Math is an empirical science

It is often claimed that math is not a science. In particular because it is not empirical. However, this is entirely wrong. Math is completely empirical. We observe certain aspects of nature such as that one and one apple brought together become two apples. We observe that there is an operation splitting an apple in two halves and that the two halves make up the total apple.

These observations lay the foundation of mathematics. But they are observations of reality and hence they make mathematics empirical.

What is amazing is that the observations are very different than what is done in e.g. physics or social sciences. The observations aim at observing some very trivial facts about nature. From these facts, everything else is deduces.

In this sense math is a (very formal) model. Models are great. They can be used to predict how a system behaves by deducing conclusions from known facts.

But it must also always be remembered that models are approximate. And they can be used to model different physical situations with differing accuracy.

This raises the question of whether math could be any different. But there is plenty of evidence that it could and can be. There are some axioms of  math that are very solid and not questioned. There there are those propositions that can not be proven with ordinary mathematics and thus can be added positive or negative. But the only difference between the axioms that we take for granted and the ones we discuss (e.g. the continuum hypothesis) is the the ones oped for discussion are harder to asses empirically if they are true or not. The same applied in geometry.

So, in conclusion; we should question every axiom and be open to others, creating other formal systems. Many of those perhaps can not be related to this world, or be used as models for phenomena in this world. Or perhaps, they all will. This is very interesting. If that is the case, that every formal system we can conceive can be used to model something in our world, it could perhaps be claimed that our world is such as it is by necessity.