the question. From this Poincaré argues that if we fail to establish the consistency of Peano's axioms for natural numbers without falling into circularity, then the principle of complete induction is not provable by general logic. Thus arithmetic and mathematics in general is not analytic but synthetic. Logicism thus rebuked and Intuition is held up. What Poincaré and the Pre-Intuitionists shared was the perception of a difference between logic and mathematics that is not a matter of language alone, but of knowledge itself. 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 (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 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, i.e., synthetic), and logic was another (analytic). Other pre-intuitionists The above examples only include the works of Poincaré, and yet Brouwer named other mathematicians as Pre-Intuitionists too; Borel and Lebesgue. Other mathematicians such as Hermann Weyl (who eventually became disenchanted with intuitionism, feeling that it places excessive strictures on mathematical progress) and Leopold Kronecker also played a role—though they are not cited by Brouwer in his definitive speech. In fact Kronecker might be the most famous of the Pre-Intuitionists for his singular and oft quoted phrase, "God made the natural numbers; all else is the work of man." Kronecker goes in almost the opposite direction from Poincaré, believing in the natural numbers but not the law of the excluded middle. He was the first mathematician to express doubt on non-constructive existence proofs that state that something must exist because it can be shown that it is "impossible" for it not to. See also Conventionalism Notes References Logical Meanderings – a brief article by Jan Sraathof on Brouwer's various attacks on