A metalinguistic and computational approach to the problem of mathematical omniscience

Abstract

In this paper, I defend the metalinguistic solution to the problem of mathematical omniscience for the possible‐worlds account of propositions by combining it with a computational model of knowledge and belief. The metalinguistic solution states that the objects of belief and ignorance in mathematics are relations between mathematical sentences and what they express. The most pressing problem for the metalinguistic strategy is that it still ascribes too much mathematical knowledge under the standard possible‐worlds model of knowledge and belief on which these are closed under entailment. I first argue that Stalnaker's fragmentation strategy is insufficient to solve this problem. I then develop an alternative, computational strategy: I propose a model of mathematical knowledge and belief adapted from the algorithmic model of Halpern et al. which, when combined with the metalinguistic strategy, entails that mathematical knowledge and belief require computational abilities to access metalinguistic information, and thus aren't closed under entailment. As I explain, the computational model generalizes beyond mathematics to a version of the functionalist theory of knowledge and belief that motivates the possible‐worlds account in the first place. I conclude that the metalinguistic and computational strategies yield an attractive functionalist, possible‐worlds account of mathematical content, knowledge, and inquiry.

An Algorithmic Impossible-Worlds Model of Belief and Knowledge

“…Our algorithmic impossible-worlds model easily satisfies this third desideratum, whereas dynamic models don't. 44 Dynamic models could provide a useful way to capture in the object language what happens after an agent performs a calculation that is obviously too long for its conclusion to count as "easily accessible" in our sense. They could thus be adapted to our algorithmic model to capture how agents can transition to new belief states.…”

“…On the algorithmic models, triviality is instead construed computationally: what is "trivial" for or "easily accessible" to an agent is what she can compute in less than units of time, given the algorithms that are available to her. Moreover, these algorithms could (and likely would) work very differently from a stepwise application of rules of inference of some background logical system R. 45 This second problem is also the reason why agents who are logically competent in Bjerring and Skipper's formal construal neither clearly fit their own intuitive 44 On the dynamic models of Solaki [41,42] and Solaki et al [43], at each state, an agent has both a set of rules that are "available" for her to use and a cognitive capacity. On their semantics, the dynamic operator " φ" captures that φ is the case after an application of the rule of inference that is both available and "affordable" given the agent's current resources.…”

“…list older applications. 8 Exceptions include [27,34,44], and mentions in literature surveys in [22, pp. 95-97; 40, p. 25].…”

“…Bjerring and Schwarz[7, pp. 23-30] also argue that permissive impossible-worlds models fail to preserve some important features of the standard possible-worlds model, viz., the recursive semantic rules of possible-worlds semantics and the idea that worlds are maximally specific ways things might be.19 That being said, Soysal[44] and Kipper, Kocurek, and Soysal[27] also develop similar algorithmic models based on a possible-worlds framework.20 This presentation follows[8, pp. 509f.]…”

The Logic of Hyperlogic. Part B: Extensions and Restrictions

“…In Part A of this series, we axiomatized a minimal logic of hyperlogic. In Part B, we extended these results to stronger logics over a restricted class 13 For discussion of this problem, see Hintikka 1975;Stalnaker 1976aStalnaker ,b, 1984Duc 1997;Alechina et al 2004;Berto 2010;Ripley 2012;Bjerring 2013;Jago 2007Jago , 2014Jago , 2015Bjerring and Schwarz 2017;Yalcin 2018;Bjerring and Skipper 2019;Hawke et al 2019;Skipper and Bjerring 2020;Elga and Rayo 2021;Hoek 2022;Soysal 2022. 14 Sedlár (2015 likewise explores a doxastic logic where the belief operator is nonclassical, though the base logic is classical.…”