Skip to main content
SpringerLink
Log in

The Future of Mathematics in Economics: A Philosophically Grounded Proposal

  • Published:
Foundations of Science Aims and scope Submit manuscript

Abstract

The use of mathematics in economics has been widely discussed. The philosophical discussion on what mathematics is remains unsettled on why it can be applied to the study of the real world. We propose to get back to some philosophical conceptions that lead to a language-like role for the mathematical analysis of economic phenomena and present some problems of interest that can be better examined in this light. Category theory provides the appropriate tools for these analytical approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The pervasiveness of this problem prompted a large initiative to address it, leaded by Brian Nosek (Baker 2015).

  2. About their close connection, see Yasugi and Passell (2003).

  3. “Economic theory may have become so abstruse that editors of the leading general journals, recognizing that very few of their readers could comprehend the theory, have cut back on publishing work of this type” (Hamermesh 2013, p. 162).

  4. The Santa Fe Institute (http://www.santafe.edu/) is actively engaged in this enterprise. Some of its members have produced interesting manifestos, describing some proposals for the unification of the economic and social disciplines, either by rethinking the notion of rationality (Gintis 2009) or by using models of complex systems for their description (Brian Arthur 2015).

  5. See Lawvere and Schanuel (2009) of Spivak (2014) for very clear and intuitive yet rigorous introductions to categories.

  6. A model of ‘ordinary mathematics’ developed in the framework of type theory. The latter, introduced by Bertrand Russell to avoid contradictions in naive set theory (Russell 2010) has become a fundamental tool in theoretical computer science, since it allows a pure symbolic manipulation by means of clear rules of how different kinds (types) of symbols (terms) interact: “It thus emerges that computational type theory is a plausible foundation for computer science as well as for computational mathematics” (Constable 2009, A foundation for computer science).

  7. An homotopy formalizes the intuitive idea of deformability of one mapping into another.

  8. See the thorough discussion of the differences between the axiomatic method proper of ZFC and category theory in Rodin (2014).

  9. See Pacuit and Roy (2015) for a description of the latter field. It can be seen that its mathematical framework departs from the classical one described as: “The current period [1960 to the late 1970s] is one of integration, in which modern mathematical economics combines elements of calculus, set theory and linear models” (Arrow 1981, p. 6).

  10. Keynes analyzes the behavior of rational agents in the market through an analogy with a newspaper contest, in which entrants are asked to choose the six most attractive faces from a hundred photographs. Those who pick the most popular faces are eligible for a prize (see Keynes 1936, p. 156).

  11. The solution adopts the form of a natural transformation, i.e. a transformation from a functor to another, preserving the inner structure of the corresponding categories. In Vassilakis (1991) it is shown that a fixed-point is the image of a succession of natural transformations, each one applied to a step of the circular process.

  12. Recall that ordinal numbers identify ordered sets. If a sequence is finite, its ordinal equals the number of its elements. If a sequence can be enumerated by the natural numbers, its corresponding ordinal number is called \(\omega \), the first infinite ordinal. But other infinite sets can be ordered in distinct ways, yielding other ordinal numbers, called transfinite (Bagaria 2014, section 3).

  13. “The term ‘supertask’ [\(\ldots \)] designates [\(\ldots \)] an infinite number of actions performed in a finite amount of time” (Pérez Laraudogoitia 2013, Introduction).

  14. A coalgebra is a categorical construction, defined in terms of a functor from a category to itself. If the category is \({\mathcal {C}}\) and \(F: {\mathcal {C}} \rightarrow {\mathcal {C}}\) a functor, a coalgebra is an object X in \({\mathcal {C}}\) together with a morphism from X to F(X) (Adámek 2005).

  15. One of the Zermelo-Fraenkel axioms of set theory is the axiom of regularity, which precludes sets like \(a=\{a\}\), which are defined in terms of themselves, called non-well founded (Barwise and Moss 1996).

  16. An adjunction between two functors is such that their composition recovers the identity, i.e. making one of the functors kind of the “inverse” of the other (Ellerman 2007).

  17. Given that a topos has a logical structure sufficiently rich to develop ‘ordinary mathematics’, it comes equipped with a way of ranking the validity of propositions (technically, a subobject classifier).

References

  • Adámek, J. (2005). Introduction to coalgebra. Theory and Applications of Categories, 14, 157–199.

    Google Scholar 

  • Aristotle. (1954). Nicomachean ethics (Sir David Ross, Trans.). London: Oxford University Press.

  • Aristotle, (1984). The complete works of Aristotle, edited by Jonathan Barnes. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Arrow, K. (1981). Historical introduction. In K. Arrow & M. Intriligator (Eds.), Handbook of mathematical economics (Vol. 1). Amsterdam: North-Holland.

