Justin Clarke-Doane

Ultrafinitism is the view that some specific Arabic numerals fail to determinately, unambiguously denote, because, for some types, no token is possible. In this article, I argue that the view is, at bottom, a thesis about modality; that it comes in at least three versions – semantic, pluralist, and physical; and that its central technical problems are to prove the key metatheorems within an ultrafinitist metatheory, including what I call the “feasible conservativeness” of ordinary science over feasible science.

Mathematical pluralism is roughly the view that any coherent foundational theory is equally correct. This is rough because a careful formulation must avoid contradiction (how could a theory and its negation both be correct?), distinguish foundational theories from algebraic ones, differ from fictionalism, and accommodate the indefinite extensibility of coherence. Even after these repairs, however, pluralism faces an insuperable problem of articulation. I consider several attempts to circumvent this problem, and argue that each fails. Their failure is parallel to the self-refutation problem for global relativism. I propose instead to analyze pluralism on the model of Carnap’s Principle of Tolerance (POT), interpreted as a non-factual policy proposal. Drawing on quasi-realism and the deflationism about truth that it presupposes, I show how such a policy can be truth-apt and embedded in complex sentences without inflating it into a factual thesis. The important point is that the considerations that support or refute it are practical, not evidential. Along the way, I isolate an underappreciated shadow of the problem of articulation, the problem of mathematical agreement. Even what follows from what in a fixed logic – indeed, even whether an explicitly given string is a proof in a logic – can depend on the foundation we use to check. So, why do we agree about mathematics to the extent that we do? I close by showing how pluralism leads to a limited non-factualist pluralism about physics. In particular, whether a discrete Ising system behaves deterministically from almost all starting states provably switches between ZF + V=L and ZFC + Large Cardinals (sufficient to imply Projective Determinacy).

In his influential book, _The Nature of Morality_, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles.” What is the epistemological relevance of this contrast, if genuine? This chapter argues that ethicists and philosophers of mathematics have misunderstood it. They have confused what the chapter calls the _justificatory challenge_ for realism about an area, D—the challenge to justify our D-beliefs—with the _reliability challenge_ for D-realism—the challenge to explain the reliability of our D-beliefs. Harman’s contrast is relevant to the first, but not, evidently, to the second. One upshot of the discussion is that genealogical debunking arguments are fallacious. Another is that indispensability considerations cannot answer the Benacerraf–Field challenge for mathematical realism.

Baroque questions of set-theoretic foundations are widely assumed to be irrelevant to physics. I argue that this is doubtful once determinism is read the way it is used in physics. At the analytic layer, determinism verdicts for possible systems can differ between canonical extensions of ZFC -- Goedel's Axiom of Constructibility (V=L) and large cardinal assumptions sufficient for Projective Determinacy (LC/PD) -- through coherence, uniqueness, and the identity of a definable initial datum. The main claim concerns a second, regularity layer. Determinism claims are meant to withstand admissible changes of gauge, mesh, coarse graining, and readout, and they are stated as claims about generic or almost sure behavior. Robust profiles of this kind land at the Sigma^1_2 level, the first level of the projective hierarchy at which regularity can fail in V=L while holding under LC/PD. Two unconditional theorems are proved. First, a fixed computable nearest neighbor Ising Hamiltonian on Z^3 with a fixed zero temperature Glauber schedule yields a Sigma^1_2-complete tail-readout profile in which only the initial microstate varies. Second, at the Kerr Cauchy horizon, canonicalizing continuous extension germs in a fixed collar contains E_0, so no universally measurable rule solves it in ZFC, while V=L supplies a projectively definable rule and PD forbids any projective one. I call the systematic study of such dependence reverse physics, on analogy with Friedman's and Simpson's reverse mathematics. One upshot is a dilemma: either physical theories must be relativized to foundational frameworks, as Carnap held for mathematics, or, with Quine, the search for new axioms is continuous with the search for new physical laws.

