Intuitionism

Intuitionism (or Neo-Intuitionism) is the approach in Logic and Philosophy of Mathematics which takes mathematics to be the constructive mental activity of humans (as opposed to the Mathematical Realism view that mathematical truths are objective, and that mathematical entities exist independently of the human mind). Thus, it holds that logic and mathematics do not consist of analytic activities wherein deep properties of existence are revealed and applied, but rather they are the application of internally consistent methods to realize more complex mental constructs.

According to Intuitionism, the truth of a statement is equivalent to the mathematician being able to intuit the statement, and not necessarily to its provability. It requires the application of intuitionistic logic (or constructivist logic), which preserves justification, rather than truth, for derived propositions. Any mathematical object is considered to be the product of a construction of a mind, so that if it can be constructed then it exists. Intuitionism is therefore a variety of Mathematical Constructivism in that it asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists.

The intuitionist interpretation of negation also differs from classical logic. In classical logic, the negation of a statement asserts that the statement is false; under Intuitionism, negation means that the statement is refutable (i.e. that there is a proof that there is no proof of it).

Intuitionism is contrasted with Pre-Intuitionism and Mathematical Realism, which take the view that mathematical theorems have an existence and exactness independent of language and logic, and that the existence of an entity can be proved by refuting its non-existence.

Intuitionism's history can perhaps be traced to the 19th Century discussions between the German mathematicians Georg Cantor (1845 - 1918) and his teacher Leopold Kronecker (1823 - 1891), and the later discussions between Gottlob Frege and Bertrand Russell. However, it was first given a detailed exposition in the early 20th Century by the Dutch mathematician L. E. J. Brouwer (1881 -1966).

The French philosopher Henri Bergson (1859 - 1941) developed his own version of Intuitionism in his "An Introduction to Metaphysics". He held that there are two distinct ways in which an object can be known: absolutely and relatively. Knowledge can be gained relatively through analysis, and absolutely through intuition. He defined intuition as a simple, indivisible experience of sympathy through which one is moved into the inner being of an object to grasp what is unique and ineffable within it.

Later, the American Stephen Kleene (1909 - 1994) brought forth a more rational consideration of Intuitionism in his "Introduction to Meta-Mathematics" of 1952.