TITLE: What is this physicist saying?
QUESTION [10 upvotes]: I do not want to poison this forum with politics. But I want to understand, precisely, what is meant by the bolded statement. It is made by a physicist who used to work at Harvard regarding the relationship between pure math and physics.

A physics-oriented question appears at the end of Chapter 3: Are we
  really talking about continuous objects themselves or about finite
  sequences of symbols that talk about continuum?
That's a good question and I am inclined to say the latter. If we talk
  about specific things, these specific things are always countable
  because they must be describable by a finite sequence of symbols. Even
  when we talk about intervals of real numbers that arguably contain an
  uncountably infinite number of real numbers, we must still specify the
  endpoints of the interval by a finite sequence of words or symbols –
  and such sequences are "discrete" i.e. "countable". 
This is why I tend to consider the very fact that real numbers are
  uncountable to be nothing else than a linguistic curiosity: the
  actual, well-defined real numbers you may ever encounter form a
  countable set! This is why the uncountability of the real numbers –
  and the whole discipline of maths based on this formally provable
  claim and similar claims – doesn't have implications for "talking
  about physics".

Here is the link to the full blog. If you want more context, I would suggest to start reading from the "Chapter 2 talks about sets, their elements..." part.
It appears he is talking about the symbols and notation we write down to describe the real numbers. But the bolded part is explicitly referring to the set of real numbers.

REPLY [3 votes]: This problem is closely related to mathematical logic and the notion of definability and truth.
One view is through the Löwenheim–Skolem theorem in mathematical logic, which states that

if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ.

This has somewhat counter-intuitive consequences, for example that any theory of real numbers (or even set theory) has a countable model. Or that any theory of natural numbers has an uncountable model.
Let's say we have a countable model of a theory of $\mathbb{R}$. Does it mean that all subsets of $\mathbb{R}$ are countable? No. We're viewing only one possible model of the theory. All subsets are countable only when this particular model is viewed from outside. But when viewed from "inside" of this model, we still can't construct a bijection between $\mathbb{N}\subset\mathbb{R}$ and $[0,1]\subset\mathbb{R}$. This is because the model doesn't contain such a bijection (even though such bijection exists when viewed from outside - it's not available inside the model). The existence of objects or their definability is different when viewed from outside of some particular model form when viewed from inside the theory.
Therefore it is a bit imprecise to say just that there is only a countable number of real numbers. What is "countable" depends on if work inside a theory or if we take a step "outside" on the meta level.

Such situations or paradoxes occur often in logic. For example Gödel's second incompleteness theorem states that a (reasonable) theory can't prove its own consistency. But of course we can prove its consistency from outside, for example by embedding the theory in some other, stronger theory.
See also Berry's paradox: the smallest positive integer not definable in fewer than twelve words. The key is that the notion of "definable" is different when observed from outside and when constructed inside the theory (for example using Gödel numbering).