On Luitzen E J Brouwer - Historical Perspectives

"So I now abandon my own attempt and join Brouwer. I tried to find solid ground in the impending dissolution of the State of analysis (which is in preparation, even though it is still only recognised by few) without forsaking the order upon which it was founded, by carrying out its fundamental principle purely and honestly. And I believe I was successful - as far as this is possible. For this order is in itself untenable, as I have now convinced myself, and Brouwer - that is the revolution!" (Hermann Weyl, 1921) 

"What Weyl and Brouwer are doing amounts in essence to taking the path once laid out by Kronecker: they seek to provide a foundation for mathematics by pitching overboard whatever discomforts them and declaring an embargo `a la Kronecker. But this would mean dismembering and mutilating our science, and, should we follow such reformers, we would run the risk of losing a large part of our most valued treasures. Weyl and Brouwer outlaw the general notion of irrational number, of function, even of number-theoretic function, Cantor’s [ordinal] numbers of higher number classes, etc. The theorem that among infinitely many natural numbers there is always a least, and even the logical law of the excluded middle, e.g., in the assertion that either there are only finitely many prime numbers or there are infinitely many: these are examples of forbidden theorems and modes of inference. I believe that impotent as Kronecker was to abolish irrational numbers (Weyl and Brouwer do permit us to retain a torso), no less impotent will their efforts prove today. No! Brouwer’s [program] is not as Weyl thinks, the revolution, but only a repetition of an attempted putsch with old methods, that in its day was undertaken with greater verve yet failed utterly. Especially today, when the state power is thoroughly armed and fortified by the work of Frege, Dedekind, and Cantor, these efforts are foredoomed to failure." (David Hilbert, [address] 1922) 

"Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs." (David Hilbert, "Die Grundlagen der Mathematik, 1934-1939)

"In Brouwer’s opinion, if you wanted certainty in mathematics you had to make a sharp distinction between what you could exhibit to the mind and what you must otherwise find meaningless. To say that something exists in mathematics without a construction being given for it is as  incomprehensible, because contradictory, as saying guilty but not proven."(Jeremy Gray, "Plato’s Ghost: The modernist transformation of mathematics", 2008)

"To Brouwer is due the first proof that the concept of dimension can be applied consistently and sensibly to a wide class of spaces. He showed in 1910 that if two manifolds are homeomorphic, then they have the same dimension. Unlike Poincaré’s naive argument, the proof of this theorem is both rigorous and difficult. Freudenthal, in editing Brouwer’s Collected Works, said that on its first appearance it must have looked like witchcraft. Intuition has been replaced by definition and proof, in this case at a forbidding price in abstractness, and even then only for a class of spaces that have a clear intuitive concept of dimension, because they are ‘locally,’ as  mathematicians say, like continua. It remained unclear if there was a large class of  topological spaces to which a generalized concept of dimension would apply." (Jeremy Gray, "Plato’s Ghost: The modernist transformation of mathematics", 2008)