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.

On scientific knowledge

I was watching this very interesting debate between some great thinkers of our time. And of course I was thinking: "no, that have to be wrong"!
The question at hand is: is all knowledge empirical? Kruass is strongly advocating thing line. Dennet and Pigliucci is a bit more vague, but at the end seem to lean to the same conclusion. I would claim that this is wrong. There is knowledge (if we even can define the term!) that is not empirical. My example of this is basically mathematical knowledge. Krauss is claiming also mathematical knowledge is empirical since the basis (the axioms) is empirical to their nature is not correct. There can be many axioms not at all connected to any empirical facts - in fact ZF is an excellent example of this. Even the designers of the damn thing did not like it since it was counter intuitive with all these axioms. Nevertheless we would all agree that "the integral of 1/x is ln(x)" is knowledge. Or an even better example, the properties of the monster group constitute knowledge.

I think I would rather go further actually and say that all knowledge is of non-empirical nature. Empirical stuff is what can be observed. However observations can never be trusted. Never. However, deductions (which might be based on observations) can be trusted and render knowledge. But all that knowledge is of the for "if a the b". If a is observed, that b can be deduced. Thus, the knowledge does not depend on the empirical fact, but is only claims something about logical consequences given the empirical fact.

On a relation between mathematics, engineering and science

There is a difference between science and mathematics in that mathematics is a bottom-up approach. That is, in mathematics we start with a set of postulates and from this builds up the structure.

Science on the other hand, is a top-down approach. We start with a given system (a part of the world) and then we try to approximate that by some description. This approach gives science the luxury to not feel bad when it is wrong (that it is wrong!). We can be safe that we will be able to perform better in the future: the underlying system do exist and hence what we are trying to describe will work out eventually (or at least, our approximation will come closer).

Mathematics on the other hand takes the bold stand and does not thrust any underlying existing system, but rather want to derive everything by them self - believing that they can formulate a consistent theory by them self. This forces mathematics to have to be absolutely correct from the starts. Since they have so solid ground to stand, but rather only their own assumptions (which mights well be contradicting and incomplete).

Now it is peculiar to note that engineering in this sense is probably closer to mathematics than to science. Engineers take postulates (circuit theory, continuous mechanics or whatever they need) and, thrusting these, try to build something. Now the engineers are in a situation similar to the mathematicians: if there is something wrong with their assumptions the bridge collapses. The scientists do not have to worry about this; the bridge is apparently standing there. The problem is instead: what makes it stand??