Wittgenstein's work remains, undeniably, now, that off one of those few philosophers who will be read by all future generations.

Remarks on the Foundations of Mathematics. By L. Wittgenstein. Edited by G. H. Von Wright, R. Rhees and G. E. M. Anscombe. Translated by G. E. M. Anscombe. Pp. 400. 37s. 6d. 1956. (Basil Blackwell, Oxford)

Part 1 Polymorphic sets: the semantics of the judgement forms general rules enumeration sets Cartesian product of a family of sets equality sets natural numbers lists cartesian product of two sets disjoint union of two sets disjoint union of a family of sets the set of small sets (the first universe) well-orderings general trees. Part 2 Subsets: subsets in the basic set theory the subset theory. Part 3 Monomorphic sets: types defining sets in terms of types. Part 4 Examples: some small examples program derivation specification of abstract data types.

Programming in Martin-Lo¨f's type theory: an introduction

data type, 179 abstraction, 14 Abstraction rule, 143 absurdity, 43 append, 68 application, 13 Application rule, 143 apply, 48, 148 arity of an expression, 18 Assumption rule, 123, 142 AUTOMATH, 8 axiom of choice, 115 Bishop, 6 Bool, 44 Bool rules, 44–45 Boolean, 44–45 Calculus of Constructions, 8 canonical expression, 26 cartesian product of a family of sets, 47–55, 148 case{i1,,in}, 41 case, 151 Church, 17 combination, 15 combined expression, 17 conjunction, 74 cons, 67, 152 Constable, 6, 167 constructive mathematics, 6 context, 29, 139 Coquand, 3, 7, 8 Curry, 10 Curry-Howard interpretation, 6 Cut rule for elements in sets, 124 Cut rule for equal elements in sets, 124 Cut rule for equal sets, 123 Cut rule for propositions, 123 Cut rule for true propositions, 124 de Bruijn, 8, 10 definiendum, 19 definiens, 19 definitional equality of expressions, 15 Dijkstra, 167, 171 disjoint union of a family of sets, 81– 82, 149 disjoint union of two sets, 79–80, 150 Dummet, 6 El formation rule, 139 elimination rule, 35 empty set, 43 enumeration sets, 41–42, 151 Eq, 57 Eq rules, 60–61 equality as a proposition, 57, 117 between canonical elements, 27 between elements in a set, 29, 31, 37, 119, 121 between expressions, 15 between sets, 28, 30, 37, 118, 121 extensional, 57 identity between objects in a type, 138 identity between types, 138 intensional, 57 equality rule, 35 equality sets, 57–62, 150 examples: Bool has only the elements true and false, 161 A partitioning problem, 171 Decidable predicates, 162 Division by 2, 155 Even or odd, 159 Module for sets with a computable equality, 182 Stack of A elements, 182 Stack of natural numbers, 179 Stronger elimination rules, 163 existential quantifier, 81, 82 expressions, 13 arity, 18 canonical —, 26 combined —, 17 saturated —, 17 single —, 17 unsaturated —, 17 extensional equality, 57 wfalse, 44 Fermat’s last theorem, 4 formation rule, 35 Frege, 17

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Programming in Martin-Löf's Type Theory

The meaning of a sentence determines how the truth of the proposition expressed by the sentence may be proved and hence one would expect proof theory to be influenced by meaning-theoretical considerations. In the present chapter we consider a proposal that also reverses the above prior-ities and determines meaning in terms of proof. The proposal originates in the criticism that Michael Dummett has voiced against a realist, truth-theoretical, conception of meaning and has been developed largely by him and Dag Prawitz, whose normalisation procedures in technical proof theory constitute the main technical basis of the proposal.

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Proof Theory and Meaning

Part I Reason, Science, and Mathematics: 1 Science as a triumph of the human spirit and science in crisis: Husserl and the Fortunes of Reason 2 Mathematics and transcendental phenomenology Part II Kurt Godel, Phenomenology and the Philosophy of Mathematics: 3 Kurt Godel and phenomenology 4 Godel's philosophical remarks on mathematics and logic 5 Godel's path from the incompleteness theorems (1931) to Phenomenology (1961) 6 Godel and the intuition of concepts 7 Godel and Quine on meaning and mathematics 8 Maddy on realism in mathematics 9 Penrose and the view that minds are not machines Part III Constructivism, Fulfilled Intentions, and Origins: 10 Intuitionism, meaning theory and cognition 11 The philosophical background of Weyl's mathematical constructivism 12 What is a proof? 13 Phenomenology and mathematical knowledge 14 Logicism, impredicativity, formalism 15 The philosophy of arithmetic: Frege and Husserl

https://assets.cambridge.org/97805218/37828/frontmatter/9780521837828_frontmatter.pdf

Phenomenology, logic and the philosophy of mathematics

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Göran Sundholm

Papers

Constructions, proofs and the meaning of logical constants