William D'Alessandro

Current LLMs can discuss philosophical ideas, evaluate arguments and perform other analytical tasks at a high level, but are conspicuously bad at producing interesting original philosophy. Why is this? Two tempting diagnoses—that LLMs can't invent new concepts, and that they can't really reason—both look unconvincing on closer inspection. I suggest that a better explanation lies in the structure of reinforcement learning for reasoning. The technique works best in domains like mathematics and coding, where good arguments follow recognizable patterns, correctness is relatively unambiguous, and large corpora of worked examples exist. Philosophy meets none of these conditions especially well, and it's unclear that either AI labs or philosophers have much incentive to develop training resources that might change this.

The field of AI safety aims to prevent increasingly capable artificially intelligent systems from causing humans harm. Research on moral alignment is widely thought to offer a promising safety strategy: if we can equip AI systems with appropriate ethical rules, according to this line of thought, they'll be unlikely to disempower, destroy or otherwise seriously harm us. Deontological morality looks like a particularly attractive candidate for an alignment target, given its popularity, relative technical tractability and commitment to harm-avoidance principles. I argue that the connection between moral alignment and safe behavior is more tenuous than many have hoped. In general, AI systems can possess either of these properties in the absence of the other, and we should favor safety when the two conflict. In particular, advanced AI systems governed by standard versions of deontology need not be especially safe.

In the face of rapidly improving AI capabilities, many AI theorists have warned about the risks of ceding power to autonomous artificial agents. In particular, going sufficiently far down this path might mean "losing control of our future", an outcome some have ranked as a worst-case catastrophe on par with human extinction. A natural way to motivate this judgment is via the thought that losing human control might lead to our domination by future AI systems. Republican political philosophers such as Pettit have held that domination characteristically involves important kinds of harm, even when the dominating party is benevolent and non-interfering. Is this thought correct? After developing an account of what it means to control humanity's future, I show that AI systems possessing such control would indeed be likely to dominate humans on our best theories of arbitrary power. However, contrary to republican orthodoxy, I argue that this need not lead to two of the three primary harms traditionally associated with domination: the psychological and social costs of subordination, and the burden of obligatory strategic behavior. Meanwhile, the third harm—pervasive uncertainty about interference—remains a serious concern even in otherwise human-friendly AI power scenarios. These results should be of interest to theoretical political philosophy, as they suggest that artificial power has unique features which are as yet underrepresented in discussions of domination.

AI tools hold considerable promise for psychological research. The precise shape of their potential uses has become clearer in recent years as machine learning models have been trained to reproduce a variety of complex human cognitive behaviors with impressive success. The prospect of AI-human performance parity, along with the advantages of AI systems in speed, cost and ease of use, has prompted psychologists to explore how science might benefit from reassigning some traditionally human research roles to machines. This chapter provides an outlook on such methods. We begin by documenting the prospective upsides of what many researchers take to be the most promising types of AI surrogacy, including the use of models to efficiently generate, screen and refine preliminary hypotheses. We then discuss such methods’ limitations and drawbacks, and more generally the methodological considerations researchers must attend to in choosing when and how to rely on machine substitutes for human behavior.

Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: What kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanatory proofs, and if so, how do they relate to the sorts of explanation encountered in philosophy of science and metaphysics?

Practice-based approaches to philosophy of mathematics have gone mainstream over the past several decades. As the paradigm has grown in popularity, however, there’s been little sustained meditation—and still less any explicit consensus—on what precisely it means for philosophy to take practice seriously. The field’s lack of a clear common methodology has begun to make itself felt in slowed and uncertain progress on core problems. Here I review the methodological situation and propose five canons to guide future research. I focus throughout on the study of explanation in mathematics as a guiding example.

The philosophy of mathematical practice (PMP) looks to evidence from working mathematics to help settle philosophical questions. One prominent program under the PMP banner is the study of explanation in mathematics, which aims to understand what sorts of proofs mathematicians consider explanatory and what role the pursuit of explanation plays in mathematical practice. PMP researchers have recently turned to corpus analysis methods as a promising alternative to small-scale case studies. Such methods stand to benefit, it would seem, from the sophisticated text-comprehension capabilities of today's frontier large language models (LLMs). This paper reports the results from such a corpus study making use of Google's Gemini 2.5 Pro. Based on a sample of 5000 mathematics preprints, the experiments yielded a dataset of hundreds of useful annotated examples. Its aim was to gain insight on questions like the following: How often do mathematicians make claims about explanation in the relevant sense? Do mathematicians’ explanatory practices vary in any noticeable way by subject matter? Which philosophical theories of explanation are most consistent with a large body of non-cherry-picked examples? How might philosophers make further use of AI tools to gain insights from large datasets of this kind? As the first PMP study making extensive use of LLM methods, it also seeks to begin a conversation about these methods as research tools in practice-oriented philosophy and to evaluate the strengths and weaknesses of current models for such work.

