Mathematics is a Natural Science
I believe that mathematics is a science, a natural science to be more precise. By science I mean the process or method of distinguishing what's true from what's false and by natural science I mean the application of science to the natural world that surrounds us.
Most people would gladly admit that physics, chemistry and biology are natural sciences. They study nature in an attempt to find out rules that govern it, to delineate what's true from what's false by means of experiment. The common view is that what really distinguishes the aforementioned fields from the study from mathematics is that they use observation, via experiment, to determine which theories about the natural world are telling the truth and which are not.
What I'm trying to argue here is that mathematics is just as much about experiment and observation of the natural as any other science. The fundamental difference is that the world described by mathematics is the natural world of information. And just like physics can have hypotheses which will at some point be refuted or confirmed (not proved!, more on this later) by experiment, the same way mathematics presents conjectures which may at some point be proved true or false. And just like natural sciences need field scientists and experimentalists that get their hands dirty and devise and conduct experiments, the same way mathematicians will conduct experiments in the form of incomplete or failed proofs, until one of them will actually prove or disprove the mathematical proposition they set out to study. One salient point here is that there's no guarantee that the mathematician may actually find such a proof, it's a process of searching that may yield some answer or not — we have absolutely no guarantee. In a similar vein, the final proof produced by the mathematician (or even a computer) is actually a natural artefact! It is embodied in the physical universe and that's probably true even if the proof hasn't escaped the mathematician's consciousness. And I would argue that even if consciousness will ultimately turn out not to be an emergent phenomenon of the physical world, but a thing in itself, it will still be a manifestation of the natural world, the world within which we exist. So, yes, a proof is a natural artefact that we need to discover before we can say with any certainty if a mathematical statement is either true or false. But that also means that a mathematical proposition can't be just true or false, it can also be "unknown as of yet" — the law of the excluded middle.
There is however something quite different between mathematics and the other natural sciences. Mathematics is free to choose some axioms and run wild with them. Every mathematical fact is contingent on those axioms — assuming the axioms, such and such is true or false. But the other sciences don't have this luxury; they can't get to decide what are the axioms of the universe, they must assume. This small difference has interesting repercussions, one being that mathematics can prove things to be true, whereas the other sciences can only prove things to be false. Here, by "things" I actually mean universal statements, of the form "for all X", and not existential statements such as "there is X". The best the "classical" sciences can do is prove a universal statement false, but they have no hope of proving it true, because there's no way to be sure that we have taken into account all the axioms of the universe — there may be laws and situations that could render our beautiful universal equations false in a blink.
Mathematics has the luxury of saying: these are our axioms, our atoms and fundamental forces and everything derives from them. Any theorem or proposition that assumes more than the given axioms is considered nonsensical. Of course, one can always explore a new set of axioms to derive new worlds of consequences based on the respective axioms.
What really fascinates me here is that these axiom sets are formed within the universe. They are not just inventions of our mind, totally detached from what is outside us — if we can imagine it, it's natural, it's part of the universe, even if that universe is just the world of information. And I think that might explain the unreasonable effectiveness of mathematics.
I'm no mathematician, but even so, during my brief encounters I've found myself wondering how mathematicians could come up with some of their theorems. But it might not be quite that remarkable if we consider the world of possibilities engendered by some axioms as a world in itself, a world that the mathematician is more than happy to explore, to study its every corner, some banal, some not, or some spuriously so. And because human beings crave order, they might see patterns and try to generalize them, not much different than what a theoretical physicist might do, I presume. So they come up with a hypothesis, only they call it conjecture, and then go on and try to prove it. As noted above, the endeavour might be fruitful or not, or it might still be ongoing.
At this point we can try to take a few steps back, maybe a couple parsecs, and we might not see the two human endeavours as quite so dissimilar — the natural scientist will take out their magnifier, pencil and pad, explore the world abord The Beagle and come up with some theory of evolution, while the mathematician might take out, or not, a pencil and a pad and explore the curious creatures formed by the atoms of its universe, the axioms. However, at the end of the day, they will both be searching for structures in their worlds, for patterns of information, for peculiar connections between the building blocks of their respective worlds. But these are not isolated worlds, they share a common home, they're informational. And both our scientists will use computation to derive meaning.
A couple resources for anyone who hasn't seen them before: