New Directions for Neo-logicism
Date
2018-07-10
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Abstract
In this dissertation I focus on a program in the philosophy of mathematics known as neo-logicism that is a direct descendant of Frege’s logicist project. That program seeks to reduce mathematical theories to logic and definitions in order to put those theories on stable epistemic and logical footing. The definitions that are of greatest importance are abstraction principles, biconditionals associating identity statements for abstract objects on one side, with equivalence classes on the other. Abstraction principles are important because they provide connections between logic on the one hand, and mathematics and its ontology on the other. Throughout this work, I advocate that the epistemic goals of neo-logicism be taken into account when we’re looking to solve problems that are of central importance to its success. Additionally, each chapter either discusses or advocates for a methodological shift, or sets up and implements a novel methodological position I believe to be broadly beneficial to the neo-logicist project. Chapter 2 traces thinking about the status of higher-order logic through the mid-twentieth century, setting the stage for issues dealt with in later chapters. Chapter 3 asks neo-logicists to look beyond set theory and consider other foundational theories, or something entirely new when looking for reductions of foundational mathematical theories. Chapter 4 is an extended argument involving non-standard analysis showing that Hume’s Principle ought not be considered analytic in Frege’s sense of the termChapters 5 and 6 move away form the (somewhat) historical work in the first three chapters, and set up new strategies for solving central neo-logicist problems by integrating formal and epistemic considerations. Chapter 5 introduces the notion of a canonical equivalence relation via a discussion of content carving. That notion is a particular way of understanding the relationship between equivalence relations and abstracts. Finally, chapter 6 makes use of canonical equivalence relations to introduce a new direction in the search for solutions to the Bad Company objection. As whole, the project can be seen as providing, as the title suggests, new directions that ought to be considered by those wishing to vindicate neo-logicism
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Logicism, Neo-logicism, Logic, Bad Company, Philosophy of Mathematics
Citation
Thomas-Bolduc, A. R. (2018). New Directions for Neo-logicism (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/32353