Encyclopedia of Humor

“First and above all he was a logician. At least thirty-five years of the half-century or so of his existence had been devoted exclusively to proving that two and two always equal four, except in unusual cases, where they equal three or five, as the case may be.” — Jacques Futrelle, “The Problem of Cell 13″

Most mathematicians are familiar with — or have at least seen references in the literature to — the equation 2 + 2 = 4. However, the less well known equation 2 + 2 = 5 also has a rich, complex history behind it. Like any other complex quantitiy, this history has a real part and an imaginary part; we shall deal exclusively with the latter here.

Many cultures, in their early mathematical development, discovered the equation 2 + 2 = 5. For example, consider the Bolb tribe, descended from the Incas of South America. The Bolbs counted by tying knots in ropes. They quickly realized that when a 2-knot rope is put together with another 2-knot rope, a 5-knot rope results.

Recent findings indicate that the Pythagorean Brotherhood discovered a proof that 2 + 2 = 5, but the proof never got written up. Contrary to what one might expect, the proof’s nonappearance was not caused by a cover-up such as the Pythagoreans attempted with the irrationality of the square root of two. Rather, they simply could not pay for the necessary scribe service. They had lost their grant money due to the protests of an oxen-rights activist who objected to the Brotherhood’s method of celebrating the discovery of theorems. Thus it was that only the equation 2 + 2 = 4 was used in Euclid’s “Elements,” and nothing more was heard of 2 + 2 = 5 for several centuries.

Around A.D. 1200 Leonardo of Pisa (Fibonacci) discovered that a few weeks after putting 2 male rabbits plus 2 female rabbits in the same cage, he ended up with considerably more than 4 rabbits. Fearing that too strong a challenge to the value 4 given in Euclid would meet with opposition, Leonardo conservatively stated, “2 + 2 is more like 5 than 4.” Even this cautious rendition of his data was roundly condemned and earned Leonardo the nickname “Blockhead.” By the way, his practice of underestimating the number of rabbits persisted; his celebrated model of rabbit populations had each birth consisting of only two babies, a gross underestimate if ever there was one.

Some 400 years later, the thread was picked up once more, this time by the French mathematicians. Descartes announced, “I think 2 + 2 = 5; therefore it does.” However, others objected that his argument was somewhat less than totally rigorous. Apparently, Fermat had a more rigorous proof which was to appear as part of a book, but it and other material were cut by the editor so that the book could be printed with wider margins.

Between the fact that no definitive proof of 2 + 2 = 5 was available and the excitement of the development of calculus, by 1700 mathematicians had again lost interest in the equation. In fact, the only known 18th-century reference to 2 + 2 = 5 is due to the philosopher Bishop Berkeley who, upon discovering it in an old manuscript, wryly commented, “Well, now I know where all the departed quantities went to — the right-hand side of this equation.” That witticism so impressed California intellectuals that they named a university town after him.

But in the early to middle 1800′s, 2 + 2 began to take on great significance. Riemann developed an arithmetic in which 2 + 2 = 5, paralleling the Euclidean 2 + 2 = 4 arithmetic. Moreover, during this period Gauss produced an arithmetic in which 2 + 2 = 3. Naturally, there ensued decades of great confusion as to the actual value of 2 + 2. Because of changing opinions on this topic, Kempe’s proof in 1880 of the 4-color theorem was deemed 11 years later to yield, instead, the 5-color theorem. Dedekind entered the debate with an article entitled “Was ist und was soll 2 + 2?”

Frege thought he had settled the question while preparing a condensed version of his “Begriffsschrift.” This condensation, entitled “Die Kleine Begriffsschrift (The Short Schrift),” contained what he considered to be a definitive proof of 2 + 2 = 5. But then Frege received a letter from Bertrand Russell, reminding him that in “Grundbeefen der Mathematik” Frege had proved that 2 + 2 = 4. This contradiction so discouraged Frege that he abandoned mathematics altogether and went into university administration.

Faced with this profound and bewildering foundational question of the value of 2 + 2, mathematicians followed the reasonable course of action: they just ignored the whole thing. And so everyone reverted to 2 + 2 = 4 with nothing being done with its rival equation during the 20th century. There had been rumors that Bourbaki was planning to devote a volume to 2 + 2 = 5 (the first forty pages taken up by the symbolic expression for the number five), but those rumor remained unconfirmed. Recently, though, there have been reported computer-assisted proofs that 2 + 2 = 5, typically involving computers belonging to utility companies. Perhaps the 21st century will see yet another revival of this historic equation.

