From 8th grade to 11th grade, I was taught mathematics by a teacher who was a tyrant. A brilliant man who left a lucrative engineering career to teach high school, he clearly was dedicated to teaching the subject. But his dedication took the form of catering to the brightest students in the class and mocking the rest. Since I was not among the brightest students, I was often the object of his ridicule. I was deeply interested in music by this point and sometimes came in late for class because I was held up in music practice; when this happened he used to mercilessly taunt me in front of everyone else and tell me that I should probably drop out of school and start a band. Sometimes he used to refuse me entry to the class. At an age where peer validation means much, this was devastating. Hatred of the man led to hatred of math and I turned into a rebel, not caring about the subject. I liked the abstract aspects of math but just wouldn’t do the problems, seeing it as an act of rebellion
The old joke goes that the world can be divided into three kinds of people – those who can count and those who can’t. Like many jokes this one has a shred of truth in it because math does seem to impose binary divisions on us. Most of us put up with it because it’s useful to us to various extents in our professions, be it accounting, biology or economics. Some of us love it. A select few are blessed with great natural ability and passion for the subject. The rest of us are not just bad at but are often proud of our mathematical deficiency. How often have we come across people or caught ourselves joking that we were always “bad at math”? The result of this attitude – as indicated by any reading of the pandemic news these days for instance – is that we live in a largely mathematically illiterate society which is both bad at and distasteful of numbers and statistics. But it doesn’t have to be that way. Read more »
Two men walking in Princeton, New Jersey on a stuffy day. One shaggy-looking with unkempt hair, avuncular, wearing a hat and suspenders, looking like an old farmer. The other an elfin man, trim, owl-like, also wearing a fedora and a slim white suit, looking like a banker. The elfin man and the shaggy man used to make their way home from work every day. Passersby and motorists would strain their heads to look. Everyone knew who the shaggy man was; almost nobody knew who his elfin companion was. And yet when asked, the shaggy man would say that his own work no longer meant much to him, and the only reason he came to work was to have the privilege of walking home with the elfin man. The shaggy man was Albert Einstein. His walking companion was Kurt Gödel.
What made Gödel, a figure unknown to the public, so revered among his colleagues? The superlatives kept coming. Einstein called him the greatest logician since Aristotle. The legendary mathematician John von Neumann who was his colleague argued for his extraction from fascism-riddled Europe, writing a letter to the director of his institute saying that “Gödel is absolutely irreplaceable; he is the only mathematician about whom I dare make this assertion.” And when I made a pilgrimage to Gödel’s house during a trip to his native Vienna a few years ago, the plaque in front of the house made his claim to posterity clear: “In this house lived from 1930-1937, the great mathematician and logician Kurt Gödel. Here he discovered his famous incompleteness theorem, the most significant mathematical discovery of the twentieth century.”
The reason Gödel drew gasps of awe from colleagues as brilliant as Einstein and von Neumann was because he revealed a seismic fissure in the foundations of that most perfect, rational and crystal-clear of all creations – mathematics. Of all the fields of human inquiry, mathematics is considered the most exact. Unlike politics or economics, or even the more quantifiable disciplines of chemistry and physics, every question in mathematics has a definite yes or no answer. The answer to a question such as whether there is an infinitude of prime numbers leaves absolutely no room for ambiguity or error – it’s a simple yes or no (yes in this case). Not surprisingly, mathematicians around the beginning of the 20th century started thinking that every mathematical question that can be posed should have a definite yes or no answer. In addition, no mathematical question should have both answers. The first requirement was called completeness, the second one was called consistency. Read more »
Mathematics and music have a pristine, otherworldly beauty that is very unlike that found in other human endeavors. Both of them seem to exhibit an internal structure, a unique concatenation of qualities that lives in a world of their own, independent of their creators. But mathematics might be so completely unique in this regard that its practitioners have seriously questioned whether mathematical facts, axioms and theorems may not simply exist on their own, simply waiting to be discovered rather than invented. Arthur Rubinstein and Andre Previn’s performance of Chopin’s second piano concerto sends unadulterated jolts of pleasure through my mind every time I listen to it, but I don’t for a moment doubt that those notes would not exist were it not for the existence of Chopin, Rubinstein and Previn. I am not sure I could say the same about Euler’s beautiful identity connecting three of the most fundamental constants in math and nature – e, pi and i. That succinct arrangement of symbols seems to simply be, waiting for Euler to chance upon it, the way a constellation of stars has waited for billions of years for an astronomer to find it.
The beauty of music and mathematics is that anyone can catch a glimpse of this timelessness of ideas, and even someone untrained in these fields can appreciate the basics. The most shattering intellectual moment of my life was when, in my last year of high school, I read in George Gamow’s “One, Two, Three, Infinity” about the fact that different infinities can actually be compared. Until then the whole concept of infinity had been a single concept to me, like the color red. The question of whether one infinity could be “larger” than another sounded as preposterous to me as whether one kind of red was better than another. But here was the story of an entire superstructure of infinities which could be compared, studied and taken apart, and whose very existence raised one of the most famous, and still unsolved, problems in math – the Continuum Hypothesis. The day I read about this fact in Gamow’s book, something changed in my mind; I got the feeling that some small combination of neuronal gears permanently shifted, altering forever a part of my perspective on the world. Read more »
On a whim I decided to visit the gently sloping hill where the universe announced itself in 1964, not with a bang but with ambient, annoying noise. It’s the static you saw when you turned on your TV, or at least used to back when analog TVs were a thing. But today there was no noise except for the occasional chirping of birds, the lone car driving off in the distance and a gentle breeze flowing through the trees. A recent trace of rain had brought verdant green colors to the grass. A deer darted into the undergrowth in the distance.
The town of Holmdel, New Jersey is about thirty miles east of Princeton. In 1964, the venerable Bell Telephone Laboratories had an installation there, on top of this gently sloping hill called Crawford Hill. It was a horn antenna, about as big as a small house, designed to bounce off signals from a communications satellite called Echo which the lab had built a few years ago. Tending to the care and feeding of this piece of electronics and machinery were Arno Penzias – a working-class refuge from Nazism who had grown up in the Garment District of New York – and Robert Wilson; one was a big picture thinker who enjoyed grand puzzles and the other an electronics whiz who could get into the weeds of circuits, mirrors and cables. The duo had been hired to work on ultra-sensitive microwave receivers for radio astronomy.
In a now famous comedy of errors, instead of simply contributing to incremental advances in radio astronomy, Penzias and Wilson ended up observing ripples from the universe’s birth – the cosmic microwave background radiation – by accident. It was a comedy of errors because others had either theorized that such a signal would exist without having the experimental know-how or, like Penzias and Wilson, were unknowingly building equipment to detect it without knowing the theoretical background. Penzias and Wilson puzzled over the ambient noise they were observing in the antenna that seemed to come from all directions, and it was only after clearing away every possible earthly source of noise including pigeon droppings, and after a conversation with a fellow Bell Labs scientist who in turn had had a chance conversation with a Princeton theoretical physicist named Robert Dicke, that Penzias and Wilson realized that they might have hit on something bigger. Dicke himself had already theorized the existence of such whispers from the past and had started building his own antenna with his student Jim Peebles; after Penzias and Wilson contacted him, he realized he and Peebles had been scooped by a few weeks or months. In 1978 Penzias and Wilson won the Nobel Prize; Dicke was among a string of theorists and experimentalists who got left out. As it turned out, Penzias and Wilson’s Nobel Prize marked the high point of what was one of the greatest, quintessentially American research institutions in history.Read more »