Two plus two is five

"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 have a rich, complex history behind it. Like any other complex quantity, 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 Blob tribe, descended from the Incas of South

America. The Blobs 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.