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The Explanatory Indispensability Of Mathematics

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The Explanatory Indispensability Of Mathematics

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Author : Nils Richards
language : en
Publisher:
Release Date : 2013

The Explanatory Indispensability Of Mathematics written by Nils Richards and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Electronic books categories.

Inference to the best explanation (IBE) is the principle of inference according to which, when faced with a set of competing hypotheses, where each hypothesis is empirically adequate for explaining the phenomena, we should infer the truth of the hypothesis that best explains the phenomena. When our theories correctly display this principle, we call them our 'best'. In this paper, I examine the explanatory role of mathematics in our best scientific theories. In particular, I will elucidate the enormous utility of mathematical structures. I argue from a reformed indispensability argument that mathematical structures are explanatorily indispensable to our best scientific theories. Therefore, IBE scientific realism entails mathematical realism. I develop a naturalistic, neo-Quinean ontology, which grounds physical and mathematical entities in structures. Mathematical structures are the truth-makers for the entities of our quantificational discourse. I also develop an 'ontic conception' of explanation, according to which explanations exist in the world, whether or not we discover and model them. I apply the ontic account to mathematical structures, arguing that these structures are the explanations for particles, forces, and even the conservation laws of physics. As such, mathematical structures provide the fundamental grounding for ontological commitment. I conclude by reviewing the evidence from modern physics for the existence of mathematical structures.

Indispensability

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Author : A. C. Paseau
language : en
Publisher: Cambridge University Press
Release Date : 2023-06-08

Indispensability written by A. C. Paseau and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-06-08 with Philosophy categories.

Our best scientific theories explain a wide range of empirical phenomena, make accurate predictions, and are widely believed. Since many of these theories make ample use of mathematics, it is natural to see them as confirming its truth. Perhaps the use of mathematics in science even gives us reason to believe in the existence of abstract mathematical objects such as numbers and sets. These issues lie at the heart of the Indispensability Argument, to which this Element is devoted. The Element's first half traces the evolution of the Indispensability Argument from its origins in Quine and Putnam's works, taking in naturalism, confirmational holism, Field's program, and the use of idealisations in science along the way. Its second half examines the explanatory version of the Indispensability Argument, and focuses on several more recent versions of easy-road and hard-road fictionalism respectively.

Explanation In Ethics And Mathematics

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Author : Uri D. Leibowitz
language : en
Publisher: Oxford University Press
Release Date : 2016-05-26

Explanation In Ethics And Mathematics written by Uri D. Leibowitz and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-05-26 with Philosophy categories.

How far should our realism extend? For many years philosophers of mathematics and philosophers of ethics have worked independently to address the question of how best to understand the entities apparently referred to by mathematical and ethical talk. But the similarities between their endeavours are not often emphasised. This book provides that emphasis. In particular, it focuses on two types of argumentative strategies that have been deployed in both areas. The first--debunking arguments--aims to put pressure on realism by emphasising the seeming redundancy of mathematical or moral entities when it comes to explaining our judgements. In the moral realm this challenge has been made by Gilbert Harman and Sharon Street; in the mathematical realm it is known as the 'Benacerraf-Field' problem. The second strategy--indispensability arguments--aims to provide support for realism by emphasising the seeming intellectual indispensability of mathematical or moral entities, for example when constructing good explanatory theories. This strategy is associated with Quine and Putnam in mathematics and with Nicholas Sturgeon and David Enoch in ethics. Explanation in Ethics and Mathematics addresses these issues through an explicitly comparative methodology which we call the 'companions in illumination' approach. By considering how argumentative strategies in the philosophy of mathematics might apply to the philosophy of ethics, and vice versa, the papers collected here break new ground in both areas. For good measure, two further companions for illumination are also broached: the philosophy of chance and the philosophy of religion. Collectively, these comparisons light up new questions, arguments, and problems of interest to scholars interested in realism in any area.

The Indispensability Of Mathematics

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Author : Mark Colyvan
language : en
Publisher: Oxford University Press
Release Date : 2001-03-22

The Indispensability Of Mathematics written by Mark Colyvan and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-03-22 with Philosophy categories.

The Quine-Putnam indispensability argument in the philosophy of mathematics urges us to place mathematical entities on the same ontological footing as other theoretical entities essential to our best scientific theories. Recently, the argument has come under serious scrutiny, with many influential philosophers unconvinced of its cogency. This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.

Mathematics And Explanation

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Author : Christopher Pincock
language : en
Publisher: Cambridge University Press
Release Date : 2023-05-25

Mathematics And Explanation written by Christopher Pincock and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-05-25 with Science categories.

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?

An Introduction To The Philosophy Of Mathematics

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Author : Mark Colyvan
language : en
Publisher: Cambridge University Press
Release Date : 2012-06-14

An Introduction To The Philosophy Of Mathematics written by Mark Colyvan and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-06-14 with Mathematics categories.

A fascinating journey through intriguing mathematical and philosophical territory - a lively introduction to this contemporary topic.

