Something about Apologetics.

[Categories+Sheaves]

Is reality a continuum or does it break down at some level and is actually made of discreet discontinuous quanta?

That was the question.

I thought we were talking about Zeno's paradoxes?

Our universe does seem to behave discretely, but Zeno's paradoxes are not proof of this of this fact. At least that's what I was arguing.

The people who favored the continuum ultimately won in terms of shaping mathematical thinking and this is why we currently have a mathematics that based on the idea of a continuum.

It's not like discrete math went extinct or anything. If anything, our physics is based on real analysis. There's still plenty of mathematicians doing research in Algebra, Combinatorics, etc.

We don't need anything on the number line but rational numbers, and they are discrete (i.e. quantized).

You probably know this already, but quantization also means our universe doesn't require the entirety of the rational numbers either.

We have no need to support these notions of "real numbers" as though they must be an integral part of our idea of absolute quantity.

This is absolutely true. We support these notions of "real numbers" because modern analysis is incredibly useful. Did I mention that it forms the backbone of QM?

According to a theory that runs on infinite-dimensional spaces of complex functions, real-world quantities are discrete. I guess that means it's time to get rid of those whimsical abstractions called real numbers!

Can you see why I think this is unreasonable?

I understand that you believe we'll get more mileage out of a system based on 'absolute' quantities. I do not understand why you think this system will afford us some problem-solving techniques that the current implementations of mathematics doesn't. A paradigm shift in mathematics would be awesome. But that sort of thing only happens when you find a system that's better than the current one.

It's various axioms of set theory that's really at the base of the problem.

I'd like you to expand on that point... but you are aware that the machinery of QM relies on that stuff too, right?

Axiom of choice? Clearly a worthless abstraction. It's not like physicists needed their hilbert spaces to have orthonormal bases or anything...

...And thus with that frame of mind they are naturally going to go off in la-la land making up mathematics on their own whim with total disregard to whether or not is means anything.

Reminds me of a guy I know.