Jim Holt’s latest book is Why Does the World Exist? (December 2015)

In the Mountains of Mathematics

Kurt Gödel and Albert Einstein, Princeton, New Jersey, 1954
“The science of pure mathematics…may claim to be the most original creation of the human spirit.” So declared the philosopher (and lapsed mathematician) Alfred North Whitehead. Strange, then, that the practitioners of this “science” still feel the need to justify their vocation—not to mention the funding that the rest of …

At the Core of Science

William Blake: Newton, 1795–circa 1805
In 1967, Steven Weinberg, then a visiting professor at MIT, published what has become one of the most frequently cited papers in physics. In it, he presented a mathematical model that “unified” two of the four fundamental forces of nature. What he showed was that these two seemingly very different …

A Mathematical Romance

Edward Frenkel, Berkeley, California, September 2010
For those who have learned something of higher mathematics, nothing could be more natural than to use the word “beautiful” in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and …

Charles Rosen’s Lost Masterpiece

Frédéric Chopin was “the greatest master of counterpoint since Mozart”—so claimed the late pianist and author Charles Rosen in a 1987 review in these pages. At the time I read this, it came as a double surprise to me. I had never thought of Chopin’s music as having a lot of contrapuntal interest. I had always imagined it to stress sonority over structure, to be more emotional—even sentimental—than intellectual: a sort of higher mood music.

He Conceived the Mathematics of Roughness

Benoit Mandelbrot, 1982. Behind him is an attempted computer simulation of a crater field. Crater fields, such as those occurring on the moon, are formed by the cumulative impact of meteorites. They have a fractal structure, one that can be mimicked by computer methods. But the program that generated this not very plausible lunar landscape contained an error, leading Mandelbrot to dub the image ‘the computer bug as artist.’
Benoit Mandelbrot, the brilliant Polish-French-American mathematician who died in 2010, had a poet’s taste for complexity and strangeness. His genius for noticing deep links among far-flung phenomena led him to create a new branch of geometry, one that has deepened our understanding of both natural forms and patterns of human behavior. The key to it is a simple yet elusive idea, that of self-similarity. To see what self-similarity means, consider a homely example: the cauliflower.

How the Computers Exploded

Robert Oppenheimer and John von Neumann in front of MANIAC, the first digital computer, at the Institute for Advanced Study, Princeton, 1952
The digital universe came into existence, physically speaking, late in 1950, in Princeton, New Jersey, at the end of Olden Lane. That was when and where the first genuine computer—a high-speed, stored-program, all-purpose digital-reckoning device—stirred into action. It had been wired together, largely out of military surplus components, in a …

A Comedy of Colors

A century and a half ago, a student who was coloring a map of England noticed that he only needed four colors to do the job—that is, to ensure that no counties sharing a border, such as Kent and Sussex, got the same color. This led him to guess that …

Geometrical Creatures

One feature of the world that few people stop to puzzle over is how many dimensions it has. Although it is a little tricky to say just what a dimension is, it does seem fairly obvious that we, the objects that surround us, and the space we move about in …

Infinitesimally Yours

When people talk about “the infinite,” they usually mean the infinitely great: inconceivable vastness, world without end, boundless power, the Absolute. There is, however, another kind of infinity that is quite different from these, though just as marvelous in its own way. That is the infinitely small, or the infinitesimal.