The Reasonable Effectiveness of Mathematics in the Natural Sciences

Nicolas Fillion
Simon Fraser University
One of the most unsettling problems in the history of philosophy examines how mathematics can be used to adequately represent the world. An influential thesis, stated by Eugene Wigner in his paper entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," claims 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." Contrary to this view, this thesis delineates and implements a strategy to show that the applicability of mathematics is very reasonable indeed. I distinguish three forms of the problem of the applicability of mathematics, and focus on one I call the problem of uncanny accuracy: Given that the construction and manipulation of mathematical representations is pervaded by uncertainty, error, approximation, and idealization, how can their apparently uncanny accuracy be explained? I argue that this question has found no satisfactory answer because our rational reconstruction of scientific practice has not involved tools rich enough to capture the logic of mathematical modelling. Thus, I characterize a general schema of mathematical analysis of real systems, focusing on the selection of modelling assumptions, on the construction of model equations, and on the extraction of information, in order to address contextually determinate questions on some behaviour of interest. A concept of selective accuracy is developed to explain the way in which qualitative and quantitative solutions should be utilized to understand systems. The qualitative methods rely on asymptotic methods and on sensitivity analysis, whereas the quantitative methods are best understood using backward error analysis. The basic underpinning of this perspective is readily understandable across scientific fields, and it thereby provides a view of mathematical tractability readily interpretable in the broader context of mathematical modelling. In addition, this perspective is used to discuss the nature of theories, the role of scaling, and the epistemological and semantic aspects of experimentation. In conclusion, we argue for a method of local and global conceptual analysis that goes beyond the reach of the tools standardly used to capture the logic of science; on their basis, the applicability of mathematics finds itself demystified.
Keywords No keywords specified (fix it)
Categories No categories specified
(categorize this paper)
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

Our Archive

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 46,425
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

How the Laws of Physics Lie.Nancy Cartwright - 1983 - Oxford University Press.
The Logic of Scientific Discovery.Karl Popper - 1959 - Studia Logica 9:262-265.
A Confutation of Convergent Realism.Larry Laudan - 1981 - Philosophy of Science 48 (1):19-49.
Computing Machinery and Intelligence.Alan M. Turing - 1950 - Mind 59 (October):433-60.
The Structure of Science.Ernest Nagel - 1961 - Les Etudes Philosophiques 17 (2):275-275.

View all 38 references / Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

The Applicabilities of Mathematics.Mark Steiner - 1995 - Philosophia Mathematica 3 (2):129-156.
The Application of Mathematics to Natural Science.Mark Steiner - 1989 - Journal of Philosophy 86 (9):449-480.
Philosophy of Mathematics.Leon Horsten - 2008 - Stanford Encyclopedia of Philosophy.
Towards a Philosophy of Applied Mathematics.Christopher Pincock - 2009 - In Otávio Bueno & Øystein Linnebo (eds.), New Waves in Philosophy of Mathematics. Palgrave-Macmillan.
Mathematical Nominalism and Measurement.Davide Rizza - 2010 - Philosophia Mathematica 18 (1):53-73.
Mathematics as an Instigator of Scientific Revolutions.Ricardo Karam - 2015 - Science & Education 24 (5-6):495-513.
The Role of Symmetry in Mathematics.Noson S. Yanofsky & Mark Zelcer - 2017 - Foundations of Science 22 (3):495-515.
The Applicability of Mathematics.Christopher Pincock - 2010 - Internet Encyclopedia of Philosophy.


Added to PP index

Total views
9 ( #824,616 of 2,286,379 )

Recent downloads (6 months)
1 ( #853,841 of 2,286,379 )

How can I increase my downloads?


My notes

Sign in to use this feature