Roughly half of my work is about bringing pure math together with important issues in general philosophy of science, metaphysics and epistemology. I’m especially interested in noncausal explanations, their role in mathematical and scientific practice, and what they can teach us about things like explanation, understanding, intertheoretic reduction, modeling and representation.
More recently I've worked on several issues in philosophy of AI, including the promise of ethical AI as a safety strategy, the use of LLMs in scientific research and the political philosophy of AI power.
I'm also interested in various issues in aesthetics and ethics, including truth in fiction and partner choice.
Scientific Progress in the Age of AI (Cambridge Elements in Philosophy of AI series, forthcoming; with Finnur Dellsén)
A mini-book in the Cambridge Elements series (~30,000 words). Looks at current, emerging and potential future uses of LLMs across science and math, and discusses the prospects for LLM-based research methods to both help and hinder scientific progress in numerous ways. My coauthor Finnur Dellsén is a leading theorist of progress, and our Element engages with and advances that debate.
"Philosophy of Mathematical Practice" (in The Blackwell Companion to the Philosophy of Mathematics, Wiley-Blackwell, forthcoming)
Gives an overview of practice-oriented research in philosophy of math. The chapter sketches the modern history of PMP, discusses what contemporary PMP aims to do and how it tries to do it, and surveys two prominent areas of PMP research: the social epistemology of proof acceptance and the theory of mathematical explanation.
“Without Impulses, Without Joy? Openness to Desire and the Virtues of Self-Control” (in The Moral Psychology of Sexual Passion, Rowman & Littlefield, forthcoming)
Defends “Cromwellianism”—the view that we should sometimes open ourselves to lust and its demands, even when these might conflict with our considered view of what’s best—against its opposite, “managerialism”. The action unfolds on the battleground of virtue ethics. I argue that managerialism-friendly accounts of self-control paint an unsatisfactory picture of the relationship between desire and virtue. I propose an alternative account which derives Cromwellian openness from autonomy, earnestness, curiosity and other components of human flourishing.
“LLMs as Philosophers: What Can They Do? Why Aren’t They Better?” (in Understanding Science with Large Language Models? Potentials for the History, Philosophy, and Sociology of Science, Transcript, 2026)
Asks why current LLMs (as of 2025) are good at discussing philosophical ideas, evaluating arguments and performing other analytical tasks at a high level, but conspicuously bad at producing interesting original philosophy. 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 the situation.
“Artificial Power, Domination and Control of Humanity’s Future” (Philosophical Quarterly, 2026)
Looks at the risk of "losing control of the future" to advanced AI, through the lens of Pettit-style republican political philosophy. 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. The third harm—pervasive uncertainty about interference—remains a serious concern even in otherwise human-friendly AI power scenarios.
“Using Large Language Models to Study Mathematical Practice” (in Mathematicians at Work: Empirically Informed Philosophy of Mathematics, Springer, forthcoming)
Reports the results from an AI-assisted corpus study conducted in summer 2025, using Google's Gemini 2.5 Pro. The study examined a sample of 5000 mathematics preprints to gather annotated examples of explanatory concerns from the mathematical literature. With a database of these examples in hand, the paper asks: 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 this large body of examples? How might philosophers make further use of AI tools to gain insights from large datasets? As the first PMP study making extensive use of LLM methods, it also seeks to start 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 Surrogacy in Psychological Research” (in The Role of Artificial Intelligence in Science: Methodological and Epistemological Studies, Routledge, forthcoming; with Jessica Thompson)
Examines how psychologists are applying (or proposing to apply) AI to traditionally human research tasks, why many are optimistic about these methods, and what we take to be the crucial methodological limitations and challenges of the AI surrogacy paradigm. We look specifically at AI-driven hypothesis generation, item piloting and instrument development, the exploration of novel experimental paradigms, general-purpose cognitive modeling, and the replacement of human study subjects with "synthetic data".
"Toward a Methodology for the Philosophy of Mathematical Practice" (Philosophy of Science, 2025)
Argues that philosophers of mathematical practice need to get clearer about what we're doing and how we should go about it: in particular, about what counts as good evidence concerning the practices we're interested in. Proposes five methodological canons to guide future PMP research, which touch on corpus analysis techniques, experimental methods, choosing source material and other issues.
