Abstract
Early in the 1900s, Russell recognizes that he and many others had been implicitly using claims like the Axiom of Choice . For such claims, Russell eventually took the position that some axioms are necessary to recovering certain results of mathematics, but may not be proven to be true absolutely. This essay traces historical roots of, and motivations for, Russell’s method of analysis and his related views about the status of mathematical axioms. I describe the position that Russell develops as “immanent logicism,” in contrast to what Irving has called “epistemic logicism.” Immanent logicism allows Russell to avoid the logocentric predicament and to propose a method for discovering structural relationships of dependence within mathematical theories.
The mathematician is only strong and true as long as he is logical, and if number rules the world, it is logic which rules number.
—William Stanley Jevons , Principles of Science (1874).
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Acknowledgements
Many thanks are due to Sean Morris for pointing me to Russell’s 1907 talk, “The Regressive Method of Discovering the Premises of Mathematics.” Soon thereafter, Richard Burian donated a number of offprints to the Virginia Tech philosophy department, and I found Larry Laudan’s 1968 paper on theories of scientific method there. I traced Jevons’s Principles following a lead from that paper, and noticed a number of similarities with the Russell talk. I am grateful to Sandra Lapointe and Chris Pincock for inviting me to contribute to this volume, and for detailed comments on an earlier draft. Professors Lapointe and Pincock organized a very productive workshop with the contributors in Hamilton, Ontario. The paper has profited from insightful suggestions at the workshop from Daniel Harris, Jeremy Heis, Colin Johnston, Alexander Klein, Marcus Rossberg, Dirk Schlimm, and Audrey Yap. I owe a debt to Nicholas Griffin for discussing the role of Jevons, for consulting the Bertrand Russell Archives at McMaster University, and for putting me on the right track in correspondence. Greg Frost-Arnold sent detailed and perceptive comments on the draft afterward. A final conversation with Christopher Pincock brought the central argument of the paper into much sharper focus. None of these are to blame for my errors or wrong turnings.
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Patton, L. (2017). Russell’s Method of Analysis and the Axioms of Mathematics. In: Lapointe, S., Pincock, C. (eds) Innovations in the History of Analytical Philosophy. Palgrave Innovations in Philosophy. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-137-40808-2_4
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