Abstract
The work of Gödel and Turing, among others, shows that there are fundamental limits to the possibility of formal quantification of natural and social phenomena . Both our knowledge and our ignorance are, to a large extent, not amenable to quantification. Disregard of these limits in the economic sphere might lead to underestimation of risk and, consequently, to excessive risk-taking. If so, this would expose markets to undue instability and turbulence. One major lesson of the Global Financial Crisis , therefore, is to reform economic methodology to expand beyond formal reasoning.
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Notes
- 1.
But see Sect. 3 below.
- 2.
We will later elaborate on that characterization.
- 3.
We use here “model” in the sense of “mathematical model”, that is, the mathematical depiction of some phenomenon; we do not use it in the model-theoretic sense.
- 4.
That is, the set of theorems of S is recursively enumerable.
- 5.
This is what we mean by “enough arithmetic”. Actually we can obtain the same results we require if we add a still weaker arithmetic condition.
- 6.
The theorem that implies such a result is a Rice-like theorem proved by da Costa and Doria in theorem 1990; see Chaitin et al. [28].
- 7.
A simple proof can be found in Chaitin et al. [28].
- 8.
Yet the Arrow–Debreu theory is undecidable; that follows from a theorem by Tsuji et al. [107].
- 9.
For a summary discussion, see http://http://www.fattails.ca.
- 10.
Personal communication to FAD.
- 11.
In binary form, zeros and ones are evenly distributed.
- 12.
Actually an injunction engraved at the Apollonic oracle at Delphos.
- 13.
Yet one must be careful here, as the so-called “reflection principles” may be interpreted as sentences of the form “X is true”.
References
Al-Suwailem, S. (2010). Behavioural complexity. Journal of Economic Surveys, 25, 481–506.
Al-Suwailem, S. (2014). Complexity and endogenous instability. Research in International Business and Finance, 30, 393–410.
Arrow, K., & Debreu, G. (1954). The existence of an equilibrium for a competitive economy. Econometrica, 22, 265–89.
Barrow, J. (1998). Impossibility: The limits of science and the science of limits. Oxford: Oxford University Press.
Barwise, J. (1989). The situation in logic. Stanford, CA: Center for the Study of Language and Information.
Barwise, J., & Moss, L. (1996). Vicious circles. Stanford, CA: Center for the Study of Language and Information.
Bays, T. (2014). Skolem’s paradox. In E. Zalta (Ed.), Stanford Encyclopedia of Philosophy. Stanford, CA: Stanford University, https://plato.stanford.edu/.
Bergstra, J. A., & Middelburg, C. A. (2011). Preliminaries to an investigation of reduced product set finance. Journal of King Abdulaziz University: Islamic Economics, 24, 175–210.
Bernstein, P. (1996). Against the Gods: The remarkable story of risk. New York: Wiley.
Bernstein, P. (2007). Capital ideas evolving. New York: Wiley.
Bernstein, P. (2007). Capital ideas: Past, present, and future. In Lecture delivered at CFA Institute Annual Conference.
Berto, F. (2009). There’s something about Gödel: The complete guide to the incompleteness theorem. Hoboken: Wiley-Blackwell.
Blaug, M. (1992). The methodology of economics. Cambridge: Cambridge University Press.
Blaug, M. (1999). The disease of formalism, or what happened to neoclassical economics after war. In R. Backhouse & J. Creedy (Eds.), From classical economics to the theory of the firm (pp. 257–80). Northampton, MA: Edward Elgar.
Blaug, M. (2002). Is there really progress in economics? In S. Boehm, C. Gehrke, H. Kurz, & R. Sturm (Eds.), Is there progress in economics? European Society for the History of Economic Thought (pp. 21–41). Northampton, MA: Edward Elgar.
Blythe, S. (2007). Risk is not a number. http://www.advisor.ca/investments/alternative-investments/risk-is-not-a-number-26881.
Bookstaber, R. (2007). A demon of our own design. New York: Wiley.
Bookstaber, R., & Langsam, J. (1985). On the optimality of coarse behavior rules. Journal of Theoretical Biology, 116, 161–193.
Bolander, T. (2002). Self-reference and logic. Φ News, 1, 9–44. http://phinews.ruc.dk/.
Bolander, T. (2014). Self-reference. In E. Zalta (Ed.), Stanford Encyclopedia of Philosophy. Stanford, CA: Stanford University, https://plato.stanford.edu/.
Bordo, M., Eichengreen, B., Klingebiel, D., & Martinez-Peria, M. S. (2001). Is the crisis problem growing more severe? Economic Policy, 32, 53–82.
