Many disciplines articulate their internal structure mathematically. Both classical thermodynamics and general competitive analysis are such, for each applies mathematics to define equilibria and to assess an equilibrium’s stability. Both fields employ strikingly similar reasoning, as opposed to mechanics, to construct stability arguments. In particular, both depend upon Liapounov’s second or direct method rather than solving for disequilibrium trajectories. By comparing these two applications, this essay demonstrates that, although the logic in both fields is the same, the meaning attached to that logic and the metaphors it voices in each diverge profoundly. Indeed, their respective allegories of natural processes are antithetical. By examining the similarities and differences in these two cases. I hope to elucidate the role of applied mathematics in each. I conclude that although the lens of mathematical argumentation tints one’s concepts—in this case most notably one’s notion of time itself-it does not impose a unique story.
As is well known, tracing the solution of a system of nonlinear differential equations is, in general, intractable. Consequently, scientists widely rely upon qualitative techniques to acquire information about systemic behavior. Liapounov’s stability theorems (1947) are prominent among these alternatives. His method is particularly helpful because it avoids the need to solve the differential equations.
Following La Salle and Lefschetz (1961), for an autonomous system of differential equations
where x is a real vector and t a real variable representing continuous time, the origin is a steady state equilibrium. Its stability within a spherical region S(A)