> The key argument that finite volume contains finite information seems potentially true but suspect. Case in point, take a unit square, it will have a diagonal who's digits have a long "infinite" string of digits but sqrt(2) isn't "infinite information", it's very finite (ie., one number!)

I share the suspicion you have, but I think your counter example is not valid because there is no unit square in (physical) reality. If you would try to measure the diagonal of the unit square you would at some (finite) point run into trouble with the uncertainty principle.

All measurements have limited precision (number of significant decimal digits gained), so the amount of data that can be gained indeed is limited. But there is no general law preventing increasing the precision with better instruments and better methods, so as measurement gets more advanced, more data (such as more decimal digits) can be obtained. The uncertainty principle postulates that uncertainties of coordinate and its conjugated momentum cannot be simultaneously arbitrarily low. It is not clear to me how that postulate would prevent arbitrary increase in precision of position measurements. In theory one could have uncertainty 10E-100m in position of an electron, if uncertainty in its momentum is of the order 10E66 or higher.

> But there is no general law preventing increasing the precision with better instruments and better methods, so as measurement gets more advanced, more data (such as more decimal digits) can be obtained.

That's not true, as at the Planck length both quantum and general relativistic effects are equally important, preventing arbitrary measurements over smaller scales - that's the case even for a continuously-divisible universe. In most quantum gravitational theories today the Planck length truly is a minimum possible distance in the same way the Planck time is the minimum possible duration, representing a quantized space-time with hard limits to the precision of measurements.

Quantum gravity "theories" are speculative extrapolations of current knowledge for extreme situations, not accepted general laws of physics. They have problems with consistency (discretization of lengths and other quantities breaks relativity postulates) and even bigger problem with lack of experience with phenomena that they describe (minimum possible lengths/times/energy/etc).

Yes I know, that's why I made the general point first, then talked about how quantum gravity theories impose even harder limits if true.

> In theory one could have uncertainty 10E-100m in position of an electron, if uncertainty in its momentum is of the order 10E66 or higher.

I'm not an expert in quantum mechanics, but what I was thinking was that to measure the diagonal of a physical unit square you would have to physically build something like a unit frame from some metal atoms for example. Would this still be an intact metal frame if the uncertainty of electron momentum would be on the order of 10E66 or higher? Naively I would think if the electron could potentially have a momentum of 10E66 due to uncertainty it is very quickly anywhere but not part of the metal frame anymore.

Edit: That part aside if you go to even smaller length scales wouldn't the Planck length be a lower bound for how precise a measurement could be?

Of course it is difficult to do, but the point was that uncertainty principle by itself does not limit precision of single measurement; there are other reasons, but those are practical (we do not have the technology and understanding to build such a setup).

> Naively I would think if the electron could potentially have a momentum of 10E66 due to uncertainty it is very quickly anywhere but not part of the metal frame anymore.

The electron presumably cannot go faster than light, so if the measurement is done fast, the electron won't move much in that time, even with immense momentum. Also, big uncertainty in momentum does not mean the electron will have large momentum for a long time. If there is some containment device, the electron can have immense momentum for a short time that however changes direction so quickly that the electron does not escape the small containment. For example, atoms are such containment devices; the more the atom has protons, the smaller the region the electron is contained in.

> Case in point, take a unit square, it will have a diagonal who's digits have a long "infinite" string of digits but sqrt(2) isn't "infinite information", it's very finite (ie., one number!)

It's tricky to talk about "information" when using arbitrary human conveniences, like decimal arabic numerals, etc. In particular:

- The side length of a unit square also has infinite digits: 1.00000...

- If adding trailing zeros seems superfluous, how about writing the side length as 0.9999...?

- The diagonal has length 1 on a ruler which measures sqrt(2)ths

- The diagonal length can be written very compactly as 'x | x^2 = 2', and there's no reason a numeral system couldn't represent that "natively"

A lot of definitions of "information" (e.g. Shannon information) depend on identifying an element out of a set. Such definitions break down when we're dealing with constants, since e.g. sqrt(2) couldn't be anything else.

It's like trying to apply computational complexity to a fixed input, e.g. it only takes constant time to sort a particular list, or to find the shortest path for a particular travelling salesman, etc.

To avoid degenerating into zero, there needs to be some set/space of possibilities which all need handling.

The author's definitions seem to do this: e.g. if we're considering the set of all possible real numbers, then they do indeed "contain infinite information"; or, a phrasing I would prefer, they would require an infinite amount of information to distinguish them exactly.

> This desperately needs to be peer reviewed by people who study the foundations of physics.

So someone like "As a theoretician, Gisin brought deep insights into quantum mechanics." (https://en.wikipedia.org/wiki/Nicolas_Gisin) ?

>The key argument that finite volume contains finite information seems potentially true but suspect.

Look into the Bekenstein Bound and maybe the Pauli Exclusion Principle. A finite volume of space certainly does have a finite limit to the amount of information (physical information, quantum states and the like) it can contain.

If sqrt(2) is a normal number then its decimal expansion would contain infinite information.

It depends on how you define information, but I'd argue that it doesn't.

In particular, those digits are not random variables which are free to take on any value; they're fixed by the definition of sqrt(2). In terms of Shannon information, the information is a measure of uncertainty in a message. If our protocol is that I'll send you digits of sqrt(2), then there's no information being transferred at all: since you could have worked them out for yourself. Alternatively, if our protocol is that I'll send you digits with uniform probability, then sending sqrt(2) would be "infinite information", but only because we're having to narrow it down out of infinitely many possibilities. I don't think this is the most useful definition though, since we can arrange for any amount of information we like: if the protocol is I will either send "5" or sqrt(2), then it contains 1 bit; and so on.

Alternatively we could use algorithmic information theory, where the information content of a message is the length of the smallest tape for a universal turing machine which outputs that message. sqrt(2) can be calculated by a very small program (cranking out digits forever), so it contains very little information. Yet even here, since it's constant, we could define our turing machine such that it emits sqrt(2) when given an empty tape, and hence it again contains no information.

Let S be an encoding of the works of Shakespeare in binary. A normal number contains S in its binary expansion. We can do this for all information. In this sense is it correct to say that a normal number contains infinite information in it?

Again, depends on your definition of information. For shannon information, you would need to define a protocol. If the protocol is "I will send you the entire works of Shakespeare" then I would be sending you no information. Likewise, if the protocol is to send you S, then S contains no information.