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God created the real numbers

ethanheilman.com

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Re: God created the real numbers

#51

Post by shkkmo »

SabrinaJewson wrote: post #26
shkkmo wrote: post #8

Earlier quoted context omitted.

> They're unphysical, and yet the very physical human mind can work with them just fine. Can it? We can only work with things we can name and the real numbers we can name are an infinitesimal fraction of the real numbers. (The nameable reals and sets of reals have the same cardinality as integers while the rest are a higher cardinality.)

We can work with unnameable things very easily. Take, for instance, every known theorem that quantifies over all real numbers. If you try to argue that proving theorems about these real numbers does not constitute “working with” them, it seems you have chosen a rather deficient definition of “working with” that does not match with how that phrase is used in the real world.

I would argue that all of those theorems work with nameable sets of real numbers but not with any unnamable real numbers themselves.

Re: God created the real numbers

#52

Post by oskaralund »

NoahZuniga wrote: post #34
IAmBroom wrote: post #28

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

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.

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?

Re: God created the real numbers

#54

Post by BalinKing »

oskaralund wrote: post #52
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.

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?

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.

Re: God created the real numbers

#55

Post by thechao »

tboyd47 wrote: post #30

> If the something under examination causes a sense of existential nausea, disorientation, and a deep feeling that is can't possibly work like that, it is divine. This is a Jewish and Christian conception of God. How can this be true when so many things that give us comfort in the natural world: fresh fruit, shade trees, sunshine and warm sand between our toes, etc., were not created by man? Even in mathematics itsel…

> This is a Muslim conception of God. How can this be true when so many things that give us comfort in the natural world: fresh fruit, shade trees, sunshine and warm sand between our toes, etc., were not created by man?

...

> The Jewish [Christian] ideal of God (YHVH) is so much more balanced.

There's enough bigotry out there. Let's not make assumptions about people's beliefs.

Re: God created the real numbers

#56

Post by dullcrisp »

NoahZuniga wrote: post #41
omnicognate wrote: post #37

Earlier quoted context omitted.

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.

You can make this statement for any dense subset of the reals, but we don’t because that would be silly.

I think the conceit is supposed to be that analysis—and therefore the reals—is the “language of nature” more so than that we can actually find the reals using scientific instruments.

To illustrate the point, using the rationals is just one way of constructing the reals. Try arguing that numbers with a finite decimal representation are the divine language of nature, for example.

Plus, maybe a hot take, but really I think there’s nothing natural about the rationals. Try using them for anything practical. If we used more base-60 instead of base-10 we could probably forget about them entirely.

Re: God created the real numbers

#58

Post by BalinKing »

tboyd47 wrote: post #30

> If the something under examination causes a sense of existential nausea, disorientation, and a deep feeling that is can't possibly work like that, it is divine. This is a Jewish and Christian conception of God. How can this be true when so many things that give us comfort in the natural world: fresh fruit, shade trees, sunshine and warm sand between our toes, etc., were not created by man? Even in mathematics itsel…

I want to push back on this, because the Christian conception of God definitely includes the idea that God created all good and comforting things, and is indeed their ultimate source. Like, just because God is transcendent[0] does not mean He cannot create things that are perfectly approachable, understandable, and enjoyable.

[0] Jesus being human changes the calculus quite a lot, of course, as elaborated in e.g. Hebrews 4:14–16. God, who was fully transcendent, became human, hence why Jesus is also called Immanuel/Emmanuel (lit. “God with us”) in the Bible.

Re: God created the real numbers

#59

Post by oskaralund »

BalinKing wrote: post #54
oskaralund wrote: post #52

Earlier quoted context omitted.

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?

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.
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