Turing’s Economics A Birth Centennial Homage K. Vela Velupillai ASSRU/Department of Economics University of Trento June 2012 ”Mathematical proofs use logical reasoning to get from assertions already accepted as true to statements called theorems .. . The work of logicians … showed how, in principle, the individual steps in such ’proofs’ could be replaced by the mechanical manipulation of symbols. This .. gave rise to the problem of finding a mechanical process, an algorithm, for deciding in advance whether from some given statements accepted as true, another desired statement could be obtained by such a sequence of steps. ..Hilbert [calling this the entscheidungsproblem declared it to be] the main problem of mathematical logic. …. But to prove that there is no algorithm to carry out some task, more was needed than the words ’explicit’ and ’mechanical’ . … Turing’s paper ’On Computable Numbers with Application to the Entscheidungsproblem’ …did all of that.” Martin Davis
20
Embed
Turing’s Economics: A Birth Centennial Homage · Turing’s Economics A Birth Centennial Homage ... results of mathematical logic may be regarded as ... by Stephen Cole Kleene ...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Turing’s Economics A Birth Centennial Homage
K. Vela Velupillai
ASSRU/Department of Economics
University of Trento
June 2012
”Mathematical proofs use logical reasoning to get from assertions already accepted as true to statements called theorems .. . The work of logicians … showed how, in principle, the individual steps in such ’proofs’ could be replaced by the mechanical manipulation of symbols. This .. gave rise to the problem of finding a mechanical process, an algorithm, for deciding in advance whether from some given statements accepted as true, another desired statement could be obtained by such a sequence of steps. ..Hilbert [calling this the entscheidungsproblem declared it to be] the main problem of mathematical logic. …. But to prove that there is no algorithm to carry out some task, more was needed than the words ’explicit’ and ’mechanical’. … Turing’s paper ’On Computable Numbers with Application to the Entscheidungsproblem’ …did all of that.” Martin Davis
Martin Davis at the Turing Centennial Conference on The Incomputable, organised by the Newton Mathematics Institute, Cambridge University at Chicheley Hall, 12-15,
June, 2012 (Photo byDebbie Ericsson-Zenith)
Martin Davis’s Computability and Undecidability (1958) remains the classic text for introducing
the mathematics of Classical Recursion Theory to practitioners of Turing’s Economics
From: The Mechanical Mind in History edited by Owen Holland & Michael Wheeler, Chapter 6, p. 124 (The Ratio Club: A Hub of British Cybernetics by Philip Husbands & Owen Holland
My personal Turing Number
(pace Erdös Number) may
well be 2.5! One of my first
papers in what I now wish to
call Turing’s Economics was
co-authored with John
Westcott (& Berc Rustem)
and published in
Automatica, in 1978. In
1980, I succeeded my
mentor, Richard Goodwin, as
the Director of Studies in
Economics at Peterhouse.
ASSRU intellectual inspirations are based on the works of Turing, Simon, Brouwer, Keynes, Sraffa & Goodwin
Turing’s Economics at the
…..‘Reason’ Unsupported by Common Sense…
The results which have been described in this article are mainly of a negative character, setting certain bounds to what we can hope to achieve purely by reasoning. These, and some other results of mathematical logic may be regarded as going some way towards a demonstration, within mathematics itself, of the inadequacy of ‘reason’ unsupported by common sense.
Alan Turing: Solvable and Unsolvable Problems (1954), p.23; emphasis added
• Orthodox Economic Theory → Theorising to show the adequacy of ‘reason’
unsupported by common sense.
• Computable Economics → Theorising to show “the inadequacy of ‘reason’
unsupported by common sense.”
Codifying the Failure of ‘Reason’ Unsupported by Common Sense in Turing’s Economics
It is undecidable whether there is an effective procedure to generate preference orderings.
Given a class of choice functions that do generate
preference orderings (pick out the set of maximal
alternatives) for any agent, there is no effective procedure to decide whether or not any arbitrary
choice function is a member of the given class.
There is no effective procedure to decide whether
given class of decision rules are “steady states of
(some) adaptive process”.
Fourteen original Results in Turing’s Economics by ASSRU Members
I. There is no effective procedure to generate preference orderings.
II. The Arrow-Debreu equilibrium is uncomputable (and its existence is proved nonconstructively).
III. The Uzawa Equivalence Theorem is uncomputable and nonconstructive.
IV. Computable General Equilibria are neither computable nor constructive.
V. The Two Fundamental Theorems of Welfare Economics are Uncomputable and Nonconstructive, respectively.
VI. The Negishi method is proved nonconstructively and the implied procedure in the method is uncomputable.
VII. Rational expectations equilibria are uncomputable and are generated by uncomputable and nonconstructive
processes.