The motivating question of this chapter is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. In this view, axioms are analogous to laws of nature. They are postulated to best systematize the data to be explained. We argue that there is a decisive difference between the cases. There is agreement on the data to be systematized in the scientific case that has no analog in the mathematical one. There is virtual consensus over observations, but conspicuous dispute over intuitions. In this respect, mathematics more closely resembles stereotypical philosophy. We conclude by distinguishing two ideas that have long been associated—realism (the idea that there is an independent reality) and objectivity (the idea that in a disagreement, only one of us can be right). We argue that, while realism is true of mathematics and philosophy, typical claims from these domains fail to be objective. One upshot of the discussion is that even many questions of fundamental physics fail to be objective in roughly the sense that the question, ‘Is the Parallel Postulate true?’, understood as one of pure mathematics, fails to be. Another is a kind of pragmatism. Factual questions in mathematics, modality, logic, and evaluative areas go proxy for non-factual practical ones.

The motivating question of this paper is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They are postulated to best systematize the data to be explained. We argue that there is a decisive difference between the cases. There is agreement on the data to be systematized in the scientific case that has no analog in the mathematical one. There is virtual consensus over observations, but conspicuous dispute over intuitions. In this respect, mathematics more closely resembles stereotypical philosophy. We conclude by distinguishing two ideas that have long been associated -- realism (the idea that there is an independent reality) and objectivity (the idea that in a disagreement, only one of us can be right). We argue that, while realism is true of mathematics and philosophy, these domains fail to be objective. One upshot of the discussion is that even questions of fundamental physics may fail to be objective in roughly the sense that the question, ‘Is the Parallel Postulate is true?’, understood as one of pure mathematics, fails to be. Another is a kind of pragmatism. Factual questions in mathematics, modality, logic, and evaluative areas go proxy for non-factual practical ones.

It is widely assumed that platonism with respect to a discourse of metaphysical interest, such as fictional or mathematical discourse, affords a better account of the semantic appearances than nominalism, other things being equal. Of course, other things may not be equal. For example, platonism is supposed to come at the cost of a plausible epistemology and ontology. But the hedged claim is often treated as a background assumption. It is motivated by the intuitively stronger one that the platonist can take the semantic appearances at 'face-value' while the nominalist must resort to ad hoc and technically problematic machinery in order to explain those appearances away. In this article, I argue that, on any natural construal of 'face-value', the platonist, like the nominalist, is not able to take the semantic appearances at face-value. And insofar as the nominalist is forced to resort to ad hoc and technically awkward devices i n order to explain those appearances away, the platonist must resort to such devices as well. One moral of the story is that the thesis that platonism affords a better account of the semantic appearances than nominalism - even other things being equal - is not trivial. Another is that we should rethink a widespread methodology in metaphysics. © 2023 Elsevier B.V., All rights reserved.

Logical monism is the view that there is ‘One True Logic’. This is the default position, against which pluralists react. If there were not ‘One True Logic’, it is hard to see how there could be one true theory of anything. A theory is closed under a logic! But what is logical monism? In this article, I consider semantic, logical, modal, scientific, and metaphysical proposals. I argue that, on no ‘factualist’ analysis (according to which ‘there is One True Logic’ expresses a factual claim, rather than an attitude like approval), does the doctrine have both metaphysical and methodological import. Metaphysically, logics abound. Methodologically, what to infer from what is not settled by the facts, even the normative ones. I conclude that the only interesting sense in which there could be -- or could fail to be -- One True Logic is noncognitive. The same may be true of the monism-pluralism debate about normative areas, like moral, epistemic, and prudential ones, generally.

This book discusses the problem of mathematical knowledge, and its broader philosophical ramifications. It argues that the problem of explaining the (defeasible) justification of our mathematical beliefs (‘the justificatory challenge’), arises insofar as disagreement over axioms bottoms out in disagreement over intuitions. And it argues that the problem of explaining their reliability (‘the reliability challenge’), arises to the extent that we could have easily had different beliefs. The book shows that mathematical facts are not, in general, empirically accessible, contra Quine, and that they cannot be dispensed with, contra Field. However, it argues that they might be so plentiful that our knowledge of them is intelligible. The book concludes with a complementary ‘pluralism’ about modality, logic and normative theory, highlighting its revisionary implications. Metaphysically, pluralism engenders a kind of perspectivalism and indeterminacy. Methodologically, it vindicates Carnap’s pragmatism, transposed to the key of realism.

