# Daniel Waxman

I'm an Assistant Professor of Philosophy at the National University of Singapore. 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, and the question of how we can obtain justification or knowledge that our best theories are consistent or coherent (especially in light of limitative results in like Gödel's incompleteness theorems).

Here is a link to my CV.

My email address is danielwaxman@gmail.com.

## 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)

*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)

**[**

*Justification as Ignorance*and Logical Omniscience__Abstract__| PDF | Published]

*Asian Journal of Philosophy*1:7 (2022)

*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)

**Stable and Unstable Theories of Truth and Syntax**with Beau Madison Mount [

__Abstract__| PDF | Published]

*Mind*130: 439-473 (2021)

**Is Mathematics Unreasonably Effective?**[

__Abstract__| PDF | Published]

*Australasian Journal of Philosophy*99: 83-99 (2021)

*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)

**A Metasemantic Challenge for Mathematical Determinacy**with Jared Warren [

__Abstract__| PDF | Published]

*Synthese*197: 477-495 (2020)

**Deflationism, Arithmetic, and the Argument from Conservativeness**[

__Abstract__| PDF | Published]

*Mind*126: 429-463 (2017)

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

__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.

__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:- Philosophy of Language
- Language and Thought
- Philosophy of Mathematics (x2) [syllabus]
- Major Political Philosophers
- Graduate Research Seminar
- Ontology and Metaontology [syllabus]
- Advanced Political Philosophy [syllabus]
- Epistemology (Lingnan) [syllabus]
- The Nature of Truth (x2: Lingnan) [syllabus]
- Modal Logic (x2: Lingnan) [syllabus]
- Critical Thinking and Logic (x2: Lingnan)
- Political Philosophy (x2: NYU, Lingnan)
- Logic (Graduate level: New School for Social Research)
- Medical Ethics (NYU) [syllabus]

- Gödel's Incompleteness Theorems (Trinity 2017)
- Frege: Foundations of Arithmetic (Trinity 2017)
- Philosophy of Mathematics (Hilary 2017)
- Philosophical Logic (Hilary 2017)

- Great Works in Philosophy (Spring 2015, instructor Harvey Lederman)
- Ancient Philosophy (Fall 2014, instructor Jessica Moss)
- Texts and Ideas: Guilt and Sin, Law and Justice (Spring 2014, instructor Tim Maudlin)
- Political Philosophy (Fall 2013, instructor Aaron James)
- Life and Death (Spring 2013, instructor Joshua Gillon)
- Great Works in Philosophy (Fall 2012, instructor Jonathan Cottrell)