ACHILLES: How very versatile of him!
ANTEATER: At the height of his creative powers, he met with a most untimely demise. One day, a very hot summer day, he was out soaking up the warmth, when a freak thundershower—the kind that hits only once every hundred years or so—appeared from out of the blue and thoroughly drenched J. S. F. Since the storm came utterly without warning, the ants got completely disoriented and confused. The intricate organization that had been so finely built up over decades all went down the drain in a matter of minutes. It was tragic.
ACHILLES: Do you mean that all the ants drowned, which obvious would spell the end of poor J. S. F.?
ANTEATER. Actually, no. The ants managed to survive, every last one them, by crawling onto various sticks and logs that floated above the raging torrents. But when the waters receded and left the ants back on their home grounds, there was no organization left. The caste distribution was utterly destroyed, and the ants themselves had no ability to reconstruct what had once before been such a finely tune organization. They were as helpless as the pieces of Humpty Dumpty in putting themselves back together again. I myself tried, like all the king’s horses and all the king’s men, to put poor Fermant together again. I faithfully put out sugar and cheese, hoping against hope that somehow Fermant would reappear… (Pulls out a handkerchief and wipes his eyes.)
ACHILLES: How valiant of you! I never knew Anteaters had such big hearts.
ANTEATER: But it was all to no avail. He was gone, beyond reconstitution. However, something very strange then began to take place, over the next few months, the ants that had been components of J.S.F. slowly regrouped, and built up a new organization. And thus was Aunt Hillary born.
CRAB: Remarkable! Aunt Hillary is composed of the very same ants Fermant was?
ANTEATER: Well, originally she was, yes. By now, some of the older an have died, and been replaced. But there are still many holdover from the J.S.F.-days.
CRAB: And can’t you recognize some of J.S.F.’s old traits coming to the fore, from time to time, in Aunt Hillary?
ANTEATER: Not a one. They have nothing in common. And there is no reason they should, as I see it. There are, after all, often sever distinct ways to rearrange a group of parts to form a “sum.” An Aunt Hillary was just a new “sum” of the old parts. Not more than the sum, mind you just that particular kind of sum.
TORTOISE: Speaking of sums, I am reminded of number theory, where occasionally, one will be able to take apart a theorem into its component symbols, rearrange them in a new order, and come up with a new theorem.
ANTEATER: I’ve never heard of such a phenomenon, although I confess to being a total ignoramus in the field.
ACHILLES: Nor have I heard of it—and I am rather well versed in the field, if I don’t say so myself. I suspect Mr. T is just setting up one of his elaborate spoofs. I know him pretty well by now.
ANTEATER: Speaking of number theory, I am reminded of J. S. F. again, for number theory is one of the domains in which he excelled. In fact he made some rather remarkable contributions to number theory Aunt Hillary, on the other hand, is remarkably dull-witted in any thing that has even the remotest connection with mathematics. Also, she has only a rather banal taste in music, whereas Sebastiant was extremely gifted in music.
ACHILLES: I am very fond of number theory. Could you possibly relate to us something of the nature of Sebastiant’s contributions?
ANTEATER: Very well, then. (Pauses for a moment to sip his tea, then resumes.) Have you heard of Fourmi’s infamous “Well-Tested Conjecture”?
ACHILLES: I’m not sure.... It sounds strangely familiar, and yet I can quite place it.
ANTEATER: It’s a very simple idea. Lierre de Fourmi, a mathematiciant by vocation but lawyer by avocation, had been reading in his copy of the classic text Arithmetica by Di of Antus, and came across a page containing the equation
He immediately realized that this equation has infinitely many solutions a, b, c, and then wrote in the margin the following notorious comment:
The equation
has solutions in positive integers a, b, c, and n only when n = 2 (and then there are infinitely many triplets a, b, c, which satisfy the equation); but there are no solutions for n > 2. I have discovered a truly marvelous proof of this statement, which, unfortunately, this margin is too small to contain.
Ever since that year, some three hundred days ago, mathematiciants have been vainly trying to do one of two things: either to prove Fourmi’s claim, and thereby vindicate Fourmi’s reputation, which, although very high, has been somewhat tarnished by skeptics who think he never really found the proof he claimed to have found—or else to refute the claim, by finding a counterexample: a set of four integers a, b, c, and n, with n > 2, which satisfy the equation. Until very recently, every attempt in either direction had met with failure. To be sure, the Conjecture has been verified for many specific values of n—in particular, all n up to 125,000. But no one had succeeded in proving it for all n—no one, that is, until Johant Sebastiant Fermant came upon the scene. It was he who found the proof that cleared Fourmi’s name. It now goes under the name “Johant Sebastiant’s Well-Tested Conjecture.”
During emigrations army ants sometimes create bridges of their own bodies. In this photograph of such a bridge (de Fourmi Lierre), the workers Ecilon burchelli colony can be seen linking their legs and, along the top of the bridge, hooking their tarsal claws together to form irregular systems of chains. A symbiotic silverfish, Trichatelura manni, is seen crossing the bridge in the center. (From E. O. Wilson, The Insect Societies. Photograph courtesy of C. W. Rettenmeyer.)
ACHILLES: Shouldn’t it be called a “Theorem” rather than a “Conjecture,” if it’s finally been given a proper proof?
ANTEATER: Strictly speaking, you’re right, but tradition has kept it this way.
TORTOISE: What sort of music did Sebastiant do?
ANTEATER: He had great gifts for composition. Unfortunately, his greatest work is shrouded in mystery, for he never reached the point of publishing it. Some believe that he had it all in his mind; others are more unkind, saying that he probably never worked it out at all, but merely blustered about it.
ACHILLES: What was the nature of this magnum opus?
ANTEATER: It was to be a giant prelude and fugue; the fugue was to have twenty-four voices, and to involve twenty-four distinct subjects, one in each of the major and minor keys.
ACHILLES: It would certainly be hard to listen to a twenty-four-voice fugue as a whole!
CRAB: Not to mention composing one!