Philosophia Mathematica Advance Access published online on March 26, 2008
Philosophia Mathematica, doi:10.1093/philmat/nkm042
Identity, Indiscernibility, and ante rem Structuralism: The Tale of i and –i
* Department of Philosophy, Ohio State University, Columbus, Ohio 43210 U.S.A.
Arché Research Centre, St Andrews University, St Andrews, Fife KY16 9AL Scotland
Correspondence: shapiro.4{at}osu.edu
Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of the other. I suggest that ‘i’ functions like a parameter in natural deduction systems.
I gave an early version of this paper at a workshop on structuralism in mathematics and science, held in the Autumn of 2006, at Bristol University. Thanks to the organizers, particularly Hannes Leitgeb, James Ladyman, and Øystein Linnebo, to my commentator Richard Pettigrew, and to the audience there. The paper also benefited considerably from a preliminary session at the Arché Research Centre at the University of St Andrews. I am indebted to my colleagues Craige Roberts, for help with the linguistics literature, and Ben Caplan and Gabriel Uzquiano, for help with the metaphysics. Thanks also to Hannes Leitgeb and Jeffrey Ketland for reading an earlier version of the manuscript and making helpful suggestions. I also benefited from conversations with RichardHeck, John Mayberry, Kevin Scharp, Jason Stanley.