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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#71

Post by Dylan16807 »

tialaramex wrote: post #68
Dylan16807 wrote: post #38

Earlier quoted context omitted.

> I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers. They're worse. Just having another dimension is significantly more relevant to reality. And the reals also ruin the word "normal".

I actually think both "real" and "normal" are helpful in building the correct intuition about how complicated the world we inhabit is rather than how simple we want it to be.

As far as we can tell, there is nothing resembling a number with infinite digits in the real world.

It's complicated in a completely different way from how the real world is complicated. I don't think that gives you useful intuition.

Re: How real are real numbers? (2004)

#72

Post by d_tr »

atoav wrote: post #70
nyc111 wrote: post #69

Earlier quoted context omitted.

> However now a neo-Pythagorian doctrine is emerging, according to which everything is 0/1 bits, and the world is built entirely out of digital information Can this be true? DNA is the information system of living creatures and as far as I know, it is not coded with 0s and 1s. So, how can we justify that "the world is built entirely out of digital information"?

For all I, a Victorian everyman, know the world is built from small pistons, gears and pulleys. Every age has their technology which they will project onto the world. A century ago one may have started to talks about everything being electrical wires and switches.

It doesn't matter whether the model assumes bits, pulleys, elves or whatever, as long as it does a better job at describing physical reality than whatever exists at the time anyway. People will try and very probably succeed in providing alternative formalism anyway.

All that matters is whether it facilitates reasoning towards the goal.

Re: How real are real numbers? (2004)

#73

Post by tialaramex »

Dylan16807 wrote: post #71
tialaramex wrote: post #68

Earlier quoted context omitted.

I actually think both "real" and "normal" are helpful in building the correct intuition about how complicated the world we inhabit is rather than how simple we want it to be.

As far as we can tell, there is nothing resembling a number with infinite digits in the real world. It's complicated in a completely different way from how the real world is complicated. I don't think that gives you useful intuition.

Of course "having infinite digits" isn't from the real world because it's about an artifice, our choosing to represent numbers with digits. But the distinction between the Rationals and the Reals isn't about those digits. The discovery that there's some fixed ratio between the diameter of a circle and its circumference is fascinating and yet though we can't (AFAIK) prove it's normal that ratio sure looks normal and across mathematics we find this ratio again, and again, and again, it's something fundamental but it clearly isn't rational.

Likewise for the square root of 2 and for Euler's Number. These numbers are ever so real and yet they sure fucking look normal to me. If you assure me they are not normal, but you can't prove it, I shall not believe you.

Re: How real are real numbers? (2004)

#74

Post by Dylan16807 »

tialaramex wrote: post #73
Dylan16807 wrote: post #71

Earlier quoted context omitted.

As far as we can tell, there is nothing resembling a number with infinite digits in the real world. It's complicated in a completely different way from how the real world is complicated. I don't think that gives you useful intuition.

Of course "having infinite digits" isn't from the real world because it's about an artifice, our choosing to represent numbers with digits. But the distinction between the Rationals and the Reals isn't about those digits. The discovery that there's some fixed ratio between the diameter of a circle and its circumference is fascinating and yet though we can't (AFAIK) prove it's normal that ratio sure looks normal and a…

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.

Re: How real are real numbers? (2004)

#75

Post by lanstin »

wongarsu wrote: post #59
andrewla wrote: post #4

Earlier quoted context omitted.

If reality is quantized and there is a smallest number that is physically relevant, you don't need the reals to break it. Take that smallest number and divide it by two, and now you have a physically meaningless number using only the rationals. This isn't fair for what quantization means in reality, but I'm just pointing out that you don't have to introduce the real numbers to get physically meaningless quantities.

Take the smallest number corresponding to a physically meaningful distance in meters, divide it by two, and that number is still a physical meaningful distance if you switch the unit to decameters or kilometers. Any rational number has some meaning if you just add the right unit, even if the units become increasingly ridiculous. But for reals that trick does not work

Sqrt(2) “any unit that is an integral multiple of the shortest interval”.

But it would get complicated, for any given allowable velocity and allowable length, you’d get more lengths from Lorentzian contraction.

There are really a couple of different ideas being combined: are there an infinite number of quantum states for the universe, are space and time continuous, is the forward direction of time resolved by computable processes.

And even bigger ones lurk: are space and time emergent properties from quantum waveforms that lack an inherent idea of space and time (but things that are highly correlated give rise to a notion of being near each other in “space time”)?

Re: How real are real numbers? (2004)

#76

Post by lanstin »

andrewla wrote: post #11
BeetleB wrote: post #5

Earlier quoted context omitted.

> What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing. It buys you the rigor of doing calculus, which buys you a lot of results that, while could be computed without calculus, would also be very difficult without it.

Agree to disagree! Doing calculus with computable numbers is totally possible and you get all the continuity you need. You need to drop the Lebesgue formulation of the integeral and either use a Reimann integral or the gauge integral (Henstock–Kurzweil) if you need a well-behaved integral in the face of very poorly-behaved functions, but in physical reality these don't exist and in abstract mathematics they are rarel…

Like the old joke: The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?

But can you do stats without measure theory? Normal distribution in the limit and all that?

There is no linearly additive measure on rationals, and therefore no way to grab a rational uniformly from (0,1). Has to be skewed to some level of complexity in the denominator.

Re: How real are real numbers? (2004)

#77

Post by tialaramex »

Dylan16807 wrote: post #74
tialaramex wrote: post #73

Earlier quoted context omitted.

Of course "having infinite digits" isn't from the real world because it's about an artifice, our choosing to represent numbers with digits. But the distinction between the Rationals and the Reals isn't about those digits. The discovery that there's some fixed ratio between the diameter of a circle and its circumference is fascinating and yet though we can't (AFAIK) prove it's normal that ratio sure looks normal and a…

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.

Re: How real are real numbers? (2004)

#78

Post by Diogenesian »

Nevermark wrote: post #34

It is too bad we don't have a pithy familiar term for the computable / definable / "nameable" / constructible reals. The most general class of undisputed numbers, consistent with the forms we actually use to represent quantities and perform numerical operations. I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers.

I don't think it makes sense to say anything except "computable real" - the computable / uncomputable distinction seems totally immaterial for the purposes of most real analysis, or even "pathological" topology and set theory involving R (except for puzzles directly involving computability). And the "interesting" transcendental computable reals are a bit of a grabbag.

The mean value theorem isn't true for the computable reals, differentiation of computable function isn't always computable, sequences tend to behave poorly, etc. There's still a lot you can say: https://en.wikipedia.org/wiki/Computable_analysis but in general calculus doesn't care about computability, that's a human problem.

Re: How real are real numbers? (2004)

#80

Post by seanhunter »

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. We want a complete ordered field, so we built the reals. Saying they don’t exist or they aren’t real or whatever seems just to be completely beside the point. They are real enough to do the thing we want them to do.

They are just as real as anything else in maths.

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