William D'Alessandro

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.

[Accepted to the 2026 Philosophy of Science Association meeting, presumptively to appear in the conference proceedings edition of Philosophy of Science] Deep-learning-based AI systems are notoriously opaque. Our tools for coping with this opacity include “explainable AI” (XAI) methods such as LIME, which aims to explain a given AI output by providing a linear model intended to locally approximate the underlying model's true decision boundary. Yet it remains controversial whether LIME models and other XAI methods can provide users with genuine understanding. One recent suggestion—defended in various forms by Fleisher (2022) and Knoks & Raleigh (2026)—is that LIME explanations function much like idealized models in science, which can successfully confer understanding in spite of their limited accuracy and lack of realistic causal detail. I argue that the analogy is a poor one. XAI explanations and idealized models are dissimilar in ways that matter. First, idealized scientific models are useful only when we have enough ancillary knowledge to tell which features of their predictions to take seriously and which to ignore; we’re rarely in this position with respect to complex AI systems. Second, we often need a “translation key” to go from the outputs of an idealized model—which might be false or nonsensical if taken at face value—to correct predictions about the target phenomenon. This also requires knowledge, experience and background theory which we typically lack in XAI settings. Third, it’s often thought that idealized models generate understanding by providing “how-possibly explanations”, which show how an event might have occurred. But how-possibly explanations are rarely useful in XAI contexts. Knoks and Raleigh attempt to shore up XAI's epistemic credentials by suggesting that it's primarily experts rather than laypeople who are in a position to gain understanding from XAI techniques. I argue that this strategy too is overly optimistic. I conclude by considering the epistemic prospects for the growing number of domain-specific variants of LIME.

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.

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.

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.