"Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analysing, to greater abstractness and logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced." -- Bertrand Russell, Introduction to Mathematical Philosophy
Many economists may be under the mistaken impression that casting economics into mathematical models ensures assumptions are explicitly stated. This is manifestly false. Here are some examples of questions about mathematical assumptions that might puzzle some economists:
- What quantity is being conserved in typical models with agents maximizing under constraints?
- Do equilibrium models with rational expectations apply when economic time series are non-ergodic?
- How would the agents in such models learn to estimate model parameters if dynamics are chaotic?
- Do models linearized around an equilibrium apply to models with multiple equilibria? Would not interesting bifurcations arise if the number of equilibria varies with model parameters?
- Have economic models been successfully tested by dimensional analysis? (One of my favorite critiques directs one to question using a measure of capital goods in numeraire units in production functions.)
- Can economic models with utility-maximizing agents handle preferences changing not randomly, but by agents imitating what they see other agents consuming?
- What more general ideas and principles (consider, for example, menu independence and the absence of lexicographic preferences) underlie utility maximization?