Linear structures, causal sets and topology

Abstract

Causal set theory and the theory of linear structures (which has recently been developed by Tim Maudlin as an alternative to standard topology) share some of their main motivations. In view of that, I raise and answer the question how these two theories are related to each other and to standard topology. I show that causal set theory can be embedded into Maudlin's more general framework and I characterise what Maudlin's topological concepts boil down to when applied to discrete linear structures that correspond to causal sets. Moreover, I show that all topological aspects of causal sets that can be described in Maudlin's theory can also be described in the framework of standard topology. Finally, I discuss why these results are relevant for evaluating Maudlin's theory. The value of this theory depends crucially on whether it is true that (a) its conceptual framework is as expressive as that of standard topology when it comes to describing well-known continuous as well as discrete models of spacetime and (b) it is even more expressive or fruitful when it comes to analysing topological aspects of discrete structures that are intended as models of spacetime. On the one hand, my theorems support (a): the theory is rich enough to incorporate causal set theory and its definitions of topological notions yield a plausible outcome in the case of causal sets. On the other hand, the results undermine (b): standard topology, too, has the conceptual resources to capture those topological aspects of causal sets that are analysable within Maudlin's framework. This fact poses a challenge for the proponents of Maudlin's theory to prove it fruitful.

Definable Categorical Equivalence

“…Philosophers of physics have applied this criterion in several case studies (cf. Barrett, 2017;Barrett et al, 2015;Hudetz, 2015;Rosenstock and Weatherall, 2016;Teh and Tsementzis, 2017;Weatherall, 2016a).…”

“…See for exampleAndréka and Németi (2014),Barrett (2017),Barrett and Halvorson (2017),Barrett, Rosenstock, and Weatherall (2015),Hudetz (2015),Lefever and Székely (2018),Rosenstock and Weatherall (2016),Teh and Tsementzis (2017) and Weatherall (2016a).2Putnam (1983, p. 40) argues that formal equivalence is even necessary for full equivalence.…”

How to count structure

“…22See, for example, Andréka, Madarász, and Németi (2005), Barrett (2017), Barrett and Halvorson (2016a, 2016b, 2017a, 2017b), Coffey (2014), Curiel (2014), Halvorson (2013), Glymour (2013), Hudetz (2015, 2017), Knox (2011, 2014), North (2009), Rosenstock et al. (2015), Rosenstock and Weatherall (2016), Teh and Tsementzis (2017), Tsementzis (2015), Van Fraassen (2014), and Weatherall (2017a).…”

Equivalent and Inequivalent Formulations of Classical Mechanics

“…3 For example, see the discussion in Abraham and Marsden (1978, §3.6). 4 In addition to the papers already cited, see Andréka et al (2005), Barrett andHalvorson (2016a,b, 2017a,b), Coffey (2014), Halvorson (2013), Glymour (2013), Hudetz (2015Hudetz ( , 2016, Knox (2011Knox ( , 2014, Rosenstock et al (2015), Rosenstock and Weatherall (2016), Tsementzis (2015), Van Fraassen (2014), and Weatherall (2017). See also the classic work by Glymour (1971Glymour ( , 1977Glymour ( , 1980, Quine (1975), and Sklar (1982).…”