Research

L.E.J. Brouwer famously took the subject’s intuition of time to be foundational and from there ventured to build up mathematics. Despite being largely critical of formal methods, Brouwer valued axiomatic systems for their use in both communication and memory. Through the Dutch Mathematical Society, Gerrit Mannoury posed a challenge in 1927 to provide an axiomatization of intuitionistic arithmetic. Arend Heyting’s 1928 axiomatization was chosen as the winner and has since enjoyed the status of being the de facto formalization of intuitionistic arithmetic. We argue that axiomatizations of intuitionistic arithmetic ought to make explicit the role of the subject’s activity in the intuitionistic arithmetical process. While Heyting Arithmetic is useful when we want to contrast constructed objects with platonistic ones, Heyting Arithmetic omits the contribution of the subject and thus falls short as a response to Mannoury’s challenge. We offer our own solution, Doxastic Heyting Arithmetic, or DHA, which we contend axiomatizes Brouwerian intuitionistic arithmetic.

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In Erkenntnis, 2023

All is vanity, we learn early in Ecclesiastes. This is motivated by the mysterious aphorism that there is nothing new under the sun. But what does it mean to say that there is nothing new under the sun? One might interpret this as a statement of the Eternal Return of the past. Alternatively, one could understand it as a statement what we call the Eternal Withering of the past. Eternal Withering is the view that the present draws from the past but not all of the past repeats. We argue that Eternal Withering motivates the claim that all is vanity. We then show how this interpretation can help us explain the positive practical suggestions in Ecclesiastes.

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In Ergo, 2023

Artemov, building upon a tradition beginning with Kolmogorov and Gödel, developed a paradigm for understanding Constructive Reasoning in terms of classical proofs. Kolmogorov–Gödel–Artemov constructivism flies in the face of the usual understanding of Constructive Reasoning as being distinguished from Classical Reasoning in terms of its theory of truth. Is there something that stands to traditional Classical Reasoning as Kolmogorov–Gödel–Artemov constructivism stands to Constructive Reasoning? In this paper, we develop an affirmative answer to this question by presenting a justification account of Classical Reasoning in terms of explicit justification. The traditional truth paradigm account of Classical Reasoning leads to the well-known paradoxes of material implication. We show that the justification account of Classical Reasoning avoids this problem.

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In Journal of Logic and Computation, 2022

Is consistency the sort of thing that could provide a guide to mathematical ontology? If so, which notion of consistency suits this purpose? Mark Balaguer holds such a view in the context of platonism, the view that mathematical objects are non-causal, non-spatiotemporal, and non-mental. For the purposes of this paper, we will examine several notions of consistency with respect to how they can provide a platon-ist epistemology of mathematics. Only a Gödelian notion, we suggest, can provide a satisfactory guide to a platonist ontology. Is consistency the sort of thing that could provide a guide to mathematical ontology? If so, which notion of consistency suits this purpose? Mark Bala-guer holds such a view in the context of platonism, the view that mathematical objects are non-causal, non-spatiotemporal, and non-mental. Balaguer's version of Platonism, Full-Blooded Platonism (FBP), is the view that "there are as many abstract mathematical objects as there could be-i.e., there actually exist abstract mathematical objects of all possible kinds" (Balaguer, 2017, p. 381). He continues: Since FBP says that there are abstract mathematical objects of all possible kinds, it follows that if FBP is true, then every purely mathematical theory that could be true-i.e., that is internally consistent-accurately describes some collection of actually existing abstract objects. Thus it follows from FBP that in order to acquire knowledge of abstract objects, all we have to do is come up with an internally consistent purely mathematical theory (and know that it is internally consistent). (Balaguer, 2017, p. 381)

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In 43rd International Wittgenstein Symposium proceedings, 2022

In his 1951 Gibbs Memorial Lecture, Kurt Gödel put forth his famous disjunction that either the power of the mind outstrips that of any machine or there are absolutely unsolvable problems. The view that there are no absolutely unsolvable problems is optimism, the view  that there are  such problems is pessimism. In his 1995—and, revised in 2013—Verificationism  Then  and  Now, Per Martin-Löf presents an illustrative argument for a constructivist form of optimism. In response to that argument, Solomon Feferman points out that Martin-Löf’s reasoning relies upon constructive understandings of key philosophical notions. In the vein of Feferman’s analysis, one might be object to Martin-Löf’s argument for either its reliance upon constructivist (as opposed to classical) considerations, or for its appeal  to  non-