    Google Scholar 

  • Backhouse, R. (1998). If mathematics is informal, then perhaps we should accept that economics must be informal too. The Economic Journal, 108, 1848–1858.

    Article  Google Scholar 

  • Bagaria, J. (2014). Set theory. In E. Zalta (Ed.), Stanford encyclopedia of philosophy (Winter 2014 Edition). http://plato.stanford.edu/archives/win2014/entries/set-theory/.

  • Baker, M. (2015). Over half of psychology studies fail reproducibility test. Nature Online,. doi:10.1038/nature.2015.18248.

    Google Scholar 

  • Barwise, J., Moss, L. (1996). Vicious circles. CSLI Lecture Notes (Vol. 60). Stanford.

  • Beaulieu, L. (1999). Bourbaki’s art of memory. Osiris (2nd series), 14, 219–251.

    Article  Google Scholar 

  • Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy, 70(19), 661–679.

    Article  Google Scholar 

  • Benacerraf, P., & Putnam, H. (1964). Introduction. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics. Prentice Hall, NJ: selected Readings.

    Google Scholar 

  • Boumans, M. (2005). How economists model the world into numbers. London and NY: Routledge.

    Book  Google Scholar 

  • Brandenburger, A., & Dekel, E. (1993). Hierarchies of beliefs and common knowledge. Journal of Economic Theory, 59, 189–198.

    Article  Google Scholar 

  • Brian Arthur, W. (2015). Complexity and the economy. New York, NY: Oxford University Press.

    Google Scholar 

  • Brouwer, L. (1999). Intuitionism and formalism (original 1913). Bulletin (New Series) of the American Mathematical Society, 37, 55–64.

    Article  Google Scholar 

  • Caramello, O. (2010). The unification of mathematics via topos theory. arXiv:1006.3930 [math.CT].

  • Carnap, R. (1964). The logicist foundation of mathematics. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics. Prentice Hall, NJ: Selected Readings.

    Google Scholar 

  • Clark, P., & Read, S. (1984). Hypertasks. Synthese, 61, 387390.

    Article  Google Scholar 

  • Constable, R. (2009). Computational type theory. Scholarpedia, 4(2), 7618.

    Article  Google Scholar 

  • Corry, L. (1997). David Hilbert and the axiomatization of physics (1894–1905). Archive for History of Exact Sciences, 51, 83–198.

    Article  Google Scholar 

  • Corry, L. (2002). David Hilbert y su Filosofa Empiricista de la Geometría. Boletín de la Asociación Matemática Venezolana, IX, 27–43.

    Google Scholar 

  • Corry, Leo. (2004). Modern algebra and the rise of mathematical structures. Basel and Boston: Birkhuser.

    Book  Google Scholar 

  • Debreu, G. (1991). The mathematization of economic theory. American Economic Review, 81, 1–7.

    Google Scholar 

  • Doering, A., & Isham, C. (2011). ‘What is a thing?’: Topos theory in the foundations of physics. Lecture Notes in Physics, 813, 753–940.

    Article  Google Scholar 

  • Düppe, T. (2011). How economic methodology became a separate science. Journal of Economic Methodology, 18, 163–176.

    Article  Google Scholar 

  • Ellerman, D. (2007). Adjoints and emergence: Applications of a new theory of adjoint functors. Axiomathes, 17, 19–39.

    Article  Google Scholar 

  • Etingof, P., Golberg, O., Hensel, S., Liu, T., Schwendner, A., Vaintrob, D., et al. (2011). Introduction to representation theory. Providence, RI: American Mathematical Society.

    Book  Google Scholar 

  • Fox, G. (1997). Reason and reality in the methodologies of economics. Cheltenham: Elgar.

    Google Scholar 

  • Franklin, J. (2009). Aristotelian realism. In A. Irvine (Ed.), Philosophy of mathematics, volume 4 in D. Gabbay, P. Thagard and J. Woods (series eds.), Handbook of the philosophy of science. Amsterdam: Elsevier.

  • Frege, G. (1964). The concept of a number (original 1884). In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics. Prentice Hall, NJ: Selected Readings.

    Google Scholar 

  • Frenkel, E. (2005). Lectures on the langlands program and conformal field theory. arXiv:hep-th/0512172.

  • Gasking, D. (1964). Mathematics and the world. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics. Prentice Hall, NJ: Selected Readings.

    Google Scholar 

  • Gintis, H. (2009). The bounds of reason: Game theory and the unification of the behavioral sciences. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Giocoli, N. (2003). Modeling rational agents. Elgar: Cheltenham and Northampton.

