“An instant later, both Professor Waxman and his time machine are obliterated, leaving the cold-blooded/warm-blooded dinosaur debate still unresolved.” (Gary Larson, The Far Side)

Daniel Waxman

I'm an Assistant Professor of Philosophy at the National University of Singapore. I am also an Associate Editor at the Australasian Journal of Philosophy. My research is mostly in the philosophy of mathematics, logic, and epistemology, and I have wide additional interests in metaphysics, metaethics, normative ethics, and political philosophy.

Before coming to NUS, I was an Assistant Professor at Lingnan University, Hong Kong. Before that, I was a Junior Research Fellow at Corpus Christi College, Oxford. I received my PhD in Philosophy from New York University in 2017, where I wrote my dissertation under the supervision of Crispin Wright, Hartry Field, and Cian Dorr. As an undergraduate, I studied Maths and Philosophy at Worcester College, Oxford.

I'm currently working on projects concerning philosophical and formal theories of truth, the objectivity and determinacy of mathematics, the possibility of justification or knowledge in the consistency of our best theories (especially in light of limitative results in like Gödel's incompleteness theorems), and the question whether there are any interesting internalist norms.

Here is a link to my CV.

My email address is [email protected].

Publications

Arithmetical Pluralism and the Objectivity of Syntax with Lavinia Picollo [Abstract▾| PDF | Published]
Noûs (forthcoming)

Arithmetical pluralism is the view that there is not one true arithmetic but rather many apparently conflicting arithmetical theories, each true in its own language. While pluralism has recently attracted considerable interest, it has also faced significant criticism. One powerful objection, which can be extracted from Parsons' work, appeals to a categoricity result to argue against the possibility of seemingly conflicting true arithmetics. Another salient objection raised by Putnam and Koellner draws upon the arithmetization of syntax to argue that arithmetical pluralism is inconsistent with the objectivity of syntax. First, we review these arguments and explain why they ultimately fail. We then offer a novel, more sophisticated argument that avoids the pitfalls of both. Our argument combines strategies from both objections to show that pluralism about arithmetic entails pluralism about syntax. Finally, we explore the viability of pluralism in light of our argument and conclude that a stable pluralist position is coherent. This position allows for the possibility of rival packages of arithmetic and syntax theories, provided that they systematically co-vary with one another.

Internalism and the Determinacy of Mathematics with Lavinia Picollo [Abstract▾| PDF | Published] ]
Mind 132: 1028-1052 (2023)

A major challenge in the philosophy of mathematics is to explain how mathematical language can pick out unique structures and acquire determinate content. In recent work, Tim Button and Sean Walsh have introduced a view they call 'internalism', according to which mathematical content is explained by internal categoricity results formulated and proven in second-order logic. In this paper, we critically examine the internalist response to the challenge and discuss the philosophical significance of internal categoricity results. Surprisingly, as we argue, while internalism arguably explains how we pick out unique mathematical structures, this does not suffice for the determinacy of mathematical discourse.

Justification and Being in a Position to Know [Abstract▾| PDF | Published]
Analysis 82: 289-298 (2022)

According to an influential recent view, S is propositionally justified in believing p iff S is in no position to know that S is in no position to know p. I argue that this view faces compelling counterexamples.

Justification as Ignorance and Logical Omniscience [Abstract▾| PDF | Published]
Asian Journal of Philosophy 1:7 (2022)

I argue that there is a tension between two of the most distinctive theses of Sven Rosenkranz's Justification as Ignorance: (i) the central thesis concerning justification, according to which an agent has propositional justification to believe p iff they are in no position to know that they are in no position to know p; and (ii) the desire to avoid logical omniscience by imposing only "realistic" idealizations on epistemic agents.

[This paper is part of a book symposium on Sven Rosenkranz's Justification as Ignorance.]

Losing Confidence in Luminosity with Simon Goldstein [Abstract▾| PDF | Published]
Noûs 55: 962-991 (2021)

A mental state is luminous if, whenever an agent is in that state, they are in a position to know that they are. Following Timothy Williamson's Knowledge and Its Limits, a wave of recent work has explored whether there are any non-trivial luminous mental states. A version of Williamson's powerful argument that none exist appeals to a safety-theoretic principle connecting knowledge and confidence: if an agent knows p, then p is true in any nearby scenario where she has a similar level of confidence in p. However, the notion of confidence in this safety principle is relatively underexplored.

This paper attempts to remedy the gap by providing a precise theory of confidence: an agent's degree of confidence in p is the objective chance they will rely on p in practical reasoning and action. This theory of confidence is then used to critically evaluate the anti-luminosity argument, leading to the surprising conclusion that although there are strong reasons for thinking that luminosity does not obtain, they are quite different from those the existing literature has considered.

Rescuing Implicit Definition from Abstractionism [Abstract▾| PDF | Published]
In Oliveri G., Ternullo C., Boscolo S. (eds) Objects, Structures, and Logics, Boston Studies in the History and Philosophy of Science, vol 339: 97-129 (2022)

Neo-Fregeans in the philosophy of mathematics hold that the key to a correct understanding of mathematics is the implicit definition of mathematical language. In this paper, I discuss and advocate the rejection of abstractionism: the constraint (implicit within much of the recent neo-Fregean tradition) according to which all acceptable implicit definitions take the form of abstraction principles. I argue if we take the axioms of mathematical theories themselves as implicit definitions, a much more attractive and unified view of mathematics results.

