TITLE: Is $1+1 = 2$ true in any base?
QUESTION [31 upvotes]: So this may be one of the stupidest questions ever, but is the identity
$$1 + 1 = 2$$
valid in any base?
It seems so, since it's like, for a given base $b$
$$1 = 000\ldots 01 = (0\cdot b^n) + \ldots (1\cdot b^0) = b^0 = 1$$
Which means $1 + 1 = 2$ no matter what base we use.
Is that right? Or is there something I miss?

REPLY [5 votes]: There are three forces at play here.
First comes our natural understanding of numbers. $2$ is a specific natural, or real, number. We understand it intuitively, and unambiguously. This is the abstraction of the idea of having two apples, two sons, two cats, or two examples for collections with two objects. As such, when we think of the number $2$, it is literally defined to be $1+1$.
Then comes the representation of a number in a particular base. This is now a question of how we dress the abstract number into a slightly more tangible form. Here $2$ is a relatively "bad" example, as most [natural-]bases are large enough so $2$ is just $2$ again. But think of the number ten. In decimal, this is $10$; in trenary this is $101$; in octal, $12$. All of these are different strings which represent the same number. How is this possible? Well, when we change base, we change how we interpret the string of symbols.
And finally comes the purely syntactical evaluation of symbols. This takes the last point and pushes it to $11$.(1) We forget that the symbols even represent the abstract quantity of how many hands a "standard" person should have. We only know that $1$, $2$ and $+$ are symbols in a mathematical language. As such we are in fact allowed to interpret them to mean pretty much anything that we want. You see this with $\pi$, which can denote a homomorphism, a projection, a function, a constant real number; but you see it less often with $2$ or $+$. Nevertheless, we are allowed to interpret those symbols to our liking, and we can easily concoct interpretations where $1+1$ and $2$ are not the same object.
In short, if you consider them as abstract numbers, they are independent of their representations, and then $1+1=2$, always. If you are asking about the interpretation of the symbols as numbers, then the answer is a qualified "usually". If you are asking in general about interpreting the symbols $1,2$ and $+$... then this is a resounding "not enough context".

(1) This can be taken in any base you prefer. Even in base $\omega$.