God, Santa, and The Tooth Fairy

[polymath257]

After first contact and we see that the aliens navigated between the stars with board games then I will admit that mathematical truths can be invented rather than having to be discovered.

Do you really expect that aliens will have the same mathematics as us? That the Peano axioms for the natural numbers are universal?  That Euclidean geometry is a priori? Do you really think that they will use the same trigonometric functions that we do? Or have the same set theory or even logic?

I don't. There are simply too many ways to construct math.

Math is, ultimately, a board game. It has rules of deduction (of play) and axioms (starting positions) and gets results from those.

On the other hand, I would suspect that their math and ours would be mutually comprehensible and that they arrive at equivalent laws of physics as we (given equivalent technology).

I would also be very, very surprised if they have anything close to the same metaphysics.