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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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blueplanet200 ◴[29 Aug 25 19:04 UTC] No.45068086[source]▶
>I'm an enthusiastic Cantor skeptic

A skeptic in what way? He said a lot.

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andrewla ◴[29 Aug 25 19:31 UTC] No.45068372[source]▶
Here I'm referring to the cloud of things that Hilbert called "Cantor's Paradise". Basically everything around the notion of cardinality of infinities.
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blueplanet200 ◴[29 Aug 25 19:46 UTC] No.45068563[source]▶
Please say more, I don't see how you can be _skeptical_ of those ideas.

Math is math, if you start with ZFC axioms you get uncountable infinites.

Maybe you don't start with those axioms. But that has nothing to do with truth, it's just a different mathematical setting.

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axolotliom ◴[29 Aug 25 20:13 UTC] No.45068878[source]▶
> I don't see how you can be _skeptical_ of those ideas.

Well you can be skeptical of anything and everything, and I would argue should be.

Addressing your issue directly, the Axiom of Choice is actively debated: https://en.wikipedia.org/wiki/Axiom_of_choice#Criticism_and_...

I understand the construction and the argument, but personally I find the argument of diagonalization should be criticized for using finities to prove statements about infinities.

You must first accept that an infinity can have any enumeration before proving its enumerations lack the specified enumeration you have constructed.

https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

> Math is math, if you start with ZFC axioms

This always bothers me. "Math is math" speaks little to the "truth" of a statement. Math is less objective as much as it rigorously defines its subjectivities.

https://news.ycombinator.com/item?id=44739315

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1. blueplanet200 ◴[30 Aug 25 06:02 UTC] No.45072250[source]▶
The axiom of choice is debated as a matter of if its inclusion into our mathematics produces useful math.

I don't think it's debated on the ground of if it's true or not.

And I was imprecise with language, but by saying "math is math" I meant that there are things that logically follow from the ZFC axioms. That is hard to debate or be skeptical of. The point I was driving was that it's strange to be skeptical of an axiom. You either accept it or not. Same as the parallel postulate in geometry, where you get flat geometry if you take it, and you get other geometries if you don't, like spherical or hyperbolic ones...

To give what I would consider to be a good counterargument, if one could produce an actual inconsistency with ZFC set theory that would be strong evidence that it is "wrong" to accept it.

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2. egorelik ◴[30 Aug 25 15:22 UTC] No.45075406[source]▶
Skepticism of a ZFC axiom in particular could just be in terms of its standard status. I don't think anyone debates that ZFC in a particular logic doesn't imply this or that, but people can get into philosophical questions about whether it is the right foundation. There are also purely mathematical reasons to care - an extra axiom may allow you to produce more useful math, but it also potentially blocks you from other interesting math by keeping you out of models where, e.g., Choice is false.