How can the infinite, a subject so remote from our finite experience, be an everyday tool for the working mathematician? Blending history, philosophy, mathematics, and logic, Shaughan Lavine answers this question with exceptional clarity. Making use of the mathematical work of Jan Mycielski, he demonstrates that knowledge of the infinite is possible, even according to strict standards that require some intuitive basis for knowledge.

An engaging account of the origins of the modern mathematical theory of the infinite, his book is also a spirited defense against the attacks and misconceptions ...

Quineans have taken the basic expression of ontological commitment to be an assertion of the form '' x '', assimilated to theEnglish ''there is something that is a ''. Here I take the existential quantifier to be introduced, not as an abbreviation for an expression of English, but via Tarskian semantics. I argue, contrary to the standard view, that Tarskian semantics in fact suggests a quite different picture: one in which quantification is of a substitutional type apparently first proposed by (...) Geach. The ontological burden is borne by constant symbols, and truth is defined separately from reference. (shrink)

Let U be an admissible structure. A cPCd(U) class is the class of all models of a sentence of the form $\neg\exists\bar{K} \bigwedge \Phi$ , where K̄ is an U-r.e. set of relation symbols and φ is an U-r.e. set of formulas of L∞ω that are in U. The main theorem is a generalization of the following: Let U be a pure countable resolvable admissible structure such that U is not Σ-elementarily embedded in HYP(U). Then a class K of countable (...) structures whose universes are sets of urelements is a cPCd(U) class if and only if for some Σ formula σ (with parameters from U), M is in K if and only if M is a countable structure with universe a set of urelements and $(\mathrm{HYP}_\mathfrak{U}(\mathfrak{M}), \mathfrak{U}, \mathfrak{M}) \models \sigma$ , where HYPU(M), the smallest admissible set above M relative to U, is a generalization of HYP to structures with similarity type Σ over U that is defined in this article. Here we just note that when Lα is admissible, HYPLα(M) is Lβ(M) for the least β ≥ α such that Lβ(M) is admissible, and so, in particular, that HYPHF(M) is just HYP(M) in the usual sense when M has a finite similarity type. The definition of HYPU(M) is most naturally formulated using Adamson's notion of a +-admissible structure (1978). We prove a generalization from admissible to +-admissible structures of the well-known truncation lemma. That generalization is a key theorem applied in the proof of the generalized Spector-Gandy theorem. (shrink)

If quantum mechanics (QM) is to be taken as an atomistic theory with the elementary particles as atoms (an ATEP), then the elementary particlcs must be individuals. There must then be, for each elementary particle a, a property being identical with a that a alone has. But according to QM, elementary particles of the same kind share all physical properties. Thus, if QM is an ATEP, identity is a metaphysical but not a physical property. That has unpalatable consequences. Dropping the (...) assumption that QM is an ATEP makes it possible to replace the assumption that elementary particles are individuals with the assumption that there are various kinds of elementary stuff that have smallest quantities — the smallest quantity of light, for example, is a photon. The problems about identity disappear, and the explanatory virtues of an ATEP are maintained. (shrink)

A system of finite mathematics is proposed that has all of the power of classical mathematics. I believe that finite mathematics is not committed to any form of infinity, actual or potential, either within its theories or in the metalanguage employed to specify them. I show in detail that its commitments to the infinite are no stronger than those of primitive recursive arithmetic. The finite mathematics of sets is comprehensible and usable on its own terms, without appeal to any form (...) of the infinite. That makes it possible to, without circularity, obtain the axioms of full Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) by extrapolating (in a precisely defined technical sense) from natural principles concerning finite sets, including indefinitely large ones. The existence of such a method of extrapolation makes it possible to give a comparatively direct account of how we obtain knowledge of the mathematical infinite. The starting point for finite mathematics is Mycielski's work on locally finite theories. (shrink)

