They're not "wrong" in tests of their real-valuedness though.
I'm somewhat confident there is an empirical test of real-valuedness in areas of physics which require infinite-valued spaces.
However, either way -- the positions of the other commenters was that *geometry* is somehow a dispensable approximation in physics!
This is an extremely radical claim with no evidence whatsoever. Rather some discrete mathematicians simply wish it were the case.
It is true that *maybe* (!) spacetime will turn out discrete, and likewise, Hilbert spaces, etc. -- and all continuous and infinite dimensional things will be discretised.
This however is a project without a single textbook. There is no such physics. There are no empirical predictions. There are no theories. This is a project within discrete mathematics.
"They're not "wrong" in tests of their real-valuedness though."
Yes, they are, or more accurate, they're not right enough for you to confidently assert the structure of space time at scales below the Planck scale. You are doing so on the basis of theories known to be broken at that scale. You are not entitled to use the theories that way.
Even the Planck scale being the limit is a mathematical number; I'm not sure we have concrete evidence of that size being the limit. I've seen a few proposed experiments that would measure at that resolution (such as certain predictions made by LQG about light traveling very long distances and different wavelengths traveling at very slightly different speeds) but I'm not aware of any that have panned out enough to have a solid result of any kind.
The real numbers are a man-made axiomatic system. They were developed to make analysis mathematically rigorous to the high standards of pure mathematicians.
The real numbers are popular outside of mathematical analysis because they provide a "kitchen sink" of every number you could possibly need.
The downside is that the reals include many numbers that you don't need. The number 0.12345678910111213... is a transcendental real number, but it is not very useful for anything. It is notoriously difficult to prove that a given number is transcendental, i.e. part of the uncountable part of the reals and not the countable algebraic subset. Which is ironic because the uncountable part is infinitely larger!
I'm not suggesting that physicists should drop their Hilbert spaces. Rather that a distinction should be drawn between mathematical model and physical reality.
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As for whether spacetime is countably infinitely divisible:
Infinity is big. Infinitely small implies that if you used all the atoms in the universe to write in scientific notation to write 10^-999..., that space would be more divisible than that. In fact for whatever absurdly tiny number you could think of, perhaps 1/(TREE iterated TREE(3) times) spacetime would be finer than that.
I'll admit it's possible, but I have trouble believing it.
Well functions have properties in virtue of being defined over the reals, eg., sin(x) --
I don't see that these properties are incidental.
Yes they obtain in virtue of /any possible "dividing" discrete sequential process/ never terminating, eg., space being "infinitely divisible".
However I dont think this is as bizarre as it appears. The issue is congition is discrete, but the world continuous.
So we are always trying to project discrete sequential processes out onto the world in order to reason about it. Iterated zooming-in will, indeed, never terminate.
I dont see that as saying anything more than continuity produces infinities when approached discretely. So, don't approach it that way, if that bothers you.
I'm somewhat confident there is an empirical test of real-valuedness in areas of physics which require infinite-valued spaces.
However, either way -- the positions of the other commenters was that *geometry* is somehow a dispensable approximation in physics!
This is an extremely radical claim with no evidence whatsoever. Rather some discrete mathematicians simply wish it were the case.
It is true that *maybe* (!) spacetime will turn out discrete, and likewise, Hilbert spaces, etc. -- and all continuous and infinite dimensional things will be discretised.
This however is a project without a single textbook. There is no such physics. There are no empirical predictions. There are no theories. This is a project within discrete mathematics.