Turing and Wittgenstein
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“The Church-Turing “Thesis” As a Special Corollary of Gödel's
“The Church-Turing “Thesis” as a Special Corollary of Gödel’s Completeness Theorem,” in Computability: Turing, Gödel, Church, and Beyond, B. J. Copeland, C. Posy, and O. Shagrir (eds.), MIT Press (Cambridge), 2013, pp. 77-104. Saul A. Kripke This is the published version of the book chapter indicated above, which can be obtained from the publisher at https://mitpress.mit.edu/books/computability. It is reproduced here by permission of the publisher who holds the copyright. © The MIT Press The Church-Turing “ Thesis ” as a Special Corollary of G ö del ’ s 4 Completeness Theorem 1 Saul A. Kripke Traditionally, many writers, following Kleene (1952) , thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing ’ s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis — what Turing (1936) calls “ argument I ” — has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing himself in what he calls “ argument II. ” The idea is that computation is a special form of math- ematical deduction. Assuming the steps of the deduction can be stated in a first- order language, the Church-Turing thesis follows as a special case of G ö del ’ s completeness theorem (first-order algorithm theorem). I propose this idea as an alternative foundation for the Church-Turing thesis, both for human and machine computation. Clearly the relevant assumptions are justified for computations pres- ently known. -
The Cyber Entscheidungsproblem: Or Why Cyber Can’T Be Secured and What Military Forces Should Do About It
THE CYBER ENTSCHEIDUNGSPROBLEM: OR WHY CYBER CAN’T BE SECURED AND WHAT MILITARY FORCES SHOULD DO ABOUT IT Maj F.J.A. Lauzier JCSP 41 PCEMI 41 Exercise Solo Flight Exercice Solo Flight Disclaimer Avertissement Opinions expressed remain those of the author and Les opinons exprimées n’engagent que leurs auteurs do not represent Department of National Defence or et ne reflètent aucunement des politiques du Canadian Forces policy. This paper may not be used Ministère de la Défense nationale ou des Forces without written permission. canadiennes. Ce papier ne peut être reproduit sans autorisation écrite. © Her Majesty the Queen in Right of Canada, as © Sa Majesté la Reine du Chef du Canada, représentée par represented by the Minister of National Defence, 2015. le ministre de la Défense nationale, 2015. CANADIAN FORCES COLLEGE – COLLÈGE DES FORCES CANADIENNES JCSP 41 – PCEMI 41 2014 – 2015 EXERCISE SOLO FLIGHT – EXERCICE SOLO FLIGHT THE CYBER ENTSCHEIDUNGSPROBLEM: OR WHY CYBER CAN’T BE SECURED AND WHAT MILITARY FORCES SHOULD DO ABOUT IT Maj F.J.A. Lauzier “This paper was written by a student “La présente étude a été rédigée par un attending the Canadian Forces College stagiaire du Collège des Forces in fulfilment of one of the requirements canadiennes pour satisfaire à l'une des of the Course of Studies. The paper is a exigences du cours. L'étude est un scholastic document, and thus contains document qui se rapporte au cours et facts and opinions, which the author contient donc des faits et des opinions alone considered appropriate and que seul l'auteur considère appropriés et correct for the subject. -
Indexed Induction-Recursion
Indexed Induction-Recursion Peter Dybjer a;? aDepartment of Computer Science and Engineering, Chalmers University of Technology, 412 96 G¨oteborg, Sweden Email: [email protected], http://www.cs.chalmers.se/∼peterd/, Tel: +46 31 772 1035, Fax: +46 31 772 3663. Anton Setzer b;1 bDepartment of Computer Science, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK, Email: [email protected], http://www.cs.swan.ac.uk/∼csetzer/, Tel: +44 1792 513368, Fax: +44 1792 295651. Abstract An indexed inductive definition (IID) is a simultaneous inductive definition of an indexed family of sets. An inductive-recursive definition (IRD) is a simultaneous inductive definition of a set and a recursive definition of a function from that set into another type. An indexed inductive-recursive