What's the definition of 2? (the number)

Syth

I've heard that for a final exam in maths involved proving that 1+1=2. Most people think it's obvious but what I was thinking of is that 2 is defined as 1+1. ie you can't really prove that 1+1=2 because all you're doing is stating a definition.

But is this correct? If not, how is 2 actually defined?

John2002

Here's a link with proof:

http://mathforum.org/library/drmath/view/51551.html

John2002

And here's proof that 2 = 1

let a = b

a² = ab Multiply both sides by a
a² + a² - 2ab = ab + a² - 2ab Add (a² - 2ab) to both sides
2(a² - ab) = a² - ab Factor the left, and collect like terms on the right
2 = 1 Divide both sides by (a² - ab)


It is flawed obviously.

Syth

The proof of 1+1=2 seems to come from the defintion of + so it's a bit of a cop out as far as I can see. It'd be better to just defin 2 as 1+1. Would this cause problems later on? Anyone an ideas on this?

John2002 You're dividing by 0.

silverside

Hi Syth

I suggest you look at a book on Analysis (I recommend Calculus by Spivak) its not as scary as it might sound and will answer a lot of your questions. Basically it will go through step by step the definition of the natural numbers, the reals, arithmetical operations, etc, etc. Either that or talk to a friendly first year maths student :-) . But you will still have to take some things as given (your starting axioms). And there will always be some questions that are outside the scope of what you have defined (Godel's incompleteness theorem (Who shaves the barber if the barber shaves everyone who doesn't shave himself).

Man U babe

Syth IS a first year maths student.

DeVore

Can we please not turn this into another argument about semantics of maths.

+ is just an operator and so needs to be defined and in that definition you'll find, probably a trite and facile, definition of 1+1.

*yawn*

DeV.

Syth

As has been said I am stuying maths. :-D

The reason I started this was because I thought that 2 was defined as 1+1, when I see the proof, I see that instead of one definition they split it into 2 definitions: That 2 is defined as the succeor of 1 and addition of 1 to a number is defined as the successor.

I felt this was a bit of a cope out, but I suppose that you can't really get some deep seated proof that 1+1=2.