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God created the real numbers

ethanheilman.com

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Re: God created the real numbers

#61

Post by AndrewDucker »

oskaralund wrote: post #59
BalinKing wrote: post #54

Earlier quoted context omitted.

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.

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.

There are limits to precision there too. The amount of available matter to build something out of and the size you can build down to before quantum effects interfere.

Re: God created the real numbers

#62

Post by kazinator »

andrewla wrote: post #3

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 that's tatamount to the belief that the minutest particle of the universe requires the equivalent of an infinite number of bits of state.

Re: God created the real numbers

#63

Post by oskaralund »

AndrewDucker wrote: post #61
oskaralund wrote: post #59

Earlier quoted context omitted.

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.

There are limits to precision there too. The amount of available matter to build something out of and the size you can build down to before quantum effects interfere.

The example was only to illustrate that measurement precision is independent of the time it takes to perform the measurement.

Re: God created the real numbers

#64

Post by cernocky »

getnormality wrote: post #36

All math is just a system of ideas, specifically rules that people made up and follow because it's useful. I'm so used to thinking this way that I don't understand what all the fuss is about, mathematical objects being "real". Ideas are real but they're not real in the way that rocks are. Whenever there's a mysterious pattern in nature, people have felt the need to assert that some immaterial "thing" makes it so. But…

Ideas are real in the way rocks are if we are concerned with their informational being. They are real informationally - ideas and math participate in forming the world. Nowadays, LLMs, Search and other apps probably affect the world even more than any common rock. Which is more real?

Re: God created the real numbers

#65

Post by CyLith »

andrewla wrote: post #39
blueplanet200 wrote: post #23

Earlier quoted context omitted.

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.

I loosely identify with the schools of intuitinalism/construtivism/finitism. Primary idea is that the Law of the Excluded Middle is not meaningful. So yes, generally not starting with ZFC. I can't speak to "truth" in that sense. The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tars…

I don't understand why you believe Banach-Tarski to be obviously false. All that BT tells me is that matter is not modeled by a continuum since matter is composed of discrete atoms. This says nothing of the falsity of BT or the continuum.

Re: God created the real numbers

#67

Post by AIPedant »

andrewla wrote: post #48
zarzavat wrote: post #21

God created the rational numbers. The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Our equations for the universe require a complete set, but that would be confusing the map with the territory. There is no physical evidence for uncountable infinities, those are purely in the imagination of man.

Why are rationals special? They represent an exactness in a similarly unphysical way as the integers. The rationals are infinitely precise. 1/3 is not the same as 0.33333 or 0.33333333 or 0.3. The real numbers exist and are approximable, either by rationals or by decimal expansion. The idea of approximability and computability are the critical things, not the specific representation.

I am confused why you think the exactness of integers and rationals is unphysical. "This egg carton has 12 eggs" is a (boring) physical statement. "You can make 1/3rd of a carton of eggs without cutting an egg" also seems perfectly physical to me. Your problem with zero-point-three-repeating is a quirk of decimal representation, not a mystical property of 1/3.

Egg cartons might sound contrived but the reals don't necessarily make sense without reference to rulers, scales, etc. And in fact the defining completeness / Dedekind cut conditions for the reals are necessary for doing calculus but any physical interpretation is both pretty abstract and probably false in reality.

Re: God created the real numbers

#68

Post by codebje »

SabrinaJewson wrote: post #47
axolotliom wrote: post #40

Earlier quoted context omitted.

> 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 t…

> Addressing your issue directly, the Axiom of Choice is actively debated: The axiom of choice is not required to prove Cantor’s theorem, that any set has strictly smaller cardinality than its powerset. Actually, I can recount the proof here: Suppose there is an injection f: Powerset(A) ↪ A from the powerset of a set A to the set A. Now consider the set S = {x ∈ A | ∃ s ⊆ A, f(s) = x and x ∉ s}, i.e. the subset of A…

Perhaps this is an ignorant question, but wouldn't you need AC to select the s ⊆ A whose existence the contradiction depends on? A constructive proof, at least the ones I'm trying to build in my head, stumbles when needing to produce that s to use in the following arguments.

Re: God created the real numbers

#69

Post by Kranar »

omnicognate wrote: post #37
NoahZuniga wrote: post #19

Earlier quoted context omitted.

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?

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.

You don't need the full set of real numbers to do physics, only the computable subset of the real numbers. Using the full reals is mostly done out of simplicity.

Re: God created the real numbers

#70

Post by AIPedant »

oskaralund wrote: post #59
BalinKing wrote: post #54

Earlier quoted context omitted.

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.

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.

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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