"I Contain Multitudes: A Typology of Digital Doppelgängers" (American Journal of Bioethics, 2025; with Trenton Ford and Michael Yankoski)
A response to Iglesias et. al's "Digital Doppelgängers and Lifespan Extension: What Matters?". Argues that a single type of AI doppelgänger system is unlikely to serve all our lifespan-extension interests, for reasons related to variability in privacy and security needs, capability desiderata, preferences for frozen vs. evolving models, and so on. We sketch the beginning of a typology of likely future doppelgänger designs, consisting of what we call family heirloom, public legacy, research archive and project surrogate systems.
"Artificial Intelligence: Approaches to Safety" (Philosophy Compass, 2025; with Cameron Domenico Kirk-Giannini)
A survey of major approaches to AI safety, covering ethical AI, scalable oversight, interpretability, and corrigibility, emphasizing philosophical dimensions of these research programs.
"Mature Intuition and Mathematical Understanding" (Journal of Mathematical Behavior, 2024; with Irma Stevens)
Details a conception of mature intuition as a capacity for fast, fluent reliable and insightful inference. We show how intuition in this sense plays important roles in mathematical practice, in both research and educational contexts. And we argue that the study of intuition has important epistemological implications. For instance, it puts considerable pressure on epistemicism about understanding, the view that possessing (objectual) understanding of some subject S can be identified with having certain S-related knowledge, beliefs or credences. Intuition, we claim, is no such epistemic state, and yet it positively contributes to understanding.
An interdisciplinary project written with a math education researcher.
The Journal of Mathematical Behavior is the 3rd-ranked math ed journal, according to this report.
"A Noetic Account of Explanation in Mathematics" (Philosophical Quarterly, 2024; with Ellen Lehet)
Defends a novel understanding-based theory of explanation in pure math. According to our account, explanatory understanding instead 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.
"Deontology and Safe Artificial Intelligence" (Philosophical Studies special issue on AI safety, 2024)
Explores some ways in which prominent deontological theories (including moderate, contractualist and non-aggregative deontology) may lead to unsafe outcomes when implemented by autonomous AI systems. Also discusses the conceptual relationship between morally alignment and safety, arguing that these properties need not coincide and that we should prefer safe systems in the event of possible conflicts. Take-home message: the ethical AI program is likely not a straightforward solution to AI risk mitigation.
"Artificial Intelligence: Arguments for Catastrophic Risk" (Philosophy Compass, 2024; with Adam Bales and Cameron Domenico Kirk-Giannini)
Some people think we should be worried about future AI systems' potential to cause us great harm. We survey discussions around two influential and interrelated arguments. One argument claims that sufficiently advanced AI systems might be expected to engage in dangerous power-seeking behavior in the course of pursuing their goals. The second argument suggests that, once we've developed human-level AI technology, subsequent systems might improve very quickly, culminating in a "singularity" far beyond human capabilities.
"Large Language Models and Biorisk" (American Journal of Bioethics, 2023; with Harry Lloyd and Nate Sharadin)
A commentary piece discussing some ways in which AI technology might facilitate the development and malicious misuse of dangerous biochemical material (e.g. by helping with the discovery of novel bioagents, or by serving as a highly effective lab assistant and thus lowering the barrier to entry for the relevant bench work). Also suggests some policy strategies for mitigating these risks.
"AI Language Models Cannot Replace Human Research Participants" (AI & Society, 2023; with Jacqueline Harding, N.G. Laskowski and Rob Long)
A short piece responding to Dillion et al.'s "Can AI Language Models Replace Human Participants?", which notes the strong correlation between average human and LLM moral judgments and speculates optimistically about the possible roles of language models in psychology research. We argue that this optimism is either overstated or misplaced.
"Unrealistic Models in Mathematics" (Philosophers' Imprint, 2023)
Shows that mathematicians, like empirical scientists, use unrealistic models to gain better understanding of the complex phenomena they study. I present a pair of case studies and draw three morals. First: that unrealistic models have important uses in pure mathematics, and their epistemic benefits include improving our understanding of their target phenomena. Second: that the understanding gained from these models (and hence from unrealistic models in general) need not flow from explanations of the target phenomena. Third: that it need not flow from counterfactual knowledge either.
"Transferable and Fixable Proofs" (Episteme, 2025)
Explores a tension between two plausible conditions on acceptable proofs: (1) that proofs containing significant mistakes may be acceptable, so long as the mistakes are fixable in a certain sense; (2) that acceptable proofs must be transferable, meaning that the information they contain is enough to convince a typical expert. I argue that these conditions conflict, and that the transferability condition is the problem. Acceptable proofs only have to satisfy a pair of weaker constraints, which I call evaluability and corrigibility.
