Symmetry & Possibility: To Reduce or not Reduce?∗

In this paper I examine the connection between symmetry and modality from the perspective of ‘reduction’ methods in geometric mechanics. I begin by setting the problem up as a choice between two opposing views: reduction and non-reduction. I then discern four views on the matter in the literature; they are distinguished by their advocation of distinct geometric spaces as representing ‘reality’. I come down in favour of non-reductive methods. 1 The Geometry of Modality. Hacking (1975) famously argued that spatially symmetric worlds could never constitute a counterexample to the principle of identity of indiscernibles [PII1]: there is always a way to redescribe the situation so that the symmetry is absent and so that there are fewer ∗ c ©D. Rickles, 2004. Draft version. Not for quotation; comments and suggestions most welcome. 1That is, the principle that any individuals, a and b, sharing all of their properties, F ∈ F, are really one and the same individual; formally, in terms of 2nd-order logic with identity: [∀F ∀ab : (Fa ≡ Fb)→ (a = b)]. Different ‘strengths’ of PII can be formulated by restricting the range of the property variable F in various ways. By having F range over all properties and relations—including the property of self-identity, a = a—PII is rendered trivial, giving a weak form of PII (a theorem of the 2nd-order predicate calculus with equality); outlawing self-identity and excluding spatiotemporal properties and relations (absolute position, for example), but including all other relations and properties, gives us a stronger form; restricting F to just intrinsic, monadic qualitative properties gives the strongest form.