Understanding: An Annotated Bibliography
This is an annotated reading path through the epistemology of understanding, ordered as a route rather than a catalogue. Each entry says what a work is for and where it is weak, with a link to the published version and to a free copy wherever one exists.
It aims to be comprehensive on understanding in mathematics and in the philosophy of science, and is deliberately selective on machine understanding: that section tracks only the work bearing directly on the epistemological questions and is reviewed rather than kept exhaustive.
Starting points
Grimm, S. R. (2025 [2021]). Understanding. In E. N. Zalta & U. Nodelman (Eds.), The Stanford Encyclopedia of Philosophy.
The Stanford Encyclopedia survey of the field, organised by analogy with the traditional analysis of knowledge: understanding is taken to have a distinctive object (connections or dependence relations, on internalist, externalist, and hybrid readings), a distinctive psychology (grasping or seeing, and the question whether this is an ability absent from ordinary knowledge), and a distinctive normative profile (how far it tolerates luck and whether it must be factive). It also lays out the philosophy-of-science disputes over explanation, idealization, and whether understanding can be transmitted by testimony. As an entry point it is comprehensive and even-handed; because it is built around the comparison with knowledge, positions that decline that framing get comparatively brief treatment.
Hannon, M. (2021). Recent Work in the Epistemology of Understanding. American Philosophical Quarterly, 58(3), 269–290.
A critical survey of the epistemology of understanding that frames the field around a single question: whether understanding reduces to a species of knowledge. It sorts the literature into reductionists, who model understanding on the belief, truth, and justification conditions familiar from knowledge, and anti-reductionists, who hold that understanding tolerates luck, false idealization, or non-propositional grasp in ways knowledge does not. As a map it is useful for locating positions, though it takes the comparison with knowledge as its organising axis and says less about approaches that decline that framing.
Gordon, E. C. (2017). Understanding in Epistemology. Internet Encyclopedia of Philosophy.
Annotation in progress.
Baumberger, C., Beisbart, C. & Brun, G. (2016). What Is Understanding? An Overview of Recent Debates in Epistemology and Philosophy of Science. In S. R. Grimm, C. Baumberger & S. Ammon (Eds.), Explaining Understanding: New Perspectives from Epistemology and Philosophy of Science. Routledge.
Annotation in progress.
What understanding is
Zagzebski, L. T. (2001). Recovering Understanding. In M. Steup (Ed.), Knowledge, Truth, and Duty: Essays on Epistemic Justification, Responsibility, and Virtue (pp. 235–251). Oxford University Press.
One of the pieces, alongside work by Elgin and Kvanvig, that put understanding back on the epistemological agenda as a state distinct from knowledge. It takes the object of understanding to be non-propositional structures of reality, a car, a piece of music, a person’s character, a causal nexus, rather than isolated true propositions, and it ties understanding to a first-person transparency that testimonial knowledge need not carry. The claim that the relevant grasping has to be achieved first-hand, and so cannot simply be handed over by testimony, is the part that later work in this list contests directly.
Kvanvig, J. L. (2003). The Value of Knowledge and the Pursuit of Understanding. Cambridge University Press.
An early and much-cited case for treating understanding as an epistemic good distinct from knowledge, on the grounds that its object is a coherent body of information rather than an isolated proposition, and that grasping the coherence-making relations within that body survives the kind of luck that defeats knowledge. The book did much to put objectual understanding on the agenda and to align it with internalist and coherentist commitments. Its central claim that understanding is immune to Gettier-style luck has been the most contested, with later work arguing that analogous luck cases can be built for understanding too.
Grimm, S. R. (2006). Is Understanding a Species of Knowledge? The British Journal for the Philosophy of Science, 57(3), 515–535.
Annotation in progress.
Elgin, C. Z. (2017). True Enough. MIT Press.
The principal sustained defence of non-factivism about understanding: idealizations and other deliberate falsehoods in science do genuine epistemic work rather than merely being tolerated, with reflective equilibrium rather than literal truth serving as the standard of acceptability. It anchors the anti-factivist side of the idealization debate and supplies many of its stock examples, from frictionless planes to the ideal gas law. Critics press that a view this permissive struggles to separate falsehoods that enable understanding from those that merely mislead, and that tightening it tends to collapse back toward a factivism about the model’s core.
Grasping and cognitive control
Strevens, M. (2024). Grasp and Scientific Understanding: A Recognition Account. Philosophical Studies, 181(4), 741–762.
Defends the explanationist claim that there is no understanding-why without grasp of a correct explanation, and then does the work of saying what grasp is: a recognitional ability, so that to grasp a property is to be able to recognise its instances and to grasp an explanation is to be able to recognise the relations it reports. It is pitched as a naturalistically respectable alternative to treating grasp as a primitive or a quasi-perceptual act. The recognition account has to answer whether it is rich enough for the abstract and mathematical items that turn up in explanations, and its prior explanationist commitment is exactly what accounts elsewhere in this list, from equivalent reformulations and from unrealistic models, are built to resist.