    Google Scholar 

  • Giocoli, N. (2009). Mathematics as the role model for neoclassical economics. In R. Arena, S. Dow, & M. Klaes (Eds.), Open economics: Economics in relation to other disciplines. London: Routledge.

    Google Scholar 

  • Goldblatt, R. (1984). Topoi: The categorical analysis of logic. North-Holland, Amsterdam.

  • Grothendieck, A. (1986). Reflections and bearing witness—The inheritors and the builder. http://www.fermentmagazine.org/rands/recoltes1.html.

  • Hamermesh, D. (2013). Six decades of top economics publishing: Who and how? Journal of Economic Literature, 51, 162–172.

    Article  Google Scholar 

  • Heckman, J. (2001). Econometrics and empirical economics. Journal of Econometrics, 100, 3.

    Article  Google Scholar 

  • Heyting, A. (1964). The intuitionist foundation of mathematics. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics. Prentice Hall, NJ: Selected Readings.

    Google Scholar 

  • Iemhoff, R. (2013). Intuitionism in the philosophy of mathematics. Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/intuitionism/.

  • Keynes, J. M. (1936). The general theory of employment, interest and money. New York: Harcourt Brace and Co.

    Google Scholar 

  • Khrennikov, A. (2010). Ubiquitous quantum structure: From psychology to finance. Berlin: Springer.

    Book  Google Scholar 

  • Kitcher, P., & Aspray, W. (1988). An opinionated introduction. In P. Kitcher & W. Aspray (Eds.), History and philosophy of modern mathematics (Vol. 11). Minneapolis: Minnesota Studies in Philosophy of Science.

    Google Scholar 

  • Kline, M. (1985). Mathematics and the search for knowledge. Oxford and New York: Oxford University Press.

    Google Scholar 

  • Kraut, R. (2006). How to justify ethical propositions: Aristotle’s method. In R. Kraut (Ed.), The Blackwell guide to Aristotle’s Nicomachean ethics. London: Blackwell Publishing.

    Chapter  Google Scholar 

  • Lambek, J. (1994). Are the traditional philosophies of mathematics incompatible? The Mathematical Intelligencer, 16, 56–62.

    Article  Google Scholar 

  • Lambek, J. (2004). What is the world of mathematics? Annals of Pure and Applied Logic, 126, 149–158.

    Article  Google Scholar 

  • Lawvere, F. W., & Schanuel, S. (2009). Conceptual mathematics: A first introduction to categories (2nd ed.). Cambridge: Cambridge University Press. 2009.

    Book  Google Scholar 

  • Leontief, W. (1971). Theoretical assumptions and nonobserved facts. American Economic Review, 61, 1–7.

    Google Scholar 

  • Linnebo, \(\emptyset \). (2013). Platonism in the philosophy of mathematics. Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/intuitionism/.

  • Lipman, B. (1991). How to decide how to decide how to\(\ldots \): Modeling limited rationality. Econometrica, 59, 1105–1125.

    Article  Google Scholar 

  • MacIntyre, A. (1984). After virtue. Notre Dame, IN: University of Notre Dame Press.

    Google Scholar 

  • Marquis, J.-P. (2015). Category theory. In E. Zalta (Ed.), The Stanford encyclopedia of philosophy (Winter 2015 Edition). http://plato.stanford.edu/archives/win2015/entries/category-theory/.

  • Mendell, H. (2004. Aristotle and mathematics. In E. Zalta (Ed.), Stanford encyclopedia of philosophy. http://plato.stanford.edu/entries/intuitionism/.

  • Mertens, J.-F., & Zamir, S. (1984). Formulation of Bayesian analysis for games with incomplete information. International Journal of Game Theory, 14, 1–29.

    Article  Google Scholar 

  • Michell, J. (2005). The logic of measurement: A realistic overview. Measurement, 38, 285–294.

    Article  Google Scholar 

  • Moss, L., & Viglizzo, I. (2004). Harsanyi type spaces and final coalgebras constructed from satisfied theories. Electronic Notes in Theoretical Computer Science, 106, 279–295.

    Article  Google Scholar 

  • Náufel do Amaral, F., & Haeusler, E. (2007). Using the internal logic of a topos to model search spaces for problems. Logic Journal of the IGPL, 15, 457–474.

    Article  Google Scholar 

  • Pacuit, E., & Roy, O. (2015). Epistemic foundations of game theory. In E. Zalta (Ed.) The Stanford Encyclopedia of Philosophy (Spring 2015 Edition). http://plato.stanford.edu/archives/spr2015/entries/epistemic-game/.