AI safety is an interdisciplinary field focused on mitigating the harms caused by AI systems. We review a range of research directions in AI safety, focusing on those to which philosophers have made or are in a position to make the most significant contributions. These include ethical AI, which seeks to instill human goals, values, and ethical principles into artificial systems, scalable oversight, which seeks to develop methods for supervising the activity of artificial systems even when they become significantly more capable than their human designers, interpretability, which seeks to render comprehensible the workings of complex machine learning models, and corrigibility, which seeks to discover ways to ensure that powerful AI systems will not resist being shut down or modified by humans.

A digital doppelgänger is an AI system trained to instantiate or imitate a particular human's personality, memories, beliefs or other personal traits, usually to be deployed after the human's death. Past discussions have assumed that, if digital doppelgängers enter wide use, it will suffice to produce at most one general-purpose doppelgänger system per human. We argue that this assumption is mistaken. For reasons related to both performance and information security, it will often be desirable to train distinct doppelgängers for distinct applications. We sketch the beginning of a typology of such systems.

Mathematicians often describe the importance of well-developed intuition to productive research and successful learning. But neither education researchers nor philosophers interested in epistemic dimensions of mathematical practice have yet given the topic the sustained attention it deserves. The trouble is partly that intuition in the relevant sense lacks a usefully clear characterization, so we begin by offering one: mature intuition, we say, is the capacity for fast, fluent, reliable and insightful inference with respect to some subject matter. We illustrate the role of mature intuition in mathematical practice with an assortment of examples, including data from a sequence of clinical interviews in which a student improves upon initially misleading covariational intuitions. Finally, we show how the study of intuition can yield insights for philosophers and education theorists. First, it contributes to a longstanding debate in epistemology by undermining epistemicism, the view that an agent’s degree of objectual understanding is determined exclusively by their knowledge, beliefs and credences. We argue on the contrary that intuition can contribute directly and independently to understanding. Second, we identify potential pedagogical avenues towards the development of mature intuition, highlighting strategies including adding imagery, developing associations, establishing confidence and generalizing concepts.

We defend a noetic account of intramathematical explanation. On this view, a piece of mathematics is explanatory just in case it produces understanding of an appropriate type. We motivate the view by presenting some appealing features of noeticism. We then discuss and criticize the most prominent extant version of noeticism, due to Inglis and Mejía Ramos, which identifies explanatory understanding with the possession of well-organized cognitive schemas. Finally, we present a novel noetic account. On our view, explanatory understanding arises from meeting specific explanatory objectives. We defend a cluster-concept account of explanatory objectives and identify four important subfamilies within the relevant network of resemblance relations. The resulting view is objectivist (in the sense that it takes explanatory success to be a matter of observer-independent fact), broader in scope than why-question-based accounts, compatible with empirical findings on experts’ explanatory judgments, and capable of generalizing (with appropriate provisos) to scientific explanation as a whole. It thus fulfills Friedman’s half-century-old demand for a general and objectivist theory which accounts for the link between explanation and understanding.

Recent progress in artificial intelligence (AI) has drawn attention to the technology’s transformative potential, including what some see as its prospects for causing large-scale harm. We review two influential arguments purporting to show how AI could pose catastrophic risks. The first argument — the Problem of Power-Seeking — claims that, under certain assumptions, advanced AI systems are likely to engage in dangerous power-seeking behavior in pursuit of their goals. We review reasons for thinking that AI systems might seek power, that they might obtain it, that this could lead to catastrophe, and that we might build and deploy such systems anyway. The second argument claims that the development of human-level AI will unlock rapid further progress, culminating in AI systems far more capable than any human — this is the Singularity Hypothesis. Power-seeking behavior on the part of such systems might be particularly dangerous. We discuss a variety of objections to both arguments and conclude by assessing the state of the debate.

We discuss potential biorisks from large language models (LLMs). AI assistants based on LLMs such as ChatGPT have been shown to significantly reduce barriers to entry for actors wishing to synthesize dangerous, potentially novel pathogens and chemical weapons. The harms from deploying such bioagents could be further magnified by AI-assisted misinformation. We endorse several policy responses to these dangers, including prerelease evaluations of biomedical AIs by subject-matter experts, enhanced surveillance and lab screening procedures, restrictions on AI training data, and access controls on biomedical AI. We conclude with a suggestion about future research directions in bioethics.