Chudnoff, E. (2014). Is Intuition Based on Understanding? Philosophy and Phenomenological Research, 89(1), 42–67.
Examines the widely held non-skeptical proposal that intuitions justify belief because they are based on the subject’s understanding of the proposition intuited, contrasting a narrow reading on which understanding is competence with the constituent concepts with a broader one that goes beyond mastery of definitions. The question pressed is whether that understanding-based story can carry the justificatory weight placed on it, or whether some intuitions instead trace to heuristics and biases that understanding does not vindicate. Like the accounts it engages, it leaves the operative notion of understanding largely unspecified.
Boghossian, P. (2016). Intuitions and the Understanding. In M. Á. Fernández Vargas (Ed.), Performance Epistemology: Foundations and Applications (pp. 137–150). Oxford University Press.
Argues that the a priori justification carried by intuitions, as in the Twin Earth case, cannot be explained either by perception or by understanding in the sense of conceptual competence, so that an appeal to intuition as a sui generis intellectual seeming is unavoidable. It works mainly by pressing Sosa’s understanding-based account, granting that intuitions are a priori but rejecting the claim that they are attractions to assent grounded in understanding of the proposition alone. The critique is largely negative: it concludes that no adequate account of what it is to grasp a proposition is currently available, and offers only a minimalist gesture in place of one.
Factivity and idealization
Mizrahi, M. (2012). Idealizations and Scientific Understanding. Philosophical Studies, 160(2), 237–252.
Defends scientific understanding against the argument from idealization by distinguishing a theory’s core tenets, its primary laws and the conditions under which they apply, from the supporting assumptions needed only to derive predictions. On this division the idealizing falsehoods sit among the supporting assumptions, so their presence does not show that understanding rests on falsehood or that a factivity requirement must be dropped. The move depends on a principled core/periphery line, and critics question whether that line can be drawn without already presupposing the verdict about understanding it is meant to support.
D’Alessandro, W. (2024). Unrealistic Models in Mathematics. Philosophers’ Imprint.
Extends the debate about idealized and unrealistic models from empirical science into pure mathematics, using cases such as Cramér’s random model of the primes to argue that mathematicians draw understanding from models that deliberately misrepresent their targets. The central claims are that this understanding is not always routed through explanation, and not always a matter of acquiring counterfactual knowledge, against accounts that treat either as the default. Because the argument leans on a handful of salient examples, how far the moral generalises across mathematical practice is left open.
Models and understanding
Weisberg, M. (2013). Simulation and Similarity: Using Models to Understand the World. Oxford University Press.
A systematic statement of the view that modelling is the indirect study of the world: modellers build and analyse interpreted structures, whether concrete, mathematical, or computational, and only then ask how these relate to a target. It develops a weighted feature-matching account of model–world similarity as an alternative to isomorphism-based accounts, and treats idealization as intentional, constrained distortion rather than mere inaccuracy. The similarity account has drawn the objection that its weighting of relevant features is fixed largely by the modeller’s purposes, leaving the model–world relation less constrained than a formal criterion would make it.
Frigg, R. & Nguyen, J. (2025). Stabilising Understanding. Philosophical Studies.
Proposes that an idealized model yields understanding of a target behaviour when the model and the target’s perfect model both fall within a class of models over which that behaviour is stable, with the indispensable shared features forming what the authors call the model’s noetic core. It is offered as a factivist account that nonetheless departs from existing factivist proposals about how models manage to get the relevant aspects of the target right, and as a way past the dilemma that non-factivism is either too permissive or collapses into factivism. Whether the stability and noetic-core notions can be specified without already presupposing which features matter is the natural pressure point.
Understanding in mathematics
Avigad, J. (2026). Mathematical Understanding. PhilSci-Archive preprint.
Argues that a theory of mathematical understanding should be a pragmatic account of the abilities that make mathematical reasoning possible, abstracted away from whichever system happens to implement them, be it a student, an expert, a symbolic algorithm, or a neural network. On this basis it reads developments in symbolic and neural AI as offering competing models of what mathematical understanding consists in, one built on explicit representations and heuristics, the other on distributed representations learned from experience. As a programmatic preprint it sets out the framing and the motivating cases rather than a worked-out inventory of abilities.
Hamami, Y. & Morris, R. L. (2024). Understanding in Mathematics: The Case of Mathematical Proofs. Noûs, 58(4), 1073–1106.
Develops an account of what it is to understand a mathematical proof, taking as its starting point the practitioners’ distinction between understanding a proof and merely verifying its steps. On the proposal, understanding a proof consists in rationally reconstructing the plan underlying it, recovering the tree of proving intentions and reproducing the practical reasoning behind each step, which also yields a graded notion on which partial understanding corresponds to the portion of the plan recovered. The account is deliberately restricted to the proof’s internal deductive content and brackets the wider context, notation, background theory, and history, that the authors grant can also bear on understanding.
Hunt, J. R. (2021). Understanding and Equivalent Reformulations. Philosophy of Science, 88(5), 810–823.
