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3.48 AVERAGE
3.48 AVERAGE
don't kid yourself...it is NOT readable if you're not a math person no matter what they say. But I liked it.
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.’
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.
Interesting autobiography, incomprehensible math. The author of this book claims that he wrote the book for people with no background in mathematics. He was a failure in that. Most people - even those with a background in mathematics, would likely not understand parts of this book. I personally found the autobiography of his difficulties getting into college in Russia because of the anti-semitism there and the help he received to advance through college and eventually end up at Harvard very interesting. I think it is great that there are people like him who understand and enjoy math and study it so that the rest of us may benefit from their knowledge. I totally did not understand most of the math that the author explained in this book and I am a college graduate, but not a math major. I did learn a few things - like the fact that the term torus refers to a donut shape - or as I prefer - bagel shape. I received this book free to review from Netgalley.
A little too mathy for non math enthusiasts and vice versa.
This book had a split personality. It was a mix of history and insights on the author's journey to becoming a world-class mathematician and solving difficult mathematical issues. However, much of it was was incomprehensible and the author does a poor job explaining the math that he is obsessed with. He uses words in atypical ways with no explanation and the reader is left puzzled. For example, what is "gauge theory" and why the word "gauge"? Why "Program" in "Langlands Program"? Some stuff is well-explained (string theory) but much isn't. I was a math major but I have trouble believing that more than a very limited # of mathematicians in the entire world can understand everything in this book - and they surely aren't bothering to read this book as they already know the material.
As a computer scientist, I was also disappointed in the passing references to the value of his field in secure banking. He says this several times but with no explanation leading me to believe he's overinflating a claim, doesn't understand it, or can't explain it. Another passing reference to Haskell, again with no explanation, was a disappointment.
In summary, I recommend the early chapters of the book. The rest is a waste of time.
As a computer scientist, I was also disappointed in the passing references to the value of his field in secure banking. He says this several times but with no explanation leading me to believe he's overinflating a claim, doesn't understand it, or can't explain it. Another passing reference to Haskell, again with no explanation, was a disappointment.
In summary, I recommend the early chapters of the book. The rest is a waste of time.
This is not really a pop math book. It's a memoir filled with some pretty complex math concepts. I was a math minor and I had to really work hard to understand most of the math here! Thank goodness for that abstract algebra class I took! But even if you skim the math parts, the story and Frenkel's delight and passion for mathematics make this a charming read. Recommended!!
I always enjoyed math well enough and took a semester of calculus, but the math in this book was mostly beyond me. I didn't devote a lot of energy to trying to understand it though -- it required a paradigm shift (which the author made clear) but the keys to make said shift were lacking.
But the author's enthusiasm was infectious and believable, and his story was interesting.
I was definitely intrigued by the fact that mathematicians have found (in a sterile semi-detached-from-reality way) the same ideas and concepts that physicists are just starting to find. I knew that math was key to explaining physics, but I didn't realize that we were learning things the other way around too.
But the author's enthusiasm was infectious and believable, and his story was interesting.
I was definitely intrigued by the fact that mathematicians have found (in a sterile semi-detached-from-reality way) the same ideas and concepts that physicists are just starting to find. I knew that math was key to explaining physics, but I didn't realize that we were learning things the other way around too.
My feelings on this book say more about my expectations than on the book itself. I only read the first 60 pages.
I bought this book hoping it was fiction. It's not.
I still had hopes that it would be about what makes mathematicians love math. It's not.
It's pretty much an autobiographical account (at least as far as I read and skimmed) of one mathematician's journey to math and all of the troubles that Jews had being accepted in Russian academia in the first part of the 1900s.
The book isn't necessarily bad.... it's just certainly not what I wanted it to be.
I bought this book hoping it was fiction. It's not.
I still had hopes that it would be about what makes mathematicians love math. It's not.
It's pretty much an autobiographical account (at least as far as I read and skimmed) of one mathematician's journey to math and all of the troubles that Jews had being accepted in Russian academia in the first part of the 1900s.
The book isn't necessarily bad.... it's just certainly not what I wanted it to be.
Interesting insight into soviet math community and subsequent exodus.