In this article, we discuss a simple argument that modal metaphysics is misconceived, and responses to it. Unlike Quine's, this argument begins with the observation that there are different candidate interpretations of the predicate ‘could have been the case’. This is analogous to the observation that there are different candidate interpretations of the predicate ‘is a member of’. The argument then infers that the search for metaphysical necessities is misguided in much the way the ‘set-theoretic pluralist’ claims that the search for the true axioms of set theory is. We show that the obvious responses to this argument fail. However, a new response has emerged that purports to prove, from higher-order logical principles, that metaphysical possibility is the broadest kind of possibility applying to propositions, and is to that extent special. We distill two lines of reasoning from the literature, and argue that their import depends on premises that any ‘modal pluralist’ should deny. Both presuppose that there is a unique typed hierarchy, which is what the modal pluralist, in the context of higher-order logic, should disavow. In other words, both presupposes that there is a unique candidate for what higher-order claims could mean. We consider the worry that, in a higher-order setting, modal pluralism faces an insuperable problem of articulation, collapses into modal monism, is vulnerable to the Russell-Myhill paradox, or even contravenes the truism that there is a unique actual world, and argue that these worries are misplaced. We also sketch the bearing of the resulting ‘Higher-Order Pluralism’ on the theory of content. One theme of the discussion is that, if Higher-Order Pluralism is correct, then there is no fixed metatheory from which to characterize higher-order reality.

It is widely agreed that the intelligibility of modal metaphysics has been vindicated. Quine's arguments to the contrary supposedly confused analyticity with metaphysical necessity, and rigid with non-rigid designators.2 But even if modal metaphysics is intelligible, it could be misconceived. It could be that metaphysical necessity is not absolute necessity – the strictest real notion of necessity – and that no proposition of traditional metaphysical interest is necessary in every real sense. If there were nothing otherwise “uniquely metaphysically significant” about metaphysical necessity, then paradigmatic metaphysical necessities would be necessary in one sense of “necessary”, not necessary in another, and that would be it. The question of whether they were necessary simpliciter would be like the question of whether the Parallel Postulate is true simpliciter – understood as a pure mathematical conjecture, rather than as a hypothesis about physical spacetime. In a sense, the latter question has no objective answer. In this article, I argue that paradigmatic questions of modal metaphysics are like the Parallel Postulate question. I then discuss the deflationary ramifications of this argument. I conclude with an alternative conception of the space of possibility. According to this conception, there is no objective boundary between possibility and impossibility. Along the way, I sketch an analogy between modal metaphysics and set theory.

Holly Smith has done more than anyone to explore and defend the importance of usability for moral theories. In Making Morality Work, she develops a moral theory that is almost universally usable. But not quite. In this article, I argue that no moral theory is universally usable, in the sense that is most immediately relevant to action, even by agents who know all the normative facts. There is no moral theory knowledge of which suffices to settle deliberation about what to do. However, this is unsurprising if the question of what to do is not a question of fact. One upshot of the discussion is that the search for a universally usable moral theory is misconceived. Another is that, contra Smith (341), agents who are radically uncertain need not lack autonomy.

Addicts are often portrayed as compelled by their addiction and thus as a paradigm of unfree action and mitigated blame. This chapter argues that our best scientific theories of addiction reveal that, psychologically, addicts are not categorically different from non-addicts. There is no pairing of contemporary accounts of addiction and of prominent theories of moral responsibility that can justify our intuitions about the mitigation of addicts but not non-addicts. Two conclusions are advanced. First, we should either treat addicts as we normally treat non-addicts (as fully culpable) or embrace the skeptical conclusion that everyone is less responsible than we thought—perhaps not responsible at all. Second, we should be doubtful that theorizing about responsibility will be advanced by focusing on particular kinds of psychopathologies.

In his influential book, The Nature of Morality, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles.” What is the epistemological relevance of this contrast, if genuine? This chapter argues that ethicists and philosophers of mathematics have misunderstood it. They have confused what the chapter calls the justificatory challenge for realism about an area, D—the challenge to justify our D-beliefs—with the reliability challenge for D-realism—the challenge to explain the reliability of our D-beliefs. Harman’s contrast is relevant to the first, but not, evidently, to the second. One upshot of the discussion is that genealogical debunking arguments are fallacious. Another is that indispensability considerations cannot answer the Benacerraf–Field challenge for mathematical realism.

