Mathematical Anti-Realism and Modal Nothingism

Mathematical Anti-Realism and Modal Nothingism

Author: Mark Balaguer

Publisher: Cambridge University Press

Published: 2023-01-05

Total Pages: 151

ISBN-13: 1009346040

DOWNLOAD EBOOK

This Element defends mathematical anti-realism against an underappreciated problem with that view-a problem having to do with modal truthmaking. Part I develops mathematical anti-realism, it defends that view against a number of well-known objections, and it raises a less widely discussed objection to anti-realism-an objection based on the fact that (a) mathematical anti-realists need to commit to the truth of certain kinds of modal claims, and (b) it's not clear that the truth of these modal claims is compatible with mathematical anti-realism. Part II considers various strategies that anti-realists might pursue in trying to solve this modal-truth problem with their view, it argues that there's only one viable view that anti-realists can endorse in order to solve the modal-truth problem, and it argues that the view in question-which is here called modal nothingism-is true.


Mathematical Pluralism

Mathematical Pluralism

Author: Graham Priest

Publisher:

Published: 2024-04-16

Total Pages: 82

ISBN-13: 1009089269

DOWNLOAD EBOOK

Mathematical pluralism is the view that there is an irreducible plurality of pure mathematical structures, each with their own internal logics; and that qua pure mathematical structures they are all equally legitimate. Mathematical pluralism is a relatively new position on the philosophical landscape. This Element provides an introduction to the position.


Mathematics and Explanation

Mathematics and Explanation

Author: Christopher Pincock

Publisher: Cambridge University Press

Published: 2023-05-25

Total Pages: 156

ISBN-13: 1009037412

DOWNLOAD EBOOK

This Element answers four questions. Can any traditional theory of scientific explanation make sense of the place of mathematics in explanation? If traditional monist theories are inadequate, is there some way to develop a more flexible, but still monist, approach that will clarify how mathematics can help to explain? What sort of pluralism about explanation is best equipped to clarify how mathematics can help to explain in science and in mathematics itself? Finally, how can the mathematical elements of an explanation be integrated into the physical world? Some of the evidence for a novel scientific posit may be traced to the explanatory power that this posit would afford, were it to exist. Can a similar kind of explanatory evidence be provided for the existence of mathematical objects, and if not, why not?


Phenomenology and Mathematics

Phenomenology and Mathematics

Author: Michael Roubach

Publisher: Cambridge University Press

Published: 2023-11-30

Total Pages: 149

ISBN-13: 1009002287

DOWNLOAD EBOOK

This Element explores the relationship between phenomenology and mathematics. Its focus is the mathematical thought of Edmund Husserl, founder of phenomenology, but other phenomenologists and phenomenologically-oriented mathematicians, including Weyl, Becker, Gödel, and Rota, are also discussed. After outlining the basic notions of Husserl's phenomenology, the author traces Husserl's journey from his early mathematical studies. Phenomenology's core concepts, such as intention and intuition, each contributed to the emergence of a phenomenological approach to mathematics. This Element examines the phenomenological conceptions of natural number, the continuum, geometry, formal systems, and the applicability of mathematics. It also situates the phenomenological approach in relation to other schools in the philosophy of mathematics-logicism, formalism, intuitionism, Platonism, the French epistemological school, and the philosophy of mathematical practice.


The Euclidean Programme

The Euclidean Programme

Author: A. C. Paseau

Publisher: Cambridge University Press

Published: 2024-02-29

Total Pages: 153

ISBN-13: 100922199X

DOWNLOAD EBOOK

The Euclidean Programme embodies a traditional sort of epistemological foundationalism, according to which knowledge – especially mathematical knowledge – is obtained by deduction from self-evident axioms or first principles. Epistemologists have examined foundationalism extensively, but neglected its historically dominant Euclidean form. By contrast, this book offers a detailed examination of Euclidean foundationalism, which, following Lakatos, the authors call the Euclidean Programme. The book rationally reconstructs the programme's key principles, showing it to be an epistemological interpretation of the axiomatic method. It then compares the reconstructed programme with select historical sources: Euclid's Elements, Aristotle's Posterior Analytics, Descartes's Discourse on Method, Pascal's On the Geometric Mind and a twentieth-century account of axiomatisation. The second half of the book philosophically assesses the programme, exploring whether various areas of contemporary mathematics conform to it. The book concludes by outlining a replacement for the Euclidean Programme.