Let $\mathfrak{U}$ be an admissible structure. A $\mathrm{cPC}_d(\mathfrak{U})$ class is the class of all models of a sentence of the form $\neg\exists\bar{K} \bigwedge \Phi$, where $\bar{K}$ is an $\mathfrak{U}$-r.e. set of relation symbols and $\phi$ is an $\mathfrak{U}$-r.e. set of formulas of $\mathscr{L}_{\infty\omega}$ that are in $\mathfrak{U}$. The main theorem is a generalization of the following: Let $\mathfrak{U}$ be a pure countable resolvable admissible structure such that $\mathfrak{U}$ is not $\Sigma$-elementarily embedded in $\mathrm{HYP}(\mathfrak{U})$. Then a class $\mathbf{K}$ of countable structures (...) whose universes are sets of urelements is a $\mathrm{cPC}_d(\mathfrak{U})$ class if and only if for some $\Sigma$ formula $\sigma$ (with parameters from $\mathfrak{U}$), $\mathfrak{M}$ is in $\mathbf{K}$ if and only if $\mathfrak{M}$ is a countable structure with universe a set of urelements and $(\mathrm{HYP}_\mathfrak{U}(\mathfrak{M}), \mathfrak{U}, \mathfrak{M}) \models \sigma$, where $\mathrm{HYP}_\mathfrak{U}(\mathfrak{M})$, the smallest admissible set above $\mathfrak{M}$ relative to $\mathfrak{U}$, is a generalization of $\mathrm{HYP}$ to structures with similarity type $\Sigma$ over $\mathfrak{U}$ that is defined in this article. Here we just note that when $\mathrm{L}_\alpha$ is admissible, $\mathrm{HYP}_{\mathrm{L}\alpha}(\mathfrak{M})$ is $\mathrm{L}_\beta(\mathfrak{M})$ for the least $\beta \geq \alpha$ such that $\mathrm{L}_\beta(\mathfrak{M})$ is admissible, and so, in particular, that $\mathrm{HYP}_{\mathbb{HF}}(\mathfrak{M})$ is just $\mathrm{HYP}(\mathfrak{M})$ in the usual sense when $\mathfrak{M}$ has a finite similarity type. The definition of $\mathrm{HYP}_\mathfrak{U}(\mathfrak{M})$ is most naturally formulated using Adamson's notion of a +-admissible structure (1978). We prove a generalization from admissible to +-admissible structures of the well-known truncation lemma. That generalization is a key theorem applied in the proof of the generalized Spector-Gandy theorem. (shrink)

It is well known that, in the terminology of Moschovakis, Descriptive set theory (1980), every adequate normed pointclass closed under ∀ω has an effective version of the generalized reduction property (GRP) called the easy uniformization property (EUP). We prove a dual result: every adequate normed pointclass closed under ∃ω has the EUP. Moschovakis was concerned with the descriptive set theory of subsets of Polish topological spaces. We set up a general framework for parts of descriptive set theory and prove results (...) that have as special cases not only the just-mentioned topological results, but also corresponding results concerning the descriptive set theory of classes of structures. Vaught (1973) asked whether the class of cPCδ classes of countable structures has the GRP. It does. A cPC(A) class is the class of all models of a sentence of the form ¬∃K̄φ, where φ is a sentence of L∞ω that is in A and K̄ is a set of relation symbols that is in A. Vaught also asked whether there is any primitive recursively closed set A such that some effective version of the GRP holds for the class of cPC(A) classes of countable structures. There is: The class of cPC(A) classes of countable structures has the EUP if ω ∈ A and A is countable and primitive recursively closed. Those results and some extensions are obtained by first showing that the relevant classes of classes of structures, which Vaught showed normed, are in a suitable sense adequate and closed under ∃ω, and then applying the dual easy uniformization theorem. (shrink)

Let A be an admissible set. A sentence of the form ∀R̄φ is a ∀1(A) (∀s 1(A),∀1(Lω1ω)) sentence if φ ∈ A (φ is $\bigvee\Phi$ , where Φ is an A-r.e. set of sentences from A; φ ∈ Lω1ω). A sentence of the form ∃R̄φ is an ∃2(A) (∃s 2(A),∃2(Lω1ω)) sentence if φ is a ∀1(A) (∀s 1(A),∀1(Lω1ω)) sentence. A class of structures is, for example, a ∀1(A) class if it is the class of models of a ∀1(A) sentence. Thus (...) ∀1(A) is a class of classes of structures, and so forth. Let Mi be the structure $\langle i, 0$. Let Γ be a class of classes of structures. We say that a sequence $J_1,\ldots,J_i,\ldots, i < \omega$, of classes of structures is a Γ sequence if $J_i \in \Gamma, i < \omega$, and there is I ∈ Γ such that M ∈ Ji if and only if [M, Mi] ∈ I, where [,] is the disjoint sum. A class Γ of classes of structures has the easy uniformization property if for every Γ sequence $J_1,\ldots,J_i, \ldots, i < \omega$, there is a Γ sequence $J'_1,\ldots,J'_i,\ldots, i < \omega$, such that $J'_i \subseteq J_i, i < \omega, \bigcup J'_i = \bigcup J_i$, and the J'i are pairwise disjoint. The easy uniformization property is an effective version of Kuratowski's generalized reduction property that is closely related to Moschovakis's (topological) easy uniformization property. We show over countable structures that ∀1(A) and ∃2(A) have the easy uniformization property if A is a countable admissible set with an infinite member, that ∀s 1(Lα) and ∃s 2(Lα) have the easy uniformization property if α is countable, admissible, and not weakly stable, and that ∀1(Lω1ω) and ∃2(Lω1ω) have the easy uniformization property. The results proved are more general. The result for ∀s 1(Lα) answers a question of Vaught (1980). (shrink)

