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Mathematics and Finance: Some Philosophical Remarks

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Abstract

I examine the role that mathematics plays in understanding and modelling finance, especially stock markets, and how philosophy affects it. To this end, I explore how mathematics penetrates finance via physics, constructing a ‘financial physics’, and I outline the philosophical backgrounds of this process, in particular the ‘philosophy of equilibrium’ and that of critical points or ‘out-of-equilibrium’. I discuss the main characteristics and a few weaknesses of these mathematizations of financial systems, notably econometrics and econophysics, and I compare the two ways, top-down and bottom-up, of building mathematical approaches to finance. The top-down approach is the most used and the most conservative. I argue that it guarantees a mathematical account, but it is less effective than the more difficult bottom-up approach, which is more appropriate and which may end up without a mathematical account of financial phenomena. I then consider two important issues raised by a mathematical approach to finance, that is, the performative and the reversing side of mathematics, and an ethics of mathematics. In the first case I argue that mathematics not only can be a performative device, but it also enables ‘reversing’ dynamics, that is, a mathematical model may become not a representational tool, but a device of social engineering—a mathematical way of investigating what are the initial conditions and processes needed to obtain an intended result. In the second case I argue that this specific relation between mathematical modelling and prediction raises an ethical question in finance, namely, a responsible construction and use of the mathematical models provided by a financial physics.

  1. 1.

    From physics to finance: the mathematical way

  2. 2.

    Finance, mathematics and the philosophy of equilibrium

  3. 3.

    Finance, mathematics and critical points

  4. 4.

    A mathematical access key?

  5. 5.

    Performativity: the mathematical forging of finance

  6. 6.

    An Ethics of mathematics in finance

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Notes

  1. The quantitative side deals with price movements (trajectories) and datapoints, while the qualitative side is about the variables (forces) that cause them (see Ippoliti 2017a, b, c).

  2. Truth be told, Peters (1994) and Keen (2011) note that FMH suggests an explanation of the conditions of stability of the markets that contradicts EMH. A market is stable when “it allows investors with different time horizons to trade smoothly. As a result, heterogeneity […] is a vital part” (Keen 2011, p. 341). In essence, for FMH the stability of the markets emerges from the fact that is that investors differ in their time horizons. This tells us also when instability is likely to occur: it happens when all investors suddenly switch to the same time horizon.

  3. A small measurement error in specifying the initial conditions of a fractal model grows exponentially in time. For instance, if your initial measure of a value is out by just 1/10th of a percent—say 456.4 instead of 456.5— after few iterations, the fractal model would be completely wrong and then useless as a means of predicting the following day’s value.

  4. See Brisset (2018), Svetlova (2012), Svetlova and Dirksen (2014).

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  1. Sapienza University of Rome, Rome, Italy

    Emiliano Ippoliti

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Ippoliti, E. Mathematics and Finance: Some Philosophical Remarks. Topoi 40, 771–781 (2021). https://doi.org/10.1007/s11245-020-09706-1

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