THE DUMPLING PARADOX: REDUCTIO-AD-ABSURDUM

In “The Dumpling Paradox,” Season 1, Episode 7, Leonard engages in reductio-ad-absurdum.

“In the space of one hundred and seventy-six years the Lower Mississippi has shortened itself two hundred and forty-two miles. That is an average of a trifle over a mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the Old Oölitic Silurian Period, just a million years ago next November, the Lower Mississippi was upwards of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-pole. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and three-quarters long, and Cairo [Illinois] and New Orleans will have joined their streets together and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.”

—Mark Twain, Life on the Mississippi (1884)

Appeal to Extremes

In “The Dumpling Paradox,” Penny’s promiscuous friend from Nebraska is not only in town, but has taken over Penny’s apartment, forcing Penny to sleep over at Sheldon and Leonard’s. The dynamics of the situation send Sheldon into disarray, not only interrupting his Saturday routine of watching Doctor Who, but also leading Sheldon to “engaging in reductio-ad-absurdum . . . the logical fallacy of extending someone’s argument to ridiculous proportions and then criticizing the result.” In logic, reductio-ad-absurdum, or the “appeal to extremes,” is a form of argument that tries to disprove something by showing it leads to an absurd or impractical conclusion. The technique can be traced back to classical Greek philosophy’s “demonstration to the impossible.” And reductio-ad-absurdum has been used ever since, both in formal philosophical and math reasoning and in situations where pedants like Sheldon wish to prove a point.

The very first and maybe best examples of reductio-ad-absurdum come from ancient Greek thinker Zeno of Elea, who flourished in the fifth century BC, probably between 490 and 430 BC. Zeno’s great innovation in thinking was the paradox, a tool to focus on the unexpected consequences of common-sense ideas, to question assumptions, and to stimulate new theories. In case you think you may never have heard of Zeno, just consider these three examples of reductio-ad-absurdum: according to Zeno’s paradoxes, motion is not possible, an arrow in flight does not move, and the fastest runner in Homer, Achilles, could never catch up with a tortoise in a race, if he gave it a head start.

For two and a half thousand years, thinkers from Aristotle to British philosopher Bertrand Russell have tried to refute Zeno’s ideas, or explain them away with varying success. Late medieval mathematicians Newton and Leibniz went some way to show up flaws in Zeno’s arguments, but the questions Zeno raised about space and time are as relevant as ever and have reemerged in quantum physics.

Zeno’s paradoxes seem to have begun in the most unusual manner. Another Greek philosopher by the name of Parmenides had written a very peculiar poem in the style of the famous Greek poet Homer, the legendary author of Iliad and Odyssey, but with very different subject matter from the battle of Troy, as in Iliad, or the journey home after the fall of Troy, as in Odyssey. Instead, the poem of Parmenides set out to prove that reality is only one thing, or entity, and that this one thing is changeless and motionless and perfect. Parmenides was reacting to traditional Greek thinking of the world in terms of earth, air, fire, and water. A plurality of things, or many things. The four elements theory tried to explain the world and how the world worked and how various transformations from one element into another might happen, and how the world was varied and differentiated. It’s all an illusion, suggested Parmenides: in fact, the world around us is one thing only.

Now, as you can well imagine, this seems like a tough argument to win. But Parmenides had Zeno and his paradoxes on his side, so one way to think of these paradoxes is that they were an attempt to undercut possible objections to the Parmenides thesis on the basis of common-sense assumptions, that there are many things and that clearly those many things do indeed move.

Achilles and the Tortoise

Probably the best-known Zeno paradox is Achilles and the Tortoise. Achilles is the fastest runner in the ancient world, and he has a race with the slowest runner in the ancient world, a tortoise. Indeed, Achilles is such a fast runner that he can give the tortoise a head start. Let’s imagine they’re racing on a hundred-meter racetrack; this is ancient Greece after all, so we won’t use yards. Imagine also that the tortoise is allowed to start ten meters in. The race begins, and first Achilles must cover the same ten meters that the tortoise was given as a head start. However, during the time that elapses as Achilles covers his first ten meters, the tortoise will have moved on. True, the tortoise wouldn’t have moved very far, as it’s a creepingly slow creature, but let’s imagine the tortoise moved on a further meter.

