there, which goes unmentioned in the book!) This is now my favorite popular math book.
I don’t know math anymore, but I regardless LOVED LOVED LOVED reading this. The man has an unbelievable gift for explaining stuff and flattering you into thinking you understand it too. I can probably now blag about SO(3) and SU(2) and SU(3) and fibres and sheaves and fundamental groups with the best of them and I feel like I own the material (which of course I don’t.)
For three days of my life I did math with Grothendieck and Gelfand, Drinfeld and Kac, basically, I sat there at the table with them. It was a massive high!
A couple months ago, to resolve a dispute with an almost equally spent former mathematician, I wasted hours scouring the Internet to remind myself what the real meaning of the cross product is, to no avail. Frenkel slips the answer en passant on page 121: the tangent space to the unity element of a Lie group is a simpler construct called a Lie algebra. The Lie algebra of an n-dimensional Lie group is a vector space of dimension n, basically. Well, the operation under which 3D space is such an algebra is the cross product. Cool, eh?
And so on. You close your eyes and you follow this boy, who grew up as a little dreamer in a city two hours out of Moscow and just wanted to know how the universe works. And you meet his teachers and take part in his search as he tries to solve the problem that consumed Einstein in the last 20 years of his life, that of coming up with a single theory that can reach beyond the “standard model” and also cover gravitation. You start with his first ever big triumph (something to do with the “knot groups,” which are explained beautifully) and you move on from there to the most intuitive ever explanation of Galois’ insight regarding the solution of polynomials (basically drawing an analogy between irrational numbers and imaginary numbers I wish somebody had told me about 30 years ago) and eventually you make it to the “Langlands problem” which I must admit went 101% over my head.
But I had enough there to cheer the author on, all the way.
The bit about the “equation of love” was beyond weird, of course, but I’m alright with it. Nobody forced me to read it. It was a bit as if somebody cooks you the best meal in history and then offers you the option of flamingo feather pie as desert: you don’t have to eat it. It did not undo the amazing first 228 pages, or that phenomenal set of notes in the back.
Also, the guy loves math. It’s so evident, it’s infectious. And he gives you a very long warning that you cannot judge math from the abject crap that passes for math in the world’s high school curricula. He hated that too, despite the fact that he could do it. Math is the amazing construct that, whether we like it or not, surrounds us, underpins the physical world and is waiting for us to go, discover it and get the answers we seek.
Chris Isham aside (who does not know me from Adam, but I once took his class), this is the most a man has ever inspired me to read math, and he did it from a book. What can I say?
I’m not sure I’ve had more fun reading a book, frankly. Did not really learn anything, it would be beyond presumptuous to say so, but It flattered me so hard, I got a proper high.
Thank you, Bernard, for suggesting this!
June 8, 2015
update : i finally gave up on this book. i just cannot make myself read any more of it.
original review in progress: So I am slogging my way through this new memoir aptly named 'Love and Math' from a Russian Jewish mathematician. I haven't gotten to any bits about love between people yet (I was hoping for some romance) - it's mostly about his love OF math with bits of details on the deep antisemitism that used to be present in Russia. Starting to realize math geeks are a bit like dirty hippies what with their "the beauty of the universe man; the melody of the interrelated number systems" etc etc. Some of the math he writes about is way over my head (I couldn't follow his discussion on braids at all this morning despite rereading it 3 times) and although he presents his ideas in a conversational tone like Brian Greene does for physics (I love how BG makes me feel like I "get" physics; it makes me feel smart!) he's not as good as translating to common man speak as Brian is. Mostly i can tell you that the guy has a deep deep passionate love for math that borders on nuts. You ever seen that movie where Ryan Gosling buys himself a RealDoll silicone woman and pretends it is real (if you haven't you should b/c it is freaky and awesome and sweet and beautiful)? Yeah this guy is like that about math. :p
original review in progress: So I am slogging my way through this new memoir aptly named 'Love and Math' from a Russian Jewish mathematician. I haven't gotten to any bits about love between people yet (I was hoping for some romance) - it's mostly about his love OF math with bits of details on the deep antisemitism that used to be present in Russia. Starting to realize math geeks are a bit like dirty hippies what with their "the beauty of the universe man; the melody of the interrelated number systems" etc etc. Some of the math he writes about is way over my head (I couldn't follow his discussion on braids at all this morning despite rereading it 3 times) and although he presents his ideas in a conversational tone like Brian Greene does for physics (I love how BG makes me feel like I "get" physics; it makes me feel smart!) he's not as good as translating to common man speak as Brian is. Mostly i can tell you that the guy has a deep deep passionate love for math that borders on nuts. You ever seen that movie where Ryan Gosling buys himself a RealDoll silicone woman and pretends it is real (if you haven't you should b/c it is freaky and awesome and sweet and beautiful)? Yeah this guy is like that about math. :p
March 23, 2019
An overly ambitious attempt by a brilliant mathematician to inculcate his love for the subject into a lay audience. The topic is modern mathematics, beyond the usual math we have mostly been exposed to in school. When he says "algebra" he means abstract algebra. Similarly for "group," "field," "category" and on up to "strings" and "sheaves." He briefly describes these objects, but sentences with ten unfamiliar terms in them are just too much for me. His topic is The Langlands Project -- also new to me -- and he vastly overestimates his ability to give an understandable overview of its content, beyond its aim of trying to unify areas of modern math and modern physics through ultra-modern research.
July 22, 2014
Edward Frenkel needs both a better agent and a better editor.
A better agent would have had a heart-to-heart talk with him about his objectives in writing this book. Frenkel claims he wants to inspire a new generation of mathematicians, particularly by bridging the "two cultures" between the arts and science by using the powers of love and math. That agent might have advised him that, since his work sorely needs the power of visualization, he hook up with a comic book artist--someone who knows the power of image and story combined--and create something along the lines of Larry Gonick's Cartoon History and Cartoon Guide series. (For an excellent example of cartoons in the service of teaching mathematical proof, see Gonick's wonderful single-pager on the irrationality of the square root of 2 in the Cartoon History of the Universe v1-7.) He could have then animated the story and placed it online or presented a summary in a TED talk, where I'm certain it would have gone viral.
A better editor would have forced Frenkel to weave his personal and professional memoirs more tightly around a unified theme, as Eric Kandel did in his stunning "In Search of Memory". That editor would have recommended that he use more accessible phrasing for some terms, like "hula hoop" instead of "ring used in rhythmic gymnastics." It turns out that Frenkel perfected his English by watching David Letterman. That's unfortunate; Letterman is a genius of the absurd, but he never showcased popular explainers and debunkers like Sagan and Randi, as Carson did. Perhaps if Frenkel had seen those masters, he would have learned better how to harness the power of imagery and pop culture references.
