TITLE: Does the Brouwer fixed point theorem admit a constructive proof?
QUESTION [31 upvotes]: Wikipedia and a few websites (and a few mathoverflow answers) say there is a constructive proof of the Brouwer fixed point theorem, some others say no. The argument for a constructive proof is always the same. The Brouwer fixed point theorem  is equivalent to some other results (Miranda, Sperner) where some algorithm produces some object (trichromatic triangle etc) related to a potential fixed point, but in fact, one also needs an additional compactness argument to conclude, and this last step  does not appear to me to be constructive.
I am not a logician, so my question is
can one give a mathematical rigorous meaning to the following statement: "there is no constructive proof to the Brouwer fixed point theorem"?

REPLY [10 votes]: One constructive version of the Brouwer fixed-point theorem which I find very elegant and illustrating runs like this:
Theorem (constructive Brouwer fixed-point theorem)
Let $B$ be the closed unit ball in $\mathbb{R}^n$ for $n\in\mathbb{N}$, and let $f$ be a uniformly continuous function from $B$ to itself. Let $G_f=\{(x,f(x))|\,x\in B\}$ be the graph of $f$ in $B\times B$, and let $D$ be the diagonal $\{(x,x)|\,x\in B\}$, also in $B\times B$. 
Then $d(G_f, D)=0$.
This version is short, and yet provides all the constructive information needed for approximative BFP-versions.

[update for clarification, oct 2019:]

For two subsets $A, B$ and $\rho\geq 0$ we write $d(A, B)=\rho$ when we can determine that $\inf(\{d(a,b)| a\in A, b\in B\})$ exists constructively and equals $\rho$.
The above phrasing of the theorem is classically equivalent to the usual phrasing of the BFPT. (That also holds for Simon Henry's answer which in essence gives the same theorem. Still, one doesn't need a localic approach to prove it, it holds in plain BISH).