Economics emerged as an autonomous science in 19th century. The founders were ambivalent with respect to the real nature of their field. It belonged on the one hand to moral and political philosophy. But the phenomena it was concerned with were all numeric: price, stock, revenue, interest, growth. We can find numbers in Ricardo, Smith and Marx, but the real problems of economy were highly philosophical. Numbers were raw materials to be morally and politically interpreted. Their theory of value conduced necessarily to class awareness. Jevons, father of the neoclassical approach attempted to fully mathematize economic phenomena. Together with the marginalist approach, mathematization completely erased the political dimension of economy. However, he was right in pointing out that a central issue in economics was to explain the transit from quality to quantity, from subjective valuation, expressed in intensities or at least ordinally, to objective prices expressed in real numbers. He was aware of the advances in non-Euclidian geometry and even polemized with Helmholtz about it. This is crucial, because since Gauss and Riemann, a space is prior to its parametrization. In other words, different metrics are possible on the same space. It is a matter of decision, according to the mathematical operations we need. Fundamental structures can be very basic and abstract. One of the most important problems in 20th century mathematics was algebraic topology. It allowed us to capture information from topological spaces through algebraic objects. Topological spaces are continuous and have no metric attached to them. Algebraic objects are discrete structures, defined by certain operations. An algebraic structure like a ring allows us to capture information about the holes of a topological space (like a torus). This is an example of mapping, but among different mathematical objects. In any case, the goal consists in reading an object through another. But this doesn’t mean that we capture the essence of any object. The other issue is the relationship global-local. Non-Euclidian spaces can be locally considered Euclidian. This has an obvious impact in the difference between micro and macroeconomics.
When we apply mathematics to real economic phenomena, we must decide which features we want to capture. The real question is thus: what do we capture when we jump from desire to objective price, from quantity to quality? Years later, the Cambridge controversy on the nature of capital faced another mathematical problem: that of counting. “Capital” is a complex entity. To be able to measure it, we need to individuate its components and assign them a value expressed in price terms. But this is what we wanted to explain in the first place: how price is determined by factors of production. The issue at stake is the same: there is a secret mathematical rule that makes counting and certain operations possible. But this rule is never explicit. We end up with Marx again: commodities in our times have both a use and an exchange value, they are inseparable; and yet there is no natural rule to go from one to the other. How much corresponds to the capitalist is decided, not mathematically deduced.