This book lacks the discipline necessary to apply the steady force to bend the topic to its narrative, jumping from trite autobiography (only because of its reliance on cliché) to strange screenplay excerpts. (Screenplay excerpts that are unintentionally hilarious; they resemble the first draft of Roman and Kent's scifi movie from the "Steve Guttenberg's Birthday" Party Down episode.)
The Langlands Program seems like an important effort in modern math. American taxpayers have funded it with millions of dollars through DARPA. It would have been great if Frenkel could have told us about it with more skill.
Ultimately, I wish that Frenkel had found that agent and that editor, because he has a lot to say. I couldn't understand most of it through the jargon-laden text that made it to final production. I wish it had been presented in such a way that I--and a much larger audience--could.
A better agent would have had a heart-to-heart talk with him about his objectives in writing this book. Frenkel claims he wants to inspire a new generation of mathematicians, particularly by bridging the "two cultures" between the arts and science by using the powers of love and math. That agent might have advised him that, since his work sorely needs the power of visualization, he hook up with a comic book artist--someone who knows the power of image and story combined--and create something along the lines of Larry Gonick's Cartoon History and Cartoon Guide series. (For an excellent example of cartoons in the service of teaching mathematical proof, see Gonick's wonderful single-pager on the irrationality of the square root of 2 in the Cartoon History of the Universe v1-7.) He could have then animated the story and placed it online or presented a summary in a TED talk, where I'm certain it would have gone viral.
A better editor would have forced Frenkel to weave his personal and professional memoirs more tightly around a unified theme, as Eric Kandel did in his stunning "In Search of Memory". That editor would have recommended that he use more accessible phrasing for some terms, like "hula hoop" instead of "ring used in rhythmic gymnastics." It turns out that Frenkel perfected his English by watching David Letterman. That's unfortunate; Letterman is a genius of the absurd, but he never showcased popular explainers and debunkers like Sagan and Randi, as Carson did. Perhaps if Frenkel had seen those masters, he would have learned better how to harness the power of imagery and pop culture references.
This book lacks the discipline necessary to apply the steady force to bend the topic to its narrative, jumping from trite autobiography (only because of its reliance on cliché) to strange screenplay excerpts. (Screenplay excerpts that are unintentionally hilarious; they resemble the first draft of Roman and Kent's scifi movie from the "Steve Guttenberg's Birthday" Party Down episode.)
The Langlands Program seems like an important effort in modern math. American taxpayers have funded it with millions of dollars through DARPA. It would have been great if Frenkel could have told us about it with more skill.
Ultimately, I wish that Frenkel had found that agent and that editor, because he has a lot to say. I couldn't understand most of it through the jargon-laden text that made it to final production. I wish it had been presented in such a way that I--and a much larger audience--could.
July 21, 2014
I enjoyed the book, but would be hard pressed to recommend it since he does explain all the details that goes into the relevant math and the listener can get lost within the weeds of the math. I did not know this branch of mathematics and was able to follow the details, but sometimes it did get overwhelming.
Math is beautiful. Behind our current different branches of abstract math there exist an ultimate theory that ties each branch together. This book explains all of this by delving into the mathematical details and stepping the listener through many abstract math concepts.
The author tells an exciting story. The description of the fundamental particles of nature are said to be described by the "eight fold path". I've often wondered what that meant. The book starts by explaining what it means to be symmetrical and how we can transform objects into mathematically equivalent systems. This leads to Evariste Galois the greatest mathematician who you probably never have heard of. On the night before he died in a duel, he connected number theory to geometry by considering the relationship of certain groups (Galois Groups) with their fields and some symmetries in order to solve quintic equations (fifth degree polynomials). Once again, I had often wondered about what was so special about solving fifth degree polynomials. The book steps me through that.
The ultimate theory of math tries to show the correspondences between different diverse areas of abstract math and then the author ties this to QED and string theory. He'll even explain what SU3 means in the standard model by analogy with constructing SO3 spaces (standard 3 dimensional ordinate systems). He'll step you through the vector spaces, function theory, and metric spaces and the functions of the metric space (sheaves) that you'll need to understand what it all means.
He really does tie all the concepts together and explains them as he presents them. You'll understand why string theorist think there could be 10 to the 500 different possible universes and so on.
Just so that any reader of this review fully understands, this is a very difficult book, and should only be listened to by someone who has wondered about some of the following topics, the meaning of the "eight fold path", the SU3 construction, and why Galois is relevant to today's physics, tying of math branches and physics together, and other just as intriguing ideas. I had, and he answers these by getting in the weeds and never talking down to the listener, but I'm guessing the typical reader hasn't wondered these topics and this book will not be as entertaining to them and might be hard to follow.
P.S. A book like this really highlights while I like audible so much. If I had read the book instead of listening to it, it would have taken me eight hours per most pages because I would have had to understand everything before preceding, but by listening I have to not dwell on a page. Another thing, the author really missed a great opportunity by making the book too complex, because he has a great math story to tell and he could have made easier analogies and talked around the jargon better.
Math is beautiful. Behind our current different branches of abstract math there exist an ultimate theory that ties each branch together. This book explains all of this by delving into the mathematical details and stepping the listener through many abstract math concepts.
The author tells an exciting story. The description of the fundamental particles of nature are said to be described by the "eight fold path". I've often wondered what that meant. The book starts by explaining what it means to be symmetrical and how we can transform objects into mathematically equivalent systems. This leads to Evariste Galois the greatest mathematician who you probably never have heard of. On the night before he died in a duel, he connected number theory to geometry by considering the relationship of certain groups (Galois Groups) with their fields and some symmetries in order to solve quintic equations (fifth degree polynomials). Once again, I had often wondered about what was so special about solving fifth degree polynomials. The book steps me through that.
The ultimate theory of math tries to show the correspondences between different diverse areas of abstract math and then the author ties this to QED and string theory. He'll even explain what SU3 means in the standard model by analogy with constructing SO3 spaces (standard 3 dimensional ordinate systems). He'll step you through the vector spaces, function theory, and metric spaces and the functions of the metric space (sheaves) that you'll need to understand what it all means.
He really does tie all the concepts together and explains them as he presents them. You'll understand why string theorist think there could be 10 to the 500 different possible universes and so on.
Just so that any reader of this review fully understands, this is a very difficult book, and should only be listened to by someone who has wondered about some of the following topics, the meaning of the "eight fold path", the SU3 construction, and why Galois is relevant to today's physics, tying of math branches and physics together, and other just as intriguing ideas. I had, and he answers these by getting in the weeds and never talking down to the listener, but I'm guessing the typical reader hasn't wondered these topics and this book will not be as entertaining to them and might be hard to follow.
P.S. A book like this really highlights while I like audible so much. If I had read the book instead of listening to it, it would have taken me eight hours per most pages because I would have had to understand everything before preceding, but by listening I have to not dwell on a page. Another thing, the author really missed a great opportunity by making the book too complex, because he has a great math story to tell and he could have made easier analogies and talked around the jargon better.
