# Publications

## PhD Thesis

My PhD Dissertation in Philosophy, entitled **The Reasonable Effectiveness of Mathematics in the Natural Sciences**, was defended on December 7th, 2012, under the supervision of Robert W. Batterman (Professor of philosophy at the University of Pittsburgh) and John L. Bell (Professor of philosophy at the University of Western Ontario).

You can download my PhD Thesis here; you can also download it from Scholarship@Western.

The referees were Robert M. Corless (Professor of applied matheamtics at the University of Western Ontario), William L. Harper (Professor Emeritus in philosophy at the University of Western Ontario), Christopher Smeenk (Professor of philosophy at the University of Western Ontario), and Mark Wilson (Professor of philosophy at the University of Pittsburgh).

#### Abstract

Inquiries into the nature of mathematics as a science of its own and into its role in empirical science have a venerable tradition. The problem that is perhaps the most unsettling examines how mathematics can be used to adequately represent the world, given that mathematics displays a kind of exactness and necessity that appears to be in sharp contrast with the contingent character of worldly facts. Many scientists and philosophers maintain that this problem---which I shall name the problem of the applicability of mathematics---is condemned to remain intrinsically mysterious. For instance, Eugene Wigner famously claimed that the "miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." It is true that, because of the resilience of the tension between considerations bearing on pure and applied mathematics, the applicability of mathematics does bear an aura of mystery.

Be that as it may, part of the mystery, in my opinion, stems from gathering many problems that require different types of solutions under the same umbrella. Thus, I will articulate the problem of the applicability of mathematics (1) by relating it to other philosophical questions about mathematics and (2) by distinguishing three aspects of the problem of the applicability of mathematics, which I call the problem of mixed sentences, the problem of unexpected applicability, and the problem of uncanny accuracy. In the natural sciences, mathematical representations of systems of things are constructed on the basis of uncertainty, measurement error, modelling error, analytical approximations, computational approximations, and other forms of guesses and ignorance that nonetheless often provide extremely accurate representations of systems. On the basis of the commonsensical rule "garbage in, garbage out," this accuracy indeed appears to be uncanny. Thus, the problem of uncanny accuracy, on which the dissertation focuses, asks this question: Given that the construction and manipulation of mathematical representation in science is pervaded by uncertainty, error, and approximation, how can their apparently uncanny accuracy be explained?

To demystify the applicability of mathematics, one needs to have the means to capture its rationality. Thus, to make correct judgements about the use of mathematics in science, it is necessary to have a correct understanding of the "logic of mathematical modelling." However, as I argue, the standard logical reconstructions fall short of this task. Thus, it is necessary to revise or supplement our "rational reconstruction toolbox," for otherwise almost all of applied mathematical sciences would be wrongly considered methodologically unsound. Hence, this dissertation is a reflective study on the methods of mathematical reasoning actually used in applied mathematics, supplemented by normative conclusions regarding our ideal image of science. In addition to having philosophical implications for our normative ideal of science, the study of the problem of uncanny accuracy has consequences for the way in which philosophers envisage conceptual analysis, much in the way pioneered by Mark Wilson.

More specifically, the dissertation explains the strategies used to manage error and uncertainty without causing an overflow of information that amounts to sacrificing the tractability of representations. Those strategies also capture how to handle the physical theories underlying mathematical modelling and what the intrinsic role of error and uncertainty in the theory of measurements is. I capture the semantic and pragmatic facets of modelling with a concept of "selective accuracy" that covers the contributions of qualitative methods such as asymptotics, bifurcation analysis, renormalization, *etc*. In order to also integrate the contributions of quantitative methods for the analysis and assessment of mathematical models, I then turn to the nature of computational mathematics in the context of physical modelling and describe a wide-ranging method known as backward error analysis. I explain in what way this method allows us to understand wide variety of approximations: discretization, truncations of asymptotic series, and discrete "computer" arithmetic. Those qualitative and quantitative methods use mathematical machinery---based on error theory, sensitivity analysis, and perturbation theory---that go far beyond the logical and probabilistic tools predominantly discussed by philosophers. On this basis, I propose a concept of differential mathematical fitness of concepts by means of which it becomes possible to philosophically systematize patterns of successful applied mathematical reasoning in order to demystify the unreasonable effectiveness of mathematics.

#### Keywords

Applied Mathematics, Asymptotics, Backward Error Analysis, Error, Exact & Numerical Solutions, Logic of Mathematical Modelling, Mathematical Tractability, Philosophy of Mathematics, Philosophy of Science, Qualitative Behaviour, Rational Reconstruction, Scientific Theories, Selective Accuracy, Uncertainty, Unreasonable Effectiveness of Mathematics