VIII. Policy rules in macroeconomic models are noneffective.
IX. Recursive Competitive Equilibria (RCE), underpinning the Real Business Cycle (RBC) model and, hence, the
Dynamic Stochastic General Equilibrium (DSGE) benchmark model of Macroeconomics, are uncomputable.
X. Dynamical systems underpinning growth theories are incapable of computation universality.
XI. There are games in which the player who in theory can always win cannot do so in practice because it is
impossible to supply him with effective instructions regarding how he/she should play in order to win. (Michael
Rabin’s Result)
XII. The theoretical benchmarks of Algorithmic Game Theory are uncomputable and non-constructive.
XIII. Nash equilibria of (even) finite games are constructively indeterminate. (Partly – and also previously - also
derived by F. Doria)
XIV. Boundedly rational agents, satisficing formalised within the framework of (metamathematical) decision
problems are capable of effective procedures of rational choice.
The Classics and the Modern Classics for students of Turing’s Economics
• Recursive Functions by Rosza Péter
([1951], 1967)
• Intorduction to Metamathematics by Stephen Cole Kleene (1952)
• Computability and Unsolvability by Martin Davis (1958)
• Theory of Algorithms by A.A. Markov
(1961)
• Computable Analysis by S. Mazur
(1963)
• Theory of Recursive Functions and Effective Computability by Hartley
Rogers, Jr. (1967)
• Foundations of Constructive Analysis by Errett Bishop (1967)
• Computation: Finite and Infinite Machines by Marvin Minsky (1967)
• Introduction to Automata Theory, Languages and
Computation by John Hopcroft & Jeffrey
Ullman (1979)
• Computability, Complexity, and Languages:
Fundamentals of Theoretical Computer Science by
Martin Davis, Ron Sigal and Elaine J.
Weyuker ([1983], 1994)
• Recursively Enumerable Sets and Degrees: A Study
of Computable Functions and Computably
Generated Sets by Robert I. Soare (1987)
• Hilbert’s Tenth Problem by Yuri Matiyasevich
(1993)
• Dynamical Systems and Numerical Analysis by
A.M. Stuart & A. R. Humphries (1996)
• An Introduction to Kolmogorov Complexity and Its
Applications by Ming Li & Paul Vitanyi (1997)
• Complexity and Real Computation by L. Blum, F.
Cucker, M. Shub and S. Smale (1998)
Those interested in pursuing research in Turing’s Economics are recommended to familiarise
themselves with some of the following classics – as well as many of the pioneering articles in The
Undecidable ed. By Martin Davis (1965) & From Frege to Gödel ed. by Jean van Heijenoort (1967)
Some Mathematical Methods used in Proving Theorems in Turing’s Economics
1. The Diagonal Method (used constructively);
2. The Halting Problem (used recursion theoretically);
3. Rice’s Theorem (used recursion theoretically);
4. The Unsolvability of Hilbert’s Tenth Problem (used Recursion theoretically)
5. The Least Fixed Point Theorem (used recursion theoretically);
7. The Incompressible Method (used in the sense of Kolmogorov Complexity);
8. The Berry Paradox (used in the sense of Algorithmic Information Theory)
9. Harrop’s Theorem (in constructive & recursion theoretic terms);
10. The Constructive Hahn-Banach Theorem;
11. The Constructive Jordan Curve Theorem;
12. Non-Reliance, as much as possible, on tertium non datur;
13. Eschewing the Axiom of Choice, Embracing the Axiom of Determinacy
Incomputability, Undecidability and Unsolvability in Turing’sEconomics
Noncomputability, Unpredictability,
Undecidability and Unsolvability in
Economic & Finance Theories
by the
ASSRU Team:
Ying-Fang Kao, V. Ragupathy, K. Vela Velupillai &
Stefano Zambelli
Forthcoming in:
Complexity, 2012
The Paradigmatic Problem Solving Example
The Trefoil Knot
These puzzles where one
is asked to separate rigid
bodies are in a way like
the ‘puzzle’ of trying to
undo a tangle, or more
generally of trying to
turn one into
another without cutting
the string. The difference
is that one is allowed to
bend the string, but not
the wire forming the
rigid bodies. In either
case, if one wants to treat
the problem seriously
and systematically one
has to replace the
physical puzzle by a
mathematical equivalent. Turing (1954), p.11
Why the Trefoil Knot?
