Histoire du "2+2=5"
by Houston Euler
"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.
Hardy's proof of the pope's identity:
The
following conversation at the Trinity High Table is recorded in Sir
Harold Jeffreys' Scientific Inference, in a note to chapter one.
Jeffreys remarks that the fact that everything followed from a single
contradiction had been noticed by Aristotle. He goes on to say that
McTaggart denied the consequence: "If 2+2=5, how can you prove that I
am the pope?" Hardy is supposed to have replied: "If 2+2=5, 4=5;
subtract 3; then 1=2; but McTaggart and the pope are two; therefore
McTaggart and the pope are one."
There are related stories like the following:
The great logician Bertrand Russell once claimed that he could prove anything if given that 1+1=1.
So one day, some smarty-pants asked him, "Ok. Prove that you're the Pope."
He thought for a while and proclaimed, "I am one. The Pope is one.