Concept
Intuitionistic type theory
Overview
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory (MLTT)) is a type theory and an alternative foundation of mathematics.
Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and philosopher, who first published it in 1972. There are multiple versions of the type theory: Martin-Löf proposed both intensional and extensional variants of the theory and early impredicative versions, shown to be inconsistent by Girard's paradox, gave way to predicative versions. However, all versions keep the core design of constructive logic using dependent types.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
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Writers on this subject have begun to show a singular unanimity as to what the laws of nature and elements of fact mean, when formulated by mathematicians, physicists and chemists. When the first mathematical, logical and natural uniformities, the first LAWS, were discovered, men were so carried away by the clearness, beauty and simplification that resulted, that they believed themselves to have deciphered…
William James, Pragmatism
States what Intuitionistic type theory is.
To say that the notion is subjective and subjective only, is so far quite correct: for the notion certainly is subjectivity itself. Not less subjective than the notion are also the judgment and syllogism: and these forms, together with the so-called Laws of Thought (the Laws of Identity, Difference, and Sufficient Ground), make up the contents of what is called the 'Elements' in the common logic. But we may go a…
Georg Wilhelm Friedrich Hegel, The Logic of Hegel
States what Intuitionistic type theory is.
Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
States what Intuitionistic type theory is.
Philosophers who discussed Intuitionistic type theory
Related ideas
- First-order logic
- Proof assistant
- Tuple
- Philosophy
- Negation
- Foundations of mathematics
- Law of excluded middle
- Proof theory
- Mathematician
- Per Martin-Löf
- Mathematical logic
- Rational number
- Reflexive relation
- Real number
- Constructivism (philosophy of mathematics)
- Natural number
- Ontology
- Principia Mathematica
- Curry–Howard correspondence
- Set theory
- Type theory
- Bertrand Russell
- Sweden
- Rocq
Place in the tradition
Part of Higher-order logic.
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