[jackhalford]

> A real number with infinite digits can’t be physically relevant.

but

> Popescu objects to the idea that digits of real numbers count as information.

I don't know where to stand, what about information encoded in geometry, like pi. If I get a spherical system, in a small enough space - no, in any space - then there's a "cutoff" to the actual number of digits to pi. Because a chunk of spacetime can't contain infinite information? sounds good.

> Quantum math bundles energy and other quantities into packets, which are more like whole numbers rather than a continuum. And infinite numbers get truncated inside black holes.

Layman here, but AFAIK concepts like black holes aren't consistent with quantum mechanics so I'm not sure it's wise to use concepts from both theories at the same time. (i.e QM predicts that wave functions evolve deterministically but GR predicts information loss in black holes, these two views conflict).

It's way beyond my grasp but some theories seems to quantize space, I wonder how those agree with the notion of "thickness".

I'm disappointed that the article doesn't point into any mathematical theory that models the "thickness" that comes from removing the empty middle theorem.

[dTal]

>then there's a "cutoff" to the actual number of digits to pi. Because a chunk of spacetime can't contain infinite information? sounds good.

It's more subtle than that. The "infinite digits" of pi isn't information, no more so than the endless decimal 1/3 = 0.333... is "infinite information". You can't use it to "store" anything. This is a distinct notion from the practical reality that real spacetime is quantized. An alternate universe with un-quantized spacetime might, or might not, allow you to store infinite information in a chunk - but every digit of pi would be relevant there.

[ineedasername]

Would it be correct to say that, under intuitionist thinking, actually constructing 1/3 = 0.333... (on paper, in a computer, whatever) would take infinite information (not to mention energy and space)?

Though if I understand correctly, intuitionist math would also hold that true infinite 0.3333.... also cannot be constructed?

[logicchains]

1/3 is a rational number, so intuitionism is fine with it. It's also fine with "computable reals": real numbers that have an algorithm for computing them (that algorithm can be represented with a finite amount of information). What cannot be constructed in intuitionism is a "non-computable real": a real number for which there is no algorithm to compute it. The vast majority of classical real numbers are non-computable.

Another way to look at it: if we have a limiting process that approaches some number X, classically we could use the law of the excluded middle to prove that it must eventually reach X, hence X "exists". Under intuitionism however we cannot say that X "exists" or construct X, all we can do is construct the process that approaches X and say this process exists.

[roywiggins]

You can write down an algorithm to generate the digits of .33..., so that set of digits exists as a "potential infinity". Same with Pi and the square root of two.

It is numbers that haven't been constructed that intuitionist mathematics doesn't generally think have been proven to exist.

[paganel]

> exists as a "potential infinity".

Not a mathematician nor a physical theorist by any means but some might regard the putting together of "exists" and "potential infinity" (even if using ") as an oxymoron. It's an endless discussion, of course, I personally think it all boils down to Zeno's paradox remaining, well, an unsolved paradox for the foreseeable future.

[shrimpx]

Imagine the lazy Fibonacci series. As long as you keep taking a number, the next one in the series is generated. It’s not incorrect to say it’s “potentially infinite”. And it exists, as the live algorithm that keeps cranking as long as you put in energy.

[paganel]

There must be a philosophical term for it but imho "potentially existing" (or having the potentiality of being constructed) is not the same thing as actually "existing" (in the reality that surrounds us).

Leaving aside the fact that we're not even sure numbers "exist", for better or worse, their "existence" is just us abstracting away some quantitates for different stuff (we've passed from counting cows or sheep on clay tablets 5000 years ago to believing that there could actually be an infinite number for us to count to).

And yes, I do believe there's a huge impedance mismatch between the world as we experience it around us and the different theoretical constructs that we now call physics or maths. I'm a Hume-ian, a guy who didn't take mathematical induction for granted (presumably not the Fibonacci series either).

[_0w8t]

Essentially Pi is the name of an algorithm that generates its digits.

[empath75]

Or 1.000000..... for that matter. Which seems absurd on its face.

[jeromebaek]

Pi is computable. What is computable, can be represented in finite information (the algorithm used to compute the number). Almost all real numbers are not computable.