A proof ${\cal P}$ of a theorem T is transferable when it's possible for a typical expert to become convinced of T solely on the basis of their prior knowledge and the information contained in ${\cal P}$. Easwaran has argued that transferability is a constraint on acceptable proof. Meanwhile, a proof ${\cal P}$ is fixable when it's possible for other experts to correct any mistakes ${\cal P}$ contains without having to develop significant new mathematics. Habgood-Coote and Tanswell have observed that some acceptable proofs are both fixable and in need of fixing, in the sense that they contain non-trivial mistakes. The claim that acceptable proofs must be transferable seems quite plausible. The claim that some acceptable proofs need fixing seems plausible too. Unfortunately, these attractive suggestions stand in tension with one another. I argue that the transferability requirement is the problem. Acceptable proofs need to only satisfy a weaker requirement I call “corrigibility.” I explain why, despite appearances, the corrigibility standard is preferable to stricter alternatives.

Models are indispensable tools of scientific inquiry, and one of their main uses is to improve our understanding of the phenomena they represent. How do models accomplish this? And what does this tell us about the nature of understanding? While much recent work has aimed at answering these questions, philosophers' focus has been squarely on models in empirical science. I aim to show that pure mathematics also deserves a seat at the table. I begin by presenting two cases: Cramér’s random model of the prime numbers and the function field model of the integers. These cases show that mathematicians, like empirical scientists, rely on unrealistic models to gain understanding of complex phenomena. They also have important implications for some much-discussed theses about scientific understanding. First, modeling practices in mathematics confirm that one can gain understanding without obtaining an explanation. Second, these cases undermine the popular thesis that unrealistic models confer understanding by imparting counterfactual knowledge.

Philosophers have rightly condemned lookism—that is, discrimination in favor of attractive people or against unattractive people—in education, the justice system, the workplace and elsewhere. Surprisingly, however, the almost universal preference for attractive romantic and sexual partners has rarely received serious ethical scrutiny. On its face, it’s unclear whether this is a form of discrimination we should reject or tolerate. I consider arguments for both views. On the one hand, a strong case can be made that preferring attractive partners is bad. The idea is that choosing partners based on looks seems essentially similar to other objectionable forms of discrimination. (In particular, the preference for attractive partners is arguably both unfair and harmful to a significant degree.) One can try to resist this conclusion in several ways. I consider three possible replies. The first has to do with the possibility of controlling our partner preferences. The second pertains to attractiveness and “good genes”. The last attempts to link certain aspects of attractiveness to a prospective partner’s personality and values. I argue that the first two replies fail conclusively, while the third only amounts to a limited defense of a particular kind of attractiveness preference. So the idea that we should often avoid preferring attractive partners is compelling.

Much recent work on mathematical explanation has presupposed that the phenomenon involves explanatory proofs in an essential way. I argue that this view, ‘proof chauvinism’, is false. I then look in some detail at the explanation of the solvability of polynomial equations provided by Galois theory, which has often been thought to revolve around an explanatory proof. The article concludes with some general worries about the effects of chauvinism on the theory of mathematical explanation. 1Introduction 2Why I Am Not a Proof Chauvinist 2.1Proof chauvinism and mathematical practice 2.2Proof chauvinism and philosophy 3An Example: Galois Theory and Explanatory Proof 4Conclusion.

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation.

According to a widespread view in metaphysics and philosophy of science, all explanations involve relations of ontic dependence between the items appearing in the explanandum and the items appearing in the explanans. I argue that a family of mathematical cases, which I call “viewing-as explanations”, are incompatible with the Dependence Thesis. These cases, I claim, feature genuine explanations that aren’t supported by ontic dependence relations. Hence the thesis isn’t true in general. The first part of the paper defends this claim and discusses its significance. The second part of the paper considers whether viewing-as explanations occur in the empirical sciences, focusing on the case of so-called fictional models. It’s sometimes suggested that fictional models can be explanatory even though they fail to represent actual worldly dependence relations. Whether or not such models explain, I suggest, depends on whether we think scientific explanations necessarily give information relevant to intervention and control. Finally, I argue that counterfactual approaches to explanation also have trouble accommodating viewing-as cases.

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences.

The problem of truth in fiction concerns how to tell whether a given proposition is true in a given fiction. Thus far, the nearly universal consensus has been that some propositions are ‘implicitly true’ in some fictions: such propositions are not expressed by any explicit statements in the relevant work, but are nevertheless held to be true in those works on the basis of some other set of criteria. I call this family of views ‘implicitism’. I argue that implicitism faces serious problems, whereas the opposite view is much more plausible than has previously been thought. After mounting a limited defence of explicitism, I explore a difficult problem for the view and discuss some possible responses.