Intuitionism (in Mathematics)

Intuitionism was first formulated by mathematician L.E.J. Brouwer. It is founded on the idea that the mind had created mathematics. Essentially, mathematics is a languageless mental construction by the mind, and the truth of a mathematical statement can only be created from a mental construction that has been proven true. Language may be utilized to communicate mathematical ideas among mathematicians, but such mathematical ideas do not depend their existence on language or communication. There are some consequences that result from intuitionism. For example, the principle of the excluded middle is no longer valid. There exists no proof nor negation of its statement, like the Riemann hypothesis, so it essentially cannot be proven to be true or false. Hence, the principle of the excluded middle is not intuitionistically valid in this way. Rather, intuitionism significantly depends on time, as statements can eventually be proven in time and hence may become intuitionistically valid, even though it has not been so before. Hence, time is the only a priori characteristic in intuitionistic mathematics. In terms of the concept of the continuum, intuitionism significantly deviates from classical mathematics. In contrast to other theories of constructive mathematics, intuitionism is fundamentally a contradiction against classical mathematics. Brouwer identifies two “acts of intuitionism” which establishes the entire theory of intuitionism. The first act is the separation of mathematics from the mathematical language. Rather than having its origin in the mathematical language, mathematics was born in the perception of the movement of time. From the first act, the natural numbers can be generated, but a severe limitation is implied on the principles of reasoning, which includes the invalidity of the principle of the excluded middle. It seems that the first act establishes intuitionistic mathematics as no area for analysis. However, the second act supplements this by re-establishing the existence of the continuum, founded on the existence of freely generated infinite sequences. The second act is the admission of the two ways of creating new mathematical entities: the first, in the form of freely generated infinite sequences from mathematical entities previously obtained; the second, in the form of mathematical species, where it possesses properties supposable of the mathematical entities previously obtained. The freedom of the second act of intuitionism in allowing the construction of infinite sequences is, in fact, what differentiates intuitionism from other constructive theories of mathematics in terms of the computability of mathematical objects and arguments. At the beginning of the 20th century, mathematical society was taken by surprise by countless propositions of paradoxes and highly nonconstructive proofs in mathematics. Philosophers and mathematicians alike were forced to acknowledge that math barely had its epistemological and ontological foundation, a problem for which Brouwer’s work with intuitionism as a philosophy of mathematics aimed to resolve.