Nice question Patrick!

1+1=2 is a true statement about the operation of counting.

Counting is a basic operation allowing us to establish consistently interpersonal relationships, as for instance by means of contracts, registers, bank-accounts, etc. Counting is a mental operation although to perform it we can help us with calculating tools, as for instance an abacus.

We discover arithmetical truths by performing such operations. So we can say that arithmetics refers to truths about the way we operate, which happens **necessarily** **step by step** .

In this sense the criticism of Platonism is justified: Numbers are not perfect archetypes that exist in a realm separated from our everyday world.

Are we the authors of these truths regarding counting operations?

On the one hand it seems we are the authors of arithmetical truths because to become aware of them we ourselves have first to assemble conveniently a finite number of items to produce a theorem or an algorithm.

So for instance Euclid arranged a theorem to prove that there are infinitely many prime numbers.

On the other hand, to date there is no theorem allowing us **to decide whether or not there are infinitely many perfect numbers.**

What is more, there are theorems (Gödel and Turing) proving that:

*Even if at a future time T we find a method to decide the above question about perfect numbers, there will be other similar questions regarding natural numbers we cannot decide with all the methods available at time T.*

These theorems are a consequence of the very fact that to count we have to proceed **step by step** , and show that we never will get hold at once of the whole truth regarding the way our mind functions.

One can better understand the nature of Arithmetics considering **a tiling puzzle** : We are not the authors of the picture that appears when the puzzle is complete. However, for the picture to appear, we have to find the way of assembling fittingly the interlocking and tessellating pieces.

Something similar holds for Arithmetics: We **discover** the arithmetical truths, even if to discover them we have to be creative and **invent** convenient assemblies of operations.

Actually what the theorems of Gödel and Turing show is that Arithmetics contains an infinite number of puzzle games we humans never will encompass once and for all!