8 - Preferences and Beliefs

Summary

INTRODUCTION

The great theorems of social mathematics discovered during the twentieth century can be separated into those that emphasize equilibrium and those that hint at chaos, inconsistency, or irrationality.

The equilibrium results all stem from Brouwer's Fixed Point theorem (Brouwer, 1910): A continuous function from the ball to itself has a fixed point. The theorem has been extended to cover correspondences (Kakutani, 1941) and infinite-dimensional spaces (Fan, 1961) and has proved the fundamental tool in showing the existence of equilibria in games (von Neumann, 1928; Nash, 1950, 1951), in competitive economies (von Neumann, 1945; Arrow and Debreu, 1954; McKenzie, 1959; Arrow and Hahn, 1971), and in coalition polities (Greenberg, 1979; Nakamura, 1979).

The first of the inconsistency results is the Gödel-Turing theorem on the decidability-halting problem in logic (Gödel, 1931; Turing, 1937): Any formal logic system (able to encompass arithmetic) will contain propositions whose validity (or truth value) cannot be determined within the system. Recently this theorem has been used by Penrose (1989, 1994) to argue against Dennett (1991, 1995) that the behavior of the mind cannot be modelled by an algorithmic computing device. A version of the Turing theorem has been used more recently to show that learning and optimization are incompatible features of games (Nachbar, 1997, 2001, 2005). There is still controversy over the meaning of the Gödel theorems, but one interpretation is that mathematical truths may be apprehended even when no formal proof is available (Yourgrau, 1999; Goldstein, 2005).