The Future of Mathematics in Economics: A Philosophically Grounded Proposal

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.

[1]  L. Brouwer,et al.  Intuitionism and formalism , 1913 .

[2]  H. Barger.  The General Theory of Employment, Interest and Money , 1936, Nature.

[3]  J. Keynes,et al.  The General Theory of Employment, Interest and Money. , 1936 .

[4]  D. Gasking,et al.  Mathematics and the world , 1940 .

[5]  E. Wigner.  The Unreasonable Effectiveness of Mathematics in the Natural Sciences (reprint) , 1960 .

[6]  A. Tarski,et al.  Logic, Methodology and Philosophy of Science Proceedings of the 1960 International Congress , 1962 .

[7]  W. Leontief.  Theoretical Assumptions and Nonobserved Facts , 1971 .

[8]  H. Putnam.  What is mathematical truth , 1975 .

[9]  Ernst Snapper,et al.  The Three Crises in Mathematics: Logicism, Intuitionism and Formalism , 1979 .

[10]  R. Goldblatt.  Topoi, the Categorial Analysis of Logic , 1979 .

[11]  R. Bhaskar,et al.  How to decide how to decide , 1983 .

[12]  J. Barnes.  The Complete Works of Aristotle , 1984 .

[13]  W. Feinberg,et al.  On After Virtue , 1984 .

[14]  Morris Kline.  Mathematics and the Search for Knowledge , 1985 .

[15]  S. Zamir,et al.  Formulation of Bayesian analysis for games with incomplete information , 1985 .

[16]  P. Kitcher,et al.  An opinionated introduction , 1988 .

[17]  Aristotle,et al.  THE NICOMACHEAN ETHICS , 1990 .

[18]  S. Vassilakis.  Rules for Changing the Rules , 1991 .

[19]  G. Debreu.  The Mathematization of Economic Theory , 1991 .

[20]  Barton L. Lipman.  How to Decide How to Decide How to. . . : Modeling Limited Rationality , 1991 .

[21]  Karl Svozil,et al.  Randomness and undecidability in physics , 1993 .

[22]  Eddie Dekel,et al.  Hierarchies of Beliefs and Common Knowledge , 1993 .

[23]  Joachim Lambek.  Are the traditional philosophies of mathematics really incompatible? , 1994 .

[24]  D. Harrison,et al.  Vicious Circles , 1995 .

[25]  Leo Corry,et al.  David Hilbert and the axiomatization of physics (1894–1905) , 1997 .

[26]  F. William Lawvere,et al.  Conceptual Mathematics: A First Introduction to Categories , 1997 .

[27]  Jan J. M. M. Rutten,et al.  Automata and Coinduction (An Exercise in Coalgebra) , 1998, CONCUR.

[28]  P. Samuelson.  How Foundations Came to Be , 1998 .

[29]  Roger E. Backhouse,et al.  If Mathematics is Informal, Then Perhaps we Should Accept That Economics Must be Informal Too , 1998 .

[30]  W. Arthur,et al.  Complexity and the economy , 2014, Science.

[31]  Bourbaki's Art of Memory , 1999, Osiris.

[32]  Jan de Leeuw,et al.  MODELS OF DATA , 2000 .

[33]  Israel Kleiner,et al.  Modern Algebra and the Rise of Mathematical Structures , 2001 .

[34]  James J. Heckman,et al.  Econometrics and empirical economics , 2001 .

[35]  Bruce J. Caldwell,et al.  How Economics Became a Mathematical Science , 2020 .

[36]  REASON AND REALITY IN THE METHODOLOGIES OF ECONOMICS - AN INTRODUCTION , 2002 .

[37]  Leo Corry.  David Hilbert y su Filosof´ia Empiricista de la Geometr´ia , 2002 .

[38]  E. Hussey.  Aristotle and Mathematics , 2002 .

[39]  Mariko Yasugi,et al.  Memoirs of a Proof Theorist: Gödel and Other Logicians , 2003 .

