# 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

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*Noûs*(forthcoming)

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.

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To appear in Oliveri (ed.),

*Objects, Structures, and Logics*

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*Mind*(forthcoming)

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*Australasian Journal of Philosophy*(forthcoming)

*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.

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*Philosophical Studies*(2019) 177: 1275-1282

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*Synthese*(2020) 197: 477-495

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*Mind*(2017) 126: 429-463

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?", has recently been published in *Philosophical Studies*.]

## Work in Progress & Under Review

__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]

A striking fact about mathematics is that it is, in a certain sense, free or unconstrained in its practice: mathematicians are free, in their capacity as mathematicians, to work within any theory provided that minimal standards of logical coherence are satisfied. How can this feature of mathematics be explained? I argue that contemporary views in the philosophy of mathematics are unable to provide a satisfying explanation. I present a novel account of mathematical truth, drawing on the doctrine of alethic pluralism, according to which (to a first approximation) mathematical truth is realized by the property of coherence. I conclude by showing that this view provides a surprisingly fruitful framework for resolving resolving issues of mathematical disagreement and the question of new set theoretic axioms.

__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.

__Abstract__]

In *Mathematical Truth*, Paul Benacerraf famously raised a dilemma for the philosophy of mathematics: very roughly, that mathematics needs a reasonable semantics and a reasonable epistemology but that it's impossible to see how these needs can be simultaneously satisfied. In this paper I argue that (i) Benacerraf's semantic constraint is in fact ambiguous between a stronger and a weaker reading; (ii) only the stronger reading is incompatible with the idea that mathematical truth is an epistemically tractable property; and (iii) the stronger reading is under-motivated.

__Abstract__]

Many who have considered the subject believe that our best mathematical theories (e.g. arithmetic, analysis, and set theory) are consistent. But, in light of Gödel's incompleteness theorems and for other reasons, it is not at all obvious how this attitude can be rationally defended. In this paper I consider one influential argument (schema) for the consistency of mathematical theories, going via the notion of truth. Roughly, it runs as follows: the axioms of the theory are true, the inference rules of logic preserve truth, so all of the theorems are true; but if the theory were inconsistent, it would have false theorems; so it must be consistent. I argue that this truth-theoretic argument is, on many plausible views of mathematics, epistemically useless: in the sense much discussed in recent epistemology, it is incapable of transmitting justification from its premises to its conclusion.

## Teaching

Lingnan:- Epistemology (Fall 2020)
- The Nature of Truth (Spring 2019 & Spring 2020) [Syllabus]
- Western Political Philosophy (Fall 2019)
- Modal Logic (Fall 2018 & Fall 2019) [Syllabus]
- Critical Thinking and Logic (Fall & Spring 2018, Spring 2019, Spring 2020)

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

- Logic (NSSR, Spring 2016)
- Political Philosophy (NYU, Summer 2015) [Syllabus]
- Medical Ethics (NYU, Summer 2013) [Syllabus]

- 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)