Why does 2+2=4?

[Scumpup]

Whenever I ask one of my math teaching coworkers this question, I usually get a peevish "It just is!" or "I don't know" as a response. So, why does 2+2=4? Just because we all agree it does?

[Lobsang]

Because certain amounts of anything were given certain numbers. It just so happens that if you had an amount that was named '2' and another amount that was named '2', and you put them together into one pile, the quantity just happens to be named '4'.

** + ** = ****

[Achernar]

The so-called Peano Axioms are the things that are true "just because" or because we agree on them. Such are axioms. 2 + 2 = 4 can be proved using them. But why are you starting with such a hard problem? Here's a proof that 1 + 1 = 2.

[Lobsang]

In other words. Mathematics models the real world. So 2 + 2 'just does' = 4 in the same way that water 'just does' flow downwards.

[C K Dexter Haven]

It's basically definitional.

We first define 1 as the unit number (or the multiplicative identity, if you prefer, meaning that 1 x a = a for any a in our number set.)

Then we define an operation called addition, and we allow that we can add the number 1 to itself repeatedly. For convenience, we define each of these, because it's clumsy as hell to say "I have 1+1+1 dogs" or "My daughter is 1+1+1+1+1+1+1+1+1+1 years old." Thus, we use the term "2" to mean "1+1" and the term "3" to mean "1+1+1" etc.

So, with those definitions out of the way:
4 = 1 + 1 + 1 + 1 by definition
= (1 + 1) + (1 + 1) because addition is commutative
= 2 + 2 by definition, Q.E.D.

[Jabba]

As Achernar has mentioned, we define addition of natural numbers using the successor. In the Peano axioms every natural number has a successor ( which is also a natural number), and every natural number except 1 is the successor of some natural number. We write x' to represent the successor of x. Now we define
2 = 1' (i.e. 2 is the successor of 1)
3 = 2'
4 = 3'
and so on.

We can now turn to addition. We define x + 1 to be the successor of x :
x + 1 = x'
(x + 2) = (x + 1)' This is legitimate because we have already defined x + 1
(x + 3) = (x + 2)'
and so on. To work out x + y, you calculate x + 1 (the successor of x) and then take the successor again to get x + 2, and so on up to x + y.

So 2 + 2 is the successor of 2 + 1, and 2 + 1 is the successor of 2. Thus 2 + 2 is the successor of the successor of 2, i.e. 4.

[plnnr]

If Bertrand Russell were still alive you could ask him. IIRC, one of his books takes about 150 pages of proof to get to the point.

[RealityChuck]

No. 2+2 = 5

-- Winston Smith

[jovan]

Originally Posted by Lobsang

Mathematics models the real world.

I don't believe this statement is true. Mathematics can be used to describe the real world, but pure math is completely separate and independant from materiality.

[Achernar]

Originally Posted by plnnr

If Bertrand Russell were still alive you could ask him. IIRC, one of his books takes about 150 pages of proof to get to the point.

Only for very large values of 150. Actually, while it is hundreds of pages into Principia Mathematica that they get to 1+1=2, I think they could have done it much faster, but that wasn't the point of the book. They were trying to codify all of mathematics from the ground up; 1+1=2 was just a tangent they spent a page on.

http://www.idt.mdh.se/~icc/1+1=2.htm

[Earthworm Jim]

Originally Posted by RealityChuck

No. 2+2 = 5

Only for large values of 2.

[the lone cashew]

So, why does 2+2=4?

As a simpleton, I thought I should field this one.

"Okay! You choose the fruit! This problem works really well with apples. Bananas you say? Okay bananas it is!"

"Now, put two bananas together. Let's call that a pile of bananas. Now here we have two banans in one pile..."

[Rune]

Originally Posted by RealityChuck

No. 2+2 = 5

-- Winston Smith

Aw. It was all O’Brien’s idea. If it hadn’t been for that ratty re-education business in 101 I’d have never agreed to that.

[maggy]

the answer which was given to me many years ago was, not so much that physically they added up but more a commonality of language - you could, theoretically, raise your kids calling what's commonly known as "1", "2". Therefore, their answer would still be the same however the word representing the figure would be different therefore not understood in a common language. Another example would be the colour blue. Again, in isolation, you could tell someone the colour of the sky is actually called bananas. It's just a name - however without a common language, no one else knows what you're talking about (except of course someone taught the same as you).

[ultrafilter]

Jabba's explanation is good, but the standard construction of the natural numbers (known as the von Neumann ordinals) starts with 0 instead of 1.

[Shepherdless]

I agree with C K Dexter Haven . You have to start with the old Principle of Identity: 1=1, or a thing is what it is. The name you give to numbers is a convention, but you first have to agree to this principle for math to be possible

Then define operators such as addition, and everything falls into place.

[duffer]

yes, I have a sense of humor, and get what the OP was getting at.

But using this convoluted thread of the value of a number, at what point is someone allowed to be photo'd for a porn site? To me 10+8 is 18. What is the curve for legal reasons? If a 10 and 8 year old is used in the same pic, can it be argued that numbers are able to be tweaked? Also, if it's an 18 year old, can it be argued that there's a form of both a 10 and 8 year old in the personality?