The above was written by Houston Euler.

A mathematician and a non-mathematician are sitting in an airport hall waiting for their flight to go. The non has terrible flight panic.

“Hey, don’t worry, it’s just every 10000th flight that crashes.”

“1:10000? So much? Then it surely will be mine!”

“Well, there is an easy way out. Simply take the next plane. It’s much more probable that you go from a crashing to a non-crashing plane than the other way round. So you are already at 1:10000 squared.”

Analysis:
1. Differentiate it and put into the refrig. Then integrate it in the refrig.
2. Redefine the measure on the referigerator (or the elephant).
3. Apply the Banach-Tarsky theorem.

Number theory:
1. First factorize, second multiply.
2. Use induction. You can always squeeze a bit more in.

Algebra:
1. Step 1. Show that the parts of it can be put into the refrig. Step 2. Show that the refrig. is closed under the addition.
2. Take the appropriate universal refrigerator and get a surjection from refrigerator to elephant.

Topology:
1. Have it swallow the refrig. and turn inside out.
2. Make a refrig. with the Klein bottle.
3. The elephant is homeomorphic to a smaller elephant.
4. The elephant is compact, so it can be put into a finite collection of refrigerators. That’s usually good enough.
5. The property of being inside the referigerator is hereditary. So, take the elephant’s mother, cremate it, and show that the ashes fit inside the refrigerator.
6. For those who object to method 3 because it’s cruel to animals. Put the elephant’s BABY in the refrigerator.

Algebraic topology:
Replace the interior of the refrigerator by its universal cover, R^3.

Linear algebra:
1. Put just its basis and span it in the refrig.
2. Show that 1% of the elephant will fit inside the refrigerator. By linearity, x% will fit for any x.

Affine geometry:
There is an affine transformation putting the elephant into the refrigerator.

Set theory:
1. It’s very easy! Refrigerator = { elephant } 2) The elephant and the interior of the refrigerator both have cardinality c.

Geometry:
Declare the following:
Axiom 1. An elephant can be put into a refrigerator.

Complex analysis:
Put the refrig. at the origin and the elephant outside the unit circle. Then get the image under the inversion.

Numerical analysis:
1. Put just its trunk and refer the rest to the error term.
2. Work it out using the Pentium.

Statistics:
1. Bright statistician. Put its tail as a sample and say “Done.”
2. Dull statistician. Repeat the experiment pushing the elephant to the refrig.
3. Our NEW study shows that you CAN’T put the elephant in the refrigerator.

Two statisticians were travelling in an airplane from LA to New York. About an hour into the flight, the pilot announced that they had lost an engine, but don’t worry, there are three left.

However, instead of 5 hours it would take 7 hours to get to New York. A little later, he announced that a second engine failed, and they still had two left, but it would take 10 hours to get to New York.

Somewhat later, the pilot again came on the intercom and announced that a third engine had died. Never fear, he announced, because the plane could fly on a single engine.

However, it would now take 18 hours to get to new York. At this point, one statistician turned to the other and said, “Gee, I hope we don’t lose that last engine, or we’ll be up here forever!”

“Psst, c’mere,” said the shifty-eyed man wearing a long black trenchcoat, as he beckoned me off the rainy street into a damp dark alley. I followed.

“What are you selling?” I asked.

“Geometrical algebra drugs.”

“Huh!?”

“Geometry drugs. Ya got your uppers, your downers, your sidewaysers, your inside-outers…”

“Stop right there,” I interrupted. “I’ve never heard of inside-outers.”

“Oh, man, you’ll love ‘em. Makes you feel like M.C. ever-lovin’ Escher on a particularly weird day.”

“Go on…”

“OK, your inside-outers, your arbitrary bilinear mappers, and here, heh, here are the best ones,” he said, pulling out a large clear bottle of orange pills.

“What are those, then?” I asked.

“Givens transformers. They’ll rotate you about more planes than you even knew existed.”

“Sounds gross. What about those bilinear mappers?”

“There’s a whole variety of them. Here’s one you’ll love — they call it ‘One Over Z’ on the street. Take one of these little bad boys and you’ll be on speaking terms with the Point of Infinity.”