June 14, 2015
I’m disappointed that so many people seem underwhelmed by the autobiographical parts of this book and feel that they are ancillary to Frenkel’s purpose. I disagree: they are, in fact, the heart and soul of Love & Math. Without them, this would be a fairly intense treatise on deep connections between abstract algebra, algebraic geometry, and quantum physics. With them, Frenkel demonstrates how the study of mathematics and a devotion to thought for thought’s sake, to fulfil human curiosity helped him personally through anti-Semitism and Soviet persecution. In some ways I was reminded of remarks Neil Turok makes in
The Universe Within
(if I am remembering correctly) about the state of education in many African countries depriving us of staggering potential intellects. How many people, poor or Jewish or otherwise unprivileged, were not as lucky as Frenkel happened to be?
Frenkel’s personal recollections are also interesting because they provide a glimpse into the lifestyle and community of professional mathematicians. This is not something most people think about, even people who are scientifically-minded. There are a few famously reclusive or otherwise lone-wolf mathematicians out there (though I think that most of them at least maintain some kind of correspondence with a few respected colleagues), but for the most part, twentieth and twenty-first century mathematics is very much a group endeavour. Frenkel describes how he helped to organize new research in the Langlands Program by gathering together mathematicians from various institutions to hear their input. Belying the stereotypes, mathematics is a very social world.
Ultimately, of course, the personal parts of the story are essential to Frenkel’s explanation of why he loves math. Again, I must disagree with those reviewers who pan this book because it doesn’t inspire them to love math … that was never the aim. Neither the book nor Frenkel are naive enough to believe that, I think. But I suspect one reason many people react the way they do when one reveals one’s mathematical inclinations is genuine bewilderment over the idea that a “normal” person could actually love math. As Frenkel points out, even when mathematical achievements are depicted in popular culture, the subject is always a social outsider.
(In a way, it’s similar to this whole idea of left brain/right brain people. “Oh, you’re a left brain person!” and, when people find out I teach both math and English, “You’ve got a weird left and right brain thing going on!” But the truth is, a lot of people in “left brain” positions that require logical reasoning are also very creative and passionate and linguistic—and a lot of “right brain” thinkers are also organized and calculating. Humans are diverse, and the stereotypes and categories we create are not that good at classifying us.)
The autobiographical elements also humanize what might otherwise be a fairly involved book. When Frenkel talks about loving math, he isn’t pulling a Cabinet of Curiosities here. Don’t get me wrong: I’m all for books explaining elementary math. But I’m pleased that Frenkel tackles much higher-concept, abstract mathematics in a nonetheless accessible and approachable way.
I’ve forgotten a lot of my undergraduate math, I am sorry to say. One day I’ll delve back into ring theory and group theory for some fun. I’m pleased by how much I do remember, however. I recognized a great deal of what Frenkel explained, even though some of it still managed to escape me. So when I say Love & Math is accessible, I’m not claiming Frenkel is going to help you comprehend abstract algebra. Rather, he demonstrates some of the concepts that power abstract algebra through some clever diagrams and explanations, and he connects abstract algebra to quantum physics.
I particularly enjoyed this latter endeavour. I knew that symmetry was one of the most significant aspects of group theory, but I didn’t understand the specific ways in which group theory actually underlies a good deal of the interactions between subatomic particles. So that was cool. There are many points where Frenkel basically explains the math behind the physics, then says, “Oh, and mathematicians figured this out long before physicists came along and discovered the math was useful.” That’s not to say math is more important than physics (that’s just, like, self-evident), but I love that we can build these models in math without any reference to the physical world … and then somehow, these models become useful in explaining the physical world. That is just mind-boggling.
As an educator, I also sympathized with another remark Frenkel makes, rather early in the book. He compares the teaching of math in high schools now to the prospect of teaching art by having students paint fences. That is, we barely get to scratch the surface of what mathematics is in high school. Frenkel speaks of quadratics with the disdain only a pure mathematician could muster. But it’s true: I don’t blame students for thinking that math is boring, because the topics we drill into them and the way we do it tends to communicate that fact. You really don’t need to know the quadratic formula—not in the days of Wolfram Alpha—but symmetry? That’s not only important but beautiful as well.
Honestly, Love & Math is not going to make you love math, and it was never supposed to. It’s not going to teach you group theory or representation theory, and you probably won’t have any clue what a Riemannian Surface or a Kac–Moody Algebra is after reading the book. (Maybe you’ll understand what a group is, in some way.) If you’re really interested in learning those things, there are books and videos and courses and wikis to help you out.
Instead, Love & Math is one mathematician’s story of how he fell in love with math, how it saved and defined his life, and how he feels honoured and awed that he has had the chance to give back to the mathematical community. Frenkel goes so far as to make a weird surrealist movie about loving math … and that is not my thing, but it’s clearly his thing, and I’m all for people doing their thing. So you go, Frenkel. And while you do that, hopefully some of the people who read this book come away with a better understanding of what it might mean to love math, even if they don’t quite share that feeling themselves.
Frenkel’s personal recollections are also interesting because they provide a glimpse into the lifestyle and community of professional mathematicians. This is not something most people think about, even people who are scientifically-minded. There are a few famously reclusive or otherwise lone-wolf mathematicians out there (though I think that most of them at least maintain some kind of correspondence with a few respected colleagues), but for the most part, twentieth and twenty-first century mathematics is very much a group endeavour. Frenkel describes how he helped to organize new research in the Langlands Program by gathering together mathematicians from various institutions to hear their input. Belying the stereotypes, mathematics is a very social world.
Ultimately, of course, the personal parts of the story are essential to Frenkel’s explanation of why he loves math. Again, I must disagree with those reviewers who pan this book because it doesn’t inspire them to love math … that was never the aim. Neither the book nor Frenkel are naive enough to believe that, I think. But I suspect one reason many people react the way they do when one reveals one’s mathematical inclinations is genuine bewilderment over the idea that a “normal” person could actually love math. As Frenkel points out, even when mathematical achievements are depicted in popular culture, the subject is always a social outsider.
(In a way, it’s similar to this whole idea of left brain/right brain people. “Oh, you’re a left brain person!” and, when people find out I teach both math and English, “You’ve got a weird left and right brain thing going on!” But the truth is, a lot of people in “left brain” positions that require logical reasoning are also very creative and passionate and linguistic—and a lot of “right brain” thinkers are also organized and calculating. Humans are diverse, and the stereotypes and categories we create are not that good at classifying us.)
The autobiographical elements also humanize what might otherwise be a fairly involved book. When Frenkel talks about loving math, he isn’t pulling a Cabinet of Curiosities here. Don’t get me wrong: I’m all for books explaining elementary math. But I’m pleased that Frenkel tackles much higher-concept, abstract mathematics in a nonetheless accessible and approachable way.
