Chapter 2
Towards an Algorithmic Revolution in Economic Theory
Book Editor(s):Stefano Zambelli, Donald A. R. George,
First published: 06 January 2012
Summary
This chapter contains sections titled:
-
A Foundational Preamble
-
Machines, Mechanisms, Computation and Algorithms
-
The Legacy of Hilbert's Dogma in Mathematical Economics
-
Reconstructing Economic Theory in the Algorithmic Mode29
-
Acknowledgments
-
Notes
-
References
References
- Aberth, Oliver (1980) Computable Analysis. New York: McGraw-Hill Book Company.
- Aberth, Oliver (2001) Computable Calculus. San Diego, CA: Academic Press.
- Beinhocker, Eric. D. (2006) The Origin of Wealth: The radical Remaking of Economics and What It Means for Business and Society. Boston, MA: Harvard Business School Press.
- Bell, John L. (1998) A Primer of Infinitesimal Analysis. Cambridge, UK: Cambridge University Press.
- Bishop, Errett. A. (1967) Foundations of Constructive Analysis. New York: McGraw-Hill Book Company.
- Bishop, Errett and Douglas Bridges (1985) Constructive Analysis. Heidelberg, Germany: Springer-Verlag.
-
Blum, Lenore, Felipe Cucker, Michael Shub and Steve Smale (1998) Complexity and Real Computation. New York: Springer Verlag.
10.1007/978-1-4612-0701-6 Google Scholar
- Bridges, Douglas and Fred, Richman (1987) Varieties of Constructive Mathematics. Cambridge, UK: Cambridge University Press.
- Brouwer, Lutizen E.J. (1907/1975) Over de grondslagen der wiskunde [On the foundations of mathematics], Academic Thesis. In Arend Heyting (ed.), L.E.J. Brouwer Collected Works: Vol. 1 – Philosophy and Foundations of Mathematics (pp. 11–104). Amsterdam, Netherlands: North-Holland; New York: American Elsevier.
- Brouwer, Lutizen E.J. (1908a/1975) Over de grondslagen der wiskunde [On the foundations of mathematics], Academic Thesis. In Arend Heyting (ed.), L.E.J. Brouwer Collected Works: Vol. 1 – Philosophy and Foundations of Mathematics (pp. 105–106). Amsterdam, Netherlands: North-Holland; New York: American Elsevier.
- Brouwer, Lutizen E.J. (1908b/1975) De onbetrouwbaarheid der logische principes [The unreliability of the logical principles]. In Arend Heyting (ed.), L.E.J. Brouwer Collected Works: Vol. 1 – Philosophy and Foundations of Mathematics (pp. 197–111). Amsterdam, Netherlands: North-Holland; New York: American Elsevier.
- Brouwer, Luitzen E.J. (1952) An intuitionist correction of the fixed-point theorem on the sphere. Proceedings of the Royal Society, London, UK, Vol. 213, 1–2, 5 June 1952.
- Clower, Robert. W. and Peter W. Howitt (1978) The transaction theory of the demand for money: a reconsideration. Journal of Political Economy 86(3): 449–466.
- Cohen, Daniel I.A. (1991) The superfluous paradigm. In J.H. Johnson and M.J. Loomes (eds), The Mathematical Revolution Inspired by Computing (pp. 323–329). Oxford: Oxford University Press.
- Davis, Martin (1978) What is a computation? In Lynn Arthur Steen (ed.), Mathematics Today – Twelve Informal Essays (pp. 242–267). New York: Springer-Verlag.
- Debreu, Gerard (1959) Theory of Value: An Axiomatic Analysis of Economic Equilibrium. London, UK: John Wiley & Sons, Inc.
- Dummett, Michael (1977) Elements of Intuitionism. Oxford: Clarendon Press.
- Euwe, Max (1929) Mengentheoretische Betrachtungen über das Schachspiel, Communicated by Prof. R. Weizenböck (May 25, 1929), Proc. Koninklijke Nederlandse Akademie Van Wetenschappen, Amsterdam, 32(5): 633–642.
- Feferman, Solomon (2009) Gödel, Nagel, minds, and machines. The Journal of Philosophy 106(4): 201–219.
