Can you scientifically prove to me that 2+2= 4?

[NJ5]

The starting point is the definition of the numbers 2, 3 and 4, and the associative property of addition:

(a + b) + c = a + (b + c)

Using the inductive definition of natural numbers, the definitions of 2, 3 and 4 are:

2 = 1 + 1
3 = 2 + 1
4 = 3 + 1

You can then replace the 3 by 2 + 1

4 = (2 + 1) + 1

Using the associative property of addition, the equation becomes:

4 = 2 + (1 + 1)

But the definition of 2 is "1+1", so:

4 = 2 + 2

That's the best I can do right now, seems like a reasonable proof relying only on one axiom. Obviously we have to rely on something to start up the proof, since nothing can be proved from nothing.