"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."
Luitzen Egbertus Jan "Bertus" Brouwer was a Dutch mathematician and philosopher who worked in topology, set theory, measure theory and complex analysis. Regarded as one of the greatest mathematicians of the 20th century, he is known as one of the founders of modern topology, particularly for establishing his fixed-point theorem and the topological invariance of dimension.
"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."
L. E. J. Brouwer"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."
L. E. J. Brouwer"[M]odern axiomaticians... have only built... linguistic systems... suitable to accompany constructible mathematical systems."
L. E. J. Brouwer"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."
L. E. J. Brouwer"[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."
L. E. J. Brouwer"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]."
L. E. J. Brouwer