Ásgeir Berg (Matthíasson)
Philosopher.
Research
I am currently running a research project developing a novel account of mathematical truth, funded by the Icelandic Research Fund (Rannís). The project builds on my previous work on the rule-following paradox and the philosophy of mathematics.
As a part of this project, I am also writing papers on inference, the nature of rule-following, and semantic dispositionalism.
I have a developing interest in the philosophy of artificial intelligence as well, particularly the philosophical questions raised by large language models.
Peer-Reviewed Journal Articles
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What Is It to Follow a Rule?
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.
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An Objective Communitarian Account of Semantic Content
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.
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The Philosophical Prospects of Large Language Models in the Future of Mathematics
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.
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How the Laws of Logic Lie about Mathematical Objects
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.
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Problems for 'Standard' Dispositionalist Accounts of Semantic Content
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's (1982) arguments against simple versions of semantic dispositionalism have inspired ever new versions. A recent account in the literature, due to Warren (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.
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Wittgenstein on Mathematical Facts
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Was Wittgenstein a Radical Conventionalist?
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Rules as Constitutive Practices Defined by Correlated Equilibria
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Contradictions and Falling Bridges: What Was Wittgenstein's Reply to Turing?
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.