The Limits of Mathematical Modeling in the Social Sciences

The following sections are included:

• Dedication
• Introduction
• About the Authors
• Contents

The following sections are included:

• Introduction
• Axiom systems: mathematics
• The Gödel phenomenon in physics and in other mathematized sciences
• Physics as an archetype for the mathematized sciences
• Axiomatics for physics: preliminary steps
• Axiomatics for physics: guidelines
• Suppes predicates
• Axiomatics for physics: the main ideas
• Axiomatics for physics: examples
• Beyond physics
• The incompleteness of analysis
• Generalized incompleteness
• Higher degrees
• θ functions and the arithmetical hierarchy
• Statement of the main undecidability and incompleteness results
• Questions settled with those techniques
• Undecidability and incompleteness in the social sciences
• Forcing and our techniques
• Evaluation of the results
• More on population dynamics
• Appendix A. Preliminary remarks
• Appendix B. Technicalities
• Appendix C. Hard stuff
• Appendix D. The counterexample function f
• Appendix F. An example
• Appendix G. Discussion and more intuitions
• Appendix H. An application: Maymin’s theorem
• Appendix I. Details
• Appendix J. The O’Donnell algorithm
• Appendix K. Almost Maymin-efficient markets
• Appendix L. A wild idea: inaccessible cardinals?
• References

What follows is based on a talk originally given by the author at the Hebrew University in Jerusalem and then, in expanded form, at the Perimeter Institute in Canada…

Karl Menger’s aim as a scholar was to find a new kind of logic to be applied to social sciences (Menger, 1979). As a mathematician, he focused his attention on finding whether the building of social order can be described from a formalistic point of view. This chapter deals with his contributes to find which kind of logic — able to include uncertainty — is used by individual and by social groups when cohesive aggregations are formed in order to get a formal way to describe ethics…

Chaitin, da Costa and Doria (2012, p. 16) say, “The Gödel–Turing phenomenon is an all-pervasive fact that has to be understood in order to be conquered and usefully mastered.” But the Gödel–Turing phenomenon is subversive of power and authority. It has, therefore, been effectively excluded from mainstream economics, which serves existing powers, at least in macroeconomics. Mainstream economists are experts, and monopoly experts have an incentive to act like the seers of Ancient Greece. But if we take the self-referential logic of Gödel–Turing seriously, the expert is put back on a plane of equality with the nonexpert. The implications of Gödel–Turing in social science, in other words, are democratic and antihierarchical. The deeply democratic message of this paper, however, is hostile to the interests of experts and thus unlikely to find a place in mainstream economics in the foreseeable future. If we do not conquer the Gödel– Turing phenomenon, we may fall into magical thinking. By “magical thinking,” I mean an argument with one or more steps that require something impossible. Unfortunately, experts often have an incentive to engage in magical thinking. Experts, including economic experts, are therefore unlikely to engage or even recognize the Gödel–Turing phenomenon. I begin with an example…

The following sections are included:

• Introduction
• Overview of complex adaptive systems
• CASs, modeling, and computation
• Applications of ABMs
• Classical logic and its (possible) discontents
• CASs and real-world decision making
• What is rationality?
• The role of heuristics
• Summary and conclusion
• References

The following sections are included:

• Introduction
• What do cosmological epistemics have to do with the social sciences?
• A review of intuitionistic logic and constructivism
• Logical contradictions
• Conclusion
• References

The following sections are included:

• Introduction
• Economics and the functions of science
• Characteristics of modern science
• Logical foundations of conservation laws
• Does economic theory respect conservation laws?
• Perpetual growth
• Synopsis
• Learning from science
• Conclusion
• Acknowledgments
• References

The following section is included:

• Index