. . . . . . . . . . . . . . . . A Materialist Dialectica Pierre-Marie Pédrot 17th September 2015 A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . . . . . 1 / 44 PPS/ π r 2 Pierre-Marie Pédrot (PPS/ π r 2 )
. . . . . . . . . . . . . . . . . Part I. « How Gödel became a computer scientist out of remorse » A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . . . . 2 / 44 Pierre-Marie Pédrot (PPS/ π r 2 )
. Thus Socrates is mortal. . . . . . . . Logic? « The LOGICIST approach » From Axioms , applying valid Rules , derive a Conclusion . Socrates is a man. All men are mortal. All cats are mortal. . Socrates is mortal. Thus Socrates is a cat. A B B C A C As long as rules are correct, you should be safe. Special emphasis on ensuring that they are indeed correct. A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 / 44 Pierre-Marie Pédrot (PPS/ π r 2 )
. Logic? . . . . . . . . . . « The LOGICIST approach » . From Axioms , applying valid Rules , derive a Conclusion . Socrates is a man. All men are mortal. Thus Socrates is mortal. All cats are mortal. Socrates is mortal. Thus Socrates is a cat. As long as rules are correct, you should be safe. Special emphasis on ensuring that they are indeed correct. A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . . . 3 / 44 . . . . . . ⊢ A → B ⊢ B → C ⊢ A → C Pierre-Marie Pédrot (PPS/ π r 2 )
. Logic? . . . . . . . . . . « The LOGICIST approach » . From Axioms , applying valid Rules , derive a Conclusion . Socrates is a man. All men are mortal. Thus Socrates is mortal. All cats are mortal. Socrates is mortal. Thus Socrates is a cat. As long as rules are correct, you should be safe. Special emphasis on ensuring that they are indeed correct. A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . . . 3 / 44 . . . . . . ⊢ A → B ⊢ B → C ⊢ A → C Pierre-Marie Pédrot (PPS/ π r 2 )
. . . . . . . . . . . . . . Logic: a long tradition of failure - 3XX. Aristotle predicts 50 years too late that Socrates had to die. Socrates is a man, all men are mortal, thus Socrates is mortal. 1641. Descartes proves that God and unicorns exist. God is perfect, perfection implies existence, thus God exists. 1901. Russell shows that there is no set of all sets. No one shall expel us from the Paradise that Cantor has created. A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . . . . . . . 4 / 44 Pierre-Marie Pédrot (PPS/ π r 2 )
. . . . . . . . . . . . . . Logic: a long tradition of failure - 3XX. Aristotle predicts 50 years too late that Socrates had to die. Socrates is a man, all men are mortal, thus Socrates is mortal. 1641. Descartes proves that God and unicorns exist. God is perfect, perfection implies existence, thus God exists. 1901. Russell shows that there is no set of all sets. No one shall expel us from the Paradise that Cantor has created. A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . . . . . . . 4 / 44 Pierre-Marie Pédrot (PPS/ π r 2 )
. . . . . . . . . . . . . . Logic: a long tradition of failure - 3XX. Aristotle predicts 50 years too late that Socrates had to die. Socrates is a man, all men are mortal, thus Socrates is mortal. 1641. Descartes proves that God and unicorns exist. God is perfect, perfection implies existence, thus God exists. 1901. Russell shows that there is no set of all sets. No one shall expel us from the Paradise that Cantor has created. A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . . . . . . . 4 / 44 Pierre-Marie Pédrot (PPS/ π r 2 )
1 There is a sentence which is neither provable nor disprovable in 2 The consistency of . . . . . . . . . . . . . . Logic: Fall of Logicism 1931: Gödel’s incompleteness theorem then is neither provable nor disprovable in Quis ipsos custodiet custodes? A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . . . . . . . 5 / 44 Assume a set of rules S which is 1 Expressive enough 2 Consistent 3 Mechanically checkable Pierre-Marie Pédrot (PPS/ π r 2 )