definition (IIRD) is a combination of both. We present a closed theory which allows us to introduce all IIRDs in a natural way without much encoding. By specialising it we also get a closed theory of IID. Our theory of IIRDs includes essentially all definitions of sets which occur in Martin-L¨of type theory. We show in particular that Martin-L¨of's computability predicates for dependent types and Palmgren's higher order universes are special kinds of IIRD and thereby clarify why they are constructively acceptable notions. We give two axiomatisations. The first and more restricted one formalises a prin- ciple for introducing meaningful IIRD by using the data-construct in the original version of the proof assistant Agda for Martin-L¨of type theory. The second one admits a more general form of introduction rule, including the introduction rule for the intensional identity relation, which is not covered by the restricted one. -
Formal Systems: Combinatory Logic and -Calculus
INTRODUCTION APPLICATIVE SYSTEMS USEFUL INFORMATION FORMAL SYSTEMS:COMBINATORY LOGIC AND λ-CALCULUS Andrew R. Plummer Department of Linguistics The Ohio State University 30 Sept., 2009 INTRODUCTION APPLICATIVE SYSTEMS USEFUL INFORMATION OUTLINE 1 INTRODUCTION 2 APPLICATIVE SYSTEMS 3 USEFUL INFORMATION INTRODUCTION APPLICATIVE SYSTEMS USEFUL INFORMATION COMBINATORY LOGIC We present the foundations of Combinatory Logic and the λ-calculus. We mean to precisely demonstrate their similarities and differences. CURRY AND FEYS (KOREAN FACE) The material discussed is drawn from: Combinatory Logic Vol. 1, (1958) Curry and Feys. Lambda-Calculus and Combinators, (2008) Hindley and Seldin. INTRODUCTION APPLICATIVE SYSTEMS USEFUL INFORMATION FORMAL SYSTEMS We begin with some definitions. FORMAL SYSTEMS A formal system is composed of: A set of terms; A set of statements about terms; A set of statements, which are true, called theorems. INTRODUCTION APPLICATIVE SYSTEMS USEFUL INFORMATION FORMAL SYSTEMS TERMS We are given a set of atomic terms, which are unanalyzed primitives. We are also given a set of operations, each of which is a mode for combining a finite sequence of terms to form a new term. Finally, we are given a set of term formation rules detailing how to use the operations to form terms. INTRODUCTION APPLICATIVE SYSTEMS USEFUL INFORMATION FORMAL SYSTEMS STATEMENTS We are given a set of predicates, each of which is a mode for forming a statement from a finite sequence of terms. We are given a set of statement formation rules detailing how to use the predicates to form statements. INTRODUCTION APPLICATIVE SYSTEMS USEFUL INFORMATION FORMAL SYSTEMS THEOREMS We are given a set of axioms, each of which is a statement that is unconditionally true (and thus a theorem). -
Wittgenstein, Turing and Gödel
Wittgenstein, Turing and Gödel Juliet Floyd Boston University Lichtenberg-Kolleg, Georg August Universität Göttingen Japan Philosophy of Science Association Meeting, Tokyo, Japan 12 June 2010 Wittgenstein on Turing (1946) RPP I 1096. Turing's 'Machines'. These machines are humans who calculate. And one might express what he says also in the form of games. And the interesting games would be such as brought one via certain rules to nonsensical instructions. I am thinking of games like the “racing game”. One has received the order "Go on in the same way" when this makes no sense, say because one has got into a circle. For that order makes sense only in certain positions. (Watson.) Talk Outline: Wittgenstein’s remarks on mathematics and logic Turing and Wittgenstein Gödel on Turing compared Wittgenstein on Mathematics and Logic . The most dismissed part of his writings {although not by Felix Mülhölzer – BGM III} . Accounting for Wittgenstein’s obsession with the intuitive (e.g. pictures, models, aspect perception) . No principled finitism in Wittgenstein . Detail the development of Wittgenstein’s remarks against background of the mathematics of his day Machine metaphors in Wittgenstein Proof in logic is a “mechanical” expedient Logical symbolisms/mathematical theories are “calculi” with “proof machinery” Proofs in mathematics (e.g. by induction) exhibit or show algorithms PR, PG, BB: “Can a machine think?” Language (thought) as a mechanism Pianola Reading Machines, the Machine as Symbolizing its own actions, “Is the human body a thinking machine?” is not an empirical question Turing Machines Turing resolved Hilbert’s Entscheidungsproblem (posed in 1928): Find a definite method by which every statement of mathematics expressed formally in an axiomatic system can be determined to be true or false based on the axioms. -
