From A Divine Language, which will be published next month by Farrar, Straus and Giroux.

1.

Prime numbers are those that can be divided cleanly only by themselves and by 1. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, and 23. Prime numbers are where imaginary mathematics begins. They are an example of our discovering properties of numbers, rather than creating them. Idealism is the name of the eighteenth- and nineteenth-century discipline that believed that the mind creates what we know and that there is nothing we can know that the mind hasn’t created. The British mathematician G. H. Hardy writes that prime numbers seemed to him the place where idealism failed: “317 is a prime not because we think so, or because our minds are shaped in one way rather than another,” he writes, “but because it is so, because mathematical reality is built that way.”

2.

Prime numbers have their own taxonomy. Twin primes are two apart. Cousin primes are four apart. And sexy primes are six apart, six being sex in Latin. From Prime Curios!, by Chris Caldwell and G. L. Honaker Jr., I know that an absolute prime is prime regardless of how it is arranged: 199, 919, 991. Palindromic primes are the same forward and backward—133020331. Tetradic primes are palindromic primes that are also prime backward and when seen in a mirror, such as 11, 101, and 1881881. A beastly prime contains 666. 700666007 is a beastly palindromic prime. A depression prime is a palindromic prime whose interior numbers are the same and smaller than the numbers on the ends: 75557, for example. Conversely, plateau primes have interior numbers that are the same and larger than the numbers on the ends, such as 1777771. Invertible primes can be turned upside down and rotated: 109 becomes 601. The only even prime number is 2. Since all other primes are odd, the interval between any two successive primes has to be even, but no one knows a rule to govern this.

3.

Prime numbers are the origin of pure mathematics—mathematics done, that is, without an interest in practical utility. Applied mathematics begins with the ability to count and measure and is procedural; pure mathematics is imaginative. That the classifications have about them a suggestion of snobbery and side-taking has a lot to do with Hardy, who sometimes called pure mathematicians “real mathematicians.” In “A Mathematician’s Apology,” Hardy wrote, “Is not the position of an ordinary applied mathematician in some ways a little pathetic? If he wants to be useful, he must work in a humdrum way, and he cannot give full play to his fancy even when he wishes to rise to the heights.” Hardy was also pleased that pure mathematics was not helpful in commerce, and especially not in war. He believed of pure mathematics that “the best of it may, like the best literature, continue to cause intense emotional satisfaction to thousands of people after thousands of years.”

4.

The aloofness of pure mathematics and its reverence for thinking infused itself into physics. In April 1969, Robert Wilson, a physicist who had worked on the Manhattan Project and was the director of the Fermi National Accelerator Laboratory in Illinois, appeared before Congress to request money for building an accelerator, which was called the 200-BeV Synchrotron. The Synchrotron was a proton accelerator that would make it possible to observe subatomic particles, some of which were theoretical. Wilson was questioned by Senator John Pastore, a Democrat from Rhode Island, who was sympathetic to science and was hoping for arguments he might use to convince the accelerator’s opponents. Pastore asked whether the accelerator involved the security of the country.

“No, sir; I do not believe so,” Wilson said.

“Nothing at all?”

“Nothing at all.”

“It has no value in that respect?”

“It only has to do with the respect with which we regard one another, the dignity of men, our love of culture,” Wilson said. “It has to do with those things. It has nothing to do with the military,” for which he added that he was sorry.

Pastore told him not to be sorry.

“I am not, but I cannot in honesty say it has any such application,” Wilson said.

Pastore tried another tack. “Is there anything here that projects us in a position of being competitive with the Russians, with regard to this race?” he asked.

Very little, Wilson said. “Otherwise, it has to do with: Are we good painters, good sculptors, great poets? I mean all the things that we really venerate and honor in our country and are patriotic about. In that sense, this new knowledge has all to do with honor and country, but it has nothing to do directly with defending our country, except to help make it worth defending.”