Intuitionism

Any view holding that some of our knowledge is got by a direct process not depending on the senses and not open to rational assessment.

The objects of such knowledge may include: moral principles (whether as the basis of duty or as ultimate values); particular moral duties on a particular occasion (sometimes called perceptual intuitionism); space and time and their contents, so far as these are presented to us independently of anything contributed by the understanding (Immanuel Kant (1724-1804)), reality as it is itself, as opposed to reality processed by us for practical purposes (investigated by Henri Bergson (1859-1941)); things known by accumulated but forgotten experience or unconscious inference (‘woman’s intuition’; but this figures less prominently in philosophy); basic truths of logic and the principles of valid inference.

An important special case of the last example cited above is the mathematical intuitionism of Luitzen Egbertus Jan Brouwer (1881-1966) and AREND HEYTING (1898-1980), a form of constructivism which insists that we should assert only what can be proved (by intuitively acceptable steps) and deny only what can be disproved.

It therefore rejects the law of excluded middle and half of the double negation principle.

Also see: anti-realism

Source:
D Pole, Conditions of Rational Inquiry (1961), ch. 1

Truth and proof

The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer’s original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that construction by intuition. The vagueness of the intuitionistic notion of truth often leads to misinterpretations about its meaning. Kleene formally defined intuitionistic truth from a realist position, yet Brouwer would likely reject this formalization as meaningless, given his rejection of the realist/Platonist position. Intuitionistic truth therefore remains somewhat ill-defined. However, because the intuitionistic notion of truth is more restrictive than that of classical mathematics, the intuitionist must reject some assumptions of classical logic to ensure that everything they prove is in fact intuitionistically true. This gives rise to intuitionistic logic.

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. For the intuitionist, this is not valid; the refutation of the non-existence does not mean that it is possible to find a construction for the putative object, as is required in order to assert its existence. As such, intuitionism is a variety of mathematical constructivism; but it is not the only kind.

The interpretation of negation is different in intuitionist logic than in classical logic. In classical logic, the negation of a statement asserts that the statement is false; to an intuitionist, it means the statement is refutable^[1] (i.e., that there is a counterexample). There is thus an asymmetry between a positive and negative statement in intuitionism. If a statement P is provable, then it is certainly impossible to prove that there is no proof of P. But even if it can be shown that no disproof of P is possible, we cannot conclude from this absence that there is a proof of P. Thus P is a stronger statement than not-not-P.

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. For example, if A is some mathematical statement that an intuitionist has not yet proved or disproved, then that intuitionist will not assert the truth of “A or not A“. However, the intuitionist will accept that “A and not A” cannot be true. Thus the connectives “and” and “or” of intuitionistic logic do not satisfy de Morgan’s laws as they do in classical logic.

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. The logical calculus preserves justification, rather than truth, across transformations yielding derived propositions. It has been taken as giving philosophical support to several schools of philosophy, most notably the Anti-realism of Michael Dummett. Thus, contrary to the first impression its name might convey, and as realized in specific approaches and disciplines (e.g. Fuzzy Sets and Systems), intuitionist mathematics is more rigorous than conventionally founded mathematics, where, ironically, the foundational elements which Intuitionism attempts to construct/refute/refound are taken as intuitively given.

Infinity

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. For example, consider the process of counting: 1, 2, 3, …

The term actual infinity refers to a completed mathematical object which contains an infinite number of elements. An example is the set of natural numbers, N = {1, 2, …}.

In Cantor’s formulation of set theory, there are many different infinite sets, some of which are larger than others. For example, the set of all real numbers R is larger than N, because any procedure that you attempt to use to put the natural numbers into one-to-one correspondence with the real numbers will always fail: there will always be an infinite number of real numbers “left over”. 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”.^[2]

Cantor’s set theory led to the axiomatic system of Zermelo–Fraenkel set theory (ZFC), now the most common foundation of modern mathematics. Intuitionism was created, in part, as a reaction to Cantor’s set theory.

Modern constructive set theory includes the axiom of infinity from ZFC (or a revised version of this axiom) and the set N of natural numbers. Most modern constructive mathematicians accept the reality of countably infinite sets (however, see Alexander Esenin-Volpin for a counter-example).

Brouwer rejected the concept of actual infinity, but admitted the idea of potential infinity.

“According to Weyl 1946, ‘Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers … the sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties, including the antinomies – a source of more fundamental nature than Russell’s vicious circle principle indicated. Brouwer opened our eyes and made us see how far classical mathematics, nourished by a belief in the ‘absolute’ that transcends all human possibilities of realization, goes beyond such statements as can claim real meaning and truth founded on evidence.”