Lambda Calculus and the Decision Problem
Lambda Calculus and the Decision Problem William Gunther November 8, 2010 Abstract In 1928, Alonzo Church formalized the λ-calculus, which was an attempt at providing a foundation for mathematics. At the heart of this system was the idea of function abstraction and application. In this talk, I will outline the early history and motivation for λ-calculus, especially in recursion theory. I will discuss the basic syntax and the primary rule: β-reduction. Then I will discuss how one would represent certain structures (numbers, pairs, boolean values) in this system. This will lead into some theorems about fixed points which will allow us have some fun and (if there is time) to supply a negative answer to Hilbert's Entscheidungsproblem: is there an effective way to give a yes or no answer to any mathematical statement. 1 History In 1928 Alonzo Church began work on his system. There were two goals of this system: to investigate the notion of 'function' and to create a powerful logical system sufficient for the foundation of mathematics. His main logical system was later (1935) found to be inconsistent by his students Stephen Kleene and J.B.Rosser, but the 'pure' system was proven consistent in 1936 with the Church-Rosser Theorem (which more details about will be discussed later). An application of λ-calculus is in the area of computability theory. There are three main notions of com- putability: Hebrand-G¨odel-Kleenerecursive functions, Turing Machines, and λ-definable functions. Church's Thesis (1935) states that λ-definable functions are exactly the functions that can be “effectively computed," in an intuitive way. -
On Synthetic Undecidability in Coq, with an Application to the Entscheidungsproblem
On Synthetic Undecidability in Coq, with an Application to the Entscheidungsproblem Yannick Forster Dominik Kirst Gert Smolka Saarland University Saarland University Saarland University Saarbrücken, Germany Saarbrücken, Germany Saarbrücken, Germany [email protected] [email protected] [email protected] Abstract like decidability, enumerability, and reductions are avail- We formalise the computational undecidability of validity, able without reference to a concrete model of computation satisfiability, and provability of first-order formulas follow- such as Turing machines, general recursive functions, or ing a synthetic approach based on the computation native the λ-calculus. For instance, representing a given decision to Coq’s constructive type theory. Concretely, we consider problem by a predicate p on a type X, a function f : X ! B Tarski and Kripke semantics as well as classical and intu- with 8x: p x $ f x = tt is a decision procedure, a function itionistic natural deduction systems and provide compact д : N ! X with 8x: p x $ ¹9n: д n = xº is an enumer- many-one reductions from the Post correspondence prob- ation, and a function h : X ! Y with 8x: p x $ q ¹h xº lem (PCP). Moreover, developing a basic framework for syn- for a predicate q on a type Y is a many-one reduction from thetic computability theory in Coq, we formalise standard p to q. Working formally with concrete models instead is results concerning decidability, enumerability, and reducibil- cumbersome, given that every defined procedure needs to ity without reference to a concrete model of computation. be shown representable by a concrete entity of the model. -
Warren Goldfarb, Notes on Metamathematics
