There's really no evidence for this, as far as we know the real numbers are a pure mathematical invention and don't have any physicality.
Even if you want to say that spacetime is dense (i.e. infinitely divisible), there's an infinite number of fields like that, the real numbers are just a convenient superset.
There's no evidence that spacetime is dense either, and many practical ways in which it is not, as an obvious upper bound if you took all the energy in the observable universe to make one photon, it would still have a finite wavelength.
In a way the real numbers are a model (or maybe a 'language') to describe physical phenomena. They work exceptionally well at that, but they are not backed by evidence and do come with (theoretical) limitations.
This bachelor's thesis is a good starting point [1], search for 'finite precision physics' or 'intuitionistic math/physics'.
Well this is a popular idea imported from comptuer science, but there's absolutely no evidence for it -- and plenty against.
Eg., QM is only linear in infinitely-dimensional real-spaces, etc.
Essentially of a physics uses real spaces indispensably. There is no evidence whatsoever that this is dispensible; other than the fever dreams of discrete mathematicians.
Remember, though: QM is wrong. Relativity also depends on continuous spaces, but it is also wrong. All the theories in physics that depend on continuous space are also wrong.
By "wrong", I mean, we know they can't predict everything correctly. QM itself can't derive relativity. Relativity doesn't have QM in it, and break down at extremes like black holes. They're both very, very, very accurate in their domains, but physics knows that neither theory has the domain of "the entire universe". This is not a wild claim by an HN commenter, this is consensus in the physics world, just perhaps not phrased in the way you're used to.
It's possible the eventual Grand Unified Theory will still have continuous space at its bottom, but it's also entirely possible it won't. Loop quantum gravity doesn't. And personally I expect some sort of new hybrid between continuous and discrete based on physics history; whenever in the past we've had a similar situation where it couldn't be X for this reason, but it couldn't be the obvious Not-X for some other reason, it has turned out to be something that had a bit of both in them, but wasn't either of them.
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 have very little understanding of physics and math and might be wrong.
But if we accept that the Planck length is the smallest possible length and the Planck time is the smallest possible time, then it seems logical that the universe is an integer lattice of these. ("Spacetime is not real-valued")
However the idea that space time is discrete is a reasonable hypothesis to test, we don’t currently have any ways to probe at those resolutions, though.
The issue isn't the reals, but that "solid object" isn't defined properly, ie., the sets under question don't have well-defined volumes.
As soon as you fix that problem, via measure theory, the paradox resolves. You dont need to ditch real numbers.