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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#81

Post by futune »

seanhunter wrote: post #80

I don’t understand constructivism at all. No numbers are real. They are all an entirely abstract construction, like lines and planes and open sets and closed balls and metric spaces and everything else. If I construct a number by saying it’s the limit of the sequence sqrt(2)-1/n as n->infinity that’s just as real as the number 1 or 1/2. It just seems really arbitrary to privilege one kind of construction over another…

The numbers you mentioned are computable numbers. Constructivists and intuitionists have no problem with them, generally. The problem is that there is only a countable number of computable reals, so what do we do about all the other reals? The ones that nobody will give an example of because it is simply not possible to do so, which is to say, the vast majority of reals?

Does it exist if it is impossible to show an example? Only if "exist" is interpreted to mean "you cannot deduce a contradiction from assuming them", which is a logically consistent position. But if you mean "they are in some sense actually there", in some describable way, then it's muddier. And you can build a logically consistent position from declaring "they are not there" as well. Precisely because no counterexample can be produced.

And then there's ultrafinitists, and yeah, they are a bit bonkers.

Re: How real are real numbers? (2004)

#82

Post by seanhunter »

futune wrote: post #81
seanhunter wrote: post #80

I don’t understand constructivism at all. No numbers are real. They are all an entirely abstract construction, like lines and planes and open sets and closed balls and metric spaces and everything else. If I construct a number by saying it’s the limit of the sequence sqrt(2)-1/n as n->infinity that’s just as real as the number 1 or 1/2. It just seems really arbitrary to privilege one kind of construction over another…

The numbers you mentioned are computable numbers. Constructivists and intuitionists have no problem with them, generally. The problem is that there is only a countable number of computable reals, so what do we do about all the other reals? The ones that nobody will give an example of because it is simply not possible to do so, which is to say, the vast majority of reals? Does it exist if it is impossible to show an e…

They are in a very meaningful sense actually there. If I draw a curve I want the line not to have holes in it, and they have to be there for that to be true.

More importantly, a function is its graph so if I want my functions to be continuous I need them to be there.

Re: How real are real numbers? (2004)

#83

Post by xscott »

seanhunter wrote: post #82
futune wrote: post #81

Earlier quoted context omitted.

The numbers you mentioned are computable numbers. Constructivists and intuitionists have no problem with them, generally. The problem is that there is only a countable number of computable reals, so what do we do about all the other reals? The ones that nobody will give an example of because it is simply not possible to do so, which is to say, the vast majority of reals? Does it exist if it is impossible to show an e…

They are in a very meaningful sense actually there. If I draw a curve I want the line not to have holes in it, and they have to be there for that to be true. More importantly, a function is its graph so if I want my functions to be continuous I need them to be there.

No matter how small you go, between every two real numbers is a computable number, and between every two computable numbers is a real number that's not computable. If you restricted yourself to computable numbers, are you sure there are any holes in your graphs and functions? Can you point to or name one?

:-)

Re: How real are real numbers? (2004)

#84

Post by seanhunter »

xscott wrote: post #83
seanhunter wrote: post #82

Earlier quoted context omitted.

They are in a very meaningful sense actually there. If I draw a curve I want the line not to have holes in it, and they have to be there for that to be true. More importantly, a function is its graph so if I want my functions to be continuous I need them to be there.

No matter how small you go, between every two real numbers is a computable number, and between every two computable numbers is a real number that's not computable. If you restricted yourself to computable numbers, are you sure there are any holes in your graphs and functions? Can you point to or name one? :-)

I understand the point about density and I'm fine with the concept of functions being continuous over a restricted domain (eg the rationals even) but if I draw a line and label one end 0 and one end 1 then I have plotted the set of all numbers in that interval, not just the ones we find computationally convenient.

The historical context about the constructivist movement is that it was a religiously-inspired objection to the work of Cantor, who some random bishop said was challenging God with his work on transfinite numbers because God owned infinity. I just find it weird that now people try to pretend that it's somehow more rigorous when really it's just an alternative axiomatic perspective that started in this shonky way and has grown to a point where it's just about respectable.

Re: How real are real numbers? (2004)

#85

Post by xscott »

seanhunter wrote: post #84
xscott wrote: post #83

Earlier quoted context omitted.

No matter how small you go, between every two real numbers is a computable number, and between every two computable numbers is a real number that's not computable. If you restricted yourself to computable numbers, are you sure there are any holes in your graphs and functions? Can you point to or name one? :-)

I understand the point about density and I'm fine with the concept of functions being continuous over a restricted domain (eg the rationals even) but if I draw a line and label one end 0 and one end 1 then I have plotted the set of all numbers in that interval, not just the ones we find computationally convenient. The historical context about the constructivist movement is that it was a religiously-inspired objection…

I can't speak to the religious bits or history. That's certainly not my motivation for thinking about this stuff. It's not about computational convenience either. Both of those seem like strawmen, but maybe they're relevant to other people.

The problem to me is that the Reals which aren't computable are absurd. We've never used any of them in all history. We can only put names on a very special few, those are countable, and we don't even know their value very well.

And the main way we prove that the Reals are uncountable is to use a proof by contradiction. It would take too long to spell it out, but they aren't really just contradicting "Reals are Countable". It's "All that other stuff we think is true AND Reals are Countable" that gets contradicted.