- Frege, Gottlob (1924/1925/1970) Sources of knowledge of mathematics and the mathematical natural sciences. In Hans Hermes, Friedrich Kambartel and Friedrich Kaulbach (eds), with the assistance of Gottfried Gabriel and Walburgs Rödding, Posthumous Writings (pp. 267–277). Oxford: Basil Blackwell.
- Gács, Péter, John T. Tromp and Paul Vitányi (2001) Algorithmic statistics. IEEE Transactions on Information Theory 47(6): 2443–2463.
-
Gandy, Robin. O. (1980) Church's thesis and principles for mechanisms. In J. Barwise, H.J. Keisler and K. Kunen (eds), The Kleene Symposium (pp. 123–148). Amsterdam, Netherlands: North-Holland.
10.1016/S0049-237X(08)71257-6 Google Scholar
- Gandy, Robin. O. (1982) Limitations to mathematical knowledge. In D. van Dalen, D. Lascar and J. Smiley (eds), Logic Colloquium'80 (pp. 129–146). Amsterdam, Netherlands: North-Holland.
- Gödel, Kurt (1951/1995) Some basic theorems on the foundations of mathematics and their implications. In Solomon Feferman, John W. Dawson, Jr., Warren Goldfarb, Charles Parsons and Robert N. Solovay (eds), Kurt Gödel – Collected Works, Volume III, Unpublished Essays and Lectures (pp. 304–323). Oxford: Oxford University Press.
-
Goodstein, Reuben. L. (1944) On the restricted ordinal theorem. Journal of Symbolic Logic 9(2): 33–41.
10.2307/2268019 Google Scholar
- Harrop, Ronald (1961) On the recursivity of finite sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, Bd, 7, pp. 136–140.
- Hayes, Brian (2003) A lucid interval. American Scientist 91(6): 484–488.
- Hilbert, David (1925 [1926]), On the infinite. In Paul Benacerraf and Hilary Putnam (eds) Philosophy of Mathematics – Selected Readings, 2nd edn (pp. 183–201), 1983. Cambridge, UK: Cambridge University Press.
- Hyland, J.M.E. (1991) Computing and foundations. In J.H. Johnson and M.J. Loomes (eds), The Mathematical Revolution Inspired by Computing (pp. 269–284). Oxford: Oxford University Press.
- Keynes, John Maynard (1936) The General Theory of Employment, Interest and Money. London, UK: Macmillan and Co., Limited.
- Kirby, Laurie and Jeff Paris (1982) Accessible independence results for Peano arithmetic. Bulletin of the London Mathematical Society 14: 285–293.
- Knuth Donald E. (1973) The Art of Computer Programming: Volume 1/Fundamental Algorithms, 2nd edn. Reading, MA: Addison-Wesley Publishing Company.
-
Knuth, Donald E. (1981) Algorithms in modern mathematics and computer science. In A.P. Ershov and Donald E Knuth (eds), Algorithms in Modern Mathematics and Computer Science (pp. 82–99). Berlin, Germany: Springer-Verlag.
10.1007/3-540-11157-3_26 Google Scholar
-
Kreisel, Georg (1974) A notion of mechanistic theory. Synthese 29: 11–26.
10.1007/BF00484949 Google Scholar
- Maietti, Maria Emilia and Giovanni Sambin (2005) Toward a minimalist foundation for constructive mathematics. In L. Crosilla and P. Schuster (eds), From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics (Chapter 6, pp. 91–114). Oxford: Clarendon Press.
- Markov, A.A. (1954/1961) Theory of Algorithms, Academy of Sciences of the USSR, Moscow and Leningrad, translated by Jacques J. Schorr-Kon and PST Staff and published for The National Science Foundation, Washington, D.C., and The Department of Commerce, USA, by The Israel Program for Scientific Translations, Jerusalem.
- Matiyasevich, Yuri M. (1993) Hilbert's Tenth Problem. Cambridge, MA: The MIT Press.
- McCulloch, Warren S. (1961/1965) What is a Number, that a Man May Know It, and a Man, that He May Know a Number, The Ninth Alfred Korzybski Memorial Lecture, reprinted in: Embodiments of Mind by Warren S. McCulloch Chapter 1, pp. 1–18. Cambridge, MA: The M.I.T. Press.