Bouchaud, J. P. (2008). Economics needs a scientific revolution. Nature, 455, 1181.
Breit, W., & Hirsch, B. (Eds.). (2009). Lives of the laureates (5th ed.). Cambridge, MA: MIT Press.
Byers, W. (2007) How mathematicians think: Using ambiguity, contradiction, and paradox to create mathematics. Princeton: Princeton University Press.
Byers, W. (2011). The blind spot: Science and the crisis of uncertainty. Princeton: Princeton University Press.
Calude, C., Jurgensen, H., & Zimand, M. (1994). Is independence an exception? Applied Mathematical Computing, 66, 63–76.
Cassidy, J. (2009). How markets fail: The logic of economic calamities. New York: Farrar, Straus, and Giroux.
Chaitin, G., Doria, F. A., & da Costa, N. C. A. (2011). Gödel’s way: Exploits into an undecidable world. Boca Raton, FL: CRC Press.
Chrystal, K. A., & Mizen, P. D. (2003). Goodhart’s law: Its origins, meaning and implications for monetary policy. In P. Mizen (Ed.), Central banking, monetary theory and practice: Essays in honour of Charles Goodhart (Vol. 1). Northampton, MA: Edward Elgar.
Clower, R. (1995). Axiomatics in economics. Southern Economic Journal, 62, 307–319.
Colander, D., Goldberg, M., Haas, A., Juselius, K., Kirman, A., Lux, T., & Sloth, B. (2009). The financial crisis and the systemic failure of the economics profession. Critical Review, 21, 249–67.
Coleman, T. (2011). A practical guide to risk management. Charottesville, VA: CFA Institute.
Collins, H. (2010). Tacit and explicit knowledge. Chicago: University of Chicago Press.
da Costa, N. C. A., & Doria, F. A. (1991). Undecidability and incompleteness in classical mechanics. International Journal of Theoretical Physics, 30, 1041–1073.
da Costa, N. C. A., & Doria, F. A. (1992). On the incompleteness of axiomatized models for the empirical sciences. Philosophica, 50, 73–100.
da Costa, N. C. A., & Doria, F. A. (2005). Computing the future. In K. Velupillai (Ed.), Computability, Complexity and Constructivity in Economic Analysis. Malden, MA: Blackwell Publishers.
Davis, M. (2000). Engines of logic. New York: W.W. Norton.
DeLong, H. (1970). A profile of mathematical logic. Reading, MA: Addison-Wesley.
Dyson, F. (2006). The scientist as rebel. New York: New York Review Books.
Erickson, J. (2015). Models of computation. University of Illinois, Urbana-Champaign. http://jeffe.cs.illinois.edu
Etzioni, A. (1988). The moral dimension: Toward a new economics. New York: The Free Press.
Ferreirós, J. (2008). The crisis in the foundations in mathematics. In T. Gowers (Ed.), The Princeton companion to mathematics (pp. 142–156). Princeton: Princeton University Press.
Feyerabend, P. (2010). Against the method (4th ed.). London: Verso.
Filimonov, V., Bicchetti, D., Maystre, N., & Sornette, D. (2013). Quantification of the high level of endogeneity and of structural regime shifts in commodity markets. The Journal of International Money and Finance, 42, 174–192.
Filimonov, V., & Sornette, D. (2012). Quantifying reflexivity in financial markets: Towards a prediction of flash crashes. Physical Review E, 85(5), 056108.
Franzen, T. (2004). Inexhaustibility. Boca Raton: CRC Press.
Frydman, R., & Goldberg, M. (2007). Imperfect knowledge economics: Exchange rates and risk. Princeton: Princeton University Press.
Frydman, R., & Goldberg, M. (2011). Beyond mechanical markets: Asset price swings, risk, and the role of state. Princeton: Princeton University Press.
FSA. (2009). The Turner review. London: Financial Services Authority.
Gleiser, M. (2014). The island of knowledge: The limits of science and the search for meaning. New York: Basic Books.
Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I (On formally undecidable propositions of Principia Mathematica and related systems I). Monatshefte für Mathematik und Physik, 38, 173–198.
Gödel, K. (1947). What is Cantor’s continuum problem? American Mathematical Monthly, 54, 515–525.
Gödel, K. (1995). Some basic theorems on the foundations of mathematics and their implications. In S. Feferman (Ed.), Kurt Gödel Collected Works, Volume III: Unpublished essays and lectures (pp. 304–323). Oxford: Oxford University Press.
Goleman, D. (2006). Social intelligence. New York: Bantam Books.
