Defending the Axioms

Defending the Axioms
Author: Penelope Maddy
Publisher: Oxford University Press
Total Pages: 161
Release: 2011-01-27
Genre: Mathematics
ISBN: 0199596182

Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. The axioms of set theory have long played this role, so the question of how they are properly judged is of central importance. Maddy discusses the appropriate methods for such evaluations and the philosophical backdrop that makes them appropriate.

Second Philosophy

Second Philosophy
Author: Penelope Maddy
Publisher: Oxford University Press
Total Pages: 461
Release: 2007-04-19
Genre: Mathematics
ISBN: 0199273669

Many philosophers these days consider themselves naturalists, but it's doubtful any two of them intend the same position by the term. In this book, Penelope Maddy describes and practises a particularly austere form of naturalism called 'Second Philosophy'. Without a definitive criterion for what counts as 'science' and what doesn't, Second Philosophy can't be specified directly - 'trust only the methods of science!' or some such thing - so Maddy proceeds instead by illustratingthe behaviours of an idealized inquirer she calls the 'Second Philosopher'. This Second Philosopher begins from perceptual common sense and progresses from there to systematic observation, active experimentation, theory formation and testing, working all the while to assess, correct and improve hermethods as she goes. Second Philosophy is then the result of the Second Philosopher's investigations.Maddy delineates the Second Philosopher's approach by tracing her reactions to various familiar skeptical and transcendental views (Descartes, Kant, Carnap, late Putnam, van Fraassen), comparing her methods to those of other self-described naturalists (especially Quine), and examining a prominent contemporary debate (between disquotationalists and correspondence theorists in the theory of truth) to extract a properly second-philosophical line of thought. She then undertakes to practise SecondPhilosophy in her reflections on the ground of logical truth, the methodology, ontology and epistemology of mathematics, and the general prospects for metaphysics naturalized.

Deleuze and the History of Mathematics

Deleuze and the History of Mathematics
Author: Simon Duffy
Publisher: A&C Black
Total Pages: 225
Release: 2013-05-09
Genre: Philosophy
ISBN: 1441113894

Gilles Deleuze's engagements with mathematics, replete in his work, rely upon the construction of alternative lineages in the history of mathematics, which challenge some of the self imposed limits that regulate the canonical concepts of the discipline. For Deleuze, these challenges provide an opportunity to reconfigure particular philosophical problems - for example, the problem of individuation - and to develop new concepts in response to them. The highly original research presented in this book explores the mathematical construction of Deleuze's philosophy, as well as addressing the undervalued and often neglected question of the mathematical thinkers who influenced his work. In the wake of Alain Badiou's recent and seemingly devastating attack on the way the relation between mathematics and philosophy is configured in Deleuze's work, Simon B.Duffy offers a robust defence of the structure of Deleuze's philosophy and, in particular, the adequacy of the mathematical problems used in its construction. By reconciling Badiou and Deleuze's seemingly incompatible engagements with mathematics, Duffy succeeds in presenting a solid foundation for Deleuze's philosophy, rebuffing the recent challenges against it.

Naturalism in Mathematics

Naturalism in Mathematics
Author: Penelope Maddy
Publisher: Clarendon Press
Total Pages: 265
Release: 1997-11-13
Genre: Philosophy
ISBN: 0191518972

Our much-valued mathematical knowledge rests on two supports: the logic of proof and the axioms from which those proofs begin. Naturalism in Mathematics investigates the status of the latter, the fundamental assumptions of mathematics. These were once held to be self-evident, but progress in work on the foundations of mathematics, especially in set theory, has rendered that comforting notion obsolete. Given that candidates for axiomatic status cannot be proved, what sorts of considerations can be offered for or against them? That is the central question addressed in this book. One answer is that mathematics aims to describe an objective world of mathematical objects, and that axiom candidates should be judged by their truth or falsity in that world. This promising view—realism—is assessed and finally rejected in favour of another—naturalism—which attends less to metaphysical considerations of objective truth and falsity, and more to practical considerations drawn from within mathematics itself. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be helpfully applied in the assessment of candidates for axiomatic status in set theory. Maddy's clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

What Do Philosophers Do?