Uses theoretically equivalent formulations of a physical theory, Lagrangian versus Hamiltonian mechanics and analogous pairs, to argue against explanationism, the view that every difference in understanding-why is a difference in grasp of an explanation. Since such formulations agree on all the facts and so share an explanation while plainly affording different understanding, it concludes that understanding does not reduce to explanation, and proposes a conceptualist supplement on which the difference lies in how the same explanatory information is organised. The argument turns on the claim that equivalent formulations really do carry the same explanation, which an explanationist can resist by assigning one formulation explanatory priority.
Proof: rigor, know-how, shareability
Burgess, J. P. & De Toffoli, S. (2022). What Is Mathematical Rigor? APhEx, 25, 1–17.
A survey of the problem of reconciling the logician’s conception of rigour, on which a proof is rigorous if it can in principle be converted into a formal derivation, with the fact that published proofs look nothing like formal derivations. It argues that the so-called standard view is really a family of views that differ on whether the required formalization is actual or merely potential and on whether the conversion counts as routine, and that neglecting these distinctions has produced cross-talk in the literature. It also opens subsidiary questions, about the function of rigour, whether it admits of degrees, and how it relates to mere acceptability, without settling them.
De Toffoli, S. (2021). Groundwork for a Fallibilist Account of Mathematics. The Philosophical Quarterly, 71(4), 823–844.
Argues that mathematical knowledge is fallible even when it rests on proof, because the warrant a proof confers depends on its being checked and accepted by the mathematical community rather than on an idealized formal object standing behind it. It ties rigour and justification to what practitioners can survey, reproduce, and share, making the social process of vetting proofs epistemically load-bearing rather than a mere sociological add-on. The account leaves open how much idealization the notion of a ‘proof that could be checked’ still smuggles in, and how to demarcate the relevant community whose acceptance counts.
Testimony and the social
Goldberg, S. C. (2010). Relying on Others: An Essay in Epistemology. Oxford University Press.
Annotation in progress.
Boyd, K. (2017). Testifying Understanding. Episteme, 14(1), 103–127.
Annotation in progress.
Machine understanding
Reviewed August 2026 — selective by design.
Grimm, S. R., Vukov, J. & Aminoff, E. (2026). Levels of Understanding, World Models, and Artificial Intelligence. Philosophical Studies.
Sets out a four-level scheme for understanding, from mapping a target’s elements to grasping their powers, identifying difference-makers, and apprehending necessitators, on which understanding deepens vertically through the levels and broadens horizontally within each, and which is meant to apply equally to human and artificial agents. It uses the scheme to reframe the question of whether AI systems that build world models thereby understand, treating a world model as a vehicle of understanding only to the extent it captures the target’s modal structure. The paper stays neutral on whether current systems clear that bar, so its contribution is a taxonomy and a research agenda rather than a verdict.
Chen, H., Grimm, S. R., Russakovsky, O. & Lombrozo, T. (2026). Machine Understanding. Trends in Cognitive Sciences.
A conceptual toolkit rather than a theory: it recasts questions about machine understanding as questions of the form ‘does system S understand target T?’ and lays out the options for specifying S, T, and the relation between them, including a ladder from base architectures to distributed human–machine systems. It sorts existing proposals into model-based accounts, which look at a system’s internal representations, and ability-based accounts, which look at its behaviour, with optional etiological and phenomenological requirements, and flags the recurring ambiguity over whether benchmark performance constitutes or merely evidences understanding. By design it defends no account, which limits how far it can adjudicate disputes about particular systems.
Ha, D. & Schmidhuber, J. (2018). World Models. arXiv:1803.10122.
The paper that popularised the term ‘world model’ in machine learning: a generative recurrent network is trained without supervision to compress an environment’s spatial and temporal regularities, and a small controller is then trained on those latent features, in part entirely within simulated rollouts before transferring back to the real task. It is included here as the technical reference point for the world-model vocabulary that later philosophical work takes up, not as a contribution to the epistemology of understanding. Its notion of a world model is tied to control performance on reinforcement-learning benchmarks and carries no claim about representation or understanding in the philosophical sense.
Understanding and inquiry
Friedman, J. (2020). The Epistemic and the Zetetic. The Philosophical Review, 129(4), 501–536.
Annotation in progress.
El Shazly, A. (2025). Inquiring to Understand. The Philosophical Quarterly.
Argues that the leading accounts, understanding as a kind of knowledge, as a static grasp of explanations, or as an ability, cannot explain how understanding is acquired, communicated, or pursued jointly, and proposes instead that understanding is a generative process of building perspectives that organise information relative to context and purpose. On this perspectival picture, understanding cannot be handed over by testimony alone but has to be reconstructed by the recipient, often through metaphor, narrative, or diagram, and joint inquiry is a matter of calibrating perspectives rather than converging on beliefs. The proposal is programmatic, and leaves open how a perspective is to be individuated or assessed.
This bibliography is under active development. Entries and annotations are being added.