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Why does 1 + 1 = 2 ?
By Anonymous on Friday, October 13, 2000 - 02:11 pm:

Does anyone know why 1 + 1 = 2 ?

This may seem a bit of a strange question, that is what I thought when asked the question.

By Brad Rodgers (P1930) on Friday, October 13, 2000 - 08:40 pm:

Actually, this is quite similar to the discussion "Where does Maths Come From".
The notion of 1+1=2 is merely a definition. however, it is odd that this should so far in all known cases apply to the real world. The simple reason why 1 chair + 1 chair = 2 chairs is generally accepted is because it is the only thing that has been observed, and it is likely that a counterexample will not be found. It is actually rather like the notion of gravity in this sense.

Brad

By Tom Hardcastle (P2477) on Friday, October 13, 2000 - 11:00 pm:

1 + 1 = 2 doesn't always apply to the real world. For example, one raindrop plus one raindrop equals one raindrop. I suppose we should derive an entirely new branch of mathematics to deal with this problem, something to do with the volume of the raindrops concerned...
:)

Isn't it the case that the axioms of arithmetic can be derived from the axioms of sets? If any of the Cambridge people can talk about this I'd be interested.

Tom.

By Dan Goodman (Dfmg2) on Saturday, October 14, 2000 - 12:00 am:

Not really my area of expertise, but I can say a few things about the definition of the natural numbers using set theory (natural numbers = 0,1,2,3,4,...). Basically, you define 0={}, 1=0 U {0}={{}}, 2=1 U {1}={{},{{}}}, etc. (Where U means set union). In general, N+1=N U {N}. Using this definition of the naturals, you can define operations like S,+ and - (S is the successor operation, Sn=n+1, or set theoretically Sn=n U {n}). [Warning, the following is speculation on my part, I'm not sure if this is how it is usually done, but it gives you the basic idea] You then inductively define a sequence of operations +b by +0(x)=x, and +Sb(x)=S(+b(x)). Then you can define the binary operator + as a+b=+b(a). From this definition, we get 1+1=+1(1)=S(+0(1))=S(1)=1 U {1}=2, by definition of 2.

Hope that helps, to define the - operation, you need to define ordered pairs, and using ordered pairs, define the integers ...,-2,-1,0,1,2,... and so forth. I can post a rough outline of defining integers, rationals and reals if anyone is interested?

By Marcus Hill (T3280) on Tuesday, October 17, 2000 - 02:54 am:

To show that "1+1=2" you need to define what you mean by "1", "+", "=" and "2". In most situations, once you have made these definitions in any way that coincides with our intuitions, then "1+1=2" follows immediately.