Is this more a philosophical question? A random, what if? If I may, sounds like an IMHO thread.

Any great epiphanies (sp?) please keep me informed.

[Airman Doors, USAF]

"What if C-A-T really spelled DOG?"

/Ogre, Revenge of the Nerds 2

It's the same kind of question. Answer: It just does.

[HPL]

Originally Posted by greenfingers

As a simpleton, I thought I should field this one.

"Okay! You choose the fruit! This problem works really well with apples. Bananas you say? Okay bananas it is!"

"Now, put two bananas together. Let's call that a pile of bananas. Now here we have two banans in one pile..."

Edmund Blackadder: Right Baldrick, let's try again shall we? This is called adding. If I have two beans, and then I add two more beans, what do I have?

Baldrick: Some beans.

Edmund Blackadder: Yes... and no. Let's try again shall we? I have two beans, then I add two more beans. What does that make?

Baldrick: A very small casserole.

Edmund Blackadder: Baldrick, the ape creatures of the Indus have mastered this. Now try again. One, two, three, four. So how many are there?

Baldrick: Three.

Edmund Blackadder: What?

Baldrick: ...and that one.

Edmund Blackadder: Three and that one. So if I add that one to the three what will I have?

Baldrick: Oh. Some beans.

Edmund Blackadder: Yes. To you Baldrick, the Renaissance was just something that happened to other people, wasn't it?

[Scumpup]

Originally Posted by Airman Doors, USAF

"What if C-A-T really spelled DOG?"

/Ogre, Revenge of the Nerds 2

It's the same kind of question. Answer: It just does.

Would you have a substantive answer if I had phrased it as "What are the logical underpinnings of the concept that 2+2=4?" or would you still have nothing to contribute?

[Airman Doors, USAF]

Excuse me. I was being silly.

Nonetheless, there is some truth to it, but I guess it doesn't fit in to the discussion. Sorry for the interruption.

[Ramanujan]

Originally Posted by jovan

I don't believe this statement is true. Mathematics can be used to describe the real world, but pure math is completely separate and independant from materiality.

i have to disagree with you here. nothing is independent of materiality. the principle of noncontradiction, for example, could just be "not so" in a different sort of universe. that we can't speak meaningfully of such a universe does not mean that no such universe is possible. this is simply how our universe operates, and we do our best to describe it.

[AtomicDog]

Originally Posted by Airman Doors, USAF

Excuse me. I was being silly.

[British General] Stop it.[/British General]

[Lawn Mower]

Keep it simple...

take 2 Apples add 2 more how many do you have?

4!

[Bippy the Beardless]

I think the maths of this problem has been thoughly covered, so I would like to take a moment to consider the language of this problem.

The = sign gets translated into several different words.
(+ sign similarly has various wordings, some I use below, but none are problamatic, unlike the synonyms for the = sign.)

equals : Good word, but doesn't really give the meaning of the sign just the name of it

becomes : This is sometimes used, but is bad, this leads to the (Platonic I think) conjecture that "How comes 2 and 2 becomes 4, what is this and (plus) such that somehow things have changed from 2 + 2 into 4?"

is the same as : This is a long old phraise, but it gets to the heart of the matter. 2 plus 2 is the same as 4, there is no sence of change from one state to the other, both states are the same.

Unfortunately, computing has taken the 'becomes' meaning and has run with it giving us coding structures such as x=x+1 just to add to confusion of the = sign.

[Achernar]

When we say x = y, we mean that every property that x has, y also has, and every property that y has, x also has. I don't think anyone is really confused by the used of = as assignment operator in programming.

[bonzer]

Originally Posted by Achernar

Actually, while it is hundreds of pages into Principia Mathematica that they get to 1+1=2, I think they could have done it much faster, but that wasn't the point of the book. They were trying to codify all of mathematics from the ground up; 1+1=2 was just a tangent they spent a page on.

Agreed. Though the myth that the book is about 1+1=2 has surely been given legs by the fact that the CUP's standard abridgement Principia Mathematica to *56, in print since 1962, more or less culminates with *54.43. From that, there's only about 20 pages to the end of the main text. While Whitehead was dead, the abridgement was presumably approved by Russell.

[Though about a quarter of the abridgement is the original introduction, which had mainly been written by Russell and which is a brilliant example of his skill at technical exposition. So he may have been biased in agreeing what to include.]

[Russ1978]

Why is there 10 basic numbers??

[GorillaMan]

Originally Posted by Russ1978

Why is there 10 basic numbers??

Because we have ten digits on our hands. It's not as stupid an answer as it seems......there had to be something as the base, and ideally something not tiny, and not massive. It was fairly predictable that the 'winner' would be the one which corresponded with what we reach when we count on our fingers.

However, IIRC, not all societies have always used Base 10.....was it the Greeks who used Base 60, hence our division of the hour into sixty minutes? Somebody who knows what they're talking about can take over....

[DarrenS]

It was the Babylonians that used the sexagismal (base 60) system.

One thing I'm not clear about - nowadays, we've formalized the natural numbers using Peano's axioms. It's presented as if that's how we started, historically; but of course we knew that 2+2=4 way before Peano was even born. Upon what we were basing that?