Tuesday, October 3, 2017

Love and Math by Edward Frenkel

Source of book: I own this.

While I didn’t go beyond algebra, trigonometry, and analytical geometry in school, I remain fascinated by math. Had I chosen a different career, I probably would have taken more advanced classes. In any event, I do find certain books about math to be fascinating, and try to read one from time to time.

Probably the one I enjoyed the most was The Nothing That Is by Robert Kaplan, about the history of the use of zero as a placeholder. Truly revolutionary. For more of the books I have reviewed with a mathematical theme, take a look at my index - specifically the section on Science. (I lump them together.) 

Love and Math is a blend of autobiography and an introduction to the area of mathematical theory called the Langlands Program. I’d try to explain it except (1) I’m not sure I really understand it myself and (2) Edward Frenkel, who is pretty good at breaking stuff down for non-mathematicians, took a whole freaking book to do it. So just read the book instead.

Frenkel grew up in Soviet Russia, as a Jew. This was problematic for him, because Jews were not (in practice) permitted to be admitted to the best schools, or even to have careers in certain areas. In his examination to get into Moscow State University, he was given a clearly bogus test, then failed on it. Nevertheless, he persisted in doing what he could. He was taken under the wing by a number of different professors and mathematicians who recognized his genius. Basically, he couldn’t study pure math, but had to look to a career as an engineer or applied math teacher. He could, however, study in his spare time, which is what he did. He eventually was invited to lecture and study at Harvard. He stayed on during the failed coup and the eventual fall of the Soviet Union, and now teaches at UC Berkeley.

The autobiographical parts of the book trace his academic and mathematical career, but don’t really get into his personal life. Instead, the biography serves to connect the mathematical concepts. Frenkel introduces each piece of the puzzle at the point where it fits in his life, and by the end, the reader can (sort of) understand at least the basics of what Frenkel has done in his field.

I must say, Frenkel does a rather outstanding job at simplifying concepts. I checked out a few different sources to try to get a better understanding, and realized that the others were even more confusing to a non-math-major sort. While I can’t say I fully understood everything, I believe I know more than when I started, and enjoyed the process. It took me a while. It was all I could do to read one chapter at at time, and then sit on it for a bit and process the information.

I won’t even attempt to get into the math itself. What is even more interesting is the way that math and the physical world work. Roger Penrose saw reality as a triad: the physical world, the mental world, and mathematics. I think he is right to a large degree. The more we learn about the physical world, the more we understand that it is written in the language of math. As a general rule, math has developed in the mind, in the realm of pure reason. But when we look for ways to describe the behavior of the universe, it turns out that the pure math we have ends up being the necessary language to describe what we see. The DNA of the universe does turn out to be math. This is fascinating and awe inspiring. While Plato’s idea of the “form” may not hold true as a general description of the world around us, math does fit that idea. There is indeed a world of ideal mathematical truth that we don’t so much invent as discover.

Einstein put it thus:

“How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably appropriate to the objects of reality?”

It seems hard to believe - and this may be one of the reasons that laymen often distrust science - but it is true.

There are a few other highlights of the book. One was Frenkel’s description of the way the Soviet system worked.

In the stagnant life of the Soviet period, talented youth could not apply their energy in business; the economy had no private sector. Instead, it was under tight government control. Likewise, communist ideology controlled intellectual pursuit in the spheres of humanities, economics, and social sciences. Every book or scholarly article in these areas had to start with quotations of Marx, Engels, and Lenin and unequivocally support the Marxist point of view of the subject.

It was hard not to laugh at this. After all, this idea was nothing new to the Soviets, and it hasn’t disappeared. You find it in many areas. Just to pick one of which I am all too familiar, within Evangelicalism here in America, most areas of study are off limits as far as intellectual honesty is concerned. You have to start with a quote from the Bible, and end up unequivocally supporting the “theologically correct” view of the subject, actual facts be damned. Likewise for [insert your favorite political dogma here.] (Maybe, say, climate science for conservatives…alternative facts, people. Or, the economic theories of Murray Rothbard - widely embraced by the Paul Ryans of the world - which dismiss empirical evidence as irrelevant to economic policy.) No matter where it happens, ossification of ideas and refusal to embrace the possibility that our beloved dogma may not be, in fact, true, leads to an inability to make further progress.

There were two exceptions to the general anti-intellectualism of Soviet academia were math and physics. It isn’t difficult to see why. They were useful. Useful for inventing and building things that blew up the enemy. And this is where many of the most talented Soviets found their outlet, even as many had reservations about the way their country was using their work.

Two particular discussions of the connection between math and physics were particularly interesting. The first was on the curved nature of space. This is hard for us to visualize. As Frenkel puts it, “We are used to thinking the space we live in is flat, and so in our everyday experience curved shapes seem to appear only within the confines of that flat space. But this is a misunderstanding, an artifact of our narrow perception of reality. And the irony is that the space we live in isn’t flat to begin with!” Perhaps one way to understand this is to think of the fact that in our everyday life, the earth appears flat. Building furniture or even an house can be done perfectly well based on that assumption. However, build a long bridge and you find very quickly that the earth is curved. Now imagine living on the scale of an ant, or a bacterium. Such a creature will never need to experience a round earth. Likewise, we cannot perceive the curvature of space, and flat space works well for our everyday experiences. But not so much when we explore larger scales.

The other discussion which was fascinating was that of duality, which underlies much of physics at the large and small scale.

It might seem strange to look for a duality in physics, but in a sense this is a concept we are all already familiar with. Take electricity and magnetism. Even though these two forces seem to be quite different, they are actually described by a single mathematical theory, called electromagnetism. This theory possesses a hidden duality that exchanges electric and magnetic forces.

This is a “theory” in the scientific sense, of course. It is abundantly clear that this duality exists. Without it, generators and electric motors, speakers and microphones, and so many other electrical and magnetic devices would be impossible.

One final bit. I have noted before that our society treats mathematics like some sort of foreign language, unnecessary for most to learn. We are math-phobic. It is socially acceptable to be illiterate in math, in a way that it is not socially acceptable to be unable to read. Even as our everyday lives are intertwined with math - any time we use technology - we are willing to accept it without bothering to understand. Frenkel quotes the poet Hans Magnus Enzensberger: Math is
“a blind spot in our culture - alien territory, in which only the elite, the initiated few have managed to entrench themselves.” We do not bother to learn the language. As Goethe put it (poets and math...who knew?): “Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.” This really is a shame. I am doing my best to make sure that all my children enter college with a solid foundation in math, and (I hope) without a fear of it or a willingness to be illiterate. This book is not easy to read, but it is fascinating. For those with a math phobia, probably better to start with something easier. For those who love math, this book will be fun and enlightening.

1 comment:

  1. I love math. In fact I've found this great website, Kahn Academy, where I can actually practice the maths that I haven't done in years. It really gets the brain working in a logical way.