[ummonk]

First of all, Pi is not an example of such a number, since it can be defined constructively without requiring physical measurements.

Second, even your proposed mechanism of picking out points would be capable of identification (albeit not computation) in a Turing-complete language. Quite simply, to be able to pick out two points you'd need to create some kind of stable structure that identifies those two points - this could be a metal bar as with the old meter bar, or some other kind of structure / apparatus which has two points in space (at a stable distance) locked in. Crucially though, atomic / quantum configuration of this stabilized apparatus would be encodable in theory, once again providing us with a way to count numbers.

There are of course numbers which are specific to our universe, such as the fine structure constant, that don't have any objective definition without reference to the world. But there are finitely many of these, and they're nameable, just not computable.

[chmod775]

> First of all, Pi is not an example of such a number

It is in its canonical form (i.e. as a numbers, not something that represents it), which is what I said.

> Quite simply, to be able to pick out two points you'd need to create some kind of stable structure that identifies those two points.

No I don't. I could pick two atoms floating through space somewhere and say "This distance, right now."

I could also say: "The distance traveled by this particle in a second."

Either number is likely to be irrational. Either number could then only be approximated in binary or whichever system your prefer - unless it just so happens to be expressible using some named irrational constants.

> There are of course numbers which are specific to our universe, such as the fine structure constant, that don't have any objective definition without reference to the world. But there are finitely many of these, and they're nameable.

I guess we'll have to agree to disagree that there are finitely many of these. I don't believe there's even countably many. And if they're not countable, they're not nameable in any common language, since in those names are countable.

It would be quite surprising to me if the universe was nice enough to limit itself to things that are neatly expressible using using the somewhat limited systems humans use, even though it has already shown that it does contain things that will never be expressible using a finite set of symbols, such as numbers or an alphabet.

[ummonk]

> It is in its canonical form (i.e. as a numbers, not something that represents it), which is what I said.

This isn't a coherent sentence. A real number is a cauchy sequence of rational numbers. There is no one canonical sequence that defines Pi.

> No I don't. I could pick two atoms floating through space somewhere and say "This distance, right now." > I could also say: "The distance traveled by this particle in a second."

That process of picking out and localizing the atoms (which are not localized to specific points) would entail performing a measurement with a macroscopic apparatus. At that point, the configuration of the apparatus can be encoded.

[btilly]

No I don't. I could pick two atoms floating through space somewhere and say "This distance, right now."

I guess that you've never heard of the Heisenberg Uncertainty Principle?

You can say that. But it isn't actually well-defined.