Mathematical models

The new term has begun, and so has my new course, Mathematical Models for the Physical, Natural and Social Sciences. This is a course offered to master students in Mathematics, in particular those enrolled in the Teaching and Science Communication curriculum.

In the first class, we discussed what a mathematical model is, and what it is not. To begin with, we had some examples of mathematical models suggested by the students: the SIR model, ever-present in these days, the optimisation of industrial processes, the optimisation (or hedging) of an investment portfolio. We continued with some questions:

  1. What are the questions that we want to answer with our model? Or, in other words, why are we considering a mathematical model in the first place?
  2. What are our variables and our data? Which kind of data can we have?
  3. Where does data come from? Why is it relevant?
  4. Who develops a mathematical model?

At this point, we went back to our quest for a definition of a mathematical model, with a tentative explanation, involving both a mathematical formulation and the real-world phenomenon, and some kind of interaction between the two.

The next order of business was errors. When is a model wrong? Where do these errors come from? How can we react to errors?

We concluded by discussing a flowchart suggested by Hugh Burkhardt , stressing the importance of the different phases and of continuous updating of the models, in response to errors made in the predictions.

Steps to a mathematical model, Hugh Burkhardt, 1979

The definition we closed with is the following one, coming from the field of mathematical models for econometry:

A mathematical model is the formal representation of ideas and knowledge of a phenomenon.

Edmond Malinvaud, Méthodes statistiques de l’économétrie, 1964

Next time: Bayes’ theorem.


Malinvaud E (1964) Méthodes statistiques de l’économétrie, Paris, Dunod.
Burkhardt H (1981) Real World and Mathematics, Blackie-Birkhäuser.
Burkhardt H (2006) Modelling in Mathematics Classrooms: reflections on past developments and the future. Zentralblatt für Didaktik der Mathematik 38: 178–195.

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