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God created the real numbers

ethanheilman.com

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Re: God created the real numbers

#181

Post by tomasson »

vessenes wrote: post #155
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?

To a mathematician saying god created the integers is the same thing as saying god created the rationals: there’s a bijection between the rationals and integers. I’m not convinced that we could have our current universe without irrationals - wouldn’t things like electromagnetism and gravity work differently if forced to be quantized between rationals? Saying ‘meh it would be close enough’ might be correct but wouldn’…

Yeah this is an understatement. Modern technology and the world economy require irrational numbers

Re: God created the real numbers

#182

Post by griffzhowl »

tomasson wrote: post #179
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…

“ Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. ” Hard disagree. This is the problem with math disconnected from physics. The real world is composed of quanta and spectra, i.e. reality is NOT continuous!

Only bound states, like electrons confined to atomic orbitals, have quantized energies. Free electrons (or any particles) can have a continuous range of energies. Quantum mechanics (and general relativity) is still based on contiuous space and time, hence a continuous range of possible velocities and (kinetic) energies

Re: God created the real numbers

#183

Post by jacquesm »

i2go wrote: post #161
akssri wrote: post #158

Theory of Computation wasn't around when all this "exciting" stuff was developed in Mathematics. Given their non-constructive nature "real" numbers are unsurprisingly totally incompatible with computation. Chaitin has a great paper on this and shows how Cantor's constructions were reflected a half-century later by Turing. https://arxiv.org/abs/math/0411418 Except of-course, while "hyper-Turing" machines that can do m…

> Given their non-constructive nature "real" numbers are unsurprisingly totally incompatible with computation. It is funny you say that when Turing defined Turing machines to compute real numbers (like π for example). In its original definition, a number was computable if its Turing machine did not stop. Which makes sense since π does not have a finite decimal expansion. Today, we usually define Turing machines to de…

This confuses the halting problem with a still running computation.

Re: God created the real numbers

#184

Post by egorelik »

blueplanet200 wrote: post #133
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…

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 a…

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.

Re: God created the real numbers

#185

Post by Eddy_Viscosity2 »

drdeca wrote: post #130
Eddy_Viscosity2 wrote: post #106

Earlier quoted context omitted.

You can hold a two in your head, but you can't hold a number with infinitely many decimal places. Any manipulations you do with the real 2 are done conceptually whereas with the natural 2, its done concretely.

The decimal places are just a way of representing it.

The infinite number of decimal places is the definitional feature of a real number. No matter how's it represented they are still there and cannot be contained in our brains. We can say pi and hold the concept of pi in our heads, but not the actual number.

Re: God created the real numbers

#186

Post by jacquesm »

tremon wrote: post #164
tomrod wrote: post #122

Earlier quoted context omitted.

Because the square root of 2 exists.

How does it exist though? Does it exist because we have a notation for it, or because we know its definition? Does the number 2 itself even exist? What does it mean to say that the number 2 exists? Calculo, ergo sum?

> Calculo, ergo sum?

Pretty much:

https://en.wikipedia.org/wiki/Church_encoding

Re: God created the real numbers

#187

Post by svnt »

griffzhowl wrote: post #182
tomasson wrote: post #179

Earlier quoted context omitted.

“ Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. ” Hard disagree. This is the problem with math disconnected from physics. The real world is composed of quanta and spectra, i.e. reality is NOT continuous!

Only bound states, like electrons confined to atomic orbitals, have quantized energies. Free electrons (or any particles) can have a continuous range of energies. Quantum mechanics (and general relativity) is still based on contiuous space and time, hence a continuous range of possible velocities and (kinetic) energies

Energies, yes, but the concept of energy quanta is inverted to e.g. time and length in that we have a maximum, not a minimum, where our understanding/models are limited, right?

Re: God created the real numbers

#188

Post by griffzhowl »

BobaFloutist wrote: post #175
NoahZuniga wrote: post #34

Earlier quoted context omitted.

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.

If I'm carrying a single apple, I can measure the number of apples I'm carrying to infinite precision. I'm carrying 1.000... apples.

You're implicitly assuming your conclusion by calling it a "single" apple, which means exactly one. "Apple" is an imprecise concept, but they're often sufficiently similar that we can neglect the differences between them and count them as if they're identical objects, but this is a simplification we impose for practical purposes.

Even for elementary particles, we can't be sure that all electrons, say, are exactly alike. They appear to be, and so we have no reason yet to treat them differently, but because of the imprecision of our measurements it could be that they have minutely different masses or charges. I'm not saying that's plausible, only that we don't know with certainty

Re: God created the real numbers

#189

Post by chopin »

tomrod wrote: post #171
tremon wrote: post #164

Earlier quoted context omitted.

How does it exist though? Does it exist because we have a notation for it, or because we know its definition? Does the number 2 itself even exist? What does it mean to say that the number 2 exists? Calculo, ergo sum?

The number 2 simply exists independent of human intervention.

I am not convinced. There are no two equal things in nature. Numbering things, say apples, is a completely human abstraction over two different things.

Re: God created the real numbers

#190

Post by griffzhowl »

speak_plainly wrote: post #174
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…

The real question is whether 1 + 1 = 2 is true independent of us recognizing it. If the answer is no, then math really is just a system of ideas, and you’ve slipped into psychologism, where truth depends on minds. But take one thing and then another: you have two things. That’s true whether or not anyone notices. Some mathematics is a human system of ideas, but some of it isn’t. Arithmetic reflects real patterns in t…

> But take one thing and then another: you have two things.

This isn't true in general, because for example you can take two equal volumes of a material and put them together, you will have less than two times the volume because of gravity. The mathematical statement that 1+1=2 follows by definition, and it's useful in applications only when the conditions are met that make it accurate, or accurate enough for the given purposes.

Mathematics is useful because the physical world exhibits regularities in its structure. Talking about logos or God adds an air of mystery to that but I don't know what more it adds

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