Once you accept the Reals, the Axiom of Choice is not simply obvious any more. And if you go down that path you get things like the Banach-Tarski paradox. To me THAT ought to be a proof by contradiction that we've made a mistake somewhere.

More interesting than that though: If the universe we live in requires non-computable Reals to describe it carefully, then it says something very weird about determinism. In order to compute a future state of a system, we need to use numbers we can't compute?!?

Re: How real are real numbers? (2004)

#86

Post by singularity2001 »

tialaramex wrote: post #77
Dylan16807 wrote: post #74

Earlier quoted context omitted.

The real world doesn't have any of those numbers, only approximate matches. My point isn't about digits, it's about precision. You can do math by hand with more precision than actually exists in the real world. And once you add any slack at all, even one part per googol, your numbers stop being normal and they can all be computed and represented in rational form.

> You can do math by hand with more precision than actually exists in the real world. This claim can't mean anything because it disappears up its own backside. If you, here in the real world, can "do math by hand" that math is in the real world.

what he probably meant is that if you write down pi with 200 digits there's nowhere in the universe where you can find pi with that precision

Re: How real are real numbers? (2004)

#87

Post by Dylan16807 »

tialaramex wrote: post #77
Dylan16807 wrote: post #74

Earlier quoted context omitted.

The real world doesn't have any of those numbers, only approximate matches. My point isn't about digits, it's about precision. You can do math by hand with more precision than actually exists in the real world. And once you add any slack at all, even one part per googol, your numbers stop being normal and they can all be computed and represented in rational form.

> You can do math by hand with more precision than actually exists in the real world. This claim can't mean anything because it disappears up its own backside. If you, here in the real world, can "do math by hand" that math is in the real world.

All you have in the real world is scribbles on paper.

I'm sorry, what did you mean by "how complicated the world we inhabit is" because I thought you were talking about physical interactions of matter and energy.

If you're including all math as "real world", then I think the claim that math teaches you something useful about the complexity of the real world is what actually disappears up its own backside.

Re: How real are real numbers? (2004)

#88

Post by singularity2001 »

there is a practical reason to get rid of the fantasy reals and restrict oneself to normal reals or some other new invention:

Since Lean has become more popular as a proving system I've stumbled upon one very annoying feature of reals: they are not computably comparable. The system says you can never know whether two arbitrary real numbers are the same because you don't have enough time to compare them.

Re: How real are real numbers? (2004)

#89

Post by seanhunter »

xscott wrote: post #85
seanhunter wrote: post #84

Earlier quoted context omitted.

I understand the point about density and I'm fine with the concept of functions being continuous over a restricted domain (eg the rationals even) but if I draw a line and label one end 0 and one end 1 then I have plotted the set of all numbers in that interval, not just the ones we find computationally convenient. The historical context about the constructivist movement is that it was a religiously-inspired objection…

I can't speak to the religious bits or history. That's certainly not my motivation for thinking about this stuff. It's not about computational convenience either. Both of those seem like strawmen, but maybe they're relevant to other people. The problem to me is that the Reals which aren't computable are absurd. We've never used any of them in all history. We can only put names on a very special few, those are countab…

> We've never used any of them in all history.

You just used them yourself in your previous post to make your argument that computable numbers are dense in incomputable numbers.[1] So presumably that makes you the first person in all of maths history to use them. Congratulations I guess? The other possibility is they get used a lot and we just don’t make a fuss about it because most of the time it’s exactly like you used them - to express an argument where it doesn’t matter whether they are computable or not.

I have no idea why you think they are absurd but as I say it’s an alternative axiomatic perspective. It just makes a huge amount of maths very inconvenient without really yielding (as far as I can see) much of anything.

I don’t have a perspective on your questions about the universe. Non-computable numbers don’t really trouble my world view. We invented the calculus and the language of continuous functions which gave rise to the rigorous construction of the real numbers precisely to better describe the universe, so it seems that they are pretty useful in that context.

[1] Which is interesting right, because this is a trivial argument in the sense that in standard first year analysis you learn that the rationals are dense in the irrationals and rationals are obviously computable so it must be that rationals are dense in the non-computable numbers. But this is a very basic argument that we can only make because we permit the Reals to be complete. It’s not possible to express the argument in the restriction to computable numbers because there is no ambient space outside the computable numbers for them to be dense into.

Re: How real are real numbers? (2004)

#90

Post by tialaramex »

singularity2001 wrote: post #86
tialaramex wrote: post #77

Earlier quoted context omitted.

> You can do math by hand with more precision than actually exists in the real world. This claim can't mean anything because it disappears up its own backside. If you, here in the real world, can "do math by hand" that math is in the real world.

what he probably meant is that if you write down pi with 200 digits there's nowhere in the universe where you can find pi with that precision

Are you sure? Naively sure, you won't find sufficiently enormous circles and even if you could such a huge circle won't have the Pi ratio here, that's a property of Euclidean space and we don't live in a Euclidean space, ours is slightly off IIRC.

But Pi shows up in other places and I'm not at all sure you can show there are no such places which distinguish some arbitrary approximation from the actual ratio.

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