What Do Philosophers Do?
Author: Penelope Maddy
Publisher: Oxford University Press
Total Pages: 265
Release: 2017
Genre: Philosophy
ISBN: 0190618698

How do you know the world around you isn't just an elaborate dream, or the creation of an evil neuroscientist? If all you have to go on are various lights, sounds, smells, tastes and tickles, how can you know what the world is really like, or even whether there is a world beyond your own mind? Questions like these -- familiar from science fiction and dorm room debates -- lie at the core of venerable philosophical arguments for radical skepticism: the stark contention that we in fact know nothing at all about the world, that we have no more reason to believe any claim -- that there are trees, that we have hands -- than we have to disbelieve it. Like non-philosophers in their sober moments, philosophers, too, find this skeptical conclusion preposterous, but they're faced with those famous arguments: the Dream Argument, the Argument from Illusion, the Infinite Regress of Justification, the more recent Closure Argument. If these can't be met, they raise a serious challenge not just to philosophers, but to anyone responsible enough to expect her beliefs to square with her evidence. What Do Philosophers Do? takes up the skeptical arguments from this everyday point of view, and ultimately concludes that they don't undermine our ordinary beliefs or our ordinary ways of finding out about the world. In the process, Maddy examines and evaluates a range of philosophical methods -- common sense, scientific naturalism, ordinary language, conceptual analysis, therapeutic approaches -- as employed by such philosophers as Thomas Reid, G. E. Moore, Ludwig Wittgenstein, and J. L. Austin. The result is a revealing portrait of what philosophers do, and perhaps a quiet suggestion for what they should do, for what they do best.

The Point of View of the Universe

The Point of View of the Universe
Author: Katarzyna de Lazari-Radek
Publisher: Oxford University Press, USA
Total Pages: 433
Release: 2014
Genre: Philosophy
ISBN: 0199603693

Tests the views and metaphor of 19th-century utilitarian philosopher Henry Sidgwick against a variety of contemporary views on ethics, determining that they are defensible and thus providing a defense of objectivism in ethics and of hedonistic utilitarianism.

Uncertain Values

Uncertain Values
Author: Stefan Riedener
Publisher: Walter de Gruyter GmbH & Co KG
Total Pages: 157
Release: 2021-10-25
Genre: Philosophy
ISBN: 3110736225

How ought you to evaluate your options if you're uncertain about what's fundamentally valuable? A prominent response is Expected Value Maximisation (EVM)—the view that under axiological uncertainty, an option is better than another if and only if it has the greater expected value across axiologies. But the expected value of an option depends on quantitative probability and value facts, and in particular on value comparisons across axiologies. We need to explain what it is for such facts to hold. Also, EVM is by no means self-evident. We need an argument to defend that it’s true. This book introduces an axiomatic approach to answer these worries. It provides an explication of what EVM means by use of representation theorems: intertheoretic comparisons can be understood in terms of facts about which options are better than which, and mutatis mutandis for intratheoretic comparisons and axiological probabilities. And it provides a systematic argument to the effect that EVM is true: the theory can be vindicated through simple axioms. The result is a formally cogent and philosophically compelling extension of standard decision theory, and original take on the problem of axiological or normative uncertainty.

Mereology: A Philosophical Introduction

Mereology: A Philosophical Introduction
Author: Giorgio Lando
Publisher: Bloomsbury Publishing
Total Pages: 249
Release: 2017-06-29
Genre: Philosophy
ISBN: 1472583671

Parthood and composition are everywhere. The leg of a table is part of the table, the word "Christmas" is part of the sentence "I wish you a merry Christmas", the 13th century is part of the Middle Ages. The Netherlands, Belgium, and Luxembourg compose Benelux, the body of a deer is composed of a huge number of cells, the Middle Ages are composed of the Early Middle Ages, High Middle Ages, and Late Middle Ages. Is there really a general theory covering every instance of parthood and composition? Is classical mereology this general theory? Are its seemingly counter-intuitive features serious defects? Mereology: A Philosophical Introduction addresses the multifaceted and lively philosophical debates surrounding these questions, and defends the idea that classical mereology is indeed the general and exhaustive theory of parthood and composition in the domain of concrete entities. Several examples of parthood and composition, involving entities of different kinds, are scrutinised in depth. Incidentally, mereology is shown to interact in a surprising way with metaontology. Presenting a well-organized and comprehensive discussion of parthood and related notions, Mereology: A Philosophical Introduction contributes to a better understanding of a subject central to contemporary metaphysics.

Mathematics and Its Applications

Mathematics and Its Applications
Author: Jairo José da Silva
Publisher: Springer
Total Pages: 274
Release: 2017-08-22
Genre: Philosophy
ISBN: 3319630733

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies.