Abstract
There has been a widening divide between two broad approaches to theoretical equivalence in physics: to what we mean when we say that two physical theories are fully equivalent, saying all the same things about the world but perhaps in different ways. On the one side is the formal approach to equivalence. Formal accounts say that physical theories are equivalent when they are formally or structurally or mathematically equivalent (in addition to being empirically equivalent). Proponents then work on figuring out which formal notion is the right one. On the other side is a growing resistance to formal approaches. Opponents note that physical theories consist of more than their formal apparatus, so that questions concerning the equivalence of theories must involve more than their formal features. They point to cases of theories that are equivalent in various formal and empirical respects, but nonetheless differ in what they say about the world. Some have gone so far as to conclude that the formal results being generated have no significance beyond pure mathematics. I advocate a middle ground. A formal equivalence of the right kind is important to questions of equivalence in physics: this is necessary (if not sufficient) for wholesale theoretical equivalence, as we can see in some familiar examples. More, it is not immediately clear, and is worth investigating, what type of formal equivalence is relevant to reasonable judgments of equivalence in physics. At the same time, since any formalism can be made to represent any kind of physical reality simply by brute stipulation, it’s also not right to claim that a formal equivalence must be physically significant in all cases, without further ado. A more nuanced position is in order. In actual scientific practice and theorizing, the choice of formalism is not completely up for stipulational grabs, in that there are better and worse choices of representational vehicle, given standard theoretical criteria: there are good scientific reasons for choosing one formalism over another. Some interpretive and physical stipulations will be made, that is, but certain choices of formalism will be more natural or well-suited than others, given those assumptions. (We should distinguish between the equivalence of descriptions, in the sense of their saying or representing the same things, and the relative naturalness or well-suitedness of descriptions, in the sense of their saying or representing those things in better or worse ways.) As a result, we can learn things of physical significance by examining a theory’s (best) formulation, and its formal relationships to other mathematical formulations. We just have to be careful to mind the stipulations we make, and to be explicit about the different respects, formal and not, in which theories can be equivalent to one another. I will give examples of familiar, reasonable judgements of equivalence and naturalness in physics to illustrate all this; suggest that, properly understood, a structural equivalence of some kind is necessary for wholesale equivalence in physics; and draw connections to the structured view of scientific theories recently advocated by Hans Halvorson.