Explanation In Ethics And Mathematics

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Author : Uri D. Leibowitz
language : en
Publisher: Oxford University Press
Release Date : 2016-05-27

How far should our realism extend? For many years philosophers of mathematics and philosophers of ethics have worked independently to address the question of how best to understand the entities apparently referred to by mathematical and ethical talk. But the similarities between their endeavours are not often emphasised. This book provides that emphasis. In particular, it focuses on two types of argumentative strategies that have been deployed in both areas. The first—debunking arguments—aims to put pressure on realism by emphasising the seeming redundancy of mathematical or moral entities when it comes to explaining our judgements. In the moral realm this challenge has been made by Gilbert Harman and Sharon Street; in the mathematical realm it is known as the 'Benacerraf-Field' problem. The second strategy—indispensability arguments—aims to provide support for realism by emphasising the seeming intellectual indispensability of mathematical or moral entities, for example when constructing good explanatory theories. This strategy is associated with Quine and Putnam in mathematics and with Nicholas Sturgeon and David Enoch in ethics. Explanation in Ethics and Mathematics addresses these issues through an explicitly comparative methodology which we call the 'companions in illumination' approach. By considering how argumentative strategies in the philosophy of mathematics might apply to the philosophy of ethics, and vice versa, the papers collected here break new ground in both areas. For good measure, two further companions for illumination are also broached: the philosophy of chance and the philosophy of religion. Collectively, these comparisons light up new questions, arguments, and problems of interest to scholars interested in realism in any area.

The Indispensability Of Mathematics Ebook

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Author : Mark Colyvan
language : en
Publisher:
Release Date : 2001

The Indispensability Of Mathematics Ebook written by Mark Colyvan and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.

Autonomy Platonism And The Indispensability Argument

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Author : Russell Marcus
language : en
Publisher: Lexington Books
Release Date : 2015-06-11

Autonomy Platonism And The Indispensability Argument written by Russell Marcus and has been published by Lexington Books this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-06-11 with Philosophy categories.

Mathematical platonism is the view that mathematical statements are true of real mathematical objects like numbers, shapes, and sets. One central problem with platonism is that numbers, shapes, sets, and the like are not perceivable by our senses. In contemporary philosophy, the most common defense of platonism uses what is known as the indispensability argument. According to the indispensabilist, we can know about mathematics because mathematics is essential to science. Platonism is among the most persistent philosophical views. Our mathematical beliefs are among our most entrenched. They have survived the demise of millennia of failed scientific theories. Once established, mathematical theories are rarely rejected, and never for reasons of their inapplicability to empirical science. Autonomy Platonism and the Indispensability Argument is a defense of an alternative to indispensability platonism. The autonomy platonist believes that mathematics is independent of empirical science: there is purely mathematical evidence for purely mathematical theories which are even more compelling to believe than empirical science. Russell Marcus begins by contrasting autonomy platonism and indispensability platonism. He then argues against a variety of indispensability arguments in the first half of the book. In the latter half, he defends a new approach to a traditional platonistic view, one which includes appeals to a priori but fallible methods of belief acquisition, including mathematical intuition, and a natural adoption of ordinary mathematical methods. In the end, Marcus defends his intuition-based autonomy platonism against charges that the autonomy of mathematics is viciously circular. This book will be useful to researchers, graduate students, and advanced undergraduates with interests in the philosophy of mathematics or in the connection between science and mathematics.

Mathematical Explanation In Science

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Author : Alexander Koo
language : en
Publisher:
Release Date : 2015

Mathematical Explanation In Science written by Alexander Koo and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with categories.

Inspired by indispensability arguments originating from Quine, mathematical realists such as Alan Baker argue that since mathematics plays a key explanatory role in our best scientific theories, then the very same reasons which convince us to be scientific realists should lead us to be mathematical realists as well. Baker's enhanced indispensability arguments (EIA) makes use of inference to the best explanation (IBE) to deliver the realist conclusion. Mathematical nominalists have resisted this argument by asserting that there is no such thing as a genuine mathematical explanation (GME) where the mathematics is playing an indispensable explanatory role. In this dissertation I will argue that the nominalist is incorrect and that GMEs do, in fact, exist. My methodology will be to develop a set of criteria that clearly defines a GME which the nominalist would gladly accept. From there, a new example of a GME will be advanced that satisfies all of the criteria. To solidify this result, I will show that Strevens' kairetic account of scientific explanation clearly points to mathematics playing an indispensable explanatory role in our supposed examples of GME. While this all bodes well for mathematical realism, I will further argue that the EIA still does not lead to mathematical realism. Baker assumes that using IBE to infer the existence of mathematical objects is unproblematic. I will challenge this assumption, and without IBE the EIA does not deliver. By no means does this result block the realist project. Ultimately, I believe that freeing ourselves from the yoke of traditional indispensability arguments and focussing on how it is that mathematics can explain physical facts explain will advance the realist position even further.

The Explanatory Indispensability Of Mathematics

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