"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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