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Semantic layering and the success of mathematical sciences

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Abstract

What are the pillars on which the success of modern science rest? Although philosophers have much discussed what is behind science’s success, this paper argues that much of the discussion is misdirected. The extant literature rightly regards the semantic and inferential tools of formal logic and probability theory as pillars of scientific rationality, in the sense that they reveal the justificatory structure of important aspects of scientific practice. As key elements of our rational reconstruction toolbox, they make a fundamental contribution to our understanding of the success of science.

At the same time, any science, however exact, is dominated by approximation, error, and uncertainty, a fact that makes one wonder how science can be so successful. This paper articulates and illustrates general themes—e.g., that truth-preserving arguments often fail to preserve approximate truth—that highlight the need for additional semantic resources. Thus, our proposal is that persistent failures to unravel the reasons behind the success of science in the face of pervasive error and uncertainty should be attributed to an insufficiently rich way of rationally reconstructing scientific and mathematical knowledge. What is missing? This paper claims that there is a third formal method of reasoning that constitutes a distinct pillar on which rests the success of science, namely, perturbation theory. The paper outlines how the representational and inferential tools of perturbation theory differ from those of logic and probability theory, and how they enable us to understand the apparently elusive aspects of the success of science.

However, compared to its peers, perturbative reasoning has not received the attention it deserves. As the paper explains, this partly results from the circumstances in which perturbation theory is taught, and partly from the fact that perturbation theory first appears to be a vaguely related collection of methods offering no systematic semantic insight. In an attempt to show that this first impression is wrong, this paper presents its contribution to the semantic dimension of scientific representation and inference in terms of what I call “semantic layering.”

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Notes

  1. A broader characterization of kinds of concepts has been developed as part of the so-called theory of measurements, starting with works such as (Scott and Suppes, 1958; Suppes & Zinnes, 1962).

  2. For readers interested in the technical details, here is how it might go. In this example, we first consider the complete ordered field \(\langle \mathbb {R},+,\times ,<\rangle \) as our reference mathematical structure and construct from it a structure \(\langle \mathbb {R},\oplus ,\otimes ,<\rangle \), where ⊕ satisfies the rules of significant-figures addition and ⊗ the rules of significant-figures multiplication. Up to the standard of accuracy typically employed in significant-figures arithmetic, the field axioms for ⊕ and ⊗ will be approximately satisfied. Thus, in the sense of ‘approximation’ relevant to this example, the structure we have constructed approximates the first. However, the important point is that it will not be true of all terms that are substitutable salva veritate in the real-number structure that they will be substitutable salva veritate approximata. I am not being too rigorous here since it is only a toy example; an example with floating-point arithmetic and computing polynomials in different bases would be more typical of the rigorous treatment that we would find in numerical analysis.

  3. Thanks to Naftali Weinberger for pointing this out.

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Acknowledgments

I would like to thank participants in the “Philosophy of Mathematics: Applicability, Practice, and Numerical Computations” session of the NUMTA 2019 conference. In particular, I would like to thank Davide Rizza for organizing the event, and for useful discussions of the version of the paper presented at the conference. I would also like to thank the Pittsburgh’s Center for Philosophy of Science fellows’ reading group, including Edouard Machery, Mike Schneider, Makmiller Pedroso, Diane O’Leary, and Edward Slowik, as well as Iman Ferestade, Travis McKenna, Milos Mihajlovic, Gilles Plante, Kesavan Thanagopal, and two anonymous reviewers for valuable feedback.

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This research was supported by SSHRC Insight Grant #435-2018-0242.

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Fillion, N. Semantic layering and the success of mathematical sciences. Euro Jnl Phil Sci 11, 91 (2021). https://doi.org/10.1007/s13194-021-00394-1

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