- Moore, Ramon E. (1966) Interval Analysis. Englewood Cliffs, NJ: Prentice-Hall.
- Moschovakis, Yiannis N. (1998) On founding the theory of algorithms. In H.G. Dales and G. Oliveri (eds), Truth in Mathematics (Chapter 4, pp. 71–104). Oxford: Clarendon Press.
-
Moschovakis, Yiannis N. (2001) What is an algorithm? In B. Engquist and W. Schmid (eds), Mathematics Unlimited – 2001 and Beyond (pp. 919–936). Berlin, Germany: Springer-Verlag.
10.1007/978-3-642-56478-9_17 Google Scholar
-
Moschoavakis, Yiannis N. and Vasilis Paschalis (2008) Elementary algorithms and their implementations. In S. Barry Cooper, Benedikt Löwe and Andrea Sorbi (eds), New Computational Paradigms: Changing Conceptions of What is Computable (pp. 87–118). New York: Springer Science and Business Media LLC.
10.1007/978-0-387-68546-5_5 Google Scholar
- Newell, Allen and Herbert A. Simon (1972) Human Problem Solving. Englewood Cliffs, NJ: Prentice-Hall, Inc.
- Newell, Allen, J.C. Shaw and Herbert A. Simon (1957) Empirical explorations of the logic theory machine: a case study in heuristics. Proceeding of the Western Joint Computer Conference 11: 218–239.
- Nisan, Noam (2007) Introduction to mechanism design (for computer scientists). In Noam Nisan, Tim Roughgarden, Éva Tardos and Vijay V. Vazirani (eds), Algorithmic Game Theory (Chapter 9, pp. 209–241). New York: Cambridge University Press.
-
Ok, Efe A. (2007) Real Analysis with Economic Applications. Princeton, NJ: Princeton University Press.
10.1515/9781400840892 Google Scholar
- Osborne, Maury (1977) The Stock Market and Finance from a Physicist's Viewpoint. Minneapolis, MN: Crossgar Press.
-
Papadimiriou, Christos H. (2007) Forward. In Noam Nisan, Tim Roughgarden, Éva Tardos and Vijay V. Vazirani (eds), Algorithmic Game Theory (pp. 29–51). New York: Cambridge University Press.
10.1017/CBO9780511800481.004 Google Scholar
- Pareto, Vilfredo (1927/1971) Manual of Political Economy, translated from the French Edition of 1927 by Ann S. Schwier, Ann S. Schwier and Alfred N. Page (eds). London: The Macmillan Press Ltd.
- Paris, Jeff and Reza Tavakol (1993) Goodstein algorithm as a super-transient dynamical system. Physics Letters A 180(1–2): 83–86.
- Posy, Carl J. (1998) Brouwer versus Hilbert: 1907–1928. Science in Context 11(2): 291–325.
- Rabin, Michael O. (1957) Effective computability of winning strategies. In M. Dresher, A.W. Tucker and P. Wolfe (eds), Annals of Mathematics Studies, No. 39: Contributions to the Theory of Games, Vol. III (pp. 147–157). Princeton, New Jersey: Princeton University Press.
-
Shafer, Glenn and Vladimir Vovk (2001) Probability and Finance: It's Only a Game. New York: John Wiley & Sons, Inc.
10.1002/0471249696 Google Scholar
-
Shankar, N. (1994) Metamathematics, Machines, and Gödel's Proof. Cambridge, UK: Cambridge University Press.
10.1017/CBO9780511569883 Google Scholar
- Shapiro, Stewart (1998) Incompleteness, mechanism, and optimism. The Bulletin of Symbolic Logic 4(3): 273–302.
- Smale, Steve (1976) Dynamics in general equilibrium theory. American Economic Review 66(2): 288–294.
- Smale, Steve (1981) Global analysis and economics. In Kenneth J. Arrow and Michael D. Intrilligator (eds), Handbook of Mathematical Economics (Vol. I, Chapter 8, pp. 331–370). Amsterdam, Netherlands: North-Holland Publishing Company.