Gray, J. (2008). Thoralf Skolem. In T. Gowers (Ed.), The Princeton companion to mathematics (pp. 806–807). Princeton: Princeton University Press.
Harcourt, G. (2003). A good servant but a bad master. In E. Fullbrook (Ed.), The crisis in economics. London: Routledge.
Hawking, S. (2002). Gödel and the end of the universe. www.hawking.org.uk.
Heiner, R. (1983). The origin of predictable behaviour. American Economic Review, 73, 560–595.
Keynes, J. M. (1936/1953). The general theory of employment, interest and money. San Diego: Harcourt Brace Jovanovich.
King, M. (2016). The end of alchemy. New York: W.W. Norton.
Kirman, A. (2012). The economy and the economic theory in crisis. In D. Coyle (Ed.), What’s the use of economics? Teaching the dismal science after the crisis. London: London Publishing Partnership.
Kline, M. (1980). Mathematics: The loss of certainty. Oxford: Oxford University Press.
Knight, F. (1921). Risk, uncertainty, and profit. New York: Houghton-Mifflin.
Kremer, P. (2014). The revision theory of truth. In E. Zalta (Ed.), Stanford Encyclopedia of Philosophy. Stanford, CA: Stanford University, https://plato.stanford.edu/.
Krugman, P. (2009, September 2). How did economists get it so wrong? New York Times.
Laplace, P. S. (1902/1951). A philosophical essay on probabilities. Translated into English from the original French 6th ed. by F.W. Truscott & F.L. Emory. New York: Dover Publications.
Lavoie, D. (1986). The market as a procedure for discovery and conveyance of inarticulate knowledge. Comparative Economic Studies, 28, 1–19.
Leonard, R. (1994). Money and language. In J. L. Di Gaetani (Ed.), Money: Lure, lore, and literature (pp. 3–13). London: Greenwood Press.
Lewis, H. (2013). Lecture notes. Harvard University, http://lewis.seas.harvard.edu/files/harrylewis/files/section8sols.pdf.
Lieberman, M. (2013). Social: Why our brains are wired to connect. New York: Crown Publishers.
Lloyd, S. (2006). Programming the universe. New York: Vintage Books.
Lloyd, S. (2013). A Turing test for free will. Cambridge, MA: MIT Press. http://arxiv.org/abs/1310.3225v1.
Machtey, M., & Young, P. (1979). An introduction to the general theory of algorithms. New York: North–Holland.
Mainelli, M., & Giffords, B. (2009). The road to long finance: A systems view of the credit scrunch. London: Centre for the Study of Financial Innovation. Heron, Dawson & Sawyer. www.csfi.org.uk.
Marks, R. (2013). Learning lessons? The global financial crisis five years on. Journal and Proceeding of the Royal Society of New South Wales, 146, 3–16, A1–A43.
McCauley, J. (2012). Equilibrium versus market efficiency: Randomness versus complexity in finance markets. In S. Zambelli & D. George (Eds.), Nonlinearity, complexity and randomness in economics (pp. 203–210). Malden, MA: Wiley-Blackwell.
Mintzberg, H. (1994). The rise and fall of strategic planning. New York: The Free Press.
Mishkin, F. (2007). The economics of money, banking, and financial markets (Alternate edition). New York: Pearson.
Nocera, J. (2009, January 2). Risk mismanagement. The New York Times.
North, D. (1990). Institutions, institutional change and economic performance. Cambridge, MA: Cambridge University Press.
Polanyi, M. (1966). The tacit dimension. New York: Doubleday.
Popper, K. (1956/1988). The open universe: An argument for indeterminism. London: Routledge.
Priest, G. (1994). The structure of the paradoxes of self-reference. Mind, 103, 25–34.
Priest, G. (2000). Beyond the limits of thought (2nd ed.). Oxford: Oxford University Press.
Putnam, H. (1967). Psychophysical predicates. In W. Capitan & D. Merrill (Eds.), Art, mind, and religion. Pittsburgh: University of Pittsburgh Press.
Rajan, U., Seru, A., & Vig, V. (2015). The failure of models that predict failures: Distance, incentives, and defaults. Journal of Financial Economics, 115, 237–260.
Reinhart, C., & Rogoff, K. (2009). This time is different: Eight centuries of financial folly. Princeton: Princeton University Press.
Rice, H. G. (1953). Classes of recursively enumerable sets and their decision problems. Transactions of the American Mathematical Society, 74, 358–366.
Rogers, H., Jr. (1967). Theory of recursive functions and effective computability. New York: McGraw-Hill.
