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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: 0s | 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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BeetleB ◴[29 Aug 25 23:36 UTC] No.45070587[source]▶
We don't need reals to make the math rigorous. Only to make the math a lot more tractable.

I've solved multiple continuous value problems by discretizing, applying combinatorics to the techniques, and then taking the limit of the result - you of course get the same result if you had simply used regular integration/differentiation, and it's a lot easier to use calculus than combinatorics.

But the point is the "rational", discretized approach will get you arbitrarily close to the answer.

It's why many analysis textbooks define a (given) real number as "a sequence of converging rational numbers" (before even defining what a limit is).

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1. Someone ◴[31 Aug 25 14:08 UTC] No.45083250[source]▶
> I've solved multiple continuous value problems by discretizing, applying combinatorics to the techniques, and then taking the limit of the result

But taking the limit of a sequence of rationals isn’t guaranteed to remain in the rationals (classic example: https://en.wikipedia.org/wiki/Basel_problem. Each partial sum is rational, but the limit of the partial sums is not)

So, how does that statement rebut “You can't do rigorous calculus (i.e. real analysis) on rationals alone.”?

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2. BeetleB ◴[31 Aug 25 17:42 UTC] No.45085158[source]▶
> But taking the limit of a sequence of rationals isn’t guaranteed to remain in the rationals

I'm not saying it does. What I'm saying is that you can make a correspondence with the reals by using only rationals.

You can define convergence without invoking the reals (Cauchy convergence). If you take any such sequence, you give that sequence a name. That name is the equivalent of a real number. You can then define addition, multiplication - any operation on the reals - with respect to those sequences (again, invoking only rational numbers).

So far, we have two distinct entities: The rationals, and the converging sequences.

Then, if you want, you can show that if you take the rationals and those entities we're calling "converging sequences" together, you can make operations involving the two (e.g. adding a rational to that converging sequence) and eventually build up what we know to be the number line.