The Belgian mathematician Jean Mawhin (born 1942) maintains that "mathematical modelling is a difficult art, akin to caricature. The greatest possible likeness must be achieved with the fewest possible strokes. One must know how to disregard a priori what will prove a posteriori negligible". This is often true. But what about physics? Existing models increasingly agree with observations, leading some scientists to regard the model as an alternative expression of reality. In their minds, the two merge, becoming equivalent expressions of the same phenomenon. The model would thus provide access to a certain form of reality. But is that really the case? Admittedly, the increasing sophistication of our mathematical representations of reality yields results that agree ever more closely with observations, and with ever greater precision. Our models are improving. But have they thereby attained some kind of "perfection", whatever that term might mean? The very notion of a model calls for closer examination…
Giving up on completeness
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Following the Italian historian of science Giorgio Israël (1945–2015), we may call "any form of mathematical description of a class of phenomena" a "mathematical model". A model is a mathematical construct designed to represent some aspect of reality by taking into account certain relevant features and all the available data. In physics, we can regard a phenomenon as understood when we can produce a consistent mathematical representation of it in hypothetico-deductive form, with the assumptions taking the form of "laws"; the representation provides a concise description which, when calibrated and applied, yields results that do not deviate "too much" from observations.
The history of science shows us that the vast majority of models share certain properties. A mathematical model is a functional representation of some element of reality using abstract mathematical objects. It is also a selective representation of reality. To implement it, scientists retain only certain features of the situation under study and disregard others. In doing so, they relinquish the idea of arriving at a complete picture of reality. A mathematical model may also be a universal representation: the same equation may adequately describe several distinct phenomena.
Thus, the differential equation applies equally well to the motion of a mass at the end of a pendulum when displaced from its resting position, to that of a slightly stretched spring, or, more generally, to the behaviour of any periodic phenomenon in which the force under consideration—equal to mass multiplied by acceleration, that is, the second derivative of the displacement function—is assumed to be proportional to that displacement. A model may therefore be viewed as a mathematical framework that unifies an entire category of phenomena, any one of which it can represent. In fact, every situation can be analysed through different models, and every model can be applied to different situations.
