[xyzzyz]

only that in a single symbolic system we can't have expressions for all of them at once.

I don't think that's true. There are only countably many different symbolic systems[1], and as we can only express countably many numbers in each, we don't leave the realm of countable.

[1] - A "symbolic system" must at least come with a procedure to tell whether a sequence is a part of it, and, unless you disbelieve the Church-Turing thesis, there are only countably many procedures.

[j2kun]

Doesn't this (and by extension, Curch-Turing) rely on the assumption that the universe is discrete? Provided it were not, and one were able to harness infinite precision, one could presumably make a new symbolic system based on it.

Not saying I believe it, just teasing out assumptions. If one is arguing whether the universe is continuous and using the Church-Turing thesis as justification for something, there's a danger of circular reasoning.

Also, I think the parent commenter is getting more at naming vs existence rather than naming versus "could be named in the future." Is the argument boiling down to that something does not exist (is not "real") if it cannot be named?

[MrMoenty]

You are right, this reasoning depends on only allowing formal systems with finitely many symbols. Turing himself actually gave justification for this in a beautiful footnote in his 1936 paper [1]:

page 249:

> I shall also suppose that the number of symbols which may be printed is finite. If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrarily small extent.

And then in the footnote:

> If we regard a symbol as literally printed on a square we may suppose that the square is 0 < x < 1, 0 < y < 1. The symbol is defined as a set of points in this square, viz. the set occupied by printer's ink. If these sets are restricted to be measurable, we can define the "distance" between two symbols as the cost of transforming one symbol into the other if the cost of moving unit area of printer's ink unit distance is unity, and there is an infinite supply of ink at x = 2. y = 0. With this topology the symbols form a conditionally compact space.

[1] https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf

[xyzzyz]

Doesn't this (and by extension, Curch-Turing) rely on the assumption that the universe is discrete? Provided it were not, and one were able to harness infinite precision, one could presumably make a new symbolic system based on it.

That's a valid point -- if you read it in a certain way, the Church-Turing hypothesis indeed states that continuous models are no more powerful than the discrete ones. In fact, we have every reason to believe it's true. See [1], and references [BCGH07], [GCB08] in that paper.

[1] - http://perso.ens-lyon.fr/yassine.hamoudi/wp-content/uploads/...

[nightcracker]

The Church-Turing hypothesis only applies to functions that apply on discrete data. If the data itself is also continuous it does not apply.

[marcosdumay]

Physics would have something to say about infinite precision.

But yes, with those assumptions, it does not hold. It is not as much that the argument is circular, but that the only tech we know how to use (symbols) limits our expressiveness.

[Retra]

The universe doesn't have to be discrete. Only our capacity to understand it rationally and form consistent linguistic models of it.