"Is It Bad to Prefer Attractive Partners?" (Journal of the American Philosophical Association, 2023)
Argues that there are prima facie similarities between the preference for physically attractive partners and various forms of wrongful discrimination, and that some obvious strategies for justifying the former don't work. I suggest that the most defensible version of the preference is one that links attractive aspects of personal style to desirable personality traits and values. Lots of discussion of the psychology and social science literature on attractiveness.
This paper was discussed on an episode of the popular philosophy and psychology podcast Very Bad Wizards.
"Proving Quadratic Reciprocity: Explanation, Disagreement, Transparency and Depth" (Synthese, 2021)
Analyzes a long-running disagreement among mathematicians about how best to explain Gauss's quadratic reciprocity theorem. I argue that some explanatory proofs of QR are "transparent", while others are "deep"; the disagreement arises because some mathematicians prefer one type of explanation over the other. Closes by raising a problem for Marc Lange's theory of mathematical explanation.
This paper is based on my math MS thesis, supervised by Ramin Takloo-Bighash.
"Viewing-as Explanations and Ontic Dependence" (Philosophical Studies, 2020)
Explores the phenomenon of “viewing one object as another”. I show that “viewing-as” lies at the heart of a distinctive type of explanation. Viewing-as cases are interesting in many ways, notably because they seem to defy the popular ontic and counterfactual conceptions of explanation. What’s needed instead, I suggest, is a “cognitivist” account that understands explanatory success in terms of benefits to thinking and reasoning.
This paper received the APA's 2019 Routledge, Taylor and Francis article prize for best publication by a pre-tenure-track philosopher. The selection committee called it "an excellent, original and important contribution to the philosophy of science".
"Mathematical Explanation beyond Explanatory Proof" (British Journal for the Philosophy of Science, 2020)
Argues that not all mathematical explanations involve proofs. Considers in detail Galois's explanation of the unsolvability of the quintic, which has been claimed to rest on an explanatory proof; I try to show that this is mistaken.
This paper was a BJPS Editor's Choice selection and is freely available to read on the journal website.
"Explanation in Mathematics: Proofs and Practice" (Philosophy Compass, 2019)
Surveys recent work on explanatory proofs and their role in mathematical practice.
I also made an accompanying Teaching and Learning Guide with suggestions for further reading and ideas for class activities and projects.
"Arithmetic, Set Theory, Reduction and Explanation" (Synthese, 2018)
Argues that viewing the natural numbers and arithmetical operations as sets has no explanatory value. Thus, contrary to received wisdom, there are bona fide intertheoretic reductions that are nevertheless unexplanatory.
"Explicitism about Truth in Fiction" (British Journal of Aesthetics, 2016)
Challenges "implicitism", a widespread view about truth in fiction according to which there are truths in some stories that aren't explicitly asserted anywhere in the relevant texts. (E.g., "Holmes is human".)
This paper was a runner-up in the British Society for Aesthetics 2014 Essay Prize contest.
Other Writing
“What’s Hot in Mathematical Philosophy” column for The Reasoner magazine
The Reasoner 16: 2 (March-April 2022, on purity of methods)
The Reasoner 16: 1 (January-February 2022, on mathematical understanding)
The Reasoner 15: 5 (September-October 2021, on the error-tolerance of proofs)
The Reasoner 15: 4 (July-August 2021, on enumerative induction and mathematical justification)
The Reasoner 15: 3 (May-June 2021, on the necessity of mathematics)
The Reasoner 15: 1 (January-February 2021, on counterfactuals and mathematical explanation)
Guest editorial and interview with Kenny Easwaran
The Reasoner 15:2 (March-April 2021), 9-12
I talk to Kenny about fractal music, Zoom conferences, journal refereeing, teaching in math and philosophy, the rationalist community and its relationship to academia, decision-theoretic pluralism, and the city of Manhattan, Kansas.
“Extraversion, Happiness, and the Pandemic”
Blog of the APA (Mar. 18, 2021)
About the nature of extraversion, the relationship between personality and happiness, and what this can tell us about the surprising fact that extraverts were happier than introverts during early COVID lockdowns.
Dissertation
Dimensions of Mathematical Explanation, UIC, 2017.
Committee: Daniel Sutherland (chair), Mahrad Almotahari, Dave Hilbert, Marc Lange, Kenny Easwaran.
I did a three-paper dissertation, the parts of which would become "Arithmetic, Set Theory, Reduction and Explanation", "Mathematical Explanation beyond Explanatory Proof", and "Viewing-as Explanations and Ontic Dependence".