I discuss methodology in epistemology. I argue that settling the facts, even the epistemic facts, fails to settle the questions of intellectual policy at the center of our epistemic lives. An upshot is that the standard methodology of analyzing concepts like knowledge, justification, rationality, and so on is misconceived. More generally, any epistemic method that seeks to issue in intellectual policy by settling the facts, whether by way of abductive theorizing or empirical investigation, no matter how reliable, is inapt. The argument is a radicalization of Moore’s Open Question Argument. I conclude by considering the ramifications of this conclusion for the debate surrounding “Modal Security”, a proposed necessary condition on undermining defeat.

I discuss Benacerraf's epistemological challenge for realism about areas like mathematics, metalogic, and modality, and describe the pluralist response to it. I explain why normative pluralism is peculiarly unsatisfactory, and use this explanation to formulate a radicalization of Moore's Open Question Argument. According to the argument, the facts -- even the normative facts -- fail to settle the practical questions at the center of our normative lives. One lesson is that the concepts of realism and objectivity, which are widely identified, are actually in tension.

Modal Security is an increasingly discussed proposed necessary condition on undermining defeat. Modal Security says, roughly, that if evidence undermines (rather than rebuts) one’s belief, then one gets reason to doubt the belief's safety or sensitivity. The primary interest of the principle is that it seems to entail that influential epistemological arguments, including Evolutionary Debunking Arguments against moral realism and the Benacerraf-Field Challenge for mathematical realism, are unsound. The purpose of this paper is to critically examine Modal Security in detail. We develop and discuss what we take to be the strongest objections to the principle. One of the aims of the paper is to expose the weakness of these objections. Another is to reveal how the debate over Modal Security interacts with core problems in epistemology — including the generality problem, and the distinction between direct and indirect evidence.

Does consciousness exist? In “The Meta-Problem of Consciousness” (MPC) David Chalmers sketches an argument for illusionism, i.e., the view that it does not. The key premise is that it would be a coincidence if our beliefs about consciousness were true, given that the explanation of those beliefs is independent of their truth. In this article, I clarify and assess this argument. I argue that our beliefs about consciousness are peculiarly invulnerable to undermining, whether or not their contents are indubitable or even obvious. However, the reason that they are peculiarly invulnerable to undermining points to a fundamental flaw in modal arguments for dualism.

Ethics and mathematics have long invited comparisons. On the one hand, both ethical and mathematical propositions can appear to be knowable a priori, if knowable at all. On the other hand, mathematical propositions seem to admit of proof, and to enter into empirical scientific theories, in a way that ethical propositions do not. In this article, I discuss apparent similarities and differences between ethical (i.e., moral) and mathematical knowledge, realistically construed -- i.e., construed as independent of human mind and languages. I argue that some are are merely apparent, while others are of little consequence. There is a difference between the cases. But it is not an epistemological difference per se. The difference, surprisingly, is that ethical knowledge, if it is practical, cannot fail to be objective in a way that mathematical knowledge can. One upshot of the discussion is radicalization of Moore’s Open Question Argument. Another is that the concepts of realism and objectivity, which are widely identified, are actually in tension.

It is widely alleged that metaphysical possibility is “absolute” possibility Conceivability and possibility, Clarendon, Oxford, 2002, p 16; Stalnaker, in: Stalnaker Ways a world might be: metaphysical and anti-metaphysical essays, Oxford University Press, Oxford, 2003, pp 201–215; Williamson in Can J Philos 46:453–492, 2016). Kripke calls metaphysical necessity “necessity in the highest degree”. Van Inwagen claims that if P is metaphysically possible, then it is possible “tout court. Possible simpliciter. Possible period…. possib without qualification.” And Stalnaker writes, “we can agree with Frank Jackson, David Chalmers, Saul Kripke, David Lewis, and most others who allow themselves to talk about possible worlds at all, that metaphysical necessity is necessity in the widest sense.” What exactly does the thesis that metaphysical possibility is absolute amount to? Is it true? In this article, I argue that, assuming that the thesis is not merely terminological, and lacking in any metaphysical interest, it is an article of faith. I conclude with the suggestion that metaphysical possibility may lack the metaphysical significance that is widely attributed to it.