In this paper, we offer a simple argument for the conclusion that the outputs of large language models (LLMs) are meaningless. Our argument is based on two key premises: (a) that certain kinds of intentions are needed in order for LLMs' outputs to have literal meanings, and (b) that LLMs cannot plausibly have the right kinds of intentions. We defend this argument from various types of responses, for example, the semantic externalist argument that deference can be assumed to take the place of intentions and the semantic internalist argument that meanings can be defined purely in terms of intrinsic relations between concepts, such as conceptual roles. We conclude the paper by discussing why, even if our argument is sound, the outputs of LLMs nevertheless seem meaningful and can be used to acquire true beliefs and even knowledge.

In a recent paper, Mandelkern & Linzen (2024) - henceforth M&L - address the question of whether language models' (LMs) words refer. Their argument draws from the externalist tradition in philosophical semantics, which views reference as the capacity of words to "achieve 'word-to-world' connections". In the externalist framework, causally uninterrupted chains of usage, tracing every occurrence of a name back to its bearer, guarantee that, for example, 'Peano' refers to the individual Peano (Kripke 1980). This account is externalist both because words pick out referents 'out there' in the world, and because what determines reference are coordinated linguistic actions by members of a community, and not individual mental states. The "central question to ask", for M&L, is whether LMs too belong to human linguistic communities, such that words by LMs may also trace back causally to their bearers. Their answer is a cautious "yes": inputs to LMs are linguistic "forms with particular histories of referential use"; "those histories ground the referents of those forms"; any occurrence of 'Peano' in LM outputs is as causally connected to the individual Peano as any other occurrence of the same proper name in human speech or text; therefore, occurrences of 'Peano' in LM outputs refer to Peano. In this commentary, we first qualify M&L's claim as applying to a narrow class of natural language expressions. Thus qualified, their claim is valid, and we emphasise an additional motivation for that in Section 2. Next, we discuss the actual scope of their claim, and we suggest that the way they formulate it may lead to unwarranted generalisations about reference in LMs. Our critique is likewise applicable to other externalist accounts of LMs (e.g., Lederman & Mahowald 2024; Mollo & Milliere 2023). Lastly, we conclude with a comment on the status of LMs as members of human linguistic communities.

What do language models (LMs) do with language? Everyone agrees that they produce sequences of (mostly) coherent sentences. But are they saying anything with those strings or simply babbling in a convincing simulacrum of language use? This is a vague question, and there are many ways of making it precise. Here we will address one aspect of the question, namely, whether LMs' words refer: that is, whether the outputs of LMs achieve "word-to-world" connections. There is prima facie reason to think they do not since LMs do not interact with the world in the way that ordinary language users do. Drawing on insights from the externalist tradition in philosophy of language, we argue that appearances are misleading and that there is good reason to think that LMs can refer.

General mathematical reasoning is computationally undecidable, but humans routinely solve new problems. Moreover, discoveries developed over centuries are taught to subsequent generations quickly. What structure enables this, and how might that inform automated mathematical reasoning? We posit that central to both puzzles is the structure of procedural abstractions underlying mathematics. We explore this idea in a case study on 5 sections of beginning algebra on the Khan Academy platform. To define a computational foundation, we introduce Peano, a theorem-proving environment where the set of valid actions at any point is finite. We use Peano to formalize introductory algebra problems and axioms, obtaining well-defined search problems. We observe existing reinforcement learning methods for symbolic reasoning to be insufficient to solve harder problems. Adding the ability to induce reusable abstractions ("tactics") from its own solutions allows an agent to make steady progress, solving all problems. Furthermore, these abstractions induce an order to the problems, seen at random during training. The recovered order has significant agreement with the expert-designed Khan Academy curriculum, and second-generation agents trained on the recovered curriculum learn significantly faster. These results illustrate the synergistic role of abstractions and curricula in the cultural transmission of mathematics.

Paul Bernays and David Hilbert carefully avoided overspecification of Hilbert's epsilon-operator and axiomatized only what was relevant for their proof-theoretic investigations. Semantically, this left the epsilon-operator underspecified. In the meanwhile, there have been several suggestions for semantics of the epsilon as a choice operator. After reviewing the literature on semantics of Hilbert's epsilon operator, we propose a new semantics with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the epsilon supports proof search optimally and is natural in the sense that it does not only mirror some cases of referential interpretation of indefinite articles in natural language, but may also contribute to philosophy of language. Finally, we ask the question whether our epsilon within our free-variable framework can serve as a paradigm useful in the specification and computation of semantics of discourses in natural language.