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Nancy Cartwright introduced an interesting distinction in the context of her study of the history of the evolution of our understanding of superconductivity. She emphasized the distinction between mathematical theories (such as the microscopic BCS theory developed in 1957) and basic mathematical models (such as the London-London model of 1935). According to Cartwright, the emergence of a full-fledged mathematical theory often depends on first developing a basic mathematical model, 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). Of course the full-fledged mathematical theory is applied not to the original phenomenon but rather to the basic mathematical model. I am wondering about the following.

Has this distinction 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.

The following useful input has been provided at the MO.SE site:

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).

Any further input would be appreciated.

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    Based on your description it's not clear why contemporary philosophers need to worry too much about this model vs (reverse) theory building iterative process which has always been a major pattern in scientific theory progress via statistical modeling fit to observed data even including paradigm shift cases, a more famous example would be the insightful phenomenal modeling of Lorentz transformation based on MM experiment results and Einstein's later special relativity theory. It would be interesting to see anything special about your case from your reading and understanding of your reference... Commented Jul 20, 2023 at 23:30
  • @DoubleKnot, thanks for your comment. I assume you are using the term "phenomenal modeling" in Cartwright's sense (rather than the generic meaning). Could you provide a link describing "phenomenal modeling of Lorentz transformation based on MM experiment results and Einstein's, etc." in more detail? Commented Jul 24, 2023 at 7:29
  • There're many references about their links in physics literature, you may take a look at this short essay Michelson, FitzGerald and Lorentz: the origins of relativity revisited. Peano's (1st order) Arithmetic theory which seems precisely and naively applied to the perennial straightforward model of natural numbers is also a famous example in math which shows their links and gave us many more different nonstandard models... Commented Jul 25, 2023 at 21:20
  • @DoubleKnot, I tried reading Brown's text but found it hard to follow, lacking sufficient background. I would like to understand your comments about phenomenal modeling of Michelson-Morley experiment better. Also, could you elaborate on your comments concerning Peano Arithmetic? Sounds intriguing. Commented Jul 26, 2023 at 8:03

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