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God created the real numbers

(www.ethanheilman.com)
136 points Bogdanp | 4 comments | 29 Aug 25 15:31 UTC | HN request time: 0.626s | 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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NoahZuniga ◴[29 Aug 25 19:33 UTC] No.45068389[source]▶
You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities).

The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?

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omnicognate ◴[29 Aug 25 20:02 UTC] No.45068743[source]▶
As in why bother using real numbers in physics? Mostly because you need them to make the maths rigorous. You can't do rigorous calculus (i.e. real analysis) on rationals alone.
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NoahZuniga ◴[29 Aug 25 20:19 UTC] No.45068921[source]▶
> You can't do rigorous calculus (i.e. real analysis) on rationals alone. Yep, but that wasn't my point.

My point was that it is possible that all values in our universe are rational, and it wouldn't be possible for us to tell the difference between this and a universe that has irrational numbers. This fact feels pretty cursed, so I wanted to point it out.

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1. dullcrisp ◴[29 Aug 25 21:08 UTC] No.45069400[source]▶
You can make this statement for any dense subset of the reals, but we don’t because that would be silly.

I think the conceit is supposed to be that analysis—and therefore the reals—is the “language of nature” more so than that we can actually find the reals using scientific instruments.

To illustrate the point, using the rationals is just one way of constructing the reals. Try arguing that numbers with a finite decimal representation are the divine language of nature, for example.

Plus, maybe a hot take, but really I think there’s nothing natural about the rationals. Try using them for anything practical. If we used more base-60 instead of base-10 we could probably forget about them entirely.

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2. Garlef ◴[30 Aug 25 12:53 UTC] No.45074217[source]▶
I think it makes much more sense to make this statement for the rational numbers: It's the smallest field inside the real numbers that contains the naturals.

So every subset that allows you to do your daily calculations contains the rationals.

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3. dullcrisp ◴[30 Aug 25 14:46 UTC] No.45075103[source]▶
They’re a field by construction, and yes, the initial field of characteristic zero, but otherwise don’t arise in any natural way. They’ll be there if you’re studying fields, but exact division by arbitrary integers doesn’t seem to be a very natural property outside the reals. Again, imagine doing any practical computations with rationals and see how far you get before resorting to decimal approximation.

I think teachers lie to children and say that decimals are just another way of representing rationals, rather than the approximation of real numbers that they are (and introduce somewhat silly things like repeating decimals to do it), which makes rationals feel central and natural. That’s certainly how it was for me until I started wondering why no programming languages come with rational number packages.

4. tomasson ◴[30 Aug 25 15:05 UTC] No.45075287[source]▶
Here’s my plug for p-adic numbers! So cool