Love and Math: The Heart of Hidden Reality

A boy who grew up in a town closer to Moscow became interested in quantum physics. Plenty of popular science books were easy for him to find to read. His parents were engineers and were quick to recognize boy's talents. A mathematics professor of the only college in the town who is also a friend of the family took the boy under his wings. The professor asked him if he knew about the group SU(3). This conversation made the boy realize that he had to learn math to answer many questions he had by reading popular science books. Thus began the journey of this boy becoming one of the premier mathematicians of today. He was introduced to great mathemati- cians from the beginning. Mathematicians gave him problems to solve. He solved his first problem on Braid groups as a freshman. He attended Israel Gelfand’s legendary seminars at Moscow State University. He was offered a visiting professorship at Harvard even before he received his bachelor’s degree. To get a permanent job he needed a degree. He enrolled in the Harvard PhD program and received his PhD in one year. He first heard about the Langlands Program at Harvard. Since then until today he has been working on the Lang- lands Program. Langlands program is like a Rosetta stone revealing similarities in three apparently unrelated fields; namely, Number the- ory & curves/finite fields, Riemann surfaces and Quantum Physics. The book "Love and Math" is Edward Frenkel’s autobiography. He has done a masterful job of trying to bring hard mathematics that he has been working on to a level of an average reader by inter-vowing it with his fascinating life experiences. It is hard not to get excited about the math that you know practically nothing about. "Love and Math" makes you wanting to read more and get to know more about this fascinating mathematics. Frenkel has included many references for an interested reader to follow through as notes at the end of the book. "Love and Math" is a must read for young and old alike.

I love this book. But, then, I teach math. Frenkel perfectly describes the frustration/thrill cycle of solving difficult problems. In that sense, he does a great job of explaining the experience of doing mathematics and why some of love it. My reservation about the book is that he's writing for those who don't 'get' the love of math, but his experience of math is at an extremely high level. The explanations of math sometimes work and sometimes require work on the reader's part. His area and level of expertise is challenging for me to comprehend fully, so the person who feels like an outsider in the field probably will still feel that way. Nonetheless, the book is worth reading for the fascinating story of Frenkel's life and experiences, even if you skim over the math. Frenkel, because of his Jewish heritage, struggled to get the education he wanted in the Soviet Union. That struggle and the generosity of the mathematics community is inspiring and heart-warming. He also tells good, human tales about other mathematicians who are writing the book of 21st century math and science.

Even if you struggled with grade school fractions, perhaps even more so because you did, this book is just the square root of 2 shy of essential. There's a whole world of magic in mathematics that eludes most of us. Our teachers fling us directly into the material without an ounce of background, application or concept. This is their colossal failure. Fascinating transcendent ideas like symmetry, congruence, dimensions and modulo primes hover like gods above the minutia. It's some of these ideas that Mr. Frenkel will introduce and that will open up a Pandora's Box of awe that will have the intelligent reader wanting more. His words to describe higher mathematics are elegance and beauty, usually associated with the arts, and once the Langlands program is introduced, it's a difficult description to deny. I would add magical or mysterious or majestic or any number of M's...The previous silly alliteration illustrates in micro form the mind boggling array of patterns that are revealed through mathematical analysis. It's a world of wonderful repetitions and shapes that reminds me of the Spyrograph toy I was fond of as a child. Edward marvels that these patterns somehow seem to exist outside of mankind and perhaps even outside the Universe itself in a Platonian Netherland. Spectacular stuff. Weaved within the narrative is Mr. Frenkel's own tale of his love for math and his struggles while growing up in the Soviet Union as recently as the 80's as a Jew. This is a remarkable story in itself which should inspire any youngsters who want to pursue difficult dreams. I highly recommend this book, not just for people with mathematical aptitude or interest, but especially for the liberal arts types out there who consider Math nothing more than a Draconian torture method for youth..