The next labor of Achilles is to cover this extra meter, which the tortoise has managed. And yet, during the time Achilles is covering this new meter, the tortoise will have moved yet farther, let’s say 10 centimeters. Again, a similar thing happens, and, even though the distance between Achilles and the tortoise gets less and less, it will never get to zero. So, according to Zeno, humiliation is brought down on old Achilles. He may be the world’s fastest runner, but he will never be able to overtake the interminable progress of the tortoise.

Rather than claiming a runner can never practically be overtaken in a race, what Zeno is actually doing is challenging other thinkers to defeat his argument. The paradox is really a challenge of the infinite. In particular, it’s a challenge to something in math known as an infinite series. That’s because if we were to find a solution to the paradox of Achilles, we would have to add up infinitely many things to find out how long it takes Achilles to do the infinite number of tasks to overtake the tortoise.

Let’s say he does the first step in half a minute, the second step in half that time, so a quarter of a minute, the third step in an eighth of a minute, the next step in a sixteenth of a minute, and so on. So, it looks like Achilles is having to do an infinite number of tasks to win the race. The way mathematicians eventually resolved this paradox was to say, well, okay, how long does it actually take for Achilles to do this infinite number of tasks? Achilles carries out the tasks because it takes him a finite amount of time. The infinite series of 1/2 plus 1/4 plus 1/8 plus 1/16, and so on, actually adds up to a small amount of time, which is likely to be less than a minute! So, Zeno was actually challenging thinkers on the question of adding up infinitely many things in math and questioning as to whether such math series had some sort of physical reality. Mathematicians did indeed solve it, but it took until the seventeenth or eighteenth century until they came up with a way to understand how to navigate these infinitely many numbers.

Mark Twain and the Eiffel Tower

My favorite “demonstration to the impossible” comes from the great American humorist, Mark Twain. In 1903, a book had been published by the English naturalist Alfred Russel Wallace. Along with Darwin, Wallace had been cofounder of the modern theory of evolution. Now, Wallace inaugurated the study of biogeography with his 1903 book Man’s Place in the Universe. In the book, Wallace argued that, given only a small change in the history of the universe would have prevented the evolution of intelligent beings, it proved that the universe was designed with the ultimate purpose of life culminating in man.

Twain’s sardonic reply to Wallace’s popular but fallacious argument begins innocently enough: “I seem to be the only scientist and theologian still remaining to be heard from on this important matter of whether the world was made for man or not. I feel that it is time for me to speak. I stand almost with the others.”

Twain then goes on to slowly and very subtly rip Wallace’s argument to pieces, beginning with astronomy and geology: “Now as far as we have got, astronomy is on our side. Mr. Wallace has clearly shown this. He has clearly shown two things: that the world was made for man, and that the universe was made for the world—to stiddy it, you know. The astronomy part is settled, and cannot be challenged. We come to the geological part. This is the one where the evidence is not all in, yet. It is coming in, hourly, daily, coming in all the time . . .”

Next, emphasizing the antiquity of the world and the universe, Twain focuses on the evolution of biota to ready the world for the coming of man, with beautiful phrases such as “The oyster being achieved, the next thing to be arranged for in the preparation of the world for man, was fish. Fish, and coal—to fry it with” and “Yes, it took thirty million years and twenty million reptiles to get one that would stick long enough to develop into something else and let the scheme proceed to the next step.”

Finally, Twain ends his riposte with sheer proof of his genius as a writer. It’s the most wonderful “appeal to extremes” example of argument: “And at last came the monkey, and anybody could see that man wasn’t far off, now. And in truth that was so. The monkey went on developing for close upon 5,000,000 years, and then turned into a man—to all appearances. Such is the history of it. Man has been here 32,000 years. That it took a hundred million years to prepare the world for him is proof that that is what it was done for. I suppose it is. I dunno. If the Eiffel Tower were now representing the world’s age, the skin of paint on the pinnacle-knob at its summit would represent man’s share of that age; and anybody would perceive that that skin was what the tower was built for. I reckon they would, I dunno.”