Is 1 + 1 = 2 a fact?

Question

One of my colleagues tried to explain me that basic arithmetic is derived from nature. And hence its a fact. Another colleague tried to argue that humans came up with basic arithmetic and then tried to correlate with nature.

All I want to know is when someone says 1 + 1 = 2, is it a fact?

If our whole civilization is started from scratch, would we still come up with same rules of mathematics?

Answer 1

Well, its a definition in some sense.

One can define the set ${\Bbb N}_0$ of natural numbers by the Peano axioms. They basically say that (1) $0\in{\Bbb N}_0$, (2) $\nu:{\Bbb N}_0\rightarrow {\Bbb N}_0$ is a one to one function with $\nu(x)\ne0$ for all $x\in {\Bbb N}_0$, where $\nu(x)$ is called the successor of $x$, and (3) the induction axiom: If a subset $X$ of ${\Bbb N}_0$ contains $0$ and with each element $x$ also $\nu(x)$, then $X={\Bbb N}_0$.

One can show that ${\Bbb N}_0=\{0,\nu(0),\nu^2(0),\ldots\}$, i.e., the elements of ${\Bbb N}_0$ are $0$ and the successors of $0$. Moreover, this set fulfilling the axioms is in some sense uniquely determined - good to know but not used later on.

From here, define $1=\nu(0)$, $2=\nu(1)=\nu^2(0)$, $3=\nu^3(0)$, and so on.

Then one can define an addition (and multiplication) operation on ${\Bbb N}_0$: $$m+0 = m$$ and $$m+\nu(n) = \nu(m+n).$$

Now $1+1 = 1+\nu(0) = \nu(1+0) = \nu(1) = 2$ as required.

Answer 2

Imagine you are in a primary class and you are being taught addition. You have one candy. Your friend gives you one more candy. How many candies do you have now?

Even if everything was going to start from scratch, you would still have 2 candies (though it might not be called 2).

So, yeah 1+1=2 is a fact and would never ever change.

Answer 3

$1+1=2$ is not a fact in general.

Consider the group $\mathbb{Z}_2$ under usual operation.

For this case, you'll get $1+1=0$. The case you are considering is of $\mathbb{Z}$ which is infinite cyclic group and yes here under usual operation $1+1=2$ is true.

This is just a simple example considering group structure there are other examples also. you need to consider the underlying set and the governing operation.

The direct use of result $1+1=2$ is just a result, assuming the structure of set and operation is that of $\mathbb{Z}$ (which is also valid for real/complex but again not in general).

Answer 4

It all depends on what you mean by 1, 2, and +. One can do something like Peano arithmetic and say that we have $0$, a $+1$ operation, and literally define $2=1+1$. This doesn't require any sort of relation to nature, and it makes the statement true regardless of anything else.

But I think the intuition people have about counting comes from counting objects, and the natural numbers and arithmetic can be viewed as a decategorification of the category of finite sets. If you have a set of things, you can combine it with another set to get a bigger set, and the "size" of this bigger set is the "sum" of the sizes of the smaller sets. Arranging things in a grid gives multiplication in a similar way.

So the question boils down to whether an alien civilizzation could develop math and not develop ideas like counting or sets. And this seems very very unlikely. There are plenty of things in math that wouldn't necessarily happen. But counting?

Answer 5

One could dispute your colleague's claims. For example, adding two drops of water together still gives you drop of water, which is also observed in nature. The question is quite philosophical and cannot be answered without more information, you could say that it depends on exactly what you mean... You may find the following article on counting from Rafael Núñez interesting: http://www.cogsci.ucsd.edu/~nunez/web/biot.pdf.