Ásgeir Berg

Communitarian accounts of semantic content have been widely rejected in the literature on the rule-following paradox and related issues, for good reason. In this paper, I offer a game-theoretic account of semantic content and focus on how it can give replies to some of the most difficult and important objections to communitarian accounts, namely (i) Boghossian’s horsey cow case, (ii) the community version of the problem of error, as well as what I will call (iii) the logical problem of error. I also show how the account can handle a recent argument in the literature, the so-called privilege argument (Guardo 2022), as well as addressing a more fundamental worry that communitarian accounts of meaning cannot result in our judgements being objective, particularly, that our judgements would be infallible on such accounts. I argue that the specifics of the account under consideration show that such worries are misguided.

What is it for an agent to follow a rule, rather than merely act in accordance with it? An intuitive and plausible answer to this question is that to follow a rule is to perform an intentional act such that S follows a rule R iff S intends to act in accordance with R and subsequently acts on that intention. This intentional acount of rule-following faces essentially two problems, widely seen as fatal: (1) Kripkean sceptical arguments, originally derived from Wittgenstein, suggesting that the requirements of a rule outstrip the possible content of our intention, and (2) a regress argument due to Boghossian, namely that the intentional view requires the agent to represent the conditional content of the rule in such a way that an inferential step is needed for the agent to move from the antecedent to the consequent, which, given some plausible assumptions about inference, leads to a regress. In this paper, I defend the intentional view of rule-following, using a game-theoretic account of semantic content outlined in Berg (2022) and Berg (forthcoming). I argue that by placing the agent in the basic constitutive practice of using the terms that figure in an expression of the rule, we can give an answer to (1). Likewise, by positing that the agents have an in-built mental mechanism which is such that it responds to the constitutive structure of the content being represented (see Quilty-Dunn and Mandelbaum 2018) and by placing the resulting movement of thought in a basic constitutive practice of inferring, we can avoid Boghossian’s regress.

In this article, we examine the philosophical implications Large Language Models might have on mathematical practice in the near future. Some prominent researchers argue that Large Language Models will soon have the ability to generate or check proofs, lifting a great burden of human mathematicians. We claim, however, that the implementation of LLM technologies in mathematics is not merely a neutral tool that assists mathematicians to continue on as before, but instead entails a radical change to the practices of mathematics with important philosophical implications. We will argue that we cannot be confident such tools will continue to work as expected, even if they become arbitrarily more reliable than they currently are, and that the kind of justification we get from LLM-generated proofs can never be properly mathematical. We will evaluate solutions to this problem involving either computer verification or human checking and argue that these cannot fix the philosophical gap to give us proper mathematical justification.

In ontological debates on the existence of mathematical objects, it has long been taken for granted that if we take our mathematical discourse at face value, it follows from the fact that our true mathematical statements refer to mathematical objects that mathematical objects exists; reference in true statements entails existence. In this paper, I argue that there are positions available in the philosophy of logic that allow us to dislodge this assumption, allowing for a nominalist position according to which statements such as ‘there are four prime numbers between 1 and 10’ are true and genuinely refer to numbers, without a corresponding statement asserting the existence of numbers following from it. Consequently, there is an overlooked nominalist position according to which objects just are what singular terms refer to and that reference is successful when those singular terms figure in true statements, with corresponding existence statements being literally false. I argue that this view is not only coherent, but that it does not entail that there are objects that do not exist.

ABSTRACT In this paper, I offer a close reading of Wittgenstein's remarks on inconsistency, mostly as they appear in the Lectures on the Foundations of Mathematics. I focus especially on an objection to Wittgenstein's view given by Alan Turing, who attended the lectures, the so called ‘falling bridges’-objection. Wittgenstein's position is that if contradictions arise in some practice of language, they are not necessarily fatal to that practice nor necessitate a revision of that practice. If we then assume that we have adopted a paraconsistent logic, Wittgenstein's answer to Turing is that if we run into trouble building our bridge, it is either because we have made a calculation mistake or our calculus does not actually describe the phenomenon it is intended to model. The possibility of either kind of error is not particular to contradictions nor to inconsistency, and thus contradictions do not have any special status as a thing to be avoided.

A popular view in metasemantics is the view that a speaker’s dispositions regarding the use of a symbol determine the meaning of that symbol for the speaker. Kripke (Wittgenstein on rules and private language, Harvard University Press, Cambridge, 1982) arguments against simple versions of semantic dispositionalism have inspired ever new versions. A recent account in the literature, due to Warren (Noûs 54(2):257–289, 2020) offers a sophisticated version of semantic dispositionalism whereby certain conditions are imposed on speaker’s dispositions to count as meaning determining—conditions we can refer to as ‘standard’. In this paper, I argue that there are a number of cases that suggest that even under such conditions, a speaker’s meaning and their dispositions can come apart, and that this suggests that dispositional accounts of semantic content presuppose that there are semantic norms, independent of a speaker’s dispositions, and thus do not explain semantic content. I conclude that, since these problems generalise to non-standard dispositionalist accounts, the dispositionalist’s prospects are dim.

In this paper, I present a game-theoretic solution to the rule-following paradox in terms of what I will call basic constitutive practices. The structure of such a practice P constitutes what it is to take part in P by defining the correctness conditions of our most basic concepts as those actions that lie on the correlated equilibrium of P itself. Accordingly, an agent S meant addition by his use of the term ‘+’ because S is taking part in a basic constitutive practice of adding where quus-like answers are ruled out by not lying on the equilibrium. The resulting picture of language will preserve the objectivity and correctness conditions of meaning, all the while providing room for the community as a whole to make a mistake. I also argue that by accepting the account presented, we have a strong reason to reject the idea that language is rule-governed.

The status of mathematical facts has long been taken to be unclear in Wittgenstein's philosophy of mathematics, and often, it seems that he wants to eliminate mathematical facts in favour of facts about our beliefs or behaviour. In this paper, I argue that by reading Wittgenstein as a radical conventionalist, we can give a reading of the relevant passages according to which Wittgenstein doesn't deny that there are mathematical facts, but rather denies that one needs a metaphysical account of what mathematical facts are and how they relate to the world that goes beyond the minimal claims of radical conventionalism—that empirical facts about how we would find natural to project our training into new cases and the constitution of concepts by our agreement are enough. At the end of the paper, I discuss the implications of this reading on how to understand a rule's determination of its own application.

This paper defends a reading of Wittgenstein’s philosophy of mathematics in the Lectures on the Foundation of Mathematics as a radical conventionalist one, whereby our agreement about the particular case is constitutive of our mathematical practice and ‘the logical necessity of any statement is a direct expression of a convention’ (Dummett 1959, p. 329). On this view, mathematical truths are conceptual truths and our practices determine directly for each mathematical proposition individually whether it is true or false. Mathematical truths are thus not consequences of a prior adoption of a convention or rules as orthodox conventionalism has it. The goal of the paper is not merely exegetical, however, and argues that radical conventionalism can withstand some of the most difficult objections that have been brought forward against it, including those of Dummett himself, and thus that radical conventionalism has been prematurely excluded from consideration by philosophers of mathematics.