Notes on Metamathematics Warren Goldfarb W.B. Pearson Professor of Modern Mathematics and Mathematical Logic Department of Philosophy Harvard University DRAFT: January 1, 2018 In Memory of Burton Dreben (1927{1999), whose spirited teaching on G¨odeliantopics provided the original inspiration for these Notes. Contents 1 Axiomatics 1 1.1 Formal languages . 1 1.2 Axioms and rules of inference . 5 1.3 Natural numbers: the successor function . 9 1.4 General notions . 13 1.5 Peano Arithmetic. 15 1.6 Basic laws of arithmetic . 18 2 G¨odel'sProof 23 2.1 G¨odelnumbering . 23 2.2 Primitive recursive functions and relations . 25 2.3 Arithmetization of syntax . 30 2.4 Numeralwise representability . 35 2.5 Proof of incompleteness . 37 2.6 `I am not derivable' . 40 3 Formalized Metamathematics 43 3.1 The Fixed Point Lemma . 43 3.2 G¨odel'sSecond Incompleteness Theorem . 47 3.3 The First Incompleteness Theorem Sharpened . 52 3.4 L¨ob'sTheorem . 55 4 Formalizing Primitive Recursion 59 4.1 ∆0,Σ1, and Π1 formulas . 59 4.2 Σ1-completeness and Σ1-soundness . 61 4.3 Proof of Representability . 63 3 5 Formalized Semantics 69 5.1 Tarski's Theorem . 69 5.2 Defining truth for LPA .......................... 72 5.3 Uses of the truth-definition . 74 5.4 Second-order Arithmetic . 76 5.5 Partial truth predicates . 79 5.6 Truth for other languages . 81 6 Computability 85 6.1 Computability . 85 6.2 Recursive and partial recursive functions . 87 6.3 The Normal Form Theorem and the Halting Problem . 91 6.4 Turing Machines . -
The Entscheidungsproblem and Alan Turing
The Entscheidungsproblem and Alan Turing Author: Laurel Brodkorb Advisor: Dr. Rachel Epstein Georgia College and State University December 18, 2019 1 Abstract Computability Theory is a branch of mathematics that was developed by Alonzo Church, Kurt G¨odel,and Alan Turing during the 1930s. This paper explores their work to formally define what it means for something to be computable. Most importantly, this paper gives an in-depth look at Turing's 1936 paper, \On Computable Numbers, with an Application to the Entscheidungsproblem." It further explores Turing's life and impact. 2 Historic Background Before 1930, not much was defined about computability theory because it was an unexplored field (Soare 4). From the late 1930s to the early 1940s, mathematicians worked to develop formal definitions of computable functions and sets, and applying those definitions to solve logic problems, but these problems were not new. It was not until the 1800s that mathematicians began to set an axiomatic system of math. This created problems like how to define computability. In the 1800s, Cantor defined what we call today \naive set theory" which was inconsistent. During the height of his mathematical period, Hilbert defended Cantor but suggested an axiomatic system be established, and characterized the Entscheidungsproblem as the \fundamental problem of mathemat- ical logic" (Soare 227). The Entscheidungsproblem was proposed by David Hilbert and Wilhelm Ackerman in 1928. The Entscheidungsproblem, or Decision Problem, states that given all the axioms of math, there is an algorithm that can tell if a proposition is provable. During the 1930s, Alonzo Church, Stephen Kleene, Kurt G¨odel,and Alan Turing worked to formalize how we compute anything from real numbers to sets of functions to solve the Entscheidungsproblem. -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Interactions between computability theory and set theory Permalink https://escholarship.org/uc/item/5hn678b1 Author Schweber, Noah Publication Date 2016 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Interactions between computability theory and set theory by Noah Schweber A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Antonio Montalban, Chair Professor Leo Harrington Professor Wesley Holliday Spring 2016 Interactions between computability theory and set theory Copyright 2016 by Noah Schweber i To Emma ii Contents Contents ii 1 Introduction1 1.1 Higher reverse mathematics ........................... 