Constructivism (philosophy of mathematics)
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In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In standard 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. This proof by contradiction is not constructively valid. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation.
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Mathematicians who have made major contributions to constructivism
- Leopold Kronecker (old constructivism, semi-intuitionism)
- L. E. J. Brouwer (forefather of intuitionism)
- A. A. Markov (forefather of Russian school of constructivism)
- Arend Heyting (formalized intuitionistic logic and theories)
- Per Martin-Löf (founder of constructive type theories)
- Errett Bishop (promoted a version of constructivism claimed to be consistent with classical mathematics)
- Paul Lorenzen (developed constructive analysis)
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Branches
- Constructive logic
- Constructive type theory
- Constructive analysis
- Constructive non-standard analysis
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See also
- Computability theory
- Constructive proof
- Finitism
- Game semantics
- Intuitionism
- Intuitionistic type theory
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