Text 3885, 113 rader
Skriven 2006-04-12 06:13:55 av George Pope (1:153/715.1275)
   Kommentar till text 3876 av Greg Sears (3:633/104)
Ärende: funnies from another BBS network
========================================
On (10 Apr 06) Greg Sears wrote to George Pope...
 GS>    How about the TWO dollar COIN!!

Sure, I'll take two, thanks!


                        The History of 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 rumors 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.


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