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);

6. Brouwerian Counter-Examples (used intuitionistically);

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

If we hurry , we can catch up to Turing on the

path he pointed out to us so many years ago.

Herbert Simon, Machine as Mind, 1995