I’ve forgotten a lot of my undergraduate math, I am sorry to say. One day I’ll delve back into ring theory and group theory for some fun. I’m pleased by how much I do remember, however. I recognized a great deal of what Frenkel explained, even though some of it still managed to escape me. So when I say Love & Math is accessible, I’m not claiming Frenkel is going to help you comprehend abstract algebra. Rather, he demonstrates some of the concepts that power abstract algebra through some clever diagrams and explanations, and he connects abstract algebra to quantum physics.
I particularly enjoyed this latter endeavour. I knew that symmetry was one of the most significant aspects of group theory, but I didn’t understand the specific ways in which group theory actually underlies a good deal of the interactions between subatomic particles. So that was cool. There are many points where Frenkel basically explains the math behind the physics, then says, “Oh, and mathematicians figured this out long before physicists came along and discovered the math was useful.” That’s not to say math is more important than physics (that’s just, like, self-evident), but I love that we can build these models in math without any reference to the physical world … and then somehow, these models become useful in explaining the physical world. That is just mind-boggling.
As an educator, I also sympathized with another remark Frenkel makes, rather early in the book. He compares the teaching of math in high schools now to the prospect of teaching art by having students paint fences. That is, we barely get to scratch the surface of what mathematics is in high school. Frenkel speaks of quadratics with the disdain only a pure mathematician could muster. But it’s true: I don’t blame students for thinking that math is boring, because the topics we drill into them and the way we do it tends to communicate that fact. You really don’t need to know the quadratic formula—not in the days of Wolfram Alpha—but symmetry? That’s not only important but beautiful as well.
Honestly, Love & Math is not going to make you love math, and it was never supposed to. It’s not going to teach you group theory or representation theory, and you probably won’t have any clue what a Riemannian Surface or a Kac–Moody Algebra is after reading the book. (Maybe you’ll understand what a group is, in some way.) If you’re really interested in learning those things, there are books and videos and courses and wikis to help you out.
Instead, Love & Math is one mathematician’s story of how he fell in love with math, how it saved and defined his life, and how he feels honoured and awed that he has had the chance to give back to the mathematical community. Frenkel goes so far as to make a weird surrealist movie about loving math … and that is not my thing, but it’s clearly his thing, and I’m all for people doing their thing. So you go, Frenkel. And while you do that, hopefully some of the people who read this book come away with a better understanding of what it might mean to love math, even if they don’t quite share that feeling themselves.
December 30, 2013
To paraphrase William Hurt from The Big Chill, sometimes you just have to let math flow over you. I realized while reading chapter 2 that I wasn't going to be able to grok and synthesize the mathematical content of this book, but this didn't stop me from enjoying it immensely. The passion of the writer for his subject is contagious, and his biography is interesting and inspiring.
Frenkel's book is a memoir of his introduction to The Other Math, the one that is creative and deep. He is diligent and obviously talented from the start, and the only hindrance to his advancement is his Jewish surname in pre-peristroika USSR. His achievements as a teenager lead to an opportunity to go to Harvard for a 3-month visit. He winds up staying 5 years, finishing a PhD and being appointed Assistant Professor. He's not yet 25 years old. The rest of the memoir follows this arc. This is the kind of guy who, while recounting a story about co-authoring a groundbreaking paper, says, "meanwhile I had become fluent in French and Portuguese ...". Yeh. Me too, buddy.
Anyway, Frenkel's narrative of his mathematical investigations is based around his involvement with something called the Langlands Program, which is a pure math initiative, as well as research in quantum physics. These two areas of research are mutually enriching to the dozen humans who understand them.
A front cover reviewer says, "If you're not a mathematician, then this book might make you want to become one." That's right but, for me, it's obvious that this wish wouldn't refer only to the opportunity, but also to the talent. I wouldn't want to become a Chicago Bull as a 49-year-old vertically challenged, poor-sighted man; I'd want to do it as a 6'6" 20-year-old with a reliable jumpshot and a sick crossover.
It is entirely possible that I will read this book again. The math smells fascinating. I can only imagine how edifying it would be if I actually understood it.
Frenkel's book is a memoir of his introduction to The Other Math, the one that is creative and deep. He is diligent and obviously talented from the start, and the only hindrance to his advancement is his Jewish surname in pre-peristroika USSR. His achievements as a teenager lead to an opportunity to go to Harvard for a 3-month visit. He winds up staying 5 years, finishing a PhD and being appointed Assistant Professor. He's not yet 25 years old. The rest of the memoir follows this arc. This is the kind of guy who, while recounting a story about co-authoring a groundbreaking paper, says, "meanwhile I had become fluent in French and Portuguese ...". Yeh. Me too, buddy.
Anyway, Frenkel's narrative of his mathematical investigations is based around his involvement with something called the Langlands Program, which is a pure math initiative, as well as research in quantum physics. These two areas of research are mutually enriching to the dozen humans who understand them.
A front cover reviewer says, "If you're not a mathematician, then this book might make you want to become one." That's right but, for me, it's obvious that this wish wouldn't refer only to the opportunity, but also to the talent. I wouldn't want to become a Chicago Bull as a 49-year-old vertically challenged, poor-sighted man; I'd want to do it as a 6'6" 20-year-old with a reliable jumpshot and a sick crossover.
It is entirely possible that I will read this book again. The math smells fascinating. I can only imagine how edifying it would be if I actually understood it.
February 21, 2016
The author certainly has a talent for explaining mathematical concepts clearly in a way that leaves you wanting to hear more, if you will excuse the cliche.
I was pleased with myself understanding the idea behind braids and also how "addition" and "identity" could be applied to them in a manner analogous to these ideas in the more conventional maths I was taught at school. I could even glimpse how these concepts might be abstracted to ever higher levels.
We were teased with notion that the symmetries of a certain mathematical set corresponded to the model for the fundamental particles of physics and the author explained how this amazing insight set him on the path to a love of maths, but unfortunately few further details were forthcoming.
So it's a shame that the author's talent for explaining maths was only on display in a few sections of the book, with a lot of the filler being name dropping of some undoubtedly great mathematical geniuses with whom he works. Unfortunately no one except the most ardent maths geek will ever have heard of most of them. A long and moderately interesting tale of persecution of Jewish mathematicians in late communist Russia padded out the rest of the book, but not much maths there.
If the author had a go at writing another book and stuck to the maths I would certainly want to give it a look. Otherwise it is difficult to see who this book is really aimed at.
I was pleased with myself understanding the idea behind braids and also how "addition" and "identity" could be applied to them in a manner analogous to these ideas in the more conventional maths I was taught at school. I could even glimpse how these concepts might be abstracted to ever higher levels.
We were teased with notion that the symmetries of a certain mathematical set corresponded to the model for the fundamental particles of physics and the author explained how this amazing insight set him on the path to a love of maths, but unfortunately few further details were forthcoming.