Frenkel's book is the best book about why math is important since professor Penrose's book, The Emperor's New Mind. How I wish Frenkel had been my math teacher in highschool. The connection made between love and math, as well as the connection in the epilogue about how math can teach us all something about our common humanity was a joy to read. Even the chapters that dealt with the Langlands Program and André Weil and his Rosetta Stone were clear and taught me something new. I even understood most of it. For the first time I came away with a clear understanding of a Reimann surface and felt the joy of elightenment as I read about dualities. Thanks, Dr. Frenkel, for a wonderful and joyful read. I too feel as you wrote that, "while we all appreciate the significance of the physical and mental worlds, many of us remain blissfully ignorant of the world of mathematics. I believe that when we awaken to this hidden reality and use its untapped powers, this will lead to a shift in our society on the order of the Industrial Revolution."Frenkel, Edward (2013-10-01). Love and Math: The Heart of Hidden Reality (p. 235). Basic Books. Kindle Edition. I happened to believe mathematics holds the key to understanding emergement properties or complexity as I explore in my recent SciFi novel, EPOCH . I hope we hear a lot more from Dr. Frankel. I highly recommend this book to anyone who has ever wondered why math is important, or who had the love of math driven out of them by an unelightened highschol math teacher.

The observation to which this book is a response is the fact that intelligent, educated, and sophisticated readers who take pride in the breadth of their knowledge, reveal with pride that they hate math. Those of us who use advanced mathematics integrally and intimately in our work have all heard this refrain. Why would someone take pride in their ignorance? Why would someone who would never say they hate art take pleasure in affirming their aversion to mathematics, which is doubtless and indisputably one of the most stunning achievements of human intellect? Frenkel's answer is that students are not taught math properly in school. Students learn how to perform calculations, but are never exposed to the deep principles of symmetry and dynamics that embody the true beauty of mathematics. His response is to show how human an enterprise modern mathematical research can be. I read this book from cover to cover with some pleasure, but I do not think he has made his case. Frenkel tell us early of the insights of Galois and even explains with some effectiveness exactly what an incredible tour de force Galois theory is. Moreover, Galois proved something with his detour through group theory that is truly remarkable: there are no general solutions to polynomial equations of degree greater than four using the operations of arithmetic operating on integers and their various integral roots. Frenkel explains that the Langlands Program is a sophisticated and ambitious extension of Galois' project, but he fails in two ways. First, he introduces a plethora of mathematical constructs to which he gives a best vague descriptions, and gives no intuition as to why they are related to one another in interesting ways. Second, he gives us no concrete examples of how all of this high-level math solves some real-world problem of the sort that Galois theory addresses. Why all the huffing and puffing? All we are told is that some of the theory Is useful in banking and cryptography. How and why? Who knows? In fact, probably most people who become mathematicians or use mathematics extensively actually like to solve rote problems. I know I did. I would spend hours and hours solving problems, deriving trigonometric identities, and integrating funny-looking rational functions. Moreover, I treat mathematics as very hard and exhausting---like running laps and lifting weights. The rewards are surely there, but it is simply hard work most of the time. Art and music are fun, not work. Of course, we must learn to appreciate art and music, but once the brain is prepared, it is all fun thereafter. Math is not like that. Math is sweat and pain all the way. Struggling through a three page paper full of symbols is excruciating effort. The reward is surely there at the end---understanding something of ineffable majesty. I still think that the beauty of mathematics can be explained to humanists, I'm not sure how, though.

This book is both an invitation to dive into Mathematics and the account of personal journey from humble beginnings and in the face of structural prejudice. Mathematics had a transforming role as a universal, objective yet dynamic discipline. Professor Frenkel, though clearly an exceptional Mathematician, faced major obstacles to pursuing his passion and profession. The book provides insights into the structural ant-semitism in Russia and the clandestine efforts of people of good will to fight against this. The Mathematical story is weaved through this personal story. This certainly has motivated me to learn about the Langland's program and modern Mathematics. I look forward to diving into this, if and when I get time. The book provides a taste focused on concepts. I think the journey through adversity and making difficult decisions along the way was poignant and inspirational. The pursuit of Mathematics, to discover the hidden deep connections of the Mathematical universe and how it relates to the remarkably comprehensible "real world", was its own reward and a lingua franca bringing together people of diverse origins, experience and passions.. Professor Frenkel appears to be a Platonist, arguing an independent universe of Mathematical Truth that humans discover (and do not create)...I think I am to but this is a deep philosophical question and I have only an extremely limited field of view. This book encourages me to broaden it.

