Less Wrong is a community blog devoted to refining the art of human rationality. Please visit our About page for more information.

Sebastian_Hagen comments on The Amanda Knox Test: How an Hour on the Internet Beats a Year in the Courtroom - Less Wrong

42 Post author: komponisto 13 December 2009 04:16AM

You are viewing a comment permalink. View the original post to see all comments and the full post content.

Comments (632)

You are viewing a single comment's thread. Show more comments above.

Comment author: Sebastian_Hagen 13 December 2009 07:11:11PM 3 points [-]

We have managed to come up with one perfect formal system of rationality: mathematics, in which you can be "absolutely certain" of a statement, as long as it can be expressed in a certain language and doesn't actually depend on any observations.

That's incorrect. As shown by Gödel's second incompleteness theorem, mathematical formal systems divide into three categories:

  1. Systems that are inconsistent.
  2. Systems that can't prove their own consistency.
  3. Systems that aren't particularly powerful.

When doing math, humans tend to assume that certain formal systems are consistent. But we can't actually prove it; it's ultimately an empirical question (and actually it would be even without the incompleteness theorem; if strong consistent formal system could prove their own consistency, that wouldn't make them any different from strong inconsistent formal systems).

Though as far as empirical questions go, the consistency of certain formal systems fundamental to human math does seem to be extremely probable.