So it's a shame that the author's talent for explaining maths was only on display in a few sections of the book, with a lot of the filler being name dropping of some undoubtedly great mathematical geniuses with whom he works. Unfortunately no one except the most ardent maths geek will ever have heard of most of them. A long and moderately interesting tale of persecution of Jewish mathematicians in late communist Russia padded out the rest of the book, but not much maths there.
If the author had a go at writing another book and stuck to the maths I would certainly want to give it a look. Otherwise it is difficult to see who this book is really aimed at.
February 15, 2014
This book would have been better if it was structured as two short novels, one about his life and the other about the mathematics he has worked on, instead of one interwoven book. The author provides suggestions for skipping chapters heavy on math but I feel that weakens the author's goal of wanting to create a book about mathematics that could be enjoyed by those terrified of math. As the book progresses the math gets more complex and the examples and analogies for it get fewer. As a reader I got lost, started skimming, and had to agree with the author that only a handful of people in the world understand what he works on. Unfortunately as a result I didn't come away with an impression or appreciation for the beauty of mathematics like I have for the double helix of DNA even though I don't understand the biology behind DNA. What I did enjoy about the book was the author's struggle and journey through life. As a persecuted Jewish boy in Russia he managed to overcome multiple roadblocks, with the help of an amazing cast of mentors, in order to pursue his love of mathematics once he discovered that was his true calling. Given the recentness of many events in the book it is an inspiring story worth reading and makes up for the book's other shortcomings.
April 4, 2020
As stated in a previous comment, I really enjoyed the first part that contains the story of the author’s first research in mathematics, including some optional mathematical details and motivations. I learned a lot from it.
Unfortunately, after that the details disappear and ideas are sketched in an, for me, incomprehensible way. Further on, the subject changes to autobiography and random musings on e.g. the idea that mathematics is not invented but discovered, supporting Plato’s thesis.
If the book only contained the ‘mathematical part’, I would give it five stars. Sadly, the second part spoiled it for me.
Unfortunately, after that the details disappear and ideas are sketched in an, for me, incomprehensible way. Further on, the subject changes to autobiography and random musings on e.g. the idea that mathematics is not invented but discovered, supporting Plato’s thesis.
If the book only contained the ‘mathematical part’, I would give it five stars. Sadly, the second part spoiled it for me.
July 15, 2014
Here's the thing ... I like math. I teach honors level math to high school students, I enjoy working on problems recreationally and I have read (and enjoyed) many math related books. So, it's not because I don't like or understand mathematics that this book did not appeal to me. I liked the autobiographical parts about learning math and pursuing his dream no matter the obstacles, but I think the math was too abstract for me (and especially for the lay audience for which it is intended). Perhaps I have the same relationship with this author as my students have with me. I like the guy as a person and respect the work he does and the passion he has, but a lot of the stuff was too far beyond my comprehension to see the beauty in it, let alone love it.
September 3, 2015
I finished reading the e-book just now.
I am greatly surprised ad the group theory widen to applying for
various science fields and math. world.
At first you should read primary "group theory" book.
there are many group theory's key words in the book.
Then please reference my some comments to the book.
Group theory widen and widen to popular science.
Currently it connects with super string theory.
for the future it will connects with popular science,math.,games,etc..............
I am greatly surprised ad the group theory widen to applying for
various science fields and math. world.
At first you should read primary "group theory" book.
there are many group theory's key words in the book.
Then please reference my some comments to the book.
Group theory widen and widen to popular science.
Currently it connects with super string theory.
for the future it will connects with popular science,math.,games,etc..............
November 18, 2013
This book is billed as being both for mathematicians and non-mathematicians, but only the first half of that is correct. The author doesn't make enough of an effort to make the language more accessible for laypeople or at least people who don't do research in math or theoretical physics for a living (which is a general problem of the style of writing in the sciences)
July 4, 2021
The "Love" in the title is the love of math. So basically this should be called "Math and More Math: A Whole Bunch of Math."
Also some auto-biography. It was interesting reading about his experience being rejected from the big university in Moscow because his grandfather had a Jewish-sounding name. Of course, they officially say that no such discrimination was taking place, but everyone knew it was. Because of that name, he had to have his test graded by a special grader, one who could claim to find an error in all answers.
The math gets pretty advanced. Most of it I had at least heard of before, and his short description of how Galois solved the problem of quintic polynomials is clearer than I've read in a whole book on that topic, but sheaves were a new, and bizarre, concept for me. He was able to give me a little feel for why this stuff is interesting, and I enjoyed the peek behind the curtain of his life in academia.
Also some auto-biography. It was interesting reading about his experience being rejected from the big university in Moscow because his grandfather had a Jewish-sounding name. Of course, they officially say that no such discrimination was taking place, but everyone knew it was. Because of that name, he had to have his test graded by a special grader, one who could claim to find an error in all answers.
The math gets pretty advanced. Most of it I had at least heard of before, and his short description of how Galois solved the problem of quintic polynomials is clearer than I've read in a whole book on that topic, but sheaves were a new, and bizarre, concept for me. He was able to give me a little feel for why this stuff is interesting, and I enjoyed the peek behind the curtain of his life in academia.
April 3, 2016
Not long after I started reading this book, I happened to have it out at a coffeeshop when an acquaintance of mine saw it, and said she had read it as well. Or rather, the first half of it.
I didn't ask why only the first half, but I think I can guess. It is Edward Frenkel's mission in life to make you love math, and not just the relatively accessible kind that you might find in newspaper puzzles. He wants to introduce you to the Langlands Program, a sort of Theory of Everything for advanced math that involves topics so abstract it takes you several chapters to even understand what they're attempting, let alone any description of what progress they have actually made.
Frenkel is a good storyteller, though, and he weaves this with his own life, as a youngster in the last days of the Soviet Union, struggling to find a way to be a mathematician in a society that closed most of the obvious paths towards that to anyone with a Jewish parent. This is partly infuriating, and partly fascinating, as he gives us a peek into a society we rarely read about in the West, one where there are almost clandestine meetings of mathematicians, meeting informally to present to each other (after getting off of work in whatever day job actually paid their rent) in a way that it is hard to imagine many Americans doing.
There is a point, perhaps about halfway through the book, where I had to take a step back from the math. Instead of actually understanding what I was being told in detail, I had to content myself with just getting the general gist of what he was talking about. I couldn't, for example, remember what exactly a Galois group was and how that compared to a Reimann surface; I could just remember they were two kinds of math that had originally appeared to be totally unrelated, that they had found surprising and deep connections between.
