Editing Intuitionism (section)

== Truth and proof ==

The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In [[Luitzen Egbertus Jan Brouwer|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 [[logical intuition|intuition]]. The vagueness of the intuitionistic notion of truth often leads to misinterpretations about its meaning. [[Stephen Cole Kleene|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 nonexistence. For the intuitionist, this is not valid; the refutation of the nonexistence 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''.{{sfn|Lakatos|2015}} There is thus an asymmetry between a positive and negative statement in intuitionism. If a statement ''P'' is provable, then ''P'' certainly cannot be refutable. But even if it can be shown that ''P'' cannot be refuted, this does not constitute a proof of ''P''. Thus ''P'' is a stronger statement than ''not-not-P''.

Similarly, to assert that ''A'' [[logical disjunction|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 [[negation|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#Truth in mathematics|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 set|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.{{Citation needed|date=June 2024}}