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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?

mlemboy
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  • This looks like a definition to me. If you said $1+2=3$, then that might be a theorem, but I don't see a sensible way not to make $1+1=2$ a definition. –  Sep 11 '19 at 07:20
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    The statement $1 + 1 = 2$ is loaded with philosophical subtleties too deep for math stack exchange. You might consider posting on philosophy stack exchange instead. – Charles Hudgins Sep 11 '19 at 07:24
  • I don't think mathematics aims to answer this question; you might have better luck on philosophy.stackexchange.com for philosophy of math. (One thing I could say is that the answer on one level is "no" since it is very unlikely they would write $1+1=2$. On another level, the answer might be "yes" because however they might write it, it will likely be a representation isomorphic to ours.) – Kyle Miller Sep 11 '19 at 07:24
  • This is a philosophical, not a mathematical question. – Angina Seng Sep 11 '19 at 07:24
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    Yes, it is an "arithmetical fact" because we prove it from arithmetical axioms, and thus it is an arithmetical theorem. This means that this "fact" must hold in every "universe" where arithmetical axioms hold. And we have a very very strong evidence that in our "current" universe with eggs and pencils and ... the arithmetical axioms hold. – Mauro ALLEGRANZA Sep 11 '19 at 07:27
  • May not quite be duplicate, but definitely related: https://math.stackexchange.com/questions/243049/how-do-i-convince-someone-that-11-2-may-not-necessarily-be-true/ – Theo Bendit Sep 11 '19 at 07:43
  • that's just an array of symbols. If you allow the expected meaning of the symbols, then yes, it's a fact. – AlvinL Sep 11 '19 at 07:54
  • I'm enjoying how the question of whether $1+1=2$ is a fact has been flagged as primarily opinion-based. (Though I suppose the irony isn't lost on whomever did it.) – Theoretical Economist Sep 11 '19 at 13:01

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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.

Wuestenfux
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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?

Aaron
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$1+1=2$ 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).

IamKnull
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    In my book, in $\Bbb Z_2$, we have $2 = 0$. The squiggle "$2$" represents "whatever you get when you add $1$ to itself". If you're in a setting where that happens to coincide with $0$, then that doesn't mean $1+1 = 2$ is wrong. It just means there are more ways to write $2$. – Arthur Sep 11 '19 at 07:31
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    In $\mathbb Z_2$, $2\equiv 0$, so $1+1=2$. – cangrejo Sep 11 '19 at 07:32
  • @Arthur You can argue that, but when I am assuming a set for e.g $\mathbb{Z}$. As in other fields of mathematics, when a set is considered and operation is defined we assume there is no element outside that set, i.e. this is our universe. For e.g. in topology. In books, this is just a way to explain. – IamKnull Sep 11 '19 at 07:33
  • @broncoAbierto if my explanation in the above comment is correct then it's not. – IamKnull Sep 11 '19 at 07:35
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    @KumarNilesh I'm not contradicting that. I also think that $\Bbb Z_2$ has exactly two elements. I'm just saying that I think $2$ and $0$ are different names for the same element. $1+1$ is a third name for that element. And so on. – Arthur Sep 11 '19 at 07:39
  • @Arthur your point is valid. But when you are taking 2 in consideration (or any 2n), what is the underlying set and operation? – IamKnull Sep 11 '19 at 07:42
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    @KumarNilesh The underlying set is ${0,1}$. The underlying operation is standard modulo $2$ addition. The element $0$ can also be called $2$ (but it's the same element), and it can also be called $4$, just as $1$ can also be called $3$ and it can be called $5$. There are two elements, but each element has many names. – Arthur Sep 11 '19 at 07:47
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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.

K. Brix
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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.

Aditya Jain
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