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Pre-intuitionism
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==Arguments over the excluded middle== It was for this assertion, among others, that [[Henri Poincaré|Poincaré]] was considered to be similar to the intuitionists. For [[Luitzen Egbertus Jan Brouwer|Brouwer]] though, the Pre-Intuitionists failed to go as far as necessary in divesting mathematics from metaphysics, for they still used ''principium tertii exclusi'' (the "[[law of excluded middle]]"). 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 [[temporal logic|situations]]. To an intuitionist this seems to rank the law of excluded middle as just as un[[rigour|rigorous]] as [[Giuseppe Peano|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, ''i.e.'', synthetic), and logic was another (analytic).
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