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Topic: (not a riddle): 2+2=5 (Read 7168 times) |
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BNC
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(not a riddle): 2+2=5
« on: Mar 30th, 2003, 11:48pm » |
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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.
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aero_guy
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Re: (not a riddle): 2+2=5
« Reply #1 on: Mar 31st, 2003, 3:16am » |
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Very nice, but my limited math/philosophy background only lets me inside a portion of it though.
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wowbagger
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Re: (not a riddle): 2+2=5
« Reply #2 on: Mar 31st, 2003, 5:20am » |
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That was a lot of fun!
I especially liked the first bit, although I could be nitpicky about actual quipus using two, three, etc. agglutinated knots for a "2", a "3" etc., but I won't.
Of course I don't understand all allusions either, but let me add a comment:
Frege's (later) work is titled "Grundgesetze der Arithmetik". That's what you get if you don't check the original.
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Icarus
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Re: (not a riddle): 2+2=5
« Reply #3 on: Mar 31st, 2003, 5:18pm » |
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Quote:| Moreover, during this period Gauss produced an arithmetic in which 2 + 2 = 3. |
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I must point out that this arithmetic was also independently developed by Johann Boylai and by Nicolai Ivanovitch Lobachevsky. Although Gauss was almost certainly the first to conceive of it, he feared to contradict the deeply rooted sentiment that 2 + 2 = 4. Even those who were willing to consider 2 + 2 = 5 would laugh at the idea of a smaller value! So Gauss never published his ideas, despite his already sterling reputation. But when the younger and still unknown Boylai proved braver as well, publishing his work, Gauss attempted to take credit for the discovery. Unbeknownst to both of them, Lobachevsky had already published in Russia 6 years earlier.
Thus it is more properly Lobachevsky's arithmetic, not Gauss's.
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Sir Col
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Re: (not a riddle): 2+2=5
« Reply #4 on: Sep 18th, 2003, 3:51pm » |
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What a superb romp through the history of mathematics. 
Thanks for that, BNC.
Perhaps it should also include the truth and fuller implications of Kurt Gödel proof, which may have been the reason for his 'retirement' from mathematics. The dilemma of mathematics remaining incomplete by not admitting inconsistencies is challenged. The result, 2+2=4, which is accepted as true, yet clearly unproved, presents us with the disturbing fact that mathematics does not just remain incomplete, but also uncertain.
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maryl
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Re: (not a riddle): 2+2=5
« Reply #5 on: Sep 18th, 2003, 4:35pm » |
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Wow! Thanks BNC, I always wanted to know the explanation for that! That sheds some light on things.
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towr
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Re: (not a riddle): 2+2=5
« Reply #6 on: Sep 19th, 2003, 1:13am » |
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on Sep 18th, 2003, 3:51pm, Sir Col wrote:| The result, 2+2=4, which is accepted as true, yet clearly unproved |
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Actually, it quite easy to prove from the definition of natural numbers (or from some definition of natural numbers) and the definition of addition.
As with all things it's only true in a certain context, that's to say you can define you're own axioms and models and create a different math-system in which the same symbols have different meaning.
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Mugwump101
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Re: (not a riddle): 2+2=5
« Reply #7 on: Jan 4th, 2004, 12:06pm » |
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I really admired your data on what 2+2=5. But today we use 2+2=4 all the time, would that be incorrect in our formulas for math? To think if 2+2 did equal 5. Then wouldn't it change our mathematical calculations? Let's use the tribe's way, 2 knots and 2 knots and tied them together (that were on a rope) we wold get two. If 2+2=5 then, 3+3 would 7? and all the numbers will be lost in place.
Was there any records of brilliant truth left by them that really does make sense that 2+2=5?
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"When I examine myself and my methods of thought, I come to the conclusion that the gift of fantasy has meant more to me than my talent for absorbing positive knowledge. "~ Albert Einstein
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aero_guy
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Re: (not a riddle): 2+2=5
« Reply #8 on: Jan 4th, 2004, 7:43pm » |
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Mug,
Having just read a post where you were getting help with homework I assume that your post was in earnest. The whole post was discussing some of the history of math through a joke. You probably wouldn't understand almost any of it as you haven't been exposed to what they are referencing, and frankly I don't get a lot of it either. Rest assured, though, most of us are pretty confident in 2+2=4.
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ThudnBlunder
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Re: (not a riddle): 2+2=5
« Reply #9 on: Jan 4th, 2004, 9:00pm » |
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Good points, aero_guy, but there's no need to call him a mug. 
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| « Last Edit: Jan 5th, 2004, 10:25am by ThudnBlunder » |
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aero_guy
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Re: (not a riddle): 2+2=5
« Reply #10 on: Jan 5th, 2004, 6:15am » |
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You're just lucky this is over the internet or I'd pop you one right in your ugly mug.
Ah, 30's vernacular.
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| « Last Edit: Jan 5th, 2004, 6:15am by aero_guy » |
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