In its latest reincarnation, these destructive ideas are being recast in the framework of
measure theory, the mathematical theory of measure spaces and measurable sets. Set theory, particularly the theory of convex sets, as well as real analysis (the analysis of relations among variables whose values are real numbers) and optimization, which used to be among the most general frameworks used by the economists a couple of generations ago (e.g. in the development of general equilibrium analysis and its derivations), have been completely absorbed as pieces within the broader mathematical framework of measure theory.
The study of stochastic processes, which underpins much of the empirical research in macroeconomics and finance economics, used to stand separate from — if not at odds with — abstract economic theory. But as things have turned out, the mathematics of stochastic processes, which resulted from the development of axiomatic probability theory, is precisely measure theory. In the language of measure theory, random variables (a generalization of the notion of a variable to account explicitly for the shifting limits of one’s cognition) are instances of measurable functions over a peculiar algebraic space, while probabilities are measures, i.e. a generalization of the intuitive geometric notion of length.
But, aside from probabilities, the concept of measure is so general — and the mathematical results established in the field are so intellectually potent — that virtually every conceivable notion in economics (e.g. space, time, quantities, prices, etc.) can be all elegantly subsumed under it. With the help of measure theory, probability theory being — again — one of its special cases, the whole mathematical paraphernalia that economists use today has now been placed within this new, unified mega-framework.
This is another one of Hegel’s historical ironies. Although the rudiments of measure theory began with the work of Borel and Lebesgue in early 20th century’s France,
Soviet mathematicians elevated it to higher levels of rigor and generality. (Let me remind you here that, originally, the
soviets — like
Occupy Wall Street today — were emergent, vibrant civic structures that plain workers and soldiers in political motion during the 1905 Russian revolution forged to collectively direct the course of history.) Building on those rudiments, and on the work of Russian mathematicians (e.g. Andrei Markov), the Soviet academic Andrei Kolmogorov developed the modern axiomatic edifice of probability theory, on which he built his analysis of stochastic processes. Kolmogorov himself, as well as Vladimir Smirnov and a bunch of lesser known Soviet mathematicians built a spectacular edifice that, paradoxically, by this Hegelian historical twist, got appropriated by, among others, Western economic theorists, who then used it in the development of modern finance and macroeconomics.
