Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

God created the real numbers

ethanheilman.com

Post reply on HN
Print view View original

41–50 of 226 posts

Re: God created the real numbers

#41

Post by NoahZuniga »

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 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.

Re: God created the real numbers

#42

Post by prmph »

athrowaway3z wrote: post #13

Can't say that I'm completely in the headspace to follow the argument, but wanted to add my 2 cents from a few years ago. Integers come into existence long before god - as the only presumption required is a difference between one thing and another (or nothing). The integers also create infinite gaps. The primes. So no - I do not think reals are closer to the divine. They require we import infinity twice to be defined…

But all numbers are abstractions, there is nothing “real” (pun unintended) about any number, so it seems strange to me to judge certain numbers on whether they map to our physical reality.

Re: God created the real numbers

#43

Post by andrewla »

NoahZuniga wrote: post #19
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…

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?

I would argue that even the rational numbers are unphysical in the same way that the integers are!

The idea that a quantity like 1/3 is meaningfully different than 333/1000 or 3333333/10000000 is not really that interesting on its own; only in the course of a physical process (a computation) would these quantities be interestingly different, and then only in the sense of the degree of approximation that is required for the computation.

The real numbers in the intuitionalist sense are the ground truth here in my opinion; the Cantorian real numbers are busted, and the rationals are too abstract.

Re: God created the real numbers

#44

Post by andrewla »

IAmBroom wrote: post #31
Dylan16807 wrote: post #24

Earlier quoted context omitted.

Can you name a physical thing that is a circle even to the baseline precision level of a 64 bit float?

A black hole.

A non-rotating black hole. Or a rotating black hole with zero charge. Or a rotating black hole with non-zero charge no external magnetic fields. Or a rotating black hole with non-zero charge with non-time-varying external magnetic fields. Or a wart on a frog on a bump on the log on a hole on the bottom of the sea.

Re: God created the real numbers

#45

Post by shonenknifefan1 »

NoahZuniga wrote: post #19
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…

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?

I think this is right. Any measurement will have finite precision, so while we might be able to discover some maximum precision that the universe uses eventually, we won't ever be able to prove that the universe has infinite precision representations from finite precision measurements.

Re: God created the real numbers

#46

Post by andrewla »

shonenknifefan1 wrote: post #45
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?

I think this is right. Any measurement will have finite precision, so while we might be able to discover some maximum precision that the universe uses eventually, we won't ever be able to prove that the universe has infinite precision representations from finite precision measurements.

Only so long as we use the rationals as an approximation. If we expect them to be exact then they are as bad as the integers.

The continuum is the reality that we have to hold to. Not the continuum in the Cantor sense, but in the intuitionalist or constructivist sense, which is continuously varying numbers that can be approximated as necessary.

Re: God created the real numbers

#47

Post by SabrinaJewson »

axolotliom wrote: post #40
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 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 that is both mapped to by f and not included in the set that maps to it. We know that f(S) ∉ S: suppose f(S) ∈ S, then we would have existence of an s ⊆ A such that f(s) = f(S) and f(S) ∉ s; by injectivity, of course s = S and therefore f(S) ∉ S, which contradicts our premise. However, we can now easily prove that there exists an s ⊆ A satisfying f(s) = f(S) and f(S) ∉ s (of course, by setting s = S), thereby showing that f(S) ∈ S, a contradiction.

Re: God created the real numbers

#48

Post by andrewla »

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.

Re: God created the real numbers

#49

Post by tromp »

IAmBroom wrote: post #31
Dylan16807 wrote: post #24

Earlier quoted context omitted.

Can you name a physical thing that is a circle even to the baseline precision level of a 64 bit float?

A black hole.

A black hole is no more a perfect sphere than a sun is. Would gravity from the nearest other black hole not have a deforming effect of at least 2^-64 ?

Re: God created the real numbers

#50

Post by zamalek »

When I truly grokked complex numbers, I felt as though real numbers were a lie - though I would now say that it was a convenient omission. There are many things that are more naturally described using complex numbers - waves (which much of reality boils down to) immediately come to mind. Even if something does align better with real numbers, it's still just x+0i. Maybe I'll change my mind ~when~ if I finally grok quaternions.
Post reply on HN