GOOSE: You should come with me to the birthday party.
DUCK: I’d rather stay home.
GOOSE: You ought to come with me. And I’ll tell you why. The day before yesterday, as I was waddling out of Swanville, another friend of mine—you don’t know him—said to me, “I’m glad you’re leaving Swanville. You’ve been shut up in your coop too much, thinking. The world is full of thoughts. There’re no thoughts left to think. You’re just thinking the thoughts others have thought. That’s a waste!”
DUCK: That’s unlikely.
GOOSE: What’s unlikely?
DUCK: That all possible thoughts have been thought.
GOOSE: Let’s do the calculation. Let’s suppose the brain has a finite capacity. . ..
DUCK: Yes, yes . . . I see what you want to do. You want to assume there’s a finite number of thoughts within the reach of the human brain. Then you want to estimate the number of people who have lived and the number of thoughts they have thought, and you want to subtract all the redundant thoughts, and leave only the unique ones, and subtract all the trivial thoughts, and leave only the interesting ones, and then you want to conclude with Epicurus (or, more accurately, with one of his superficial followers) that we ought to go to birthday parties as often as we can because carpe diem before the ship goes down with its fowls.
GOOSE: That’s my conclusion. I suppose there could be a few new thoughts within the reach of someone in the world this year. But I’m not a mathematician. I don’t know if my ideas about math are new or wrong or interesting or worthless.
DUCK: Well, let’s hear them. This will be amusing!
GOOSE: Don’t laugh until I’m finished.
DUCK: I’m listening.
GOOSE: Is mathematics necessarily as it is? Could we invent a mathematics where the number 2 vanishes, and 4 becomes prime? Now we have a new distribution of the prime numbers. How would this distribution relate to the zeta function? . . . Now let’s suppose that after another hour, 3 vanishes too, and now 12 becomes prime. We could compare the distribution of the prime numbers when 2 is removed with the distribution of the prime numbers when 3 is removed with the distribution of the prime numbers when 5 is removed, etc., with both sequential removals and concurrent removals. Then we could see how these different distributions of primes relate to the zeta function or to some other function; then we could group the different distributions of primes according to any shared characteristics.
DUCK: There’s probably something obviously wrong with that idea.
GOOSE: But if we invent mathematics, then we should be able to remove any number we wish. . .. Let’s suppose that wherever we looked, there were never thirteen objects. Never thirteen of anything, ever! Never thirteen ponds, never thirteen nests, never thirteen trees. The numbers of things always jump from twelve to fourteen. Then . . . would thirteen still exist in the abstract? Could we imagine thirteen in that case?
DUCK: You have strange questions.
GOOSE: Once upon a time, mathematicians didn’t use the negative numbers. Well . . . if we can calculate without the negative numbers, couldn’t we calculate without the number 2? Or without the number 1? Let’s start with 1. Let’s remove 1 from the integers. What would be the impact? We’d lose the multiplicative identity function. We’d have to redefine the successor function. The answer would be undefined whenever 1 would have been the result of a calculation. . .. The impacts would be different if we removed the number 2.
DUCK: Well . . . would that be interesting?
GOOSE: Maybe. Let’s remove a prime number such as 13 and obtain a new set of primes. Then we ask if Euclid’s famous proof that there is no largest prime still holds. . .. Imagine aliens who—due to some quirk of their brains—start counting at 60. Could they rewrite our quantum mechanics using only their integers? Would they use only their prime numbers when creating the keys for their cryptographies?
DUCK: These are interesting speculations—but probably interesting to me only because I’m not a mathematician.
GOOSE: Why should the second natural number always be 2? Let it be 11 or 23!
DUCK: You’re proposing the removal of the number 2—but you say “the second number” because the numbers exist necessarily.
GOOSE: Children sometimes forget 7 when counting on their fingers. Couldn’t everyone—and forever—forget 7 all at once? Would we notice that we’ve forgotten 7?
DUCK: We’d notice when we wanted to cut a birthday cake into sevenths.
GOOSE: So, the numbers exist necessarily? We haven’t invented them? We couldn’t have invented them differently? Is this what we mean when we say “math is discovered”—that we couldn’t have invented it differently?
* Orlando Bartro is the author of Toward Two Words, a comical & surreal novel about a man who finds yet another woman he never knew, regularly available at Amazon for $4.91.