L. E. J. Brouwer

“Profile at Wolfram Research”

“"Luitzen Egbertus Jan Brouwer" by Mark van Atten in Stanford Encyclopedia of Philosophy”

“In 1908 Brouwer introduced for the first time "weak counterexamples", for the purpose of showing that certain classically acceptable statements are constructively unacceptable (Brouwer 1908). Too much emphasis on these examples has sometimes created the false impression that refuting claims of classical mathematics is the principle aim of intuitionism.”

“Cantor and his disciples... think they have knowledge of all sorts of further sets; their fundamental principle... comes to about the same as the axiomaticians. ...[T]his principle is unjustified and... we assert that the several paradoxes of the 'Mengenlehre' [Set theory]... have no right to exist... [I]t would have been the duty of Cantorians, immediately to reject a notion which gives rise to contradictions, because it is... not built... mathematically.”

“[T]here exist no other sets than finite and denumerably infinite sets and continua... [I]n mathematics we can create only finite sequences, further by means of... 'and so on' the order type ω, but only consisting of equal elements... but no other sets.”

“[S]uppose... we have proved... that the logical system, built... of... linguistic axioms... is consistent [and] we find a mathematical interpretation... [D]oes it follow... that such a mathematical system exists? Such... has never been proved... Thus... it is nowhere proved that a finite number, subjected to a provably consistent system of conditions, must always exist.”

“[M]odern axiomaticians... have only built... linguistic systems... suitable to accompany constructible mathematical systems.”

“[W]heresoever in logic the word all or every is used mathematics... tacitly involves the restriction: insofar as belonging to a mathematical structure which is supposed to be constructed beforehand.”

“[T]he idea that by... such linguistic structures we can obtain knowledge of mathematics apart from that... constructed by direct intuition, is mistaken. And more so is the idea that we can lay in this way the foundations of mathematics.”

“[T]he proposition:A function is either differentiable or not differentiable.says nothing; it expresses the same as...If a function is not differentiable, then it is not differentiable.But the logician... projects a mathematical system, and calls such... an application of the tertium non datur.”

“Man, inclined to... a mathematical view.., has... applied this bias to mathematical language, and in former centuries exclusively to the language of logical reasonings: the science arising from this... is theoretical logic.”

“[I]t is easily conceivable that, given the same organization of the human intellect and... the same mathematics, a different language would have been formed, into which the language of logical reasoning... would not fit. Probably there are still peoples... isolated... for which this is... the case. And no more is it excluded that in a later... development the logical reasonings will lose their present position in the languages of the cultured peoples.”

“With which mathematical notions a spoken or written symbol will be made to correspond... will... differ according to the milieu. ...[W]hich domains of mathematics will be accompanied by a language ...will depend ...upon ...which domains ...have found most applications to the guidance of action or as a means of understanding about action.”

“Logic depends upon mathematics. ...[I]ntuitive logical reasoning ...remains if ...one restricts oneself to relations of whole and part; the mathematical structures ...do not justify any priority of logical reasoning over ordinary mathematical reasoning.”

“The words of... mathematical demonstration merely accompany a mathematical construction that is effected without words. At the point where you enounce... contradiction ...the construction no longer goes ...the required structure cannot be embedded in the given basic structure. ...[T]his observation, I do not think of as a principium contradictionis [principle of contradiction].”

“Mathematics is independent of logic. ...Where mathematical objects are given by their relations to the ...parts of a mathematical structure, we transform these ...by a sequence of tautologies and thus proceed to the relations of the object to other components of the structure. ...The fact that ...a theorem is ...only understood after a chain of tautologies, proves merely that we build our structures too complicated to be comprehended in one view.”

“Life is a magic garden. With wondrous softly shining flowers, but between the flowers there are the little gnomes, they frighten me so much, they stand on their heads, and the worst is, they call out to me that I should also stand on my head, every once in a while I try, and I die of embarrassment; but sometimes the gnomes shout that I am doing very well, and that I’m indeed a real gnome myself after all. But on no account I will ever fall for that.”