unproblematically  mathematical  premises.  We  argue  that  both  of  these  responses fall short. On one hand, to be critical of Martin-Löf’s reasoning for its constructiveness is to reject what would otherwise be a scientific advance on the basis of the assumption of constructivism’s falsehood or implausibility, which is of course uncharitable at best. On the other hand, to object to the argument for its use of non-unproblematically mathematical premises is to assume that there is some philosophically neutral mathematics, which is im-

plausible. Martin-Löf’s argument relies upon his

third law, the claim that from the impossibility of a proof of a proposition we can construct a proof of its negation. We close with a discussion of some ways in which this claim can be criticized from the constructive point of view. Specifically, we contend that Martin-Löf’s third law is incompatible with what has been called “Poincaré’s  Principle  of  Epistemic  Conservation”, the thesis that  genuine  increase in mathematical knowledge requires subject-specific insight.

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In Semiotic Studies, 2020

In this paper, we investigate epistemic predicates in extensions of arithmetic. We use as our case study Kurt Gödel’s 1951 thesis that either the power of the human mind surpasses that of any finite machine or there are absolutely unsolvable problems. Because Gödel also claimed that his disjunction was a mathematically established fact, we must ask the following: what sort of syntactical object should formalize human reason? In this paper, we lay the foundations for a predicate treatment of this epistemic feature. We begin with a very general examination of the Gödel sentence in the arithmetical context. We then discuss two systems of modal predicates over arithmetic. The first, called coreflective arithmetic or 𝖢𝗈𝖯𝖠, extends 𝖯𝖠 with a coreflective modal predicate but does not contain a consistency statement. The second, called doxastic arithmetic or 𝖣𝖠⁠, has as its characteristic feature the consistency statement but does not contain coreflection or its instance, the 𝟦 axiom. We examine the logical properties of, motivations for and criticisms of both systems. We close with a brief comparison of the systems in the context of Gödel’s disjunction.

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In Journal of Logic and Computation, 2020

The logical analysis of epistemic paradoxes—including, for example, the Moore and Gödel-Buridan paradoxes—has traditionally been performed assuming the whole range of corresponding modal logic principles: {𝖣,𝖳,𝟦,𝟧}{D,T,4,5}. In this paper, it is discovered precisely which of those principles (including also the law of excluded middle, LEM) are responsible for the paradoxical behavior of the Moore, Gödel-Buridan, Dual Moore, and Commissive Moore sentences. Further, by reproducing these paradoxes intuitionistically we reject a conjecture that these paradoxes are caused by the LEM. An exploration of the Gödel-Buridan sentence prompts the inquiry into a system, Doxastic Arithmetic, 𝖣𝖠, designed to represent the arithmetical beliefs of an agent who accepts all specific arithmetical proofs and yet believes in the consistency of their own beliefs. For these reasons, 𝖣𝖠 may be regarded as an epistemic way to circumvent limitations of Gödel’s Second Incompleteness Theorem.

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In Logical Foundations of Computer Science, 2018

Given some doxastic logic, Hintikka paradoxical sentences are those sentences that are satisfiable but refutable if believed. The standard examples of doxastic paradoxes are the Moore and Buridan-Gödel sentences. Considering the standard logics for modeling doxastic agents, should we prefer logics in which the believed versions of these sentences are refutable? One might answer affirmatively assuming that to refute the believed sentence is to avoid paradox. One motivation for this approach is the thought that these sentences are intuitively unbelievable for rational agents. In this note, we examine the extent to which this intuition motivates the preference for logics in which the believed versions of these sentences are refutable. We argue that such a motivation is weak by providing a theory (namely, the recently introduced Doxastic Arithmetic (DA)) within which a rational agent believes a Buridan-Gödel sentence. 

Link to Wittgenstein Symposium Program

Quine's translation argumnent figures centrally in his views on logic. The goal of this paper is to get clear on that argument. It can be interpreted as an argument to the effect that one should never translate somebody’s speech as going against a law of the translator’s logic. Key to this reading of the translation argument is the premise that one should never translate somebody's speech such that their speech is unintelligible. Ultimately, it is my aim to reject this reading. I argue that only a weaker conclusion—one that says “not most of the time” instead of the stronger “never”—should be attributed to Quine. Accordingly, I propose and defend a weaker version of the first premise that better coheres with the weaker conclusion of the translation argument. Instead of the claim that one should never translate somebody’s speech such that their speech is unintelligible I argue that we should only ascribe to Quine the claim that one should not most of the time translate somebody’s speech in a way that makes it unintelligible. I go on to  sum up the results of my discussion and respond to a criticism of my reading.

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In Topoi, 2016