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.
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Notes
The pervasiveness of this problem prompted a large initiative to address it, leaded by Brian Nosek (Baker 2015).
About their close connection, see Yasugi and Passell (2003).
“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).
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).
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).
An homotopy formalizes the intuitive idea of deformability of one mapping into another.
See the thorough discussion of the differences between the axiomatic method proper of ZFC and category theory in Rodin (2014).
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).
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).
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.
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).
“The term ‘supertask’ [\(\ldots \)] designates [\(\ldots \)] an infinite number of actions performed in a finite amount of time” (Pérez Laraudogoitia 2013, Introduction).
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).
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).
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).
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).
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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.
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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
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DOI: https://doi.org/10.1007/s10699-016-9492-9