Rosser, J. B. (2001) Alternative Keynesian and post Keynesian perspectives on uncertainty and expectations. Journal of Post Keynesian Economics, 23, 545–66.
Rubinstein, A. (1998). Modeling bounded rationality. Cambridge, MA: MIT Press.
Savin, N., & Whitman, C. (1992). Lucas critique. In P. Newman, M. Milgate, & J. Eatwell (Eds.), New Palgrave dictionary of money and finance. New York: Palgrave.
Shiller, R. (2009). Irrational exuberance. Princeton: Princeton University Press.
Simon, H. (1978). Rationality as process and as product of thought. American Economic Review, 68, 1–16.
Simon, H. (1978/1992). Rational decision-making in business organizations. In A. Lindbeck (Ed.), Nobel Lectures, Economics 1969–1980 (pp. 343–371). Singapore: World Scientific Publishing Co. https://www.nobelprize.org.
Soros, G. (2003). Alchemy of finance. Hoboken: Wiley.
Soros, G. (2009, October 26). The general theory of reflexivity. Financial Times.
Sornette, D. (2003). Why stock markets crash: Critical events in complex financial systems. Princeton: Princeton University Press.
Sornette, D., & Cauwels, P. (2012). The illusion of the perpetual money machine. Notenstein Academy White Paper Series. Zurich: Swiss Federal Institute of Technology. http://arxiv.org/pdf/1212.2833.pdf.
Stewart, I. (1991). Deciding the undecidable. Nature, 352, 664–665.
Stuart, T. (2013). Understanding computation. Beijing: O’Reilly.
Svozil, K. (1993). Randomness and undecidability in physics. Singapore: World Scientific.
Svozil, K. (1996). Undecidability everywhere? In A. Karlqvist & J.L. Casti (Eds.), Boundaries and barriers: On the limits to scientific knowledge. Reading, MA: Addison-Wesley, pp. 215–237. http://arxiv.org/abs/chao-dyn/9509023v1.
Taleb, N. (2007). The black swan. New York: Random House.
Taleb, N. (2012). Antifragile. New York: Random House.
Trichet, J.-C. (2009). A paradigm change for the global financial system. BIS Review, 4/2009.
Tsuji, M., Da Costa, N. C. A., & Doria, F. A. (1998). The incompleteness of theories of games. Journal of Philosophical Logic, 27, 553–568.
Tuckett, D. (2011). Minding the market: An emotional finance view of financial instability. Basingstoke: Palgrave MacMillan.
Turing, A. (1936). On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, Series, 2(42), 230–265.
Velupillai, K. V. (2000). Computable economics. Oxford: Oxford University Press.
Velupillai, K. V. (2012). Introduction. In K. V. Velupillai et al. (Eds.), Computable economics. International Library of Critical Writings in Economics. Northampton, MA: Edward Elgar.
Velupillai, K. V., Zambelli, S., & Kinsella, S. (Eds.). (2012). Computable economics. International Library of Critical Writings in Economics. Northampton, MA: Edward Elgar.
Wiltbank, R., Dew, N., Read, S., & Sarasvathy, S. D. (2006). What to do next? A case for non-predictive strategy. Strategic Management Journal, 27, 981–998.
Winrich, J. S. (1984). Self-reference and the incomplete structure of neoclassical economics. Journal of Economic Issues, 18, 987–1005.
Wolpert, D. (2008). Physical limits of inference. Physica, 237, 1257–1281.
Yanofsky, N. (2003). A universal approach to self-referential paradoxes, incompleteness and fixed points. The Bulletin of Symbolic Logic, 9, 362–386.
Yanofsky, N. (2013). The outer limits of reason: What science, mathematics, and logic cannot tell us. Cambridge, MA: MIT Press.
Acknowledgements
We are grateful to the editors and an anonymous referee for constructive comments and suggestions that greatly improved the readability of this text. FAD: wishes to acknowledge research grant no. 4339819902073398 from CNPq/Brazil, and the support of the Production Engineering Program, coppe/UFRJ, Brazil. SA: wishes to acknowledge the valuable discussions with the co-authors, particularly FAD. The views expressed in this chapter do not necessarily represent the views of the Islamic Development Bank Group.
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Al-Suwailem, S., Doria, F.A., Kamel, M. (2018). Is Risk Quantifiable?. In: Chen, SH., Kao, YF., Venkatachalam, R., Du, YR. (eds) Complex Systems Modeling and Simulation in Economics and Finance. CEF 2015. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-319-99624-0_14
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