A quotation by L E J Brouwer
The subject for which I am asking your attention deals with the foundations of mathematics. To understand the development of the opposing theories existing in this field one must first gain a clear understnding of the concept "science"; for it is as a part of science that mathematics originally took its place in human thought.
Intuitionism and formalism (Inaugural address, 1912)
...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.
Beyond the natural numbers, addition, multiplication, and mathematical induction are intuitively clear.
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.
Mathematics is nothing more, nothing less, than the exact part of our thinking.
The construction itself is an art, its application to the world an evil parasite.