How to count structure

Abstract

There is sometimes a sense in which one theory posits 'less structure' than another. Philosophers of science have recently appealed to this idea both in the debate about equivalence of theories and in discussions about structural parsimony. But there are a number of different proposals currently on the table for how to compare the 'amount of structure' that different theories posit. The aim of this paper is to compare these proposals against one another and evaluate them on their own merits.

On Automorphism Criteria for Comparing Amounts of Mathematical Structure

“…We begin by discussing why SYM * works in the cases where it makes judgments. This argument has been before: see Barrett (2021) and the references therein. The basic idea is that there is a sense in which X has at least as much structure as Y according to SYM * just in case X actually has all of the structures that Y has.…”

“…It is also the only conception of structural comparison that has been explicitly articulated and defended (e.g. in Barrett, 2018Barrett, , 2021.…”

“…This problem is gestured at in the discussion of a criterion called SYM * * byBarrett (2015a), and it is mentioned explicitly byBarrett (2015b, p. 3). It is also discussed byBarrett (2021).5 Subgroup differs from SYM * * , introduced and rejected byBarrett (2015a), in that SYM * * says "more" where Subgroup says "at least as much". Wilhelm's modification avoids the problem that some objects have more structure than themselves.…”

“…Triviality has been discussed byBarrett (2021); see alsoWeatherall (2021), who raises a related triviality concern for category theoretic approaches.12 Barrett (2021) offers an example to make the same point in connection with SYM * .…”

What are empirical consequences? On dispensability and composite objects

“…Second is the theoretical equivalence question, which asks whether the two theories are fully equivalent; it may be that two theories seem to have different ontological commitments, but that this is a merely apparent difference (cf. North, 2009;Curiel, 2014;Barrett, 2015Barrett, , 2019Barrett, , 2020Weatherall, 2019a). Third is the dispensability question, which asks whether we have reasons to prefer one theory's ontology to another; it may be that we have reasons similar to parsimony to accept one of two empirically equivalent theories.…”