[40]  N. Giocoli.  Modeling Rational Agents , 2003 .

[41]  Marcel Boumans,et al.  How economists model the world into numbers , 2004 .

[42]  Lawrence S. Moss,et al.  Harsanyi Type Spaces and Final Coalgebras Constructed from Satisfied Theories , 2004, CMCS.

[43]  Joachim Lambek,et al.  What is the world of mathematics? , 2004, Ann. Pure Appl. Log..

[44]  Lectures on the Langlands program and conformal field theory , 2005, hep-th/0512172.

[45]  Joel Michell,et al.  The logic of measurement: A realist overview , 2005 .

[46]  N. Giocoli.  Blanqui Lecture - in the Sign of the Axiomatic Method: Mathematics as the Role Model for Neoclassical Economics , 2006 .

[47]  David Ellerman,et al.  Adjoints and emergence: applications of a new theory of adjoint functors , 2007 .

[48]  F. Stephan,et al.  Set theory , 2018, Mathematical Statistics with Applications in R.

[49]  Edward Hermann Haeusler,et al.  Using the Internal Logic of a Topos to Model Search Spaces for Problems , 2007, Log. J. IGPL.

[50]  John Earman,et al.  Handbook of philosophy of science , 2007 .

[51]  R. Kraut.  The Blackwell Guide to Aristotle's Nicomachean Ethics , 2008 .

[52]  Yoav Shoham,et al.  Multiagent Systems - Algorithmic, Game-Theoretic, and Logical Foundations , 2009 .

[53]  How to Justify Ethical Propositions: Aristotle's Method , 2008 .

[54]  Andreas Doering,et al.  “What is a Thing?”: Topos Theory in the Foundations of Physics , 2008, 0803.0417.

[55]  Robert L. Constable.  Computational type theory , 2009, Scholarpedia.

[56]  H. Gintis.  The Bounds of Reason: Game Theory and the Unification of the Behavioral Sciences , 2014 .

[57]  The unification of Mathematics via Topos Theory , 2010, 1006.3930.

[58]  Andrei Khrennikov,et al.  Ubiquitous Quantum Structure: From Psychology to Finance , 2010 .

[59]  Andrei Khrennikov,et al.  Ubiquitous Quantum Structure , 2010 .

[60]  Alexander Katovsky,et al.  Category Theory , 2010, Arch. Formal Proofs.

[61]  How economic methodology became a separate science , 2011 .

[62]  Till Düppe.  How economic methodology became a separate science , 2011 .

[63]  Giacomo Bonanno,et al.  Epistemic Foundations of Game Theory , 2012 .

[64]  A. Subramaniam.  The Mathematician , 2012 .

[65]  Economic Theory and Mathematics—An Appraisal , 2012 .

[66]  D. Hamermesh.  Six Decades of Top Economics Publishing: Who and How? , 2012 .

[67]  Andrei Rodin,et al.  Axiomatic Method and Category Theory , 2012, 1210.1478.

[68]  Fernando A. Tohmé,et al.  Abduction in economics: a conceptual framework and its model , 2013, Synthese.

[69]  Ilarion V. Melnikov,et al.  Introduction to representation theory , 2014 .

[70]  C. Flori,et al.  Homotopy Type Theory : Univalent Foundations of Mathematics , 2014 .

[71]  David I. Spivak.  Category Theory for the Sciences , 2014 .

[72]  Eric Pacuit,et al.  Epistemic Foundations of Game Theory , 2015 .

[73]  Fernando A. Tohmé,et al.  Abduction: A categorical characterization , 2015, J. Appl. Log..

[74]  Monya Baker,et al.  Over half of psychology studies fail reproducibility test , 2015, Nature.

[75]  W. Quine,et al.  On What There Will Be , 1948, Realisms Interlinked.

[76]  C. Juchem-Grundmann,et al.  Introduction , 2019 .

[77]  T. Shakya.,et al.  Historical Introduction , 2002, Jewish Identities: Fifty Intellectuals Answer Ben-Gurion.