Preintuitionism - Arguments Over The Excluded Middle

Arguments Over The Excluded Middle

It was for this assertion, among others, that Poincaré was considered to be similar to the intuitionists. For Brouwer though, the Pre-Intuitionists failed to go as far as necessary in divesting mathematics from metaphysics, for they still used principium tertii exclusi or the "Law of excluded middle". (Note: It actually reads "principle of the excluded third", but it is not commonly known by that name.)

The principle of the excluded middle does lead to some strange situations. For instance, statements about the future such as "There will be a naval battle tomorrow" do not seem to be either true or false, yet. So there is some question whether statements must be either true or false in some situations. To an intuitionist this seems to rank the law of excluded middle as just as unrigorous as Peano's vicious circle.

Yet to the Pre-Intuitionists this is mixing apples and oranges. For them mathematics was one thing (a muddled invention of the human mind (aka. synthetic)), and logic was another (analytic).