Alan Turing’s (& Warren McCulloch’s) Influence on Visions for Herbert Simon’s Classical Behavioural Economics
On Modelling Mind, Intelligence, Thought, underpinned by Computability, via a study of Complex
Human Problem Solving in the context of ‘Finite’ Tasks & Combinatorial Games
Turing
On Computable Numbers with an Application to the Entscheidungsproblem (1936/7) Computing Machinery and Intelligence (1950) Solvable and Unsolvable Problems (1954) McCulloch
What is a Number, that a Man May Know it, and a Man, that He May Know a Number? (1961) A Logical Calculus of the Ideas Immanent in Nervous Activity (with Walter H. Pitts) (1943)
Human Problem Solving by Newell & Simon
From: Alan M. Turing by Sara Turing (facing p. 50)
Why GO?
He showed [Joan Clarke ] a
book on GO, and lay on the
floor in his room at the Crown
Inn demonstrating some of the
situations in the game.
“[Turing and Clarke] shared
many interests, both were keen
chess players and, as Clarke had
studied Botany at school, she
could become involved with
Turing’s life long enthusiasm
of the growth and form of
plant life.” (From the Obituary of
Joan Clarke by Lynsey Ann Lord)
Nonlinear & Coupled, Constructive, Dynamics as a Foundation for Emergent Complex Evolution in Turing’s Economics
• Turing on Morphogenesis • The Chemical Basis of Morphogenesis, Phil. Tran. Royal Society, 1952
• Brouwer on Choice Sequences • Brouwer’s Cambridge Lectures on Intuitionism, CUP, 1981
• Goodwin on Nonlinear & Coupled Dynamics • Dynamical Coupling with Especial Reference to Markets Having Production Lags, Econometrica, 1947 • The Nonlinear Accelerator and the Persisitence of Business Cycles, Econometrica, 1951
• Sraffa on Intersectoral Monetary Production Economics • Production of Commodities by Means of Commodities, CUP, 1960
• Keynes on Monetary Production Economics • A Monetary Theory of Production, Spiethoff Festschrift, 1933
• The Fall & Rise of British Emergentism: Mill to Sperry • Four Traditions of Emergence: Morphogenesis, Ulam-von Neumann Cellular Automatas, the Fermi-Pasta-Ulam Problem
and British Emergentism by Vela Velupillai, in: Alan Turing – His Work and Impact, edited by B. Cooper & J. van Leeuwen, Elsevier, 2012.
• Analogies with the Fermi-Pasta-Ulam Paradoxes • Studies in Nonlinear Problems: I, by E.Fermi, J. Pasta & S. Ulam, Los Alamos Preprint, LA-1940, 7 November, 1955
Varieties of Theories of Complexity in Turing’s Economics
Randomness and Algorithmic Probabilities (From Richard von
Mises ….)
Finance Theory without Probabilities ( From Ville …)
Learning as Induction (From Putnam ….)
Complexity of Human Problem Solving (From Simon …)
Computational, Algorithmic, Dynamic and Diophantine Complexities (From Kolmogorov and Matiyasevich ….)
The Complexity of Combinatorial, Constructive and Arithmetical Games (From Steinhaus, Euwe and Gödel …)
The Computational Complexity of Approximation Processes (From Smale….)
Emergent Complexity (from Mill, Lloyd Morgan …)
Turing’s Economics: Current Frontiers
• Diophantine Decision Problems instead of Optimizations • Hilbert’s Tenth Problem as the Paradigmatic Exercise in
showing Problem Unsolvability in Turing’s Economics • Undecidabilities, Unsolvabilities and Uncomputabilities instead
of Probabilistic Indeterminacies • Towards Natural Number – Goodstein – Dynamics with
Undecidable Attractors, underpinned by Finite Axiomatics instead of Real Number Dynamical Systems
• Constructive Combinatorial and Arithmetic Games with Degrees of Solvability instead of Non-Constructive, Non-Algorithmic Games
• Proof as Constructions and the Computational Complexity of Proofs
• The Challenges of the Church-Turing Thesis: Is it necessary? Is it desirable? How do constructivists do without it?
• Does Relative Computability Solve the Halting Problem?
Andrew Hodges Characterizing the Personality of Alan Turing
It was typical for him … to seek to outdo Bell Telephone Laboratories with his single brain, and to build a better system with his own hands. .. Turing’s wording [for his computer design of 1945] indicates authoritative judgement, and not the submitting of a proposal for the approval of superiors. .. It might be more true to say that Turing had resisted this Cambridge classification [between applied and pure mathematics] from the outset. He attacked every kind of problem – from arguing with Wittgenstein, to the characteristics of electronic components, to the petals of a daisy.*
This is, in a nutshell, the Lesson of Turing’s Economics
*The Logical and the Physical by Andrew Hodges, in: New Computational Paradigms – Changing Conceptions of What is Computable, edited by S. Barry Cooper, Benedikt Löwe & Andrea Sorbi, p. 4