2 + 2 = 5

[Captain Chaos]

Can anyone prove this?

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Humor in der Mathematik

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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 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. Therefore, the Pope and I are one."

 

[Selta]

2+2=5, for very large values of 2.

 

[Baryonyx]

Absolutely!

 

[velvetkevorkian]

Are you taking this at face value, and if so did you miss the "Humor in der Mathematik" heading? (Humour in mathematics)

If you knew it was a joke all along, just ignore me.

 

[Brad Barker]

"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...."

 

[pigpen02]

Poor Winston learned this the hard way....(1984 reference)

 

[AmazingGracePlayer]

That's reminds me of something my teacher taught me...

Since 1/3 is 0.33333333 (3 repeating), then 1/3 +1/3 +1/3 should equal 0.9999999 (9 repeating) but not 1. It does equal 1 when you round up, but we are talking about exact values here. It's impossible for three 1/3's to add up to one unless one of the 1/3 is 0.333333333334. But 1/3 is not 0.33333333334, it's 0.333333333333 with three repeating, there's no fours involved in there. So therefore 1/3 + 1/3 + 1/3 = 0.9999999 (9 repeating). Anyone care to prove that wrong?

 

[Selta]

That's reminds me of something my teacher taught me...

Since 1/3 is 0.33333333 (3 repeating), then 1/3 +1/3 +1/3 should equal 0.9999999 (9 repeating) but not 1. It does equal 1 when you round up, but we are talking about exact values here. It's impossible for three 1/3's to add up to one unless one of the 1/3 is 0.333333333334. But 1/3 is not 0.33333333334, it's 0.333333333333 with three repeating, there's no fours involved in there. So therefore 1/3 + 1/3 + 1/3 = 0.9999999 (9 repeating). Anyone care to prove that wrong?

I've been saying that forever. But, it's not true.
1/3 is .333 repeating forever. In essence, it doesn't matter, since you get so close to 1, that it is 1. Basically, it's a summation notation. I proved it in calc 3 at one point using an integral .
And, you also have to realize, when you go from fractions to decimals, there's rounding error . 1/3 + 1/3 + 1/3 = 1. 0.333333 x 3 does not, excatly equal one.

 

[lemur821]

That's reminds me of something my teacher taught me...

Since 1/3 is 0.33333333 (3 repeating), then 1/3 +1/3 +1/3 should equal 0.9999999 (9 repeating) but not 1. It does equal 1 when you round up, but we are talking about exact values here. It's impossible for three 1/3's to add up to one unless one of the 1/3 is 0.333333333334. But 1/3 is not 0.33333333334, it's 0.333333333333 with three repeating, there's no fours involved in there. So therefore 1/3 + 1/3 + 1/3 = 0.9999999 (9 repeating). Anyone care to prove that wrong?

But .99... is 1.

 

[jenderfazz]

That's reminds me of something my teacher taught me...

Since 1/3 is 0.33333333 (3 repeating), then 1/3 +1/3 +1/3 should equal 0.9999999 (9 repeating) but not 1. It does equal 1 when you round up, but we are talking about exact values here. It's impossible for three 1/3's to add up to one unless one of the 1/3 is 0.333333333334. But 1/3 is not 0.33333333334, it's 0.333333333333 with three repeating, there's no fours involved in there. So therefore 1/3 + 1/3 + 1/3 = 0.9999999 (9 repeating). Anyone care to prove that wrong?

That issue has been beaten to death, and there is no way to resolve it cleanly. This will be debated forever......

The idea is that 1/3 = 0.3333333 with an infinite amount of 3's. As far as I know, 0.999999 with an infinite amount of 9's = 1. However, some people insist that 1 = 3 x 0.33333(infinite number of 3's...)4. Which makes sense, yet it doesn't, because infinity is infinity. You can't pretend there's a 4 at the end, since infinity means the 3's never end, ever. Infinity screws everything up. That's the problem with decimal numbers. Fractions get it right, and decimals approximate them. It all makes sense with fractions, or on a number line, etc.

 

[AmazingGracePlayer]

That issue has been beaten to death, and there is no way to resolve it cleanly. This will be debated forever......

The idea is that 1/3 = 0.3333333 with an infinite amount of 3's. As far as I know, 0.999999 with an infinite amount of 9's = 1. However, some people insist that 1 = 3 x 0.33333(infinite number of 3's...)4. Which makes sense, yet it doesn't, because infinity is infinity. You can't pretend there's a 4 at the end, since infinity means the 3's never end, ever. Infinity screws everything up.

No, no, no, 0.999999 with an infinite number of 9's does not equal 1, but it roughly equals (that squiggly equal sign) to 1. 0.9999999 is 0.9999999, 1 is 1.
Also that infinite hotel problem is a brain twister... It's pretty much there's a hotel with infinite number of rooms, and each room is filled with a guest, and 5 new guests came to the hotel, and they took the first 5 rooms, and other guests each moved down 5 rooms, will there be enough rooms? That's not as philosophical as the 0.999 IMO.

