Constructivism (philosophy of mathematics)

Definition

In the philosophy of mathematics, constructivism asserts that it is necessary to find a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation.

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Ambiguity Andrey Markov (Soviet mathematician)Anne Sjerp Troelstra Antithesis Arend Heyting Argument Axiom Axiom of choice Axiom of dependent choice BHK interpretation Belief Bias Bijection CZF Cantor's diagonal argument Cantor–Bernstein–Schroeder theorem Cardinal number Categorical logic Cauchy sequence Classical logic Computability theory Computable function Computable number Computer science Construction of the real numbers Constructive analysis Constructive logic Constructive non-standard analysis Constructive proof Constructive set theory Constructive type theory Constructivism (disambiguation)Contradiction Countable Credibility Critical thinking David Hilbert Deductive reasoning Diaconescu-Goodman-Myhill theorem Dialectic Dialetheism Dirk van Dalen Effective topos Equivalence class Errett Bishop Euler's number Evidence Existence Existence property Existential quantification Existential quantifier Explanation Explanatory power Fact Fallacy Fictionalism Finite set Finitism Formalism (mathematics)Free choice sequences Free variables Function (mathematics)Game semantics Goldbach's conjecture Grundlagen der Mathematik Harold Edwards (mathematician)Heyting arithmetic History of topos theory Homomorphism Hopf algebra Hypothesis Infinity Informal logic Inhabited set Inquiry Internal language Internet Encyclopedia of Philosophy Intuitionism Intuitionistic logic Intuitionistic type theory L. E. J. Brouwer Law of bivalence Law of non-contradiction Law of the excluded middle Lebesgue measure Lee Smolin Leopold Kronecker List of fallacies Logical atomism Logical system Logicism Luitzen Egbertus Jan Brouwer Martin Hyland Measure theory Modulus of convergence Nikolai Aleksandrovich Shanin Nominalism Occam's razor Onto Opinion Partial function Paul Bernays Paul Lorenzen Per Martin-Löf Philosophical analysis Philosophical logic Philosophical razor Philosophical realism Philosophy of mathematics Platonic realism Pragmatism Premise Prime number Proof by contradiction Propaganda Prudence Quantifiers (logic)Rational number Real analysis Realizability Reason Recursively enumerable Relevance Rhetoric Rigour Set (mathematics)Socratic method Solomon Feferman Stanford Encyclopedia of Philosophy Theory Theory of justification Three Roads to Quantum Gravity Topos Topos theory Trivial (mathematics)Type theory Typed lambda calculus Unity of opposites Vagueness Witness (mathematics)ZF set theory

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