"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." (Luitzen E J Brouwer, [letter to C S Adama van Scheltema] , 1906)
"It is well to notice in this connection [the mutual relations between the results of counting and measuring] that a natural law, in the statement of which measurable magnitudes occur, can only be understood to hold in nature with a certain degree of approximation; indeed natural laws as a rule are not proof against sufficient refinement of the measuring tools." (Luitzen E J Brouwer, "Intuitionism and Formalism", Bulletin of the American Mathematical Society, Vol. 20, 1913)
“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." (Luitzen E J Brouwer, 1913)
"[...] an incorrect theory, even if it cannot be inhibited by any contradiction that would refute it, is none the less incorrect, just as a criminal policy is none the less criminal even if it cannot be inhibited by any court that would curb it." (Luitzen E J Brouwer)
"Beyond the natural numbers, addition, multiplication, and mathematical induction are intuitively clear." (Luitzen E J Brouwer)
"Mathematics is nothing more, nothing less, than the exact part of our thinking." (Luitzen E J Brouwer)
"One cannot inquire into the foundations and nature of mathematics without delving into the question of the operations by which the mathematical activity of the mind is conducted. If one failed to take that into account, then one would be left studying only the language in which mathematics is represented rather than the essence of mathematics." (Luitzen E J Brouwer)
"The construction itself is an art, its application to the world an evil parasite." (Luitzen E J Brouwer)
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