Abstract
Dimensional analysis can offer us explanations by allowing us to answer What-if–things-had-been-different? questions rather than in virtue of, say, unifying diverse phenomena, important as that is. Additionally, it is argued that dimensional analysis is a form of modelling as it involves several of the aspects crucial in modelling, such as misrepresenting aspects of a target system. By highlighting the continuities dimensional analysis has with forms of modelling we are able to describe more precisely what makes dimensional analysis explanatory and understand otherwise puzzling aspects of dimensional reasoning, such as introducing fictitious dimensions and excluding dimensionally relevant information to characterise some systems. Finally, thinking of dimensional arguments as a form of modelling allows an explication of the role abstraction and multiple realisability; not as compatibility with other possible worlds but as compatibility with different fictional descriptions of our own world.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Stellar structure is the internal structure of stars: the composition, size, pressure gradients, and energy transport mechanisms, and how these interact to produce the properties of the star.
Essentially we assume that \(\rho (\hbox {r}){/}\!<\rho >\) the density as a function of radius as a fraction of average density is independent of total mass.
I am grateful to Marc Lange (private communication) for clarification of this point.
Noether’s theorem is a classic example of finding independent theoretical justification for a symmetry constraint on laws. Consider a symmetry assumption that physics should be spatially translationally invariant. In other words, all things being equal, the laws of physics shouldn’t alter because we are in Leeds rather than London. Noether was able to show that conservation of linear momentum follows from translational symmetry, likewise temporal symmetry leads to the principle of conservation of energy and rotational symmetry leads to conservation of angular momentum. So when we say both Newton’s and Einstein’s laws adhere to conservation of energy and momentum, we are constraining them via a very basic symmetry assumption. Similarly dimensional explanations offer explanations by locating relational facts about the dimensional constraints inherent within a system.
References
Batterman, R. (2002). The devil in the details. Asymptotic reasoning in explanation, reduction, and emergence. Oxford: Oxford University Press.
Bokulich, A. (2008). Reexamining the quantum-classical relation: Beyond reductionism and pluralism. Cambridge: Cambridge University Press.
Butterfield, J., & Bouatta, N. (2011). Emergence and reduction combined in phase transitions. http://philsci-archive.pitt.edu/id/eprint/8554.
Chandrasekhar, S. (1958). An introduction to the study of stellar structure. New York: Dover. [1939].
Frigg, R. (2010). Models and fiction. Synthese, 172, 251–268.
Giere, R. N. (2006). Scientific perspectivism. Chicago: University of Chicago Press.
Hughes, R. I. G. (2010). The theoretical practices of physics: Philosophical essays. Oxford: Oxford University Press.
Huntley, H. E. (1967). Dimensional analysis. New York: Dover.
Kitcher, P. (1989). Explanatory unification and the causal structure of the world. In Kitcher & Salmon (Eds.), Scientific explanation. Minneapolis: University of Minnesota Press.
Knuuttila, T., & Boon, M. (2011). How do models give us knowledge? The case of Carnot’s ideal heat engine. European Journal of Philosophy of Science, 1, 309–334.
Lange, M. (2009). Dimensional explanations. Nous, 43(4), 742–775.
Morrison, M. (2000). Unifying scientific theories: Physical concepts and mathematical structures. Cambridge: Cambridge University Press.
Pexton, M. (2013). Non-causal explanation, Ph.D. Thesis University of Leeds.
Rayleigh, L. (1915). The principle of similitude. Nature, 95, 66–68.
Riabouchinsky, D. (1915). The principle of similitude. Nature, 95, 591.
Weslake, B. (2010). Explanatory depth. Philosophy of Science, 77(2), 273–294.
Woodward, J. (2003). Making things happen: A theory of causal explanation. Oxford: Oxford University Press.
Acknowledgments
I would like to thank Juha Saatsi without who’s supervision this work would have not been possible, and Marc Lange for helpful comments, as well as the comments of the editors and referees. This work was supported by the Arts and Humanities Research Council and Templeton Foundation.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pexton, M. How dimensional analysis can explain. Synthese 191, 2333–2351 (2014). https://doi.org/10.1007/s11229-014-0401-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-014-0401-x