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God created the real numbers

(www.ethanheilman.com)
136 points Bogdanp | 2 comments | 29 Aug 25 15:31 UTC | HN request time: 0.401s | source
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andrewla ◴[29 Aug 25 18:33 UTC] No.45067770[source]▶
I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct.

Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but representing the state even of a very simple system involves continuous quantities.

But the Cantor vision of the real numbers is just wrong and completely unphysical. The idea of arbitrary precision is intrinsically broken in physical reality. Instead I am off the opinion that computation is the relevant process in the physical universe, so approximations to continuous quantities are where the "Eternal Nature" line lies, and the abstraction of the continuum is just that -- an abstraction of the idea of having perfect knowledge of the state of anything in the universe.

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chasd00 ◴[29 Aug 25 19:20 UTC] No.45068269[source]▶
> The idea of arbitrary precision is intrinsically broken in physical reality.

you said a lot and i probably don't understand but doesn't pi contradict this? pi definitely exists in physical reality, wherever there is a circle, and seems to be have a never ending supply of decimal points.

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Dylan16807 ◴[29 Aug 25 19:47 UTC] No.45068570[source]▶
Can you name a physical thing that is a circle even to the baseline precision level of a 64 bit float?
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1. adrian_b ◴[31 Aug 25 10:36 UTC] No.45082104[source]▶
The most perfect things from this POV that have been made by humans are spheres of monocrystalline silicon, which have been made for the purpose of counting how many atoms they contain, for an extremely accurate determination of the mass of silicon atoms.

The accuracy of their volume and radius did not reach the level of a 64-bit float, but it was several orders of magnitude better that of 32-bit FP numbers.

While you cannot build a thing made of molecules with an accuracy better than that of a FP64 number, you can have a standing wave in a resonator, which stays in a cryostat, where the accuracy of its wavelength is 4 orders of magnitude better than the accuracy of a FP64 number, and where the resonator is actively tuned, typically with piezoelectric actuators, so that its length stays at a precise multiple of the wavelength, i.e. with the same accuracy. Only the average length of the resonator has that accuracy, the thermal movements of the atoms cause variations of length superposed over the average length, which are big in comparison with the desired precision, which is why the resonator must be cooled for the best results.

However, it does not really matter whether we can build a perfect sphere or circle. What it matters that modelling everything while using a geometry that supposes the existence of perfect circles we have never seen errors that could be explained by the falseness of this supposition.

The alternative of supposing that there are no perfect circles is not simpler, but much more complicated, so why bother with it?

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2. Dylan16807 ◴[31 Aug 25 14:54 UTC] No.45083617[source]▶
> However, it does not really matter whether we can build a perfect sphere or circle.

When talking about whether arbitrarily precise numbers are real in the universe, it extremely matters.

Sadly, atoms exist. In some ways that makes things more complicated, but it's the truth. Anything made of discrete chunks in a grid can't have arbitrarily precise dimensions.