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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: 0.752s | 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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IAmBroom ◴[29 Aug 25 19:56 UTC] No.45068658[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).

Citation needed.

Especially since there are well-established math proofs of irrational numbers.

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NoahZuniga ◴[29 Aug 25 20:00 UTC] No.45068714[source]▶
The argument is essentially that you can only measure things to finite precision. And for any measurement you've made at this finite precision, there exist both infinitely rational and irrational numbers. So it's impossible to rule out that the actual value you measured is one of those infinitely many rational numbers.
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oskaralund ◴[29 Aug 25 20:38 UTC] No.45069122[source]▶
This argument feels like it's assuming the conclusion. If in principle it is only possible to measure quantities to finite precision, then it follows logically that we couldn't tell the difference between a rational and real universe. The question is, is the premise true here?
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BalinKing ◴[29 Aug 25 20:57 UTC] No.45069310[source]▶
AFAIK it would take an infinite amount of time to measure something to infinite precision, at least by the usual ways we’d think to do so…. I suppose one could assume a universe where that somehow isn’t the case, but (to my knowledge) that’s firmly in science-fiction territory.
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oskaralund ◴[29 Aug 25 21:15 UTC] No.45069474[source]▶
I don't think time and measurement precision are necessarily related in that way. You can measure weight with increased precision by using a more precise scale, without increasing the time it takes to do the measurement.
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1. AIPedant ◴[29 Aug 25 22:13 UTC] No.45070024[source]▶
The real point is that it takes infinite energy to get infinite precision.

Let me add that we have no clue how to do a measurement that doesn't involve a photon somewhere, which means that it's pure science fiction to think of infinite precision for anything small enough to be disturbed by a low-energy photon.

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2. oskaralund ◴[29 Aug 25 22:32 UTC] No.45070155[source]▶
I'm not making the case that it is possible to make measurements with infinite precision. I'm making the case that the argument "It is not possible to make measurements with infinite precision, therefore we cannot tell if we live in a rational or a real world." is begging the question. The conclusion follows logically from the premise. Unless the argument is just "we can't currently distinguish between a rational and a real world", but that seems trivial.