Is there a metric space where 2 + 2 = 5? First I want to say, sorry if my question doesn't make sense, I just started getting in the world of topology and metric spaces, and I'm not sure I fully understand the concepts yet.
Here's my understanding so far: a metric space is defined by its distance function, which assign a real number $d(x,y)$ to every pair $(x,y)$.
The space we are "used" to and mostly work with is the Euclidean space, defined by $d (x,y) = |y - x|$.
If we move to a different space with a different distance formula, will that change the arithmetic that we are used to? So for example, $1 + 1$ would no longer be equal to $2$?
This lead to my real question for this: is there a space where $2 + 2 = 5$? can we create one? what would be the definition of the distance function for that space?
Again, that's my whole understanding from today's session. I might be totally off with what I grasped, sorry if that's the case
Thank you very much.
 A: Even though $2+2$ won't be equal to $5$, you could use a different metric on the real line ( $d(x,y)=\frac54|x-y|$ instead of $d(x,y)=|x-y|$ ) so that $2+2=4$ is "$5$ units from $0$", distance-wise. 
A: It's a good question.  But it does come down to what we think 2 + 2 means.  I can easily invent a metric where $d(0,2) = 2$ and $d(2,4) = 2$ and $d(0,4) = 3$ (this obeys the triangle inequality that $d(0,4) \le d(0,2) + d(2,4)$) is if you squint and get very fuzzy in your definition of what "2 + 2" means, I can say with my pants not entirely on fire, that 2 + 2 =3.  
But this is a trick.  If I define arithmetic by the metric, I pretty much can't declare the label by which I called "4" to "really" be actually 4.  Or if I define arithmetic to be a sort of "counting stick" then this metric doesn't really represent "adding".  In a way I'm defining my "counting stick" to be curved.
In fact this is precisely how the old fashioned slide-rules worked.  Here's a metric:  d(a,b) = |log (b) - log(a)|.  (I hope I did that right; I just thought of it just now.) Then for 1 < b< c, Then d(1,b) + d(b,c) = d(1, b*c).  This turns a + b into a*b.  Or as the old joke about the snakes on Noah's Ark being placed on wooden tables to get them to reproduce ....