What I REALLY like about this book is that Frenkel talks about real math without talking down to the reader. I don't follow it all, since I did not read it as carefully as it needs to be read, but I will take another pass at it soon. This is in contrast to textbooks, which 1. I wouldn't be able to follow at all since I don't have the background nor will I spend the time to get it, and 2. I would probably never re-read a math textbook. Come on, really? Nope. I read "My Brain is Open" about Paul Erdos a few years ago, and loaned it out, so I can't remember for sure how it compares to this one, but I think I was left a little frustrated. Paul Erdos collaborated in many mathematical ventures, but as I remember, it is assumed the reader couldn't possibly follow what they were. Edward Frenkel makes the attempt. I am flattered and delighted. He is skillful enough in this so that I want to rise to the occasion. In addition, he tells the story of his own life and career in an engaging way. One of the things that I find remarkable in Frenkel's account is the view into a mostly-male field that isn't about soldiering or something similarly physical. As a woman, I did take four years of math in college but didn't continue. It wasn't because I didn't like it - I loved it! It was because of the toxic posturing that accompanies the field, or at least, in my college it seemed to. We students were subtly and not-so-subtly evaluated by our professors and classmates to see if we had "what it takes" to be a great mathematician. One of my professors actually called the other female student in my year to his office and told her that, while she would never be a first-tier mathematician, she had what it took to become a second-tier thinker and she should consider going into math as a career. When she told me about this, she was crying. She did math because she liked to do it, and it had not occurred to her that such rankings were part of the winnowing process. I, of course, was also devastated because clearly I was not even going to be second-tier. Neither of us continued in the field, though, looking back, I think that we should have. Female brains are different than male brains, and we both frequently noted that we found some classes hard and others easy, and that seemed to be in inverse relationship to what our male classmates found. If we had gone on in the "difficult" fields that we thought were easy, who knows where it might have led? With what we know now about the relationship between focused practice and achievement, it seems insane to worry so much about what minds look like in the moment. If my friend had wanted to become a first-tier mathematician, she ought to have been advised to do what Frenkel does, that is, think about math all the time and poke around in the literature and talk to other mathematicians and think about it some more and talk with colleagues some more and do that with passion and verve. I personally work very well in collaboration, but collaboration, while Frenkel talks about it a LOT, was not encouraged in my college. It was too important to evaluate how brilliant people were as they stood alone. The one time I did hang out with a mathematician who was in a class further on than mine, I did so well in the class that we had been talking about, that I went to the professor and confessed that I had an unfair advantage. He said that cheating was frowned upon but he would let it pass this time. I certainly never made the mistake of cheating by talking to another mathematician again! And now, as I read Frenkel, I see how crucial that is. Of course.

Recently, books have been appearing in which an applied mathematician will mix stories about his own research with an introduction to some field such as applications of nonlinear dynamics. This book is a departure from this format. First, there is an absorbing embedded story of how the author, growing up in the USSR, managed to surmount the obstacles placed before Jewish mathematicians in attaining a top position in the system. When he relocated to the USA, his life changed and his mathematical creativity was recognized in a new field of pure mathematics, which takes me to the second element of the book: an introduction to the excitement of doing pure mathematics. I'm trained in mathematics, although not with a Ph.d., and I immersed myself in his presentation of his field, finding it very interesting indeed -- although it can be tough reading even with some advanced math background. I enjoyed this book very much and, by the way, I agree with the author's critical review of a book by Max Tegmark that concludes that the universe is made of purely mathematical objects. This absurd idea shows that it's possible to have mastered theoretical physics and cosmology and still not grasp elementary metaphysics. Frenkel avoids that trap because he seems to have a good understanding of the philosophical interpretation of natural science as well as a powerful command of the difficult terrain of the frontiers of pure mathematical research today. I very much recommend this book for those who love mathematics as such even if its motivation in internal to math and not (initially) linked to a scientific problem.