Thread: Mathematics

Originally Posted by Xei

A question I can't answer is why reality outside of human scales seems to be based upon mathematics. I tend to view all human mental constructs as approximations of patterns which will break down when taken outside of the realm in which we formulated them (for instance, our conception of space and geometry as being rectilinear breaks down at scales outside of our experience, yet before knowledge of this many would have claimed that such a conception was 'inherently obvious' and 'an intrinsic part of reality'. Even more fundamental constructs like cause and effect have the same problem).

Yet with mathematics it was totally backwards, in one specific circumstance: we initially came up with complex numbers (two-dimensional numbers involving the square root of minus 1) more than two centuries ago when trying to solve totally prosaic problems (namely cubic equations, which can correspond to real questions about volumes, etc.), but all they were was an intermediary step on the way to the real answer which corresponded to something physical. These numbers turned out to provide a lot of insight into things like polynomials and limits (which were abstracted from intuitive, physical things), even though they themselves were originally an abstraction not corresponding to anything real.

The really weird thing is that quantum mechanics turns out to be pretty much intrinsically based on them. It's all formulated in complex numbers, and the underlying mechanism is based on how these numbers behave; whenever you want an answer, you end up measuring the 'size' of the underlying complex numbers (the size being a normal real number giving a real answer), yet the engine underneath the vehicle is all based on complex stuff.

Why on Earth is it that an obscure academic abstraction that came purely from consideration of macroscopic intuitions ended up being the basis of reality on the most fundamental scales, totally removed from the macroscopic world? How did we find the basis of reality without first having to explore down to this level??

It's a red herring to say that complex numbers are intrinsic to quantum mechanics. You could do all of it with vectors. Complex numbers are just more convenient. Furthermore, if they were essential to quantum mechanics, then that still doesn't make them the "basis of reality on the most fundamental scales". It would just mean that they are an intrinsic part of one part of one model of reality that happens to be very interesting and occasionally useful.

So you're making a strong assumption in claiming that we've discovered the "basis of reality" without even having to explore it. Hilbert space happened to be a convenient model because all the solutions for various wave equations live in a hilbert space. With other equations it would be a symplectic manifold or something.

What it boils down to is that the space of mathematical systems is just so large that it would be very surprising to find a system that couldn't be modeled to any degree of accuracy using it. The vast majority of mathematics is only ever used indirectly (if at all) in modeling reality.

Again, no big thing. The other way around would be the surprise.

Originally Posted by PhilosopherStoned

It's a red herring to say that complex numbers are intrinsic to quantum mechanics. You could do all of it with vectors. ...

Thank you. I was going to point out the same thing but getting into an argument with Xei (who we all know has issues with admitting he's wrong) was not something I had the energy to do. Good on you, sir.

The OP mentioned Pythagoras. That's interesting, because Pythagoras is the perfect example of why mathematics is indeed a series of mental models, not reality. You cannot find a perfect right triangle in this universe. But, you can find situations where using certain geometrical rules is beneficial. For example, I would imagine that civil engineering would have been *slightly* more difficult without Pythag.

Another line of evidence is the fact that the mathematics get more complex, not simpler, as the physics gets deeper. This is consistent with humans trying to impose mental models on aspects of reality that are further and further removed from regular experience.

Uh yeah, 2D 'vectors' equipped with a multiplication rule isomorphic to that of complex numbers.

Otherwise known as... complex numbers.

Originally Posted by PhilosopherStoned

What it boils down to is that the space of mathematical systems is just so large that it would be very surprising to find a system that couldn't be modeled to any degree of accuracy using it. The vast majority of mathematics is only ever used indirectly (if at all) in modeling reality.

But we're not talking about obscure systems with layers upon layers of complicated rules and refinements. We're talking about a very simple structure that was already ubiquitous to mathematicians due to its potency in pure maths research - a structure discovered by extending abstractions from the familiar physical world (like the axioms of algebra) to realms with no physical meaning in that world, which turned out to be the natural language of phenomena entirely outside of human experience.