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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 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.
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