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Is there a problem with mathematics?

Since the call was launched, the debate has been rerouted on mathematics. So as to treat us as idiots: my dear children, you haven’t understood that mathematics are neutral, that they have nothing to do with ideology, that they are not questionable in themselves, that there are the very model of rigor.

Obviously, the problem does not lie in mathematics, which are the way they are. The problem lies in the RELEVANCE of the models and theories that use them, and it is this relevance that we contest - and we have not been answered.

Let us take a simple example: the growth models and models of "real" business cycles (RBC), which now hold an important place in the teaching of macroeconomics. These models all presuppose that the economy is narrowed down to a single individual - sometimes called Robinson, sometimes "representative agent" . Not only does he have to choose how much he is going to produce, consume and invest - but also how he is going to divide his time, between work and leisure - and thus make decisions for his entire life once and for all.

The (inter-temporal) tastes of this individual are represented by a mathematical function (called "utility function"). So are the techniques he has at his disposal (in the present and in the future), by another function (this is the "production function"). Given these assumptions, the problem is of a mathematical nature: finding the "production-consumption-work (or leisure)" trajectory that maximizes our individual’s (inter-temporal) utility function, given the limited number of resources and techniques that he has at his disposal. This kind of problem is treated by a branch of mathematics - called either "variation calculus", or "optimal control" - which is used, for example, when one seeks to determine the best trajectories to put a satellite into orbit, minimizing fuel expenses.

Optimal control resorts to relatively complex mathematical objects (functionals, Hamiltonians, differential equations); it generally only allows one to obtain approximate solutions - generated by more or less complicated algorithms. The confrontation of these solutions with available data also raises problems of a statistical nature (calibration, construction of tests).

Of course, NOBODY QUESTIONS THE VALIDITY OF THESE VARIOUS MATHEMATICAL TECHNIQUES IN AND OF THEMSELVES - doing so would be ridiculous ! The question we raise is completely different: it is that of the relevance of the model studied. Is it relevant to study the actual evolution of a whole country’s economy as if it reflected the choices of a lonely individual? The answer is obvious: NO. In that case, what is the point of overburdening the student with the methods of "optimal control", the statistical techniques of confrontation of Robinson’s "intertemporal choices" with the available data on the GDP evolution of such and such country? According to us, there is none.

If we are wrong on this subject, then let somebody prove it: IT IS AT THIS LEVEL THAT THE DEBATE MUST BE SITUATED, and not at any other.

We have taken the example of the "representative agent", very fashionable these days. But we could also have taken the consumer and producer theories, in microeconomics, which speculate as much on fictive entities: let someone prove to us that these speculations are relevant, and we are ready to accept their mathematical developments. But, please, do not let them be used to divert the students’ attention, in leading them to believe that they are the problem, whereas they pose none.

 
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