Is temperature a continuous variable?

Philosophers of science (and even scientists) often endorse the continuum picture, that is, they argue that continuity is essential for many accounts of scientific representation, explanation and understanding: this is called the continuum fallacy.

One version of the fallacy has it that continuum models are indispensable, ‘in principle’, because many macroscale phenomena cannot be explained in purely reductive, microphysical terms, and can instead be explained only by positing continuum models at the macroscale. That is, continuum models play an indispensable role in scientific explanations despite being effectively decoupled from the microphysical details of the systems they describe.

Another version of the fallacy relates to a pragmatic view of continuum models, that is, the justification for such models comes from their mathematical convenience and empirical adequacy, as discontinuous representations are generally intractable. A further version of the fallacy has it that the discontinuities apparent in scientific representations may not really be there when the associated physics is parsed out carefully, and, thus, all things considered, continuity seems to be the norm.

By focusing on the anomalous case of temperature discontinuities, we argue that the continuum view is fallacious because:

i) continuum models at the macro level are not necessarily decoupled from the microscopic features of a physical system, making continuum models empirically inadequate and inapplicable in many situations of current scientific interest;

ii) the evidence of temperature discontinuities runs from the macroscopic to the microscopic – that is, they are present in both the data (experiments and simulations) and the phenomena (theories and models) pertaining to thermal systems; and

iii) the discontinuous modelling of temperature profiles can not only be more empirically adequate than continuum modelling but also provide us with a better scientific understanding of the underlying thermal phenomena.

The upshot is that: a) continuum models are not indispensable in describing physical phenomena; b) temperature is not necessarily a continuous function in our best scientific representations of the world; and c) that its continuity, where applicable, depends on various microphysical factors. A note of caution is that although the mathematical models of these variables may show discontinuities, these need not indicate any deep metaphysical results