L. E. J. Brouwer

"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."

"The viewpoint of the formalist must lead to the conviction that if other symbolic formulas should be substituted for the ones that now represent the fundamental mathematical relations and the mathematical-logical laws, the absence of the sensation of delight, called "consciousness of legitimacy," which might be the result of such substitution would not in the least invalidate its mathematical exactness. To the philosopher or to the anthropologist, but not to the mathematician, belongs the task of investigating why certain systems of symbolic logic rather than others may be effectively projected upon nature. Not to the mathematician, but to the psychologist, belongs the task of explaining why we believe in certain systems of symbolic logic and not in others, in particular why we are averse to the so-called contradictory systems in which the negative as well as the positive of certain propositions are valid."

"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."

"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]."

"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."

"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."

"[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."

"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."

"[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."

"[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."

"[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."

"Riemann was the first to show the right way for research on the by starting from the idea that space is a Zahlenmannigfaltigkeit, thus a system built... by ourselves. ...Pasch, Hilbert and others, because they considered ...[t]his hypothesis as arbitrary, resumed the logical foundation ...trying to better Euclid by constructing ...linguistic structures ...solely by ...logical principles. ...Such disturbing consequences follow when language, ...a means ...for the communication of mathematics, but which has nothing to do with mathematics ...except as an accompaniment, is considered essential, and when the laws governing the succession of sentences ...are seen as directives for acts of mathematical construction."

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

"[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."

"[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."

"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."

"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."