If you're able to content yourself with this level of understanding, Frenkel's book is an engaging read. If you don't, and you're not ready to do a lot of heavy lifting to pound some rather abstract topics into your head just to read this book, then you may not. But if nothing else, Frenkel does a great job of convincing you that mathematicians are not, on the whole, less passionate and more coldly logical than the rest of us. On the contrary, a large part of becoming a great mathematician seems to consist of immersing yourself in it long enough to acquire an intuition about what to try when attacking a new problem. Moreover, in the current labor market, people with good analytical skills usually could be making more money doing something else, like engineering or computer programming. The ones who nonetheless choose to pursue what they view as the purest form of knowledge, are self-selected for being a little bit on the idealistic side.
I was never tempted to stop, and I'm kind of sorry it's finished. Reading this book is kind of like spending a few hours with that super-smart friend of yours who is excited about something that you cannot really understand, but it's still fun to hear him talk about it because he's so into it. Life is full of people trying to pretend they are not impressed. This guy is impressed, and he wants to share what he's found. It's worth the effort to let him tell you about it.
I didn't ask why only the first half, but I think I can guess. It is Edward Frenkel's mission in life to make you love math, and not just the relatively accessible kind that you might find in newspaper puzzles. He wants to introduce you to the Langlands Program, a sort of Theory of Everything for advanced math that involves topics so abstract it takes you several chapters to even understand what they're attempting, let alone any description of what progress they have actually made.
Frenkel is a good storyteller, though, and he weaves this with his own life, as a youngster in the last days of the Soviet Union, struggling to find a way to be a mathematician in a society that closed most of the obvious paths towards that to anyone with a Jewish parent. This is partly infuriating, and partly fascinating, as he gives us a peek into a society we rarely read about in the West, one where there are almost clandestine meetings of mathematicians, meeting informally to present to each other (after getting off of work in whatever day job actually paid their rent) in a way that it is hard to imagine many Americans doing.
There is a point, perhaps about halfway through the book, where I had to take a step back from the math. Instead of actually understanding what I was being told in detail, I had to content myself with just getting the general gist of what he was talking about. I couldn't, for example, remember what exactly a Galois group was and how that compared to a Reimann surface; I could just remember they were two kinds of math that had originally appeared to be totally unrelated, that they had found surprising and deep connections between.
If you're able to content yourself with this level of understanding, Frenkel's book is an engaging read. If you don't, and you're not ready to do a lot of heavy lifting to pound some rather abstract topics into your head just to read this book, then you may not. But if nothing else, Frenkel does a great job of convincing you that mathematicians are not, on the whole, less passionate and more coldly logical than the rest of us. On the contrary, a large part of becoming a great mathematician seems to consist of immersing yourself in it long enough to acquire an intuition about what to try when attacking a new problem. Moreover, in the current labor market, people with good analytical skills usually could be making more money doing something else, like engineering or computer programming. The ones who nonetheless choose to pursue what they view as the purest form of knowledge, are self-selected for being a little bit on the idealistic side.
I was never tempted to stop, and I'm kind of sorry it's finished. Reading this book is kind of like spending a few hours with that super-smart friend of yours who is excited about something that you cannot really understand, but it's still fun to hear him talk about it because he's so into it. Life is full of people trying to pretend they are not impressed. This guy is impressed, and he wants to share what he's found. It's worth the effort to let him tell you about it.
August 14, 2016
Disclaimer: I don't think I was the target audience for this book. I'm a physicist. I use math a lot, and many of the concepts in the book were either already familiar to me or at least on the periphery of my awareness. I got a lot out of the math explanations here. I really appreciated that, but I think that maybe Frenkel fell short of his goal of making these concepts accessible to the average reader.
Maybe the goal, though, isn't for everyone to understand the concepts, but appreciate his passion for them. In that, he definitely succeeds. His life story is captivating, and the conviction with which he tells us how interesting all these topics are is overwhelming. I generally don't like math -- that's what I have in common with the average reader, I guess. I do lots of math -- I use it, I need it, and I'm pretty good at it -- but pure math has never interested me. I'm interested in its applications, which is why I'm a physicist and not a mathematician. My vision generally starts clouding over whenever branes are mentioned. I found myself being drawn into his explanations of riemann surfaces and Lie algebras, though. He actually got me interested in some pure math. And so yes, he succeeded. He made someone who isn't crazy about math interested in what he's working on. The Langlands program actually is fascinating. Isn't that amazing? Connections between big ideas are startling and exciting.
Ultimately, I can't analyze this book from anyone's perspective but my own. So I'll just say that I though that was most assuredly a 5-star amazing book. I'll put all the usual caveats on there that people are different and maybe others won't like it as much, but that's true of any book I like.
The writing of this book was a bold, ambitious move. I applaud Frenkel for the effort and thoroughly enjoyed reading it.
Maybe the goal, though, isn't for everyone to understand the concepts, but appreciate his passion for them. In that, he definitely succeeds. His life story is captivating, and the conviction with which he tells us how interesting all these topics are is overwhelming. I generally don't like math -- that's what I have in common with the average reader, I guess. I do lots of math -- I use it, I need it, and I'm pretty good at it -- but pure math has never interested me. I'm interested in its applications, which is why I'm a physicist and not a mathematician. My vision generally starts clouding over whenever branes are mentioned. I found myself being drawn into his explanations of riemann surfaces and Lie algebras, though. He actually got me interested in some pure math. And so yes, he succeeded. He made someone who isn't crazy about math interested in what he's working on. The Langlands program actually is fascinating. Isn't that amazing? Connections between big ideas are startling and exciting.
Ultimately, I can't analyze this book from anyone's perspective but my own. So I'll just say that I though that was most assuredly a 5-star amazing book. I'll put all the usual caveats on there that people are different and maybe others won't like it as much, but that's true of any book I like.
The writing of this book was a bold, ambitious move. I applaud Frenkel for the effort and thoroughly enjoyed reading it.
January 12, 2015
Frenkel takes the reader through the world of modern math in an expression of great passion for an often maligned subject. He interlaces the story of mathematical developments in the last fifty or so years with his personal story of growing up in Soviet Russia and emigrating to the United States. The stories of anti-Semitism he encountered in Russia are cringe worthy. Imagine being singled out with especially difficult testing for a college entrance exam, which you still pass, and then being failed by racist examiners. Despite this discrimination, Frenkel's pursuit of mathematical frontiers led him to top of his field. There he found his work contributing to what's known as the Langland's Program, which seeks to unite seemingly separate fields of math (like number theory and geometry). Connecting these worlds of math have contributed to an increased understanding of quantum physics and formulated cryptography methods used every day on the internet. Even if I can't understand some of the denser language, it's exciting to read the passion researchers have for pushing on the boundaries of human knowledge.