1 1.2 Computability in generic extensions....................... 3 1.3 Further results .................................. 5 2 Higher reverse mathematics, 1/28 2.1 Introduction.................................... 8 2.2 Reverse mathematics beyond type 1....................... 11 2.3 Separating clopen and open determinacy.................... 25 2.4 Conclusion..................................... 38 3 Higher reverse mathematics, 2/2 41 3 3.1 The strength of RCA0 ............................... 41 3.2 Choice principles ................................. 50 4 Computable structures in generic extensions 54 4.1 Introduction................................... -
15-819 Homotopy Type Theory Lecture Notes
15-819 Homotopy Type Theory Lecture Notes Nathan Fulton October 9 and 11, 2013 1 Contents These notes summarize and extend two lectures from Bob Harper’s Homotopy Type Theory course. The cumulative hierarchy of type universes, Extensional Type theory, the ∞-groupoid structure of types and iterated identity types are presented. 2 Motivation and Overview Recall from previous lectures the definitions of functionality and transport. Function- ality states that functions preserve identity; that is, domain elements equal in their type map to equal elements in the codomain. Transportation states the same for 0 0 type families. Traditionally, this means that if a =A a , then B[a] true iff B[a ] true. In proof-relevant mathematics, this logical equivalence is generalized to a statement 0 0 about identity in the family: if a =A a , then B[a] =B B[a ]. Transportation can be thought of in terms of functional extensionality. Un- fortunately, extensionality fails in ITT. One way to recover extensionality, which comports with traditional mathematics, is to reduce all identity to reflexivity. This approach, called Extensional Type theory (ETT), provides a natural setting for set-level mathematics. The HoTT perspective on ETT is that the path structure of types need not be limited to that of strict sets. The richer path structure of an ∞-groupoid is induced by the induction principle for identity types. Finding a type-theoretic description of this behavior (that is, introduction, elimination and computation rules which comport with Gentzen’s Inversion Principle) is an open problem. 1 Homotopy Type Theory 3 The Cumulative Hierarchy of Universes In previous formulations of ITT, we used the judgement A type when forming types. -
Church–Turing Thesis CSCI 3130 Formal Languages and Automata Theory
Church–Turing Thesis CSCI 3130 Formal Languages and Automata Theory Siu On CHAN Chinese University of Hong Kong Fall 2015 1/20 What is a computer? program computer output input A computer is a machine that manipulates data according to a list of instructions 2/20 What is a computer? def hello(name): print(”Hello ”, name) Hello world world Google Now No, rain is not expected today Will it rain today? at Ma Liu Shui Machine Learning algorithm Environment self driving 3/20 Turing machines head state control a b a … | {z }| {z } input blanks Can both read from and write to the tape Head can move both left and right Unlimited tape space Has two special states accept and reject 4/20 Example ∗ L1 = fw#w j w 2 fa; bg g Strategy: Read and remember the first symbol abbaa#abbaa Cross it off xbbaa#abbaa Read the first symbol past # xbbaa#abbaa If they don’t match, reject If they do, cross it off xbbaa#xbbaa 5/20 Example ∗ L1 = fw#w j w 2 fa; bg g Strategy: Look for and remember the first uncrossed symbol xbbaa#xbbaa Cross it off xxbaa#xbbaa Read the first symbol past # xxbaa#xbbaa If they do, cross it off, else reject xxbaa#xxbaa At the end, there should be only x’s xxxxx#xxxxx if so, accept; otherwise reject 6/20 How Turing machines operate current state: q1 a/bL q1 q2 a b a … Replace a with b, and move head left new state: q2 a/bL q1 q2 a b b … 7/20 Computing devices: from practice to theory 8/20 Brief history of computing devices Antikythera Mechanism (~100BC) Its reproduction Abacus (Sumer 2700-2300BC, Babbage Difference engine China 1200) (1840s)