“People think they don’t understand math, but it’s all about how you explain it to them. If you ask a drunkard what number is larger, 2/3 or 3/5, he won’t be able to tell you. But if you rephrase the question: what is better, 2 bottles of vodka for 3 people or 3 bottles of vodka for 5 people, he will tell you right away: 2 bottles for 3 people, of course.”
Edward Frenkel does a good job highlighting the pivotal importance of math. He asserts that instead of being a dull academic subject, it’s a universal language, free from bias, that can make apparent the deeper mysteries of the universe. If only it was taught different in schools.
Frenkel is a professor of mathematics at UC Berkeley. Even though he is under fifty, he has had a rich and distinguished career that traces back to the Soviet Union of the 1980’s, when Communist oppression was in full force. His story is my favorite part of this book, and his struggle just to get a decent education as a Jew facing blatant bureaucratic discrimination made me more appreciative of the freedoms I enjoy. indeed, this oppression was a source of strength for him, going on to say that, “In this environment, mathematics and theoretical physics were oases of freedom. Through communist apparatchiks wanted to control every aspect of life, these areas were just too abstract and difficult for them to understand.” Mathematics set him free, and his passion for the subject is infectious because he’s also a clear and thoughtful writer.
Slate piece by the author, illuminating the political importance of a mathematically literate society.
Farnam Street review, more in depth than my own.
What the professionals had to say: The New York Times review.
Buy from Amazon: Love and Math: The Heart of Hidden Reality
While our perception of the physical world can always be distorted, our perception of mathematical truths can’t be. They are objective, persistent, necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere—no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand years from now. And what’s also amazing is that we own all of them. No one can patent a mathematical formula, it’s ours to share. There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It’s too precious to be given aways to the “initiated few.” It belongs to all of us.
Consider this: in 1996, a commission appointed by the U.S. government gathered in secret and altered a formula for the Consumer Price Index, the measure of inflation that determines the tax brackets, Social Security, Medicare, and other indexed payments. Tens of millions of Americans were affected, but there was little public discussion of the new formula and its consequences. And recently there was another attempt to exploit this arcane formula as a backdoor on the U.S. economy.
Far fewer of these sorts of backroom deals could be made in a mathematically literate society. Mathematics equals rigor plus intellectual integrity times reliance on facts. We should all have access to the mathematical knowledge and tools needed to protect us from arbitrary decisions made by the powerful few in an increasingly math-driven world. Where there is no mathematics, there is no freedom.
There is a common fallacy that one has to study mathematics for years to appreciate it. Some even think that most people have an innate learning disability when it comes to math. I disagree: most of us have heard of and have at least a rudimentary understanding of such concepts as the solar system, atoms and elementary particles, the double helix of DNA, and much more, without taking courses in physics and biology. And nobody is surprised that these sophisticated ideas are part of our culture, our collective consciousness. Likewise, everybody can grasp key mathematical concepts and ideas, if they are explained in the right way. To do this, it is not necessary to study math for years; in many cases, we can cut right to the point and jump over tedious steps.
The problem is: while the world at large is always talking about planets, atoms, and DNA, chances are no one has ever talked to you about the fascinating ideas of modern math, such as symmetry groups, novel numerical systems in which 2 and 2 isn’t always 4, and beautiful geometric shapes like Riemann surfaces.
One of my teachers, the great Israel Gelfand, used to say: “People think they don’t understand math, but it’s all about how you explain it to them. If you ask a drunkard what number is larger, 2/3 or 3/5, he won’t be able to tell you. But if you rephrase the question: what is better, 2 bottles of vodka for 3 people or 3 bottles of vodka for 5 people, he will tell you right away: 2 bottles for 3 people, of course.
Perhaps, a bigger point is that it is perfectly OK if something is unclear. That’s how I feel 90 percent of the time when I do mathematics, so welcome to my world! The feeling of confusion (even frustration, sometimes) is an essential part of being a mathematician. But look at the bright side: how boring would life be if everything in it could be understood with little effort! What makes doing mathematics so exciting is our desire to overcome this confusion; to understand; to lift the veil on the unknown. And the feeling of personal triumph when we do understand something makes it all worthwhile.
Each of us was born in a particular year, lives in a house that has a particular number on the street, has a phone number, a PIN to access a bank account at the ATM, and so forth. All of these numbers have something in common: each of them is obtained by adding number 1 to itself a certain number of times: 1+1 is 2, 1+1+1 is 3, and so on. These are called the natural numbers.
