# 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
of 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.

## Resources

A couple resources for anyone who hasn't seen them before:

- Peter J. Denning — Computing is a Natural Science
- Ian Stewart — A Subway Named Turing (I have actually read a version titled "Commuters and Computers: The Intelligent Subway", which I can't find online any longer)