Why does 2+2 = 4?

Date: 6/4/96 at 22:6:35
From: Garikai Campbell
Subject: 2+2 =? 4

I have a question. I was nosing around your site and saw the question
why does 2+2 = 4? 
(in http://mathforum.org/dr.math/problems/gora10.27.html   )
In your discussion you say that the hard thing to show is that 1+1 = 2, 
but you say that 4 is just another name for 1+1+1+1. 

Isn't this a little incongruent? Also, I am not sure I know what takes 
so long to prove. Doesn't this all (if you are saying that 2 is a name 
for 1+1 and 4 is a name for 1+1+1+1) boil down to the construction of 
integers as a rank 1 free abelian group?

Garikai "KAI" Campbell

P.S. Who is Dr. Math really?

Date: 6/5/96 at 1:25:38
From: Doctor Pete
Subject: 2+2 =? 4

First, I'd like to address your question of "Who is Dr. Math"....

"Dr. Math," in short, is not a single individual, but a collective 
group of people (many of them Swarthmore undergraduates) who provide 
the service of answering math questions via E-mail and the World Wide 
Web.  So perhaps one should ask, "Who *are* Dr. Math?"  (In 
particular, I am an undergrad at Caltech with an avid interest in 
geometry, Galois theory, and in getting a job that wouldn't result in 
my becoming a Fed or being locked up in an ivory tower.)

As for the question about the construction of the integers, it is more 
one of set-theoretical result than an algebraic one.  Remember that 
"hard" is a relative term; for a college student, "hard" might entail 
finding the Galois group of a polynomial; for a child in the 3rd 
grade, it would probably mean something more on the level of 
multiplying two ten-digit numbers.  So when kids are told by their 
math teachers that proving why 1+1 = 2 is "hard," it usually means two 
things:  (1) The familiarity with concepts required to construct the 
integers is not attainable by most elementary schoolchildren, and 
(2) aside from this, the teacher him/herself either doesn't know 
remember/know how to show it, or doesn't want to talk about it with 
the person who asked the question.

In any case, there are a few things to be seen from this - primarily, 
that the notion of "1+1 = 2" is not necessarily a trivial one, nor is 
it an axiom which children are told to assume; rather, it is a 
(constructible/"provable") result.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 6/5/96 at 9:15:23
From: Garikai Campbell
Subject: 2+2 =? 4

The reason I asked is: I am a graduate of Swarthmore and am now 
finishing up (or at least trying to) at Rutgers. I thought Dr. Math 
might be Swat faculty/students.

> As for the question about the construction of the integers, it is 
> more of set-theoretical result than an algebraic one.  

I don't know if I buy that, especially given what you were willing to 
assume in your answer. Certainly if you start from nothing and ask how 
can you get something, you may want to start set-theoretically by 
saying that 1 (namely something) is the set consisting of nothing and 
from there the construction is really algebraic in nature. All you 
really do is construct words (as in the construction of free groups). 
I hope this makes sense, it is not exactly the clearest exposition.

I guess what I was saying was that in your answer, you asked the 
reader to agree that 4 was just a name for 1+1+1+1. Isn't that what 
the symbol 2 is also? And so I must not understand the following 
comment that 

> the notion of "1+1=2" is not necessarily a trivial one, nor is it an 
> axiom which children are told to assume; rather, it is a 
> (constructible/"provable") result.

Let me just expand on exactly what I mean, but with a few notes:
1. I will not consider inverses in this discussion and
2. I will assume that 1 has been defined.
Consider that the only "number" you had was 1 and you had an operation 
called +. Other numbers are defined to be just strings of 1's 
seperated by the + symbol.And that's it! Isn't this more accurate and 
just as easy to convey (using a few more words) to a third grader.

Garikai "KAI" Campbell

Date: 6/5/96 at 17:17:53
From: Doctor Ce
Subject: Re: 2+2 =? 4

Hello,

Let me add the following to Dr. Pete's answer.

Construction of the natural numbers does not boil down to the 
construction of the integers as a rank 1 free abelian group.

The reason is that a rank 1 free abelian group is a stronger concept 
than the natural numbers, which (with 0) are a monoid with identity 
(there are no inverses).

In any case, before you can say 4 = 1+1+1+1, you have to show that 
addition is associative.  This is a "difficult" theorem starting with 
the Peano axioms. It is necessary to show this so that you can even 
write 1+1+1+1 and make sense (no parentheses).

-- Doctor Ce

Date: 6/6/96 at 8:30:1
From: Garikai Campbell
Subject: Re: 2+2 =? 4

I understand that. My point was not to construct the natural numbers 
but rather to use enough of a piece of the construction of the 
integers to illustrate that showing 1+1+1+1=1+1 + 1+1 may be easier 
using this construction. 

> In any case, before you can say 4 = 1+1+1+1, you have to show that 
> addition is associative.  

I sincerely do not understand why - what I am saying is that 4 is a 
name given to the string of symbols 1+1+1+1. The confusion may be in 
the fact that I am using this well known symbol +. I think what I am 
saying is better said using no plus sign. In other words, what if I 
say you have the symbol A and the group operation is concatenation. 
Throw in the symbols B and 0 (where B = A^(-1)). The group is the set 
of all reduced words (i.e. a free group with one generator). We then 
call 4 the word AAAA and 2 the word AA. Then it is more easy to see 
that 2 "+" 2 = 4 since what this really means is does AA AA = AAAA. In 
this explanation I never need to talk about many of the hard issues 
that come up in showing in general that the object I am talking about 
is in fact a group.

Part of my original point was that the answer asked the reader to 
believe that showing 1+1 = 2 is hard but to assume that 1+1+1+1 is 
just a name for 4. This seemed  terribly incongruent, and hence all 
the hooplah. 

Thanks for your quick response Doctor Ce. I am very interested to know 
your thoughts.

Garikai Campbell

Date: 6/14/96 at 2:14:30
From: Doctor Ethan
Subject: Re: 2+2 =? 4

Hey,

I am Ethan, a very recent graduate of Swat and very good friend of the 
answerer of the question under discussion.

I hate to jump into a conversation where I am not exactly 
qualified but I do want to say that you are right. In that answer 
there is some confusion about whether we are just calling 4 a label
for 1+1+1+1 or if there is something else going on.

Particularly since it is 2:15 and I have been doing Doctor Math for 5 
hours I can't tell what is going on in this problem

I can tell you though that after this conversation is over I will take 
what I can from here and try to update that answer to make it a little 
more right.

Perhaps you would like to help by replying to this with how you would 
answer the question 

Why does 2+2 = 4?

-Doctor Ethan,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/