May 6, 2016
Edward Frenkel, Love and Math [2013] 306 pages [Kindle]
I read this amazing book more or less by accident; I was testing the direct download link in the library catalogue and this is the book I happened to download to my Kindle. I started looking through it and the preface convinced me to read it. Frenkel is a mathematician, born in the U.S.S.R. and now at Berkeley, who is working on something called the "Langlands Project". He begins by discussing the fact that otherwise well-educated people who would never brag that they hated literature or art or music seem proud to say they "hate math", and admit they know nothing about it; and he suggests that the reason is the way it is taught. He uses the analogy, suppose you took a course in Art, and instead of showing you the works of Rembrandt and Van Gogh they spent the entire class teaching you to paint -- fences. Would you be interested in pursuing art? But this is the way that math is generally taught -- as a practical collection of tools for doing other (mostly boring) things. The "masterpieces" of math are never mentioned. He also points out that most of the math which is taught in high school and even the early years of college dates from the ancient Greeks or the early modern period -- current work is never mentioned; it's as if a physics course ended with the work of Galileo and Newton, and gave the impression that everything in physics was already discovered and all that was left was to use it for practical engineering. Then he goes on to talk about popular science writers, such as Stephen Hawking or Brian Greene, who present the exciting developments of contemporary work in physics for the layperson, and asks why there are so few if any popularizations of work at the frontiers of contemporary math. He warns of the political dangers of allowing a small elite to monopolize knowledge of mathematics in an age when all our lives depend more than ever before on applications of math, from the Internet to Wall Street; he suggests that a mathematically literate population would not be so easily taken in by the doubletalk of the bankers and politicians.
That is the gap he is trying to fill with this book. The focus is on his own work on the Langlands Project, and like the physics books of Kip Thorne or Leonard Susskind, he includes much of his own experiences. In the first few chapters, he explains how he became interested in math as a high school student; how he was denied entry into the more prestigious schools because he was Jewish, and the official academic world in the era of the final decay of the Soviet bureaucracy was highly anti-Semitic; how he was privately mentored by some of the great Soviet mathematicians, who were opposed to anti-Semitism, including the legendary Israel Gelfand, who was himself Jewish; how he worked in private and more or less secretly on group theory, and sent his first papers abroad; how he was invited to Harvard in the first months of perestroika under Gorbachev, where he became a Visiting Professor before he had even become a grad student; and how he came in contact with the Langlands Project. After that, the book is largely a popular account of the Langlands Project itself.
While popular physics books try to avoid math, a popular math book of course has to be about math, and I learned much from this book. At the beginning, his explanations are very clear, and make ideas like modulo arithmetic, finite fields, Lie groups and Riemann surfaces understandable (some of the very subjects I had trouble with in reading Penrose's The Road to Reality). Later on, the book becomes more difficult to follow and his explanations of vector spaces and representations were briefer and less clear; by the end he is mentioning things like fiber bundles and automorphic functions without any explanation at all. If we keep in mind the comparison to someone like Brian Greene, though, it is hard to fault this too much; no one expects to actually learn quantum theory or string theory from a popular science book, and we shouldn't expect to learn advanced math from a book like this -- what we want to learn is what in general it is about, what the questions are and what sort of research is going on, and he is very good at that. His observations that sets, functions and numbers have been left behind for categories, sheaves and vector spaces was interesting; something like the way classical physics gave way to quantum theory. Quantum theory and string theory are here, as the "fourth column" of the Langlands Program; the author has collaborated with physicist-mathematicians such as Edward Witten on the connections between the two fields.
So what is the Langlands Program? Essentially, it is a program to discover patterns common to various seemingly unrelated branches of mathematics (and recently physics), in particular to relate group theory and curves over finite fields to subjects like harmonic analysis and Riemann surfaces. If you don't know what these things are -- that's evidence for his thesis.
I read this amazing book more or less by accident; I was testing the direct download link in the library catalogue and this is the book I happened to download to my Kindle. I started looking through it and the preface convinced me to read it. Frenkel is a mathematician, born in the U.S.S.R. and now at Berkeley, who is working on something called the "Langlands Project". He begins by discussing the fact that otherwise well-educated people who would never brag that they hated literature or art or music seem proud to say they "hate math", and admit they know nothing about it; and he suggests that the reason is the way it is taught. He uses the analogy, suppose you took a course in Art, and instead of showing you the works of Rembrandt and Van Gogh they spent the entire class teaching you to paint -- fences. Would you be interested in pursuing art? But this is the way that math is generally taught -- as a practical collection of tools for doing other (mostly boring) things. The "masterpieces" of math are never mentioned. He also points out that most of the math which is taught in high school and even the early years of college dates from the ancient Greeks or the early modern period -- current work is never mentioned; it's as if a physics course ended with the work of Galileo and Newton, and gave the impression that everything in physics was already discovered and all that was left was to use it for practical engineering. Then he goes on to talk about popular science writers, such as Stephen Hawking or Brian Greene, who present the exciting developments of contemporary work in physics for the layperson, and asks why there are so few if any popularizations of work at the frontiers of contemporary math. He warns of the political dangers of allowing a small elite to monopolize knowledge of mathematics in an age when all our lives depend more than ever before on applications of math, from the Internet to Wall Street; he suggests that a mathematically literate population would not be so easily taken in by the doubletalk of the bankers and politicians.
That is the gap he is trying to fill with this book. The focus is on his own work on the Langlands Project, and like the physics books of Kip Thorne or Leonard Susskind, he includes much of his own experiences. In the first few chapters, he explains how he became interested in math as a high school student; how he was denied entry into the more prestigious schools because he was Jewish, and the official academic world in the era of the final decay of the Soviet bureaucracy was highly anti-Semitic; how he was privately mentored by some of the great Soviet mathematicians, who were opposed to anti-Semitism, including the legendary Israel Gelfand, who was himself Jewish; how he worked in private and more or less secretly on group theory, and sent his first papers abroad; how he was invited to Harvard in the first months of perestroika under Gorbachev, where he became a Visiting Professor before he had even become a grad student; and how he came in contact with the Langlands Project. After that, the book is largely a popular account of the Langlands Project itself.
While popular physics books try to avoid math, a popular math book of course has to be about math, and I learned much from this book. At the beginning, his explanations are very clear, and make ideas like modulo arithmetic, finite fields, Lie groups and Riemann surfaces understandable (some of the very subjects I had trouble with in reading Penrose's The Road to Reality). Later on, the book becomes more difficult to follow and his explanations of vector spaces and representations were briefer and less clear; by the end he is mentioning things like fiber bundles and automorphic functions without any explanation at all. If we keep in mind the comparison to someone like Brian Greene, though, it is hard to fault this too much; no one expects to actually learn quantum theory or string theory from a popular science book, and we shouldn't expect to learn advanced math from a book like this -- what we want to learn is what in general it is about, what the questions are and what sort of research is going on, and he is very good at that. His observations that sets, functions and numbers have been left behind for categories, sheaves and vector spaces was interesting; something like the way classical physics gave way to quantum theory. Quantum theory and string theory are here, as the "fourth column" of the Langlands Program; the author has collaborated with physicist-mathematicians such as Edward Witten on the connections between the two fields.