. . . . . . . . . . . . . . . . Logic: Fall of Logicism 1931: Gödel’s incompleteness theorem then Quis ipsos custodiet custodes? A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . 5 / 44 . . . . Assume a set of rules S which is 1 Expressive enough 2 Consistent 3 Mechanically checkable 1 There is a sentence which is neither provable nor disprovable in S 2 The consistency of S is neither provable nor disprovable in S Pierre-Marie Pédrot (PPS/ π r 2 )
. . . . . . . . . . . . . . . . Logic: Fall of Logicism 1931: Gödel’s incompleteness theorem then Quis ipsos custodiet custodes? A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . 5 / 44 . . . . Assume a set of rules S which is 1 Expressive enough 2 Consistent 3 Mechanically checkable 1 There is a sentence which is neither provable nor disprovable in S 2 The consistency of S is neither provable nor disprovable in S Pierre-Marie Pédrot (PPS/ π r 2 )
. You need constructive logic . . . . . . . . . Logic: Rise of Computer Science I « Rather than trusting rules, let us trust experiments. » Suspicious principles . Excluded Middle Reductio ad Absurdum Peirce’s Law From 1931, Gödel tried to atone for his incompleteness theorem Constructivizing non-constructive principles 1 Double-negation translation (1933) 2 Dialectica (’30s, published in 1958) A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 / 44 From a proof of ∃ x . A [ x ] be able to recover a witness t and a proof of A [ t ] . A ∨ ¬ A ¬¬ A → A (( A → B ) → A ) → A Pierre-Marie Pédrot (PPS/ π r 2 )
. Without the aforementioned . . . . . . Logic: Rise of Computer Science II Gödel focussed on intuitionistic logic. The mathematician A constructive logic Advocated by Brouwer for philosophical reasons (1920’s) suspicious axioms . The computer scientist Proofs are dynamic objects rather than static applications of rules (Gentzen ’33, Prawitz ’65) Witness extraction algorithmically recoverable From a proof one can extract a program (Kleene ’49) Curry-Howard isomorphism (1960’s) Intuitionistic proofs are -calculus programs A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 / 44 Pierre-Marie Pédrot (PPS/ π r 2 )
. philosophical reasons (1920’s) . . . . . . . Logic: Rise of Computer Science II Gödel focussed on intuitionistic logic. The mathematician A constructive logic Advocated by Brouwer for Without the aforementioned . suspicious axioms The computer scientist Proofs are dynamic objects rather than static applications of rules (Gentzen ’33, Prawitz ’65) Witness extraction algorithmically recoverable From a proof one can extract a program (Kleene ’49) Curry-Howard isomorphism (1960’s) A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 / 44 Intuitionistic proofs are λ -calculus programs Pierre-Marie Pédrot (PPS/ π r 2 )
. . . . . . . . . . . . . . . . The niftiest programming language of them all! Terms Types A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . 8 / 44 . . . . The beloved λ -calculus t ::= x | λ x . t | t u | . . . A ::= α | A → B | . . . ( x : A ) ∈ Γ Γ , x : A ⊢ B Γ ⊢ t : A → B Γ ⊢ u : A Γ ⊢ t u : B Γ ⊢ λ x . t : A → B Γ ⊢ x : A ( λ x . t ) u → β t [ x := u ] Type derivations are proofs, compatible with β -reduction: If Γ ⊢ t : A and t → β r then Γ ⊢ r : A . Pierre-Marie Pédrot (PPS/ π r 2 )
. prenex form. . . . Which logic for which programs? Standard members of each community will complain. The mathematician “This logic is crappy, it does not feature the following principles I am acquainted with.” Either A or not A hold. Every bounded monotone sequence has a limit. Two sets with same elements are equal. Every formula is equivalent to its … . The computer scientist “This language is crappy, it does not feature the following structures I am ac- quainted with.” printf("Hello world") x <- 42 while true { … } goto #considered_harmful fork() … INTUITIONISTIC LOGIC FUNCTIONAL LANGUAGE A Materialist Dialectica 17/09/2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 / 44 Pierre-Marie Pédrot (PPS/ π r 2 )
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