This article is about Intuitionism in mathematics and philosophical logic. For the term in moral epistemology, see Circle of viewpoints pdf intuitionism. This article includes a list of references, but its sources remain unclear because it has insufficient inline citations.
The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. To an intuitionist, the claim that an object with certain properties exists is a claim that an object with those properties can be constructed. Any mathematical object is considered to be a product of a construction of a mind, and therefore, the existence of an object is equivalent to the possibility of its construction. This contrasts with the classical approach, which states that the existence of an entity can be proved by refuting its non-existence.
The interpretation of negation is different in intuitionist logic than in classical logic. Similarly, to assert that A or B holds, to an intuitionist, is to claim that either A or B can be proved. In particular, the law of excluded middle, “A or not A”, is not accepted as a valid principle. Intuitionistic logic substitutes constructability for abstract truth and is associated with a transition from the proof of model theory to abstract truth in modern mathematics. Among the different formulations of intuitionism, there are several different positions on the meaning and reality of infinity. The term potential infinity refers to a mathematical procedure in which there is an unending series of steps.
After each step has been completed, there is always another step to be performed. The term actual infinity refers to a completed mathematical object which contains an infinite number of elements. In Cantor’s formulation of set theory, there are many different infinite sets, some of which are larger than others. Any infinite set that can be placed in one-to-one correspondence with the natural numbers is said to be “countable” or “denumerable”.
Infinite sets larger than this are said to be “uncountable”. Intuitionism was created, in part, as a reaction to Cantor’s set theory. Brouwer rejected the concept of actual infinity, but admitted the idea of potential infinity. Intuitionism’s history can be traced to two controversies in nineteenth century mathematics.