Based on her earlier ground-breaking axiomatization of quantified modal logic, the papers collected here by the distinguished philosopher Ruth Barcan Marcus cover much ground in the development of her thought, including influential essays on moral conflict, on belief and rationality, and on some historical figures.

A critique of Shaughan Lavine's attempt in /Understanding the Infinite/ to reduce talk about the infinite to finitely comprehensible terms.

It is common in metaphysical discourse to make claims like “Everything is self-identical” in which “everything” is intended to range over everything. This sort of “unrestricted” generality appears central to metaphysical discourse. But there is debate whether such generality, which appears to involve quantification over an all-inclusive domain, is even meaningful. To address this concern, Shaughan Lavine and Vann McGee supply competing accounts of the generality expressed by this use of “everything.” I argue that, from the perspective of (...) the metaphysician, neither of their proposals is entirely suitable. But their central insights, as well as their shortcomings, suggest an account which is suitable for metaphysical discourse, what I call “genuinely unrestricted quantification.” For while genuinely unrestricted quantification does not directly supply an account of the metaphysician's “everything,” it disentangles what are properly metaphysical issues from issues concerning how we ought to understand the quantifiers. It makes clear that whether the metaphysician's “everything” is meaningful cannot simply be resolved by supplying an appropriate account of our understanding of quantification. It depends on addressing certain substantive ontological questions. (shrink)

Guido Löhr's recent article makes several insightful and productive suggestions about how to proceed with the empirical study of collective action. However, their critique of the conclusions drawn in Gomez-Lavin & Rachar (2022) is undermined by some issues with the interpretation of the debate and paper. This discussion article clears up those issues, presents new findings from experiments developed in response to Löhr's critiques, reflects on the role of experimental research in the development and refinement of philosophical theories, and adds (...) to Löhr's suggestions about the path forward. (shrink)

The debate regarding the nature of joint action has come to a stalemate due to a dependence on intuitional methods. Normativists, such as Margaret Gilbert, argue that action-relative normative relations are inherent in joint action, while non-normativists, such as Michael Bratman, claim that there are minimal cases of joint action without normative relations. In this work, we describe the first experimental examinations of these intuitions, and report the results of six studies that weigh in favor of the normativist paradigm. Philosophical (...) ramifications and further extensions of this work are then discussed. (shrink)

What do we owe each other when we act together? According to normativists about collective action, necessarily something and potentially quite a bit. They contend that collective action inherently involves a special normative status amongst participants, which may, for example, involve mutual obligations to receive the concurrence of the others before leaving. We build on recent empirical work whose results lend plausibility to a normativist account by further investigating the specific package of mutual obligations associated with collective action according to (...) our everyday understanding. However, our results cast doubt on a proposed obligation to seek the permission of co-actors before exiting a collective action, and suggest instead that this obligation is a function of explicit promising. We then discuss how our results pave the path for a new normativism, a theory that neither under- nor overshoots the target given by our common conception of the interpersonal obligations present in collective action.*. (shrink)

‘Ulysses contracts’ are an instrument through which a psychiatric patient may prearrange involuntary commitments to be put into effect if the patient satisfies certain diagnostic criteria in the future. Proposals for Ulysses contracts typically impose numerous safeguards. This paper argues against the intuitively plausible safeguard which permits only presently remitted patients to contract. Instead of requiring a patient's remission, it is argued that the appropriate safeguard is the patient's ability, whether remitted or not, to offer good reasons for wishing to (...) contract. In short, what matters is not an executive's character, but an executive's reasons, and a bad executive may have good reasons. Attempts to deny the accessibility of good reasons in unremitted patients are rejected on the ground that psychiatric diagnosis requires psychiatrists to be able to distinguish between good and bad reasons in both remitted and unremitted patients. If psychiatrists cannot do that, psychiatric diagnosis is impossible. (shrink)