So what is the Langlands Program? Essentially, it is a program to discover patterns common to various seemingly unrelated branches of mathematics (and recently physics), in particular to relate group theory and curves over finite fields to subjects like harmonic analysis and Riemann surfaces. If you don't know what these things are -- that's evidence for his thesis.
August 14, 2024
Yes, most of the mathematics described in this book is hard. Edward Frenkel's work is cutting-edge mathematical research, after all, with applications in quantum mechanics, so it's no walk in the park. I give him credit for even attempting to explain it to a lay audience. As he states in "A Guide for the Reader", you can skip the sections and chapters that deal with the math. I think most readers of this book will end up doing so at some point, mainly enjoying the biographical parts.
Frenkel's point in writing this book, as he explains in the preface, is to share his love for mathematics with an audience who doesn't get to see much of its hidden beauty. He does this by taking the reader on the journey that led him to mathematics, starting with a friend of his father's who taught Frenkle about symmetries and braid groups when he was in his late teens. This sparked his lifelong passion for math, and we learn all about the obstacles thrown into his path by the Soviet system that made it next to impossible for a son of a Jew to study or work in research mathematics. Luckily, he was the right age to take advantage of Perestroika, and then the collapse of the Soviet Union, and was able to both finish his studies and find an academic position in the United States.
We get introduced to the main project in Frankel's career, the Langlands Program, in chapter 8, "Magic Numbers," where a lot of background is given. The Langlands Program, if finished, will link together several fields of mathematics that seem unrelated on the surface. This is difficult to do and pretty rare. The background leading up to this is wide-ranging, and Frenkle tries to simplify his explanations as much as possible, but this is where the abstract nature of advanced mathematics is probably going to confuse most of his audience. In a mere 15 pages, you get a high-level whirlwind tour of the Galois groups, how the Shimura-Taniyama-Weil conjecture was used to prove Fermat's Last Theorem, how this conjecture is a general result for finding the generating function of cubic equations (using the generating function of the Fibonacci numbers as an analogy), and how it is a special case of the Langlands Program. I had to read chapter 8 very carefully and slowly, but I got the gist of it, if not a deep understanding. By the next chapter, where he introduces Lie (pronounced "lee") groups and algebras, I admit I was lost.
The remaining math sections left me feeling like Frenkel was Beethoven, explaining how he had composed his symphonies, while I was the novice student who had finished a basic course in music theory, and knew how to play "Für Elise" but not much else. While his enthusiasm for mathematics and his life's work are all obvious, I just cannot appreciate the depth or the details on the same level. I found this discouraging and more than a little frustrating, because I wanted to see the same beauty that he did. I did appreciate the journey he has taken to get to this point, and the real-life depiction of how mathematical research is actually done—sometimes painfully slowly. Perhaps after some years of study I will be able to appreciate the mathematical sections just as much as the biographical ones.
December 17, 2016
“Love and Math” is an unusual mash-up of three different stories. In one, a young Jewish prodigy transcends the antisemitism of Soviet Russia to take a preeminent place in the world of professional mathematics. The second story is love for the mathematics itself—a radical unification of different, apparently distinct, fields of pure mathematics. The third story is a reflection on how erotic love is built on symmetries, much like the mathematics. I found this third bit silly and strained.
The author’s love of mathematics comes through very strongly, so an autobiography would have been a terrific book if it unified these first two parts more clearly. Unfortunately, the author loves the math so much that he commits the cardinal sin of those who love too much: assuming the reader is as infatuated as he is. Sometimes one want to interrupt “no, her smile really doesn't outshine the sun, and her walk really isn’t more graceful than a doe.”
That said, I personally share the author’s love of this kind of mathematics. In fact, after finishing this book, I took down some of my college math texts (I was in an honors math program at a top university many years ago) an re-experienced the arcane joy of doing pure math.
To be fair, he tried to keep the difficult stuff in (long) end notes, so as not to interrupt the narrative. And it may have been my own fault that I couldn’t resist spending a lot of time working through those end notes. But without the end notes, the narrative would be a lot of “math speak” that would make no sense to most readers. But I think he could have conveyed the joy of abstract thought without requiring the reader to choose between wading into or skimming over the details. We really don’t need to know what a Lie group, or even a group, is in order to appreciate that the author was extremely fortunate in his mentors.
My bottom line: if you really want to know how modern efforts to unify algebra, geometry, and number theory arose, read the book (except for the last chapter). Otherwise, hope that the author comes out with an autobiography that dwells more on his inter-personal relationships.
August 22, 2019
I can't in good faith give this book a 5 star rating, despite my immense enjoyment of it. The introduction frames the book as an exploration of modern math and how to appreciate it--with particular focus on the layperson. Unfortunately, this promise is never really delivered upon. The book is far more a memoir--which is far from a bad thing. Dr. Frenkel lived an interesting life and it was a compelling tale of his voyage through mathematics. However, the math quickly flies off the rails. I am a computational biologist in a biostats type field and I was still largely left in the dust in chapters 14 and 17. I imagine complete laymen would lose their way even earlier. Dr. Frenkel does poise that it is ok to skip math heavy sections; however, some chapters (like 14 and 17) consist majorly of mathematics.
If you can make it to Chapter 18, Dr. Frenkel does delve a bit more into the 'love' aspect that is this book's namesake. Generally, I really enjoyed the book. If he had written strictly as a memoir, it would have gotten a 5 star rating. Likewise, if he had managed to keep the mathematics at a lower level (he succeeds early in the book, but loses it later, in my opinion) I would also give it a 5. But it ends up being slightly muddied by the two together, as it breaks the smoothness of the narrative. Still, I think Frenkel is the man. If you want a sense of his personality, he has done a few videos in conjunction with Numberphile on Youtube that I highly recommend.
If you can make it to Chapter 18, Dr. Frenkel does delve a bit more into the 'love' aspect that is this book's namesake. Generally, I really enjoyed the book. If he had written strictly as a memoir, it would have gotten a 5 star rating. Likewise, if he had managed to keep the mathematics at a lower level (he succeeds early in the book, but loses it later, in my opinion) I would also give it a 5. But it ends up being slightly muddied by the two together, as it breaks the smoothness of the narrative. Still, I think Frenkel is the man. If you want a sense of his personality, he has done a few videos in conjunction with Numberphile on Youtube that I highly recommend.
January 5, 2015
This book will leave you breathless. While it's not an "easy" read, Frenkel does a great job of making complicated concepts understandable and interesting. He exposed me to a world I never even knew existed (even though I've taken over a dozen applied math courses in university). If you don't finish this book with a burning desire to learn more about the beautiful world of mathematics, I'd be willing to bet you didn't give this book the time or attention it needs to be fully appreciated. This is by far one of the best books I've ever read.
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