Number Concepts

Number Concepts

Author: Richard Samuels

Publisher: Cambridge University Press

Published: 2024-02-07

Total Pages: 100

ISBN-13: 100905967X

DOWNLOAD EBOOK

This Element, written for researchers and students in philosophy and the behavioral sciences, reviews and critically assesses extant work on number concepts in developmental psychology and cognitive science. It has four main aims. First, it characterizes the core commitments of mainstream number cognition research, including the commitment to representationalism, the hypothesis that there exist certain number-specific cognitive systems, and the key milestones in the development of number cognition. Second, it provides a taxonomy of influential views within mainstream number cognition research, along with the central challenges these views face. Third, it identifies and critically assesses a series of core philosophical assumptions often adopted by number cognition researchers. Finally, the Element articulates and defends a novel version of pluralism about number concepts.


Philosophical Uses of Categoricity Arguments

Philosophical Uses of Categoricity Arguments

Author: Penelope Maddy

Publisher: Cambridge University Press

Published: 2023-12-21

Total Pages: 75

ISBN-13: 1009432915

DOWNLOAD EBOOK

This Element addresses the viability of categoricity arguments in philosophy by focusing with some care on the specific conclusions that a sampling of prominent figures have attempted to draw – the same theorem might successfully support one such conclusion while failing to support another. It begins with Dedekind, Zermelo, and Kreisel, casting doubt on received readings of the latter two and highlighting the success of all three in achieving what are argued to be their actual goals. These earlier uses of categoricity arguments are then compared and contrasted with more recent work of Parsons and the co-authors Button and Walsh. Highlighting the roles of first- and second-order theorems, of external and internal theorems, the Element concludes that categoricity arguments have been more effective in historical cases that reflect philosophically on internal mathematical matters than in recent questions of pre-theoretic metaphysics.


Iterative Conceptions of Set

Iterative Conceptions of Set

Author: Neil Barton

Publisher: Cambridge University Press

Published: 2024-06-30

Total Pages: 122

ISBN-13: 1009227254

DOWNLOAD EBOOK

Many philosophers are aware of the paradoxes of set theory (e.g. Russell's paradox). For many people, these were solved by the iterative conception of set which holds that sets are formed in stages by collecting sets available at previous stages. This Element will examine possibilities for articulating this solution. In particular, the author argues that there are different kinds of iterative conception, and it's open which of them (if any) is the best. Along the way, the author hopes to make some of the underlying mathematical and philosophical ideas behind tricky bits of the philosophy of set theory clear for philosophers more widely and make their relationships to some other questions in philosophy perspicuous.


Fictionalism in Philosophy

Fictionalism in Philosophy

Author: Bradley Armour-Garb

Publisher:

Published: 2020

Total Pages: 257

ISBN-13: 0190689609

DOWNLOAD EBOOK

This volume collects some of the most up-to-date work on philosophical fictionalism-the idea that a notion of pretense or fiction can help resolve certain puzzles or problems in philosophy. After a detailed discussion in the book's introductory chapter of how philosophers should think of fictionalism and its connection to metaontology more generally, the remaining chapters provide readers with arguments for and against this view from leading scholars in the fields of epistemology, ethics, metaphysics, philosophy of science, philosophy of language, and others.


Metaphysics, Sophistry, and Illusion

Metaphysics, Sophistry, and Illusion

Author: Mark Balaguer

Publisher: Oxford University Press

Published: 2021-01-26

Total Pages: 336

ISBN-13: 0192638831

DOWNLOAD EBOOK

Metaphysics, Sophistry, and Illusion does two things. First, it introduces a novel kind of non-factualist view, and argues that we should endorse views of this kind in connection with a wide class of metaphysical questions, most notably, the abstract-object question and the composite-object question. (More specifically, Mark Balaguer argues that there's no fact of the matter whether there are any such things as abstract objects or composite objects—or material objects of any other kind.) Second, Metaphysics, Sophistry, and Illusion explains how these non-factualist views fit into a general anti-metaphysical view called neo-positivism, and explains how we could argue that neo-positivism is true. Neo-positivism is the view that every metaphysical question decomposes into some subquestions—call them Q1, Q2, Q3, etc.—such that, for each of these subquestions, one of the following three anti-metaphysical views is true of it: non-factualism, or scientism, or metaphysically innocent modal-truth-ism. These three views can be defined (very roughly) as follows: non-factualism about a question Q is the view that there's no fact of the matter about the answer to Q. Scientism about Q is the view that Q is an ordinary empirical-scientific question about some contingent aspect of physical reality, and Q can't be settled with an a priori philosophical argument. And metaphysically innocent modal-truth-ism about Q is the view that Q asks about the truth value of a modal sentence that's metaphysically innocent in the sense that it doesn't say anything about reality and, if it's true, isn't made true by reality