Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Thought about this recently, and was a bit stuck.

Is all mathematics based on the concept that $1+1=2$?

For example, if $1+1\ne2$, then all arithmetic won't work, right?

share|improve this question
    
No, some more mathematics is based on the concept that $(a+b)^2=a^2 +2ab +b^2$. In fact, the third part of it is based on the craziest concept of all that $(x^2)^{'}_{x}=2x$ –  Alex Feb 10 '13 at 3:03
1  
It might be cheating the question a little, but arithmetic mod 2 has no concept of 2. In fact, $1+1=0 \bmod{2}$. (Also, keep in mind $1$ and $2$ are just symbols! We could just as easily use a different number system, where $\theta + \theta = \upsilon$ and define $\theta$ and $\upsilon$ just as we define $1$ and $2$.) –  George V. Williams Feb 10 '13 at 3:14
3  
This question is pretty much in the makes-no-sense ballpark. Wittgenstein pretty much nailed it. –  Mariano Suárez-Alvarez Feb 10 '13 at 3:21
1  
I think the confusion here is we don't know whether @think123 is asking "does $1+1=2$ hold for all of mathematics" or "does all of mathematics stem from the definition $1+1=2$". –  Alexander Gruber Feb 10 '13 at 3:22
1  
I think that the tagging is a bit off here. –  Asaf Karagila Feb 10 '13 at 3:45
show 7 more comments

2 Answers 2

up vote 7 down vote accepted

Sometimes $1+1=0$, when you're working in a unital ring with characteristic $2$!

But those are the only possibilities. $2$ is defined as $1+1$. (So actually in the characteristic $2$ case, $1+1=2$ still- it's just that $0=2$.)

share|improve this answer
add comment

If $1+1=5$ just declare $5$ as $2$ and move on with it. You are mixing the idea of $2$ with its symbol.

We usually define the meaning of the symbol $2$ as a shorthand for $1+1$, but it might as well be any other symbol. It should be noted that if we assume the "usual" axioms of mathematics and we manage to prove that $1+1\neq2$ then our axioms have an inherent contradiction and we need to find it. But so far we're good.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.