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Nancy Cartwright introduced an interesting distinction with regard to modeling of physical phenomena. According to Cartwright, a mathematical theory is not applied directly to such phenomena. Rather, one first builds a basic mathematical model of the phenomenon in question, in a step Cartwright refers to as phenomenological model-building (I think she uses "phenomenological" in the sense of "empirical" but I may be wrong). Only then does one exploit a full-fledged mathematical theory, applied not to the original phenomenon but rather to the basic mathematical model. Since I am not familiar with the relevant literature, I am wondering if this distinction has been explored further in recent work.

The relevant papers are the following:

Cartwright, Nancy; Shomar, Towfic; Su'arez, Mauricio. The tool box of science. Tools for the building of models with a superconductivity example. Poznan Studies in the Philosophy of the Sciences and the Humanities 44 (1995), 137--149.

Su'arez, Mauricio; Cartwright, Nancy. Theories: Tools versus models. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, Vol. 39, Issue 1 (2008), 62--81. https://doi.org/10.1016/j.shpsb.2007.05.004.

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    $\begingroup$ I'm not down-voting or voting to close, but to me this line of inquiry seems like a part of the philosophy of science, and not a part of mathematics per se. $\endgroup$
    – Sam Hopkins
    Commented Jul 6, 2023 at 18:45
  • $\begingroup$ Well it certainly has to do with mathematical modeling in mathematical physics. We have both a tag for mathematical physics, and a tag for reference request, which is what this question is. $\endgroup$
    – Mikhail Katz
    Commented Jul 6, 2023 at 18:47
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    $\begingroup$ @SamHopkins, I myself tend to think that mildly philosophical questions about mathematics can be part of mathematics, and needn't be partitioned-off... since mathematicians' attitudes/philosophies about things certainly have an effect on the course of events ... $\endgroup$
    – paul garrett
    Commented Jul 8, 2023 at 19:21
  • $\begingroup$ The distinction is interesting but I might point out that physicists themselves are usually not aware of philosophical approaches to physics and model building and prefer to figure out how things work by getting their hands dirty (they are the ones that know where the shoe bites because they wear it, to paraphrase something Weinberg said). $\endgroup$
    – Hollis Williams
    Commented Aug 31 at 19:49
  • $\begingroup$ An example is Popper’s “falsifiability criterion” which was created by a philosopher but does not really work in practice. $\endgroup$
    – Hollis Williams
    Commented Aug 31 at 19:57

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Cartwright's case study, model building for the theory of superconductivity, has been explored further in the Ph.D.thesis The Role of Concrete Models in the Revolution in Superconductivity (A. Chattoraj, 2015).

More generally, one of Cartwright's former students, Roman Frigg, has developed this approach in a monograph, Models and theories: A philosophical inquiry (2022).

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  • $\begingroup$ Is it correct to describe Cartwright's insight as follows? London and London developed a phenomenological model of superconductivity. Eventually (two decades later?), what is considered a proper theoretical explanation at the molecular level was developed. However, Cartwright stresses the importance of the model, and notes that sometimes the significance of models is not sufficiently appreciated when people think of theory as explaining the phenomena directly. ?? $\endgroup$
    – Mikhail Katz
    Commented Jul 13, 2023 at 8:49
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    $\begingroup$ as a physicist, this distinction between a "model" and a "theory" does not resonate with me; what we seek is an "explanation"; the fact that a superconductor has zero resistance was known since 1911; we had to wait until 1957 for an explanation why (BCS theory); London&London in 1935 described the 1911 observations in terms of mathematical equations, without connecting these equations with established physical foundations; their equations "came out of the blue"; this is why their work is not considered an explanation of superconductivity. $\endgroup$
    – Carlo Beenakker
    Commented Jul 13, 2023 at 8:59
  • $\begingroup$ I see, thanks. Is it generally thought that the London-London "model" (if you allow me to use that term) was helpful in eventually motivating the BCS theory two decades later? $\endgroup$
    – Mikhail Katz
    Commented Jul 13, 2023 at 9:03
  • $\begingroup$ P.S. I really know nothing about this, but the point of the London-London 1935 thing seems to have been not to explain zero resistance (known since 1911), but rather to explain zero magnetic field (known only since 1933). Is this accurate? $\endgroup$
    – Mikhail Katz
    Commented Jul 13, 2023 at 9:51
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    $\begingroup$ yes, this is accurate (although the two are intimately related: the magnetic field is "expelled" from the superconductor by an electrical current which does not decay because the resistance is zero); the point I wish to make is that the London equation describes the field expulsion by adding "by hand" an equation to the Maxwell equations of electrodynamics, without explaining where this equation comes from; the BCS theory derives the London equation, which is why for a physicist the BCS theory explains superconductivity while the London equation does not. $\endgroup$
    – Carlo Beenakker
    Commented Jul 13, 2023 at 10:05

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