This paper uses the tools of experimental philosophy to examine the nature of interpersonal normativity in collective action, focusing on cases of immoral collective action and collective action by friends. The results of our two studies, which expand on recent empirical interventions into longstanding debates in social ontology, demonstrate that according to our everyday judgments there are interpersonal obligations in cases of collective action, even when immoral, and that, while friendship elicits judgments of togetherness, it does not affect the norms (...) of collective action. We also situate these results in the philosophical literature and discuss their consequences for a “pluralism” about sociality. (shrink)

A Zoom discussion about racism and ableism in philosophy.

Working memory is a foundational construct of cognitive psychology, where it is thought to be a capacity that enables us to keep information in mind and to use that information to support goal directed behavior. Philosophers have recently employed working memory to explain central cognitive processes, from consciousness to reasoning. In this paper, I show that working memory cannot meet even a minimal account of natural kindhood, as the functions of maintenance and manipulation of information that tie working memory models (...) and theories together do not have a coherent or univocal realizer in the brain. As such, working memory cannot explain central cognition. Rather, I argue that working memory merely redescribes its target phenomenon, and in doing so it obfuscates relevant distinctions amongst the many ways that brains like ours retain and transform information in the service of cognition. While this project ultimately erodes the explanatory role that working memory has played in our understanding of cognition, it simultaneously prompts us to evaluate the function of natural kinds within cognitive science, and signals the need for a productive pessimism to frame our future study of cognitive categories. (shrink)

In their target article, Lynch, Parke, and O’Malley argue against the quick application of causal, interventionist explanatory frameworks to microbiomes and their purported role in many disparate states, from obesity to anxiety. I think the authors have undersold the force of their argument. A careful consideration of the scope of their claims, made easier by a parallel drawn from the history of explanation in neuroscience, yields a productive pessimism: that causal explanations likely operate at the wrong level of analysis for (...) dynamic, distributed, Quineian entities like the microbiome. That is, we shouldn’t expect causal explanations for microbiomes at all—and this includes the authors’ own “microbiome success story” of C. difficile. Neuroscience, with its own computationally challenging, dynamic entity—the brain—may provide lessons for how to approach something like predictive control over the microbiome. (shrink)

To what extent do aesthetic taste and our interest in the arts constitute who we are? In this paper, we present a series of empirical findings that suggest an Aesthetic Self Effect supporting the claim that our aesthetic engagements are a central component of our identity. Counterfactual changes in aesthetic preferences, for example, moving from liking classical music to liking pop, are perceived as altering us as a person. The Aesthetic Self Effect is as strong as the impact of moral (...) changes, such as altering political partisanship or religious orientation, and significantly stronger than for other categories of taste, such as food preferences. Using a multidimensional scaling technique to map perceived aesthetic similarities among musical genres, we determined that aesthetic distances between genres correlate highly with the perceived difference in identity. Further studies generalize the Aesthetic Self Effect beyond the musical domain: general changes in visual art preferences, for example from more traditional to abstract art, also elicited a strong Self Effect. Exploring the breadth of this effect we also found an Anaesthetic Self Effect. That is, hypothetical changes from aesthetic indifference to caring about music, art, or beauty are judged to have a significant impact on identity. This effect on identity is stronger for aesthetic fields compared to leisure activities, such as hiking or playing video games. Across our studies, the Anaesthetic Self Effect turns out to be stronger than the Aesthetic Self Effect. Taken together, we found evidence for a link between aesthetics and identity: we are aesthetic selves. When our tastes in music and the arts or our aesthetic interests change we take these to be transformative changes. (shrink)

This paper examines a parallel between Aristotle’s account of phantasia and contemporary psychological models of working memory, a capacity that enables the temporary maintenance and manipulation of information used in many behaviors. These two capacities, though developed within two distinct scientific paradigms, share a common strategy of psychological explanation, Aristotelian Faculty Psychology. This strategy individuates psychological components by their target-domains and functional roles. Working memory and phantasia result from an attempt to individuate the psychological components responsible for flexible thought and (...) are thus implicated in most of our robust cognitive processes, from reading comprehension to problem solving. We then present two novel objections which suggest that these capacities cannot explain our ability to engage in flexible thought. To escape the resultant impasse, we survey alternatives and argue that most promising strategies depend on identifying the behaviors attributed to intelligent thought and action. (shrink)