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/ |
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