TITLE: Examples of theorems that are true conceptually, but false computationally
QUESTION [31 upvotes]: I am currently reading Theorem Proving in Lean, a document dealing with how to use Lean (an open source theorem prover and programming language). My question stems from chapter 11.
The particular section of the chapter that I'm interested in discusses the history and philosophical context of an initially "essentially computational" style of mathematics, until the 19th century, when a "more "conceptual"" understanding of mathematics was required.
The following quote is taken from the second paragraph of section 11.1 of the linked document.

The goal was to obtain a powerful “conceptual” understanding without getting bogged down in computational details, but this had the effect of admitting mathematical theorems that are simply false on a direct computational reading.

This is by no means essential to the understanding of the contents of the document itself, but I'm curious to ask what examples of theorems are false in a "direct computational reading".
In other words, are there any examples of theorems that are true conceptually, but false computationally?

REPLY [3 votes]: This paper by Potgieter has examples to show that, in a suitable computational sense, Brouwer's Fixed point theorem is false. Quoting from the abstract:

The main results, the counter-examples of Orevkov and
  Baigger, imply that there is no procedure for finding the fixed
  point in general by giving an example of a computable function which
  does not fix any computable point.