- Sraffa, Piero (1960) Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory. Cambridge, UK: Cambridge University Press.
- Steinhaus, Hugo (1965) Games, an informal talk. The American Mathematical Monthly 72(5): 457–468.
-
Timpson, Christopher G. (2004) Quantum computers: the Church-Turing hypothesis versus the turing principle. In Christof Teuscher (ed.), Alan Turing – Life and Legacy of aGreat Thinker (pp. 213–240). Berlin, Germany: Springer-Verlag.
10.1007/978-3-662-05642-4_9 Google Scholar
- van Dalen, Dirk (1999) Mystic, geometer and intuitionist: the life of L.E.J. Brouwer – Volume 2: Hope and Disillusion. Oxford: Clarendon Press.
- van Lambalgen, Michiel (1987) Random sequences, Doctoral Dissertation, University of Amsterdam, 16 September, 1987.
- Velupillai, K. Vela (1997) Expository notes on computability and complexity in (arithmetical) games. Journal of Economic Dynamics and Control 21(6): 955–979.
-
Velupillai, K. Vela (2000) Computable Economics. Oxford: Oxford University Press.
10.1093/0198295278.001.0001 Google Scholar
- Velupillai, K. Vela (2005) The unreasonable ineffectiveness of mathematics in economics. Cambridge Journal of Economics 29(6): 849–872.
- Velupillai, K. Vela (2006) The algorithmic foundations of computable general equilibrium theory. Applied Mathematics and Computation 179(1): 360–369.
- Velupillai, K. Vela (2009) Uncomputability and undecidability in economic theory. Applied Mathematics and Computation 215(4): 1404–1416.
- Velupillai, K. Vela (2009a) A computable economist's perspective on computational complexity, J. Barkley Rosser, Jr. (ed.), The Handbook of Complexity Research (Chapter 4, pp. 36–83). Cheltenham, Gloucestershire, UK: Edward Elgar Publishing Ltd.
- Velupillai, K. Vela (2009b) The Algorithmic Revolution in the Social Sciences: Mathematical Economics, Game Theory and Statistics. Invited Lecture, presented at the Workshop on Information Theoretic Methods in Science and Engineering, Tampere, Finland, August, 17/19, 2009. Published in the Proceedings of WITMSE 2009.
- Velupillai, K. Vela (2010a) In praise of anarchy in research and teaching. Economic and Political Weekly XLV(14): 51–55.
-
Velupillai, K. Vela (2010b) Foundations of boundedly rational choice and satisfying decisions. Advances in Decision Sciences, April, 2010.
10.1155/2010/798030 Google Scholar
- Velupillai, K. Vela (2010c) Reflections on mathematical economics in the algorithmic mode. New Mathematics and Natural Computation Forthcoming in, Vol. 6.
- Velupillai, K. Vela and Stefano Zambelli (2010) Computation in economics. Forthcoming In John Davis and Wade Hands (eds), The Elgar Companion to Recent Economic Methodology, Cheltenham: Edward Elgar, 2011.
- von Neumann, John (1928) Zur theorie der gesellsschaftsspiele. Mathematische Annalen 100: 295–320.
- von Neumann, John (1938/1945–6) A model of general economic equilibrium. Review of Economic Studies XIII(1): 1–9.
- Wang, Hao (1960/1970) Toward mechanical mathematics, In Logic, Computers and Sets (Chapter IX, pp. 224–268). New York, NY: Chelsea Publishing Company.
- Whitehead, Alfred North and Bertrand Russell (1927) Principia Mathematica (Vol. I–III). Cambridge, UK: Cambridge University Press.
- Wittgenstein, Ludwig (1939) Wittgenstein's Lectures on the Foundations of Mathematics – Cambridge, 1939. In Cora Diamond (ed.). From the Notes of R.G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smuthies. Chicago, IL: The University of Chicago Press.
- Zermelo, Ernst (1913) Über ein anwendung der mengenlehre auf die theorie des schachspiels. In E.W. Hobson and A.E.H. Love (eds). Proceedings of the Fifth International Congress of Mathematicians (11–28 August, 1912, Vol. 2, pp. 501–504). Cambridge, UK: Cambridge University Press.