We also have the number 0, and the negative numbers: -1,-2,-3,… As we discussed in Chapter 5, these numbers go by the name “integers.” So an integer is a natural number, or number 0, or the negative of a natural number.
We also encounter slightly more general numbers. A price, in dollars and cents, is often represented like this: $2.59, meaning two dollars and fifty-nine cents. This is the same as 2 plus the fraction 59/100, or 59 times 1/100. Here 1/100 means the quantity that being added to itself 100 times gives us 1. Numbers of this kind are called rational numbers, or fractions.
You may be wondering: how could one come up with these kinds of conjectures in the first place?
This is really a question about the nature of of mathematical insight. The ability to see patterns and connections that no one had seen before does not come easily. It is usually the product of months, if not years, of hard work. Little by little, the inkling of a new phenomenon or a theory emerges, and at first you don’t believe in yourself. But then you say: “what if it’s true?” You try to test the idea by doing sample calculations. Sometimes these calculations are hard, and you have to navigate through mountains of heavy formulas. The probability of making a mistake is very high, and if it does not work at first, you if it does not work at first, you try to redo it, over and over again.
More often than not, at the end of the day (a month, or a year), you realize that your initial idea was wrong, and you have to try something else. These are the moments of frustration and despair. You feel that you have wasted an enormous amount of time, with nothing new to show for it. This is hard to stomach. But you can never give up. You go back to the drawing board, you analyze more data, you learn from your previous mistakes, you try to come up with a better idea. And every once in a while, suddenly, your idea starts to work. It’s as if you had spend a fruitless day surfing, when you finally catch a wave: you try to hold on to it and ride it for as long as possible. At moments like this, you have to free your imagination and let the wave take you as far as it can. Even if the idea sounds totally crazy at first.
Thinking about a Lie group, or any manifold, of more than three dimensions can be very challenging. Our brain is wired in such a way that we can only imagine geometric shapes, or manifolds, in dimensions up to three. Even imagining the four-dimensional combination of space and time is a strenuous task: we just don’t perceive the time (which constitutes the fourth dimension) as an equivalent of a spatial dimension. What about higher dimensions? How can we analyze five- or six- or hundred-dimension manifolds?
Think about this in terms of the following analogy:works of art give us two-dimensional renderings of three-dimensional objects. Artists paint two-dimensional projections of these objects on the canvas and use the technique of perspective to create the illusion of depth, the third dimension, in their paintings. Likewise, we can imagine four-dimensional objects by analyzing their three-dimensional projections.
Another, more efficient way to imagine a forth dimension is to think of a four-dimensional object as a collection of its three-dimensional “slices”. This would be similar to slicing a loaf of bread, which is three-dimensional, into slices so thin that we could think of them as being two dimensional.
If the forth dimension represents time, then this four-dimensional “slicing” is known as photography. Indeed, snapping a picture of a moving person gives us a three-dimensional slice of a four-dimensional object representing that person in the four-dimensional space-time (this slice is then projected onto a plane).
Though we cannot imagine a four-dimensional space, we can actualize it mathematically.
In this environment, mathematics and theoretical physics were oases of freedom. Through communist apparatchiks wanted to control every aspect of life, these areas were just too abstract and difficult for them to understand. Stalin, for one, never dared to make any pronouncements about math. At the same time, Soviet leaders also realized the importance of these seemingly obscure and esoteric areas for the development of nuclear weapons, and that’s why they did not want to “mess” with these areas. As the result, mathematicians and theoretical physicists who worked on the atomic bomb project (many of them reluctantly, I might add) were tolerated, and some even treated well, by Big Brother.
I believe that this was the main reason why so many talented young students chose mathematics as their profession. This was the one area in which they could engage in free intellectual pursuit.
When I talk about a connection between love and math, I don’t mean to say that love can me reduced to math. Rather, my point is that there is a lot more to math than most of us realize. Among other things, mathematics gives us a rationale and an additional capacity to love each other and the world around us. A mathematical formula does not explain love, but it can carry a charge of love.
As poet Norma Farber wrote,
Make me no lazy love…
Move me from case to case.
Mathematics moves us “from case to case,” and herein lies its deep and largely untapperd spiritual function.
Albert Einstein wrote: “Every one who is seriously involved in the pursuit of science becomes convinced that some spirit is manifest in the laws of the Universe—a spirit vastly superior to that of man, and one in the face of which we with our modest powers must feel humble.” And Issac Newton expressed his feelings this way: “to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lie all undiscovered before me.”