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Profs Duke It Out in Big Number Duel

Two Philosophers Vie to Write Largest Possible Finite Number on Chalkboard

By Mandana T. Manzari

Dr. Evil clutched his heart as though it had been pierced by an arrow. Trembling, he fell to his knees on the floor of the crowded stuffy room, all eyes watching him. The Mexican Multiplier threw up his hands in victory, smiling, as Dr. Evil whispered, “I’ve been crushed.” The battle was finally over.

On Friday, Jan. 26, two philosophers, MIT Associate Professor Agustin Rayo (The Mexican Multiplier) and Princeton Associate Professor Adam N. Elga (Dr. Evil) engaged in the Large Number Duel, in which they attempted to one-up each other by inscribing the largest finite number ever to be written on an ordinary-sized chalkboard. The feat, if successfully accomplished, would be worthy of a note in the Guinness Book of World Records.

The Large Number Duel required the use of logic, numerical tricks, and philosophical wit. The use of philosophy is “crucial,” Rayo said. “The limit of math ability was reached at the end. Knowing a bit of philosophy, that was the key.”

“The philosophical debate-turned-boxing-match really opened my eyes to some of the cool math-related things that the Philosophy Department here does,” Quentin E. Smith ’10 said. “They were able to invoke all sorts of esoteric constructions in their quest to write a bigger number. I also enjoyed when they referenced the Busy Beaver function.”

The rules of the duel gave free rein to the contestants’ creativity and humor, maintaining only a ban on the use of infinity, and restricting statements about the number proposed to a primitive semantic vocabulary. The battle itself was intense and the room in the Dreyfoos wing of the Stata Center was packed, with people standing on chairs and at least 20 students craning their necks from the doorway.

The contest opened in the style of a boxing match, with competitors presented “in the red corner” and “in the blue corner.” Elga went first, writing the number one. “Ha!” announced Rayo, as he countered with a string of ones across the board. Elga retaliated with a clever trick, erasing a line through the base of half of the ones to turn them into factorials.

As the battle continued, the contestants began defining their own functions. Moments into their definitions, a student raised her hand and asked Elga if the operation he had written on the board was even computable. Elga cleared his throat, smiled and succinctly replied, “No.”

Functions became more and more complicated, at one point prompting the announcer to proclaim, “It looks like there are words in your number.”

Near the end of the duel, Rayo furiously scribbled on the whiteboard: “The smallest number bigger than any number that can be named by an expression in the language of first order set-theory with less than a googol (10100) symbols.”

Although this definition took a bit of tweaking, including what Rayo described as his “second order logic trick,” it soon won him the duel.

As Elga collapsed, slain, the referee closed the ceremony. “It was a great game,” Elga said. “Heated at times, but nevertheless, a really great game.”

After the battle, when asked if he thought he set the Guinness record, Dr. Rayo said “It’s hard to be sure, but the number is bigger than any number I have ever seen.”

Although it is unclear whether the competitors succeeded in setting a new record, they did succeed in entertaining audience members with their antics.

“My favorite part of the battle was, I have to say, the melodramatics of the two competitors,” Smith said. “They played up their toil. At one point, the MIT contestant walked away and leaned his head against the wall in shame. The referee started to count down, and then the contestant turned around and said “wait a minute!” and then turned back and forth as he attempted to figure out whether he could truly write a larger number.”

Despite the fact that his large number led him to victory against his graduate school friend Elga, Rayo said in an interview after the game that his favorite number is 19. “It has to be prime, of course,” he said matter-of-factly.

Staff Reporter Nick Semenkovich contributed to the reporting of this article.