 

[Selta]

No, no, no, 0.999999 with an infinite number of 9's does not equal 1, but it roughly equals (that squiggly equal sign) to 1. 0.9999999 is 0.9999999, 1 is 1.
Also that infinite hotel problem is a brain twister... It's pretty much there's a hotel with infinite number of rooms, and each room is filled with a guest, and 5 new guests came to the hotel, and they took the first 5 rooms, and other guests each moved down 5 rooms, will there be enough rooms? That's not as philosophical as the 0.999 IMO.

Again, you're rounding a fraction to a decimal, and that's where the error occurs. Just like with 1/6, where .16666666666666666666666666666666666666 x 6 doesn't equal 1, but 1/6 x 6 does.
You lose a lot in the rounding, and that's why I always ALWAYS use fractions only. Plus, they're easier to work with.
Beyond that, wait until you get to calc. If 1/3 x 3 not equaling one is blowing your mind, wait until you get to real math .

 

[lemur821]

No, no, no, 0.999999 with an infinite number of 9's does not equal 1, but it roughly equals (that squiggly equal sign) to 1. 0.9999999 is 0.9999999, 1 is 1.

For all intents and purposes it's 1, I should say. If it's not close enough for your purpose, just add digits until it is. I don't see any reason to treat it differently.

 

[IconBasser]

curse you infinity... Its all your fault that we'll never reach the end of the universe...

 

[AmazingGracePlayer]

Again, you're rounding a fraction to a decimal, and that's where the error occurs. Just like with 1/6, where .16666666666666666666666666666666666666 x 6 doesn't equal 1, but 1/6 x 6 does.
You lose a lot in the rounding, and that's why I always ALWAYS use fractions only. Plus, they're easier to work with.
Beyond that, wait until you get to calc. If 1/3 x 3 not equaling one is blowing your mind, wait until you get to real math .

I got a 5 on the BC Calc exam my junior year (last year).
1/3 does not equal 0.3333333, it equals 0.3333 with an infinite number of 3's. And 3+3+3=9, so that means.

0.3333....
+0.3333....
+0.3333....
=0.9999....

 

[Selta]

I got a 5 on the BC Calc exam my junior year (last year).
1/3 does not equal 0.3333333, it equals 0.3333 with an infinite number of 3's.

And that's where you're getting yourself in a knot, bud.
Good for your five by the way, I got straight A's through three years of College Calculus. What's that prove?

 

[Ripper]

i still like the one kind of math me and my friend made...

1+1=0... 1+0=2...0+0=1...2+2=3...3+3=4...etc...

that and... 9/9 should = .999999999 right? not 1?

 

[AmazingGracePlayer]

And that's where you're getting yourself in a knot, bud.
Good for your five by the way, I got straight A's through three years of College Calculus. What's that prove?

It proves that it doesn't take any advanced calculus theories for someone to argue that 1/3 + 1/3 +1/3 = 0.999999999999.

 

[Selta]

It proves that it doesn't take any advanced calculus theories for someone to argue that 1/3 + 1/3 +1/3 = 0.999999999999.

I'm not saying it does take any. I'm saying that when you get there, you'll be able to better argue it, especially once you see integration.
1/3 + 1/3 + 1/3 = 3/3 = 1.
0.3333 + 0.3333 + 0.3333 does not = 1.
Again
You LOST information when you ROUND from fractions to decimals. And that is where you're getting screwed up, and, frankly, you're putting too much effort into it. It doesn't effect anything in any case.

 

[AmazingGracePlayer]

I'm not saying it does take any. I'm saying that when you get there, you'll be able to better argue it, especially once you see integration.
1/3 + 1/3 + 1/3 = 3/3 = 1.
0.3333 + 0.3333 + 0.3333 does not = 1.
Again
You LOST information when you ROUND from fractions to decimals. And that is where you're getting screwed up, and, frankly, you're putting too much effort into it. It doesn't effect anything in any case.

You lose information with you round 1/3 to 0.333 but not 0.333333 with infinite number of 3's. Because 1/3 IS 0.3333 w/ infinite number of threes. Unless you can argue that 1/3 is 0.33333333.....4.

And if 1/3 = 0.333333 w/ infinite number of threes, then that means 1/3 + 1/3 +1/3 should also equal to 0.333333 + 0.33333 + 0.33333 each with infinite amount of 3's. Because, like I said, 1/3 IS 0.333333 w/ infinite number of threes.

I know that 1/3 + 1/3 + 1/3 = 3/3, and anything divided by itself (other than 0 and infinity) is 1. So 3/3 = 1, thus 1/3 + 1/3 + 1/3 = 1. But one can also argue, see above paragraph, how 1/3 is exactly same as 0.333 w/ infinite number of 3's. And 0.3333 + 0.3333 + 0.3333 (each number w/ infinite number of 3's) is equall to 0.9999 (w/ infinite number of 9's), which does not equal one.

Enlighten me, how would you apply integration to that adding numbers? (no sarcasm)