“Dead Putting Society” (1990) tells the story of a miniature golf math, in which Bart simpson is playing Todd Flanders, the son of neighbor Ned Flanders. It is a very high-stakes confrontation, because the father of the loser faces a terrible fate. He will have to mow the winner’s lawn in his wife’s dress.
During a tense exchange between the two fathers, Homer and Ned invoke infinity to reinforce their positions:
Homer: This time tomorrow, you’ll be wearing high heels!
Ned: Nope, you will.
Homer: ‘Fraid not.
Ned: ‘Fraid so!
Homer: ‘Fraid not.
Ned: ‘Fraid so!
Homer: ‘Fraid not infinity!
Ned: ‘Fraid not infinity plus one!
Homer: D’oh!
I asked which of the writers had suggested this piece of dialogue, but nobody was able to remember. This is not surprising, as the script was written more than two decades ago. However, there was general agreement that Homer and Ned’s petty argument would have derailed the scriptwriting process, as it would have triggered a debate over the nature of infinity. So, is infinity plus one more than infinity? Is it a meaningful statement or just gobbledegook? Can it be proved?
In their efforts to answers these questions, the mathematicians around the scripting table would doubtless have mentioned the name of George Cantor, who was born in St. Petersburg, Russia, in 1845. Cantor was the first mathematician to really grapple with the meaning of infinity. However, his explanations were always deeply technical, so it was left to the eminent German mathematician David Hilbert (1862-1943) to convey Cantor’s research. He had a knack for finding analogies that made Cantor’s ideas about infinity more palatable and digestible.
One of Hilbert’s most celebrated explanations of infinity involved an imaginary building known as Hilbert’s Hotel — a rather grand hotel with an infinite number of rooms and each door marked 1, 2, 3, and so on. One particularly busy evening, when all the rooms are occupied, a new guest turns up without a reservation. Fortunately, Dr. Hilbert, who owns the hotel, has a solution. He asks all his guests to move from their current rooms to the next one in the hotel. So the guest in room 1 moves to room 2, the guest in room 2 moves to room 3, and so on. Everyone still has a room, but room 1 is now empty and available for the new guest. This scenario suggests (and it can be proven more rigorously) that infinity plus one is equal to infinity; a paradoxical conclusion, perhaps, but one that is undeniable.
This means that Ned Flanders is wrong when he thinks he can trump Homer’s infinity with infinity plus one. In fact, Flanders would have been wrong even if he tried to win the argument with “infinity plus infinity,” as proved by another vignette about Hilbert’s Hotel.
The hotel is full again when an infinitely large coach arrives. The coach driver asks Dr. Hilbert if the hotel can accommodate his infinite number of passengers. Hilbert is unfazed. He asks all his current guests to move to a room number that is double their current room number, so the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, and so on. The existing infinity of guests now occupy only the even-numbered rooms, and an equally infinite number of odd-numbered rooms are now vacant. In this way, the hotel is able to provide rooms for the infinite number of coach passengers.
Once more, this appears to be paradoxical. You might even suspect that it is nonsense, perhaps nothing more than the result of ivory tower philosophizing. Nevertheless, these conclusions about infinity are more than mere sophistry. Mathematicians reach these conclusions about infinity, or any other concept, by building rigorously, step-by-step, upon solid foundations.
This point is well made by an anecdote in which a university vice chancellor complains to the head of his physics department: “Why do physicists always need so much money for laboratories and equipment? Why can’t you be like the mathematics department? Mathematicians only need money for pencils, paper, and wastepaper baskets. Or even better, why can’t you be like the philosophy department? All they need is pencils and paper.”
The anecdote is a dig at philosophers, who lack the rigor of mathematicians. Mathematics is a meticulous search for the truth, because each new proposal can be ruthlessly tested and then either accepted into the framework of knowledge or discarded into the wastepaper basket. Although mathematical concepts might sometimes be abstract and arcane, they must still pass a process of intense scrutiny. Thus, Hilbert’s Hotel has clearly demonstrated that
infinity = infinity + 1
infinity = infinity + infinity
Although Hilbert’s explanation avoids technical mathematics, Cantor was forced to delve deep into the mathematical architecture of numbers in order to reach his paradoxical conclusions about infinity, and his intellectual struggles took their toll on him. He suffered severe bouts of depression, spent extended periods in a sanatorium, and grew to believe that he was in direct communication with God. Indeed, he credited God for helping him to develop his ideas and believed that infinity was synonymous with God: “It is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute.” Cantor’s mental state was partly the result of being criticized and mocked by more conservative mathematicians who could not come to terms with his radical conclusions about infinity. Tragically, Cantor died malnourished and impoverished in 1918.
After Cantor’s death, Hilbert commended his colleague’s attempt to address the mathematics of infinity, stating: “The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.”
He made it very clear that he sat in Cantor’s corner in the battle to comprehend infinity: “No one shall drive us from the paradise Cantor has created for us.”
