Proving 1 + 1 = 2

Mathematics is the science where we always prove things. This has even entered into popular rhetorics: If we claim that some fact or other is “mathematically proven” it is the highest kind of authority. Usually this is sloppy phrasing or simply a lie, because sociological, psychological, political, even physical facts cannot be mathematically proven, because you can only mathematically prove mathematic statements, and they are not. Mostly they are, if at all, juristically proven (“true beyond a reasonable doubt”). But in our own sphere we mathematicians prove everything, and if we can’t we don’t assume that thing as true (be that painful as it may).

I have a few times explained this to students. Usually the consequence is that someone ask me how you can prove that 1 + 1 = 2, or something along that line. In these cases I have to admit I generally just answer that this is indeed possible to prove, but don’t show how. The reason is that it simply would take too long. But here in this medium I have all the time I need.

If you read this article (and others in the same vein, which are sure to come soon) you will know what lies at the heart of mathematics. This is the stuff they don’t teach at school and I promise, it won’t contain any of the stuff you learned to hate when you were 14. (Fractions, for example: A neat and immensely useful method, but how on earth could anybody like doing them? It’s the results and the proofs that give satisfaction, the nuts-and-bolts calculations is just work for performing monkeys.)

If you feel that despite this you cannot be interested just skip this article! Mathematics is just one of my topics in this blog.

We start by asking: what are 1 and 2 anyway, or to be more exact: what are numbers?

For quite some time mathematicians have been using a tool called the axiomatic method. This lies at the roots of all mathematics, and certainly deserves to be talked about in detail. But I want to keep this article as short as possible and so I am forced to defer this undertaking to some other time. But I promise I will make good for it soon!

The basic idea is this: We might think we all share a common notion of numbers (that is natural numbers, 1,2,3,4 etc.). But is it true? In prehistoric times people used to count “one–two–many…” (allegedly there is still a tribe in Brazil that does that), and I read somewhere that in Indo-European languages the words “nine” and “new” are etymologically related—because when nine was invented after centuries of people only knowing 1 to 8 it became the new number (about 5000 years ago).

So I think we certainly do not have an instinctive notion of numbers. We merely learn it at a young age because our world is filled with numbers.

Learning numbers that way resembles learning a game by watching other people play and trying for yourself until you are sure you are doing it right. The downside is that in this way you can never be sure you are doing it right. Maybe in the next match a situation might arise where you do not know what to do? Some special case you are not prepared for?

But mathematicians want to be sure. In this analogy, it would mean that we want to thoroughly read the rules of the game before we play.

So mathematicians (actually it was an Italian, Giuseppe Peano) wrote down a set of rules (or axioms) for natural numbers. They clearly state which (natural) numbers exist and some properties of them. Here are Peano’s axioms:

1. There is a natural number, call it 1.
2. For every natural number n there is a natural number n′ which is the successor of n.
3. The successor of any natural number is not 1.
4. If two natural numbers have the same successors, they are the same.
5. If a set of natural numbers contains 1 and, for every number n in it also contains its successor n′, then that set is the set of all natural numbers.

Once we have these rules we will never give a thought to what natural numbers actually are. We don’t know and we don’t need to know. All we need to know is that we can use these rules freely. We have no idea what numbers are, but we know what they do. (If I write any more on this, it is bound to get philosophical and I am going to reserve that for some other occasion. Suffice to say that however you may look on the matter, if you accept these rules as true you are still with me—and we will not use anything other than these rules.)

Adding up

Now can define a few numbers and give them names:

• 1 we already know.
• 2 = 1′
• 3 = 2′ = 1″

and so on …

Now we know what 1 and 2 are, what what do we mean by “+”?

The task at hand is to define addition using only the Peano rules of natural numbers. This is done by stating two simple laws of addition, which I am sure will concur with your “instinctive” notion of Addition. We state:

“+” is an operator between two natural numbers that satisfies the following laws:

1. n + 1 = n′ for all natural numbers n.
2. n + m′ = n′ + m for all natural numbers n and m.

Now what is 1 + 1? We simply use the first rule, setting n = 1 and get

1 + 1 = 1′ = 2.

That is a proof. If we consent that natural numbers satisfy the Peano rules and that addition satisfies the two rules stated above and 2 is the successor of 1 then 1 + 1 = 2 is mathematically proven.

Now that was disappointingly simple. Let’s try for something harder. What’s 2 + 2 (that is 1′ + 1′)?

This time we start with addition rule (2). We find 1′ + 1′ = 1″ + 1. After that we once again use rule (1) and obtain

2 + 2 = 1′ + 1′ = 1″ + 1 = 1′′′ = 4.

I think you will now have no problem calculating 5 + 9 or whatever takes your fancy.