Stable and Unstable Theories of Truth and Syntax with Beau Madison Mount [Abstract▾| PDF | Published]
Mind 130: 439-473 (2021)

Recent work on formal theories of truth has revived an approach, due originally to Tarski, on which syntax and truth theories are sharply distinguished—`disentangled'—from mathematical base theories. In this paper, we defend a novel philosophical constraint on disentangled theories: we argue that these theories must be epistemically stable in that they must possess an intrinsic motivation justifying no strictly stronger theory. We argue that in a disentangled setting, even if the base and the syntax theory are individually stable, they may nevertheless be jointly unstable. We contend that this flaw afflicts many proposals discussed in the literature. We go on to defend a new, stable disentangled theory: double second-order arithmetic.

Is Mathematics Unreasonably Effective? [Abstract▾| PDF | Published]
Australasian Journal of Philosophy 99: 83-99 (2021)

Many mathematicians, physicists, and philosophers have suggested that the fact that mathematics—an a priori discipline informed substantially by aesthetic considerations—can be applied to natural science is mysterious. This paper sharpens and responds to a challenge to this effect. I argue that the aesthetic considerations used to evaluate and motivate mathematics are much more closely connected with the physical world than one might presume, and (with reference to case-studies within Galois theory and probabilistic number theory) show that they are correlated with generally recognized theoretical virtues, such as explanatory depth, unifying power, fruitfulness, and importance.

Supertasks and Arithmetical Truth with Jared Warren [Abstract▾| PDF | Published]
Philosophical Studies 177: 1275-1282 (2019)

This paper discusses the relevance of supertask computation for the determinacy of arithmetic. Recent work in the philosophy of physics has made plausible the possibility of supertask computers, capable of running through infinitely many individual computations in a finite time. A natural thought is that, if true, this implies that arithmetical truth is determinate (at least for e.g. sentences saying that every number has a certain decidable property). In this paper we argue, via a careful analysis of putative arguments from supertask computations to determinacy, that this natural thought is mistaken: supertasks are of no help in explaining arithmetical determinacy.

A Metasemantic Challenge for Mathematical Determinacy with Jared Warren [Abstract▾| PDF | Published]
Synthese 197: 477-495 (2020)

This paper presents a challenge for the view that mathematical truth is determinate. We begin by discussing the bearing of famous independence results on the issue, arguing that there is no straightforward argument from independence to indeterminacy. Nevertheless, we present a serious challenge for advocates of determinacy: fundamentally, to give a metasemantic explanation of how it can arise. We defend two constraints on any acceptable explanation and show that they can be used to develop a powerful argument against determinacy. We believe our discussion poses a challenge for many significant philosophical theories of mathematics, applying even to those which accept the determinacy of basic arithmetic.

Deflationism, Arithmetic, and the Argument from Conservativeness [Abstract▾| PDF | Published]
Mind 126: 429-463 (2017)

Many philosophers believe that a deflationary truth theory must conservatively extend any base theory to which it is added (put roughly: talking about truth shouldn't allow us to establish any new claims about any subject-matter that doesn't involve truth). But when applied to arithmetic, it's argued, the imposition of a conservativeness requirement leads to a serious objection to deflationism: for the Gödel sentence for Peano Arithmetic (PA) is not a theorem of PA, but becomes a theorem when PA is extended by adding certain appealing principles governing truth.

I argue in this paper that no such objection succeeds. The issue turns on how we understand the notion of logical consequence implicit in any conservativeness requirement, and whether or not we possess a categorical conception of the natural numbers (i.e. whether we can rule out so-called "non-standard models"). I offer a disjunctive response: if we do possess a categorical conception of arithmetic, then deflationists have principled reason to accept a conservativeness requirement stated in terms of a rich notion of logical consequence according to which the Gödel sentence follows from PA. But if we do not, then the reasons for requiring the derivation of the Gödel sentence lapse, and deflationists are free to accept a conservativeness requirement stated proof-theoretically. Either way, deflationism is in the clear.

[A reply by Julian Murzi and Lorenzo Rossi, entitled "Conservative Deflationism?", appears in Philosophical Studies.]

Drafts

Imagining the Infinite [Abstract▾| Draft PDF]

This paper investigates the procedure of conceiving of a model of mathematical structures, where this involves a distinctive kind of visual mathematical thinking. I argue that (i) conceiving of structures in this way is best understood as an exercise of the imagination (and not, as many philosophers would contend, an exercise of rational intuition); (ii) once the relevant kind of imaginative capacity is clarified, it becomes apparent that we can in fact conceive of infinite mathematical structures; and (iii) by doing so, we obtain justification in the consistency or coherence of certain mathematical theories.

Did Gentzen Prove the Consistency of Arithmetic? [Abstract▾| Draft PDF]

In 1936, Gerhard Gentzen famously gave a proof of the consistency of Peano arithmetic. There is no disputing that Gentzen provided us with a mathematically valid argument. This paper addresses the distinct question of whether Gentzen's result is properly viewed as a proof in the epistemic sense: an argument that can be used to obtain or enhance justification in its conclusion. Although Gentzen himself believed that he had provided a “real vindication” of Peano arithmetic, many subsequent mathematicians and philosophers have disagreed, on the basis that the proof is epistemically circular or otherwise inert. After gently sketching the outlines of Gentzen's proof, I investigate whether there is any epistemically stable foundational framework on which the proof is informative. In light of this discussion, I argue that the truth lies somewhere in between the claims of Gentzen and his critics: although the proof is indeed epistemically non-trivial, it falls short of constituting a real vindication of the consistency of Peano arithmetic.

Teaching

As primary instructor: Oxford (tutorials/classes): NYU (as TA):