How did L.E.J. Brouwer develop intuitionism?

L.E.J. Brouwer developed intuitionism as a response to the early 20th-century quest to secure a firm foundation of mathematics following the discovery of contradictions in naive set theory.31

His development of the school was marked by several key intellectual shifts:

• Mental Construction: Brouwer posited that mathematical objects are not independent entities but mental constructions governed by self-evident laws.23
• Rejection of Logical Principles: In his 1907 doctoral thesis and subsequent 1908 work, he attacked the logical foundations proposed by contemporaries like David Hilbert and Bertrand Russell. He specifically rejected the law of excluded middle—the principle that every statement must be either true or false—arguing that it is invalid when applied to infinite sets where a proof may not be possible.34
• Constructivism: He contended that mathematical assertions are only meaningful if they can be proven through constructive arguments. Consequently, he developed theories of sets, measure, and functions that avoided nonconstructive proofs.314

While Brouwer’s rejection of classical logic was controversial, his work fundamentally challenged how mathematicians define truth and existence in mathematics.24