Diversity Reading List

Abstract: This article offers a logical, linguistic, and philosophical account of modern quantification theory. Contrasting the standard approach to quantifiers (according to which logical quantifiers are defined by enumeration) with the generalized approach (according to which quantifiers are defined systematically), the article begins with a brief history of standard quantifier theory and identifies some of its logical, linguistic, and philosophical strengths and weaknesses. It then proceeds to a brief history of generalized quantifier theory and explains how it overcomes the weaknesses of the standard theory. One of the main philosophical advantages of the generalized theory is its philosophically informative criterion of logicality. The paper describes the work done so far in this theory, highlights some of its central logical results, offers an overview of its main linguistic contributions, and discusses its philosophical significance.

From the Introduction: "Modern mathematics is based on the axiomatic method. We choose axioms and a deductive system---rules for deducing theorems from the axioms. This methodology is designed to guarantee that we can proceed from "obviously" true premises to true conclusions, via inferences which are "obviously" truth-preserving. [...] New and interesting questions arise if we give up as myth the claim that our theorizing can ever be separated out from the complex dynamic of interwoven social/political/historical/cultural forces that shape our experiences and views. Considering mathematics as a set of stories produced according to strict rules one can read these stories for what they tell us about the very real human desires, ambitions, and values of the authors (who understands) and listen to the authors as spokespersons for their cultures (where and when). This paper is the self-respective and self-conscious attempt of a mathematician to retell a story of mathematics that attends to the relationships between who we are and what we know."

Abstract:
We seem to have a good grasp of how the subject matters of truth-functional composites depends on their components: it’s simply fusion (Hawke in Australas J Philos 96:697–723, 2018, Fine in Philos Studies 177:129–171, 2020, Plebani and Spolaore in Philos Q 71:605–622, 2021, Plebani and Spolaore in Philos Stud 181:247–265, 2021, Berto in Topics of Thought, Oxford University Press, Oxford, 2022). But what relation should the subject matter of subsentential components bear to the subject matter of the sentences they feature in, and what to say about the quantified sentences of first-order predicate logic? Given how well we seem to understand sentential subject matter in the context of propositional logic, I propose a reduction of the subject matter of subsentential components and of quantified sentences to the subject matter of quantifier-free sentences. I argue that the view squares with the Fregean intuitions that gave rise to the construction of first-order logic as we know it today, motivating its adequacy as a theory of subject matter for first-order languages. I then show that this first-order aboutness theory has predictive and explanatory power, leading us to accept a modified version of Yablo’s (Aboutness, Princeton University Press, Princeton, 2014) principle of immanent closure, as well as a new conception of arbitrary objects/reference. Finally, I propose some potential developments and restrictions.

Abstract:

Despite emerging attention to Indigenous philosophies both within and outside of feminism, Indigenous logics remain relatively underexplored and underappreciated. By amplifying the voices of recent Indigenous philosophies and literatures, I seek to demonstrate that Indigenous logic is a crucial aspect of Indigenous resurgence as well as political and ethical resistance. Indigenous philosophies provide alternatives to the colonial, masculinist tendencies of classical logic in the form of paraconsistent—many-valued—logics. Specifically, when Indigenous logics embrace the possibility of true contradictions, they highlight aspects of the world rejected and ignored by classical logic and inspire a relational, decolonial imaginary. To demonstrate this, I look to biology, from which Indigenous logics are often explicitly excluded, and consider one problem that would benefit from an Indigenous, paraconsistent analysis: that of the biological individual. This article is an effort to expand the arenas in which allied feminists can responsibly take up and deploy these decolonial logics.

Abstract: Newton's Principia introduces four rules of reasoning for natural philosophy. Although useful, there is a concern about whether Newton's rules guarantee truth. After redirecting the discussion from truth to validity, I show that these rules are valid insofar as they fulfill Goodman's criteria for inductive rules and Newton's own methodological program of experimental philosophy; provided that cross-checks are used prior to applications of rule 4 and immediately after applications of rule 2 the following activities are pursued: (1) research addressing observations that systematically deviate from theoretical idealizations and (2) applications of theory that safeguard ongoing research from proceeding down a garden path.

Abstract:
"It must be the desire of every reasonable person to know how to justify a contention which is of sufficient importance to be seriously questioned. The explicit formulation of the principles of sound reasoning is the concern of Logic". This book discusses the habit of sound reasoning which is acquired by consciously attending to the logical principles of sound reasoning, in order to apply them to test the soundness of arguments. It isn’t an introduction to logic but it encourages the practice of logic, of deciding whether reasons in argument are sound or unsound. Stress is laid upon the importance of considering language, which is a key instrument of our thinking and is imperfect.

Abstract:
Over a period of more than 30 years, more than 100 mathematicians worked on a project to classify mathematical objects known as finite simple groups. The Classification, when officially declared completed in 1981, ranged between 300 and 500 articles and ran somewhere between 5,000 and 10,000 journal pages. Mathematicians have hailed the project as one of the greatest mathematical achievements of the 20th century, and it surpasses, both in scale and scope, any other mathematical proof of the 20th century. The history of the Classification points to the importance of face-to-face interaction and close teaching relationships in the production and transformation of theoretical knowledge. The techniques and methods that governed much of the work in finite simple group theory circulated via personal, often informal, communication, rather than in published proofs. Consequently, the printed proofs that would constitute the Classification Theorem functioned as a sort of shorthand for and formalization of proofs that had already been established during personal interactions among mathematicians. The proof of the Classification was at once both a material artifact and a crystallization of one community’s shared practices, values, histories, and expertise. However, beginning in the 1980s, the original proof of the Classification faced the threat of ‘uninvention’. The papers that constituted it could still be found scattered throughout the mathematical literature, but no one other than the dwindling community of group theorists would know how to find them or how to piece them together. Faced with this problem, finite group theorists resolved to produce a ‘second-generation proof’ to streamline and centralize the Classification. This project highlights that the proof and the community of finite simple groups theorists who produced it were co-constitutive–one formed and reformed by the other.

Abstract:

Janina Hosiasson-Lindenbaum is a known figure in philosophy of probability of the 1930s. A previously unpublished manuscript fills in the blanks in the full picture of her work on inductive reasoning by analogy, until now only accessible through a single publication. In this paper, I present Hosiasson’s work on analogical reasoning, bringing together her early publications that were never translated from Polish, and the recently discovered unpublished work. I then show how her late work relates to Rudolf Carnap’s approach to “analogy by similarity” developed in the 1960s. Hosiasson turns out to be a predecessor of the line of research that models analogical influence as inductive relevance. A translation of Hosiasson’s manuscript concludes the paper.

Abstract:

According to a popular interpretation, Carnap’s interpretation of probability had evolved from a logical towards a subjective conception. However Carnap himself insisted that his basic philosophical view of probability was always the same. I address this apparent clash between Carnap's self-identification and the subsequent interpretations of his work. Following its original intentions, I reconstruct inductive logic as an explication. The emerging picture is of a versatile linguistic framework, whose main function is not the discovery of objective logical relations in the object language, but the stipulation of conceptual possibilities. Within this representation, I map out the changes that the project went through. Seen from such an explication-based perspective, inductive logic becomes quite hard to categorize using the standard labels.

Abstract:
Some personal thoughts and opinions on what “good quality mathematics” is and whether one should try to define this term rigorously. As a case study, the story of Szemer´edi’s theorem is presented.