Rayo on logical space

Wed 25 Jun 2014 11:58 AM

Over at NDPR, there is an interesting review by Cian Dorr of Augustin Rayo's The Construction of Logical Space

Rayo argues that the structure of logical space depends on practical considerations about which system would best support the enquiry that querists want to do. Logical space is primarily characterized by claims of the form "PHI just is PSI", which can be glossed as "Necessarily, PHI iff PSI". Dorr focusses on applications in philosophy of math. For example, by saying the "For the number of hands I have to be two just is for me to have exactly two hands", one can embrace talk about the number two without accepting TWO as a full-blooded thing. This Trivialist Platonism steers between the dire reefs of nominalism and the dread monster of substantive Platonism.

Dorr worries that trying to apply the strategy to cases in natural science stretches it thin enough to reveal its absurdity. However, Rayo does consider scientific examples: both real examples (like "heat just is mean kinetic energy" and "to be a species just is to be a maximal group of interbreeding organisms") and toy examples (like a chemist who thinks that part of what it is to be water is to be composed of hydrogen and oxygen versus a crank who thinks water can sometimes be made of gold).

Accepting a 'just is' sentence exacts costs and offers benefits. Rayo writes (and Dorr quotes), "The cost... is a decrease in the range of theoretical resources one has at one's disposal. ... The benefit... is that one is relieved from the need to answer certain questions. ... And the relevant questions can be very awkward indeed: they don't lend themselves to satisfying answers from the perspective of one's current theorizing, and extensions of one's theorizing that might deliver better answers seem ad hoc." This is rather like Carnap's picture of metaphysics, on which we accept frameworks for pragmatic reasons; Rayo's addition is to limn the structure of a framework with the 'just is' sentences that it accepts.

The identical rivals response to underdetermination fits nicely into Rayo's framework. Scientists begin with what seem to be to distinct theories. The rivalry between them is dissolved by positing that any state of affairs described in one theory just is a state of affairs described in the other, that the difference is one of representational form rather than substantive disagreement. Greg Frost-Arnold and I argue that this move was crucial in twentieth-century physics. Considering a magnet moving relative to a conductor and a conductor moving relative to a magnet, one might distinguish a moving magnet generating an electric field (in the first case) from an electromotive force arising the conductor (in the latter). Einstein's key move is to deny that there is any different. The first just is the second, described from a different frame of reference.

Greg and I argue that this is more a strategic move than a discovery. The case of quantum mechanics illustrates this more clearly. In the late 1920s, Schrödinger's wave mechanics Heisenberg's matrix mechanics were prima facie incompatible. Yet they agreed in application, and physicists took them to be describing the same state of affairs. The decision to develop the theories in that way led to von Neumann offering a more general framework which subsumed both. There is an alternate outcome which was compatible with everything that physicists knew in the 20s. To put it in Rayo's terms, the conception of logical space that physicists accepted could have been unable to fruitfully support enquiry. The formal differences in the details between waves and matrixes might have mattered in application.

So (I think) natural science provides other nice illustrations rather than refutations of Rayo's claims.

Dorr concludes by both endorsing and attributing to Rayo the view that "in metaphysics the most important work is often done, not in arguing for a claim or defending it against objections, but simply in getting it onto the table in such a way that we can understand it and appreciate its appeal." However, this strikes me as being most sensible given Rayo's Carnapian sensibilities. Making a picture comprehensible allows us to see what it would mean to use it, to use it experimentally, and to reckon its costs and benefits.


The Construction of Logical Space, Rayo's book

Dorr's review at NDPR

The Identical Rivals Response to Underdetermination, my paper with Greg Frost-Arnold