  • Pérez Laraudogoitia, J. (2013). Supertasks. In E. Zalta (Ed.), The Stanford encyclopedia of philosophy (Fall 2013 edition), Edward N. Zalta (ed.). http://plato.stanford.edu/archives/fall2013/entries/spacetime-supertasks/.

  • Quine, W. V. O. (1964). On what there is. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics. Prentice Hall, NJ: Selected Readings.

    Google Scholar 

  • Rodin, A. (2014). Axiomatic method and category theory. Berlin: Springer.

    Book  Google Scholar 

  • Russell, B. (2010). The principles of mathematics (original 1903). London: Routledge.

    Google Scholar 

  • Rutten, J. J. (1998). Automata and coinduction (an exercise in coalgebra). Lecture Notes in Computer Science, 1466, 193–217.

    Google Scholar 

  • Samuelson, P. (1952). Economic theory and mathematics: An appraisal. American Economic Review (Vol.42). Papers and Proceedings of the Sixty-fourth Annual Meeting of the American Economic Association, pp. 56–66.

  • Samuelson, P. (1998). How foundations came to be. Journal of Economic Literature, 36, 1375–1386.

    Google Scholar 

  • Shoham, Y., & Leyton-Brown, K. (2009). Multiagent systems: Algorithmic, game-theoretic, and logical foundations. New York: Cambridge University Press.

    Google Scholar 

  • Snapper, E. (1979). The three crises in mathematics: Logicism. Intuitionism and Formalism, Mathematics Magazine, 52, 207–216.

    Article  Google Scholar 

  • Spivak, D. I. (2014). Category theory for the sciences. Cambridge, MA: MIT Press.

    Google Scholar 

  • Suppes, P. (1962). Models of data. In E. Nagel, P. Suppes, & A. Tarski (Eds.), Logic, methodology and philosophy of science: Proceedings of the 1960 congress. Stanford, CA: Stanford University Press.

    Google Scholar 

  • Svozil, K. (2003). Randomness and undecidability in physics. Singapore: World Scientific.

    Google Scholar 

  • The Univalent Foundations Program. (2013). Homotopy type theory: Univalent foundations of mathematics. http://homotopytypetheory.org/book/.

  • Tohmé, F., Caterina, G., & Gangle, R. (2015). Abduction: A categorical characterization. Journal of Applied Logic, 13, 78–90.

    Article  Google Scholar 

  • Tohmé, F., & Crespo, R. (2013). Abduction in economics: A conceptual framework and its model. Synthese, 190, 4215–4237.

    Article  Google Scholar 

  • Vassilakis, S. (1991). Functorial fixed points: A non-technical introduction. Working Paper 266, Department of economics, University of Pittsburgh.

  • Vassilakis, S. (1991). Rules for changing the rules. Technical Report 32, Stanford Institute for Theoretical economics. Stanford University.

  • von Neumann, J. (1960). The mathematician. In J. Newman (Ed.), The world of mathematics (Vol. IV). London: Allen & Unwin.

    Google Scholar 

  • Weintraub, E. R. (2002). How economics became a mathematical science. Durham and London: Duke University Press.

    Book  Google Scholar 

  • Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and applied mathematics, 13, 1–14.

    Article  Google Scholar 

  • Yasugi, M., & Passell, N. (2003). Memoirs of a proof theorist: Gdel and other logicians. Singapore: Word Scientific.

    Book  Google Scholar 

Download references

Acknowledgments

Previous versions of this work (partially funded by Conicet, through Grant PIP 11220110100804) were presented at the XLVIII Annual Meeting of the Asociación Argentina de Economía Política (Rosario, Argentina, November 13, 2013) in a round table on “Mathematics in Economics” and at the CHESS Seminar, in the Department of Philosophy of the University of Durham (UK) on October 15, 2014. We are deeply grateful for the comments and criticisms we received from Nancy Cartwright, Enrique Kawamura, Erin Nash, Wendy Parker, Julian Reiss and Jorge Roetti, as well as from two anonymous referees, which allowed us to improve the quality of the paper. Of course, any remaining faults are our own responsibility.

Author information

Authors and Affiliations

  1. IAE-Universidad Austral and CONICET, Mariano Acosta s/n, 1630, Pilar, Argentina

    Ricardo Crespo

  2. INMABB (CONICET) and Universidad Nacional del Sur, 12 de Octubre y San Juan, 8000, Bahía Blanca, Argentina

    Fernando Tohmé

Authors
  1. Ricardo Crespo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fernando Tohmé
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ricardo Crespo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Crespo, R., Tohmé, F. The Future of Mathematics in Economics: A Philosophically Grounded Proposal. Found Sci 22, 677–693 (2017). https://doi.org/10.1007/s10699-016-9492-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10699-016-9492-9

Keywords

Search

Navigation