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Unrealistic Models in Mathematics

D'Alessandro, William (2022) Unrealistic Models in Mathematics. [Preprint]

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Abstract

Models are indispensable tools of scientific inquiry, and one of their main uses is to improve our understanding of the phenomena they represent. How do models accomplish this? And what does this tell us about the nature of understanding? While much recent work has aimed at answering these questions, philosophers' focus has been squarely on models in empirical science. I aim to show that pure mathematics also deserves a seat at the table. I begin by presenting two cases: Cramér’s random model of the prime numbers and the function field model of the integers. These cases show that mathematicians, like empirical scientists, rely on unrealistic models to gain understanding of complex phenomena. They also have important implications for some much-discussed theses about scientific understanding. First, modeling practices in mathematics confirm that one can gain understanding without obtaining an explanation. Second, these cases undermine the popular thesis that unrealistic models confer understanding by imparting counterfactual knowledge.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
D'Alessandro, Williamd.william@lmu.de0000-0002-5451-079X
Keywords: models, modeling, unrealistic models, understanding, explanation, mathematical practice, number theory
Subjects: Specific Sciences > Mathematics > Epistemology
Specific Sciences > Mathematics > Methodology
Specific Sciences > Mathematics > Practice
General Issues > Explanation
Specific Sciences > Mathematics
General Issues > Models and Idealization
Depositing User: Dr. William D'Alessandro
Date Deposited: 02 Oct 2022 16:57
Last Modified: 02 Oct 2022 16:57
Item ID: 21225
DOI or Unique Handle: https://doi.org/10.3998/phimp.1712
Subjects: Specific Sciences > Mathematics > Epistemology
Specific Sciences > Mathematics > Methodology
Specific Sciences > Mathematics > Practice
General Issues > Explanation
Specific Sciences > Mathematics
General Issues > Models and Idealization
Date: 2022
URI: https://philsci-archive-dev.library.pitt.edu/id/eprint/21225

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