**Author**: Mark Balaguer

**Publisher:** Oxford University Press on Demand

**ISBN:** 9780195143980

**Category:** Mathematics

**Page:** 217

**View:** 7299

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## Platonism and Anti-Platonism in Mathematics

In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He establishes that both platonism and anti-platonism are defensible views and introduces a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, proceeding to defend anti-platonism (in particular, mathematical fictionalism) against various attacks--most notably the Quine-Putnam indispensability attack.
## Platonism and Anti-Platonism in Mathematics

In this highly absorbing work, Balaguer demonstrates that no good arguments exist either for or against mathematical platonism-for example, the view that abstract mathematical objects do exist and that mathematical theories are descriptions of such objects. Balaguer does this by establishing that both platonism and anti-platonism are justifiable views. Introducing a form of platonism, called "full-blooded platonism," that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks-most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good arguments for or against platonism but that we could never have such an argument. This lucid and accessible book breaks new ground in its area of engagement and makes vital reading for both specialists and all those intrigued by the philosophy of mathematics, or metaphysics in general.
## Platonism and Anti-Platonism in Mathematics

In this highly absorbing work, Balaguer demonstrates that no good arguments exist either for or against mathematical platonism-for example, the view that abstract mathematical objects do exist and that mathematical theories are descriptions of such objects. Balaguer does this by establishing that both platonism and anti-platonism are justifiable views. Introducing a form of platonism, called "full-blooded platonism," that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks-most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good arguments for or against platonism but that we could never have such an argument. This lucid and accessible book breaks new ground in its area of engagement and makes vital reading for both specialists and all those intrigued by the philosophy of mathematics, or metaphysics in general.
## Plato's Problem

What is mathematics about? And how can we have access to the reality it is supposed to describe? The book tells the story of this problem, first raised by Plato, through the views of Aristotle, Proclus, Kant, Frege, Gödel, Benacerraf, up to the most recent debate on mathematical platonism.
## What is Mathematics, Really?

Reflecting an insider's view of mathematical life, the author argues that mathematics must be historically evolved, and intelligible only in a social context.
## Platonism, Naturalism, and Mathematical Knowledge

This study addresses a central theme in current philosophy: Platonism vs Naturalism and provides accounts of both approaches to mathematics, crucially discussing Quine, Maddy, Kitcher, Lakoff, Colyvan, and many others. Beginning with accounts of both approaches, Brown defends Platonism by arguing that only a Platonistic approach can account for concept acquisition in a number of special cases in the sciences. He also argues for a particular view of applied mathematics, a view that supports Platonism against Naturalist alternatives. Not only does this engaging book present the Platonist-Naturalist debate over mathematics in a comprehensive fashion, but it also sheds considerable light on non-mathematical aspects of a dispute that is central to contemporary philosophy.
## Philosophy of Mathematics

Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians.
## God Over All

God Over All: Divine Aseity and the Challenge of Platonism is a defense of God's aseity and unique status as the Creator of all things apart from Himself in the face of the challenge posed by mathematical Platonism. After providing the biblical, theological, and philosophical basis for the traditional doctrine of divine aseity, William Lane Craig explains the challenge presented to that doctrine by the Indispensability Argument for Platonism, which postulates the existence of uncreated abstract objects. Craig provides detailed examination of a wide range of responses to that argument, both realist and anti-realist, with a view toward assessing the most promising options for the theist. A synoptic work in analytic philosophy of religion, this groundbreaking volume engages discussions in philosophy of mathematics, philosophy of language, metaphysics, and metaontology.
## Free Will as an Open Scientific Problem

In this largely antimetaphysical treatment of free will and determinism, Mark Balaguer argues that the philosophical problem of free will boils down to an open scientific question about the causal histories of certain kinds of neural events. In the course of his argument, Balaguer provides a naturalistic defense of the libertarian view of free will. The metaphysical component of the problem of free will, Balaguer argues, essentially boils down to the question of whether humans possess libertarian free will. Furthermore, he argues that, contrary to the traditional wisdom, the libertarian question reduces to a question about indeterminacy--in particular, to a straightforward empirical question about whether certain neural events in our heads are causally undetermined in a certain specific way; in other words, Balaguer argues that the right kind of indeterminacy would bring with it all of the other requirements for libertarian free will. Finally, he argues that because there is no good evidence as to whether or not the relevant neural events are undetermined in the way that's required, the question of whether human beings possess libertarian free will is a wide-open empirical question.
## Free Will

A philosopher considers whether the scientific and philosophical arguments against free will are reason enough to give up our belief in it.
## Truth in Mathematics

The nature of truth in mathematics is a problem which has exercised the minds of thinkers from at least the time of the ancient Greeks. This book is an overview of the most recent work undertaken in this subject, and is unique in being the result of interactions between researchers from both philosophy and mathematics. The articles are written by world leaders in their respective fields and are of interest to researchers in both disciplines.
## Why Is There Philosophy of Mathematics At All?

This truly philosophical book takes us back to fundamentals - the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as 'what makes mathematics mathematics?', 'where did proof come from and how did it evolve?', and 'how did the distinction between pure and applied mathematics come into being?' In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that proof and other forms of mathematical exploration continue to be living, evolving practices - responsive to new technologies, yet embedded in permanent (and astonishing) facts about human beings. It distinguishes several distinct types of application of mathematics, and shows how each leads to a different philosophical conundrum. Here is a remarkable body of new philosophical thinking about proofs, applications, and other mathematical activities.
## After Gödel

Richard Tieszen analyzes, develops, and defends the writings of Kurt Gödel (1906-1978) on the philosophy and foundations of mathematics and logic. Gödel's relation to the work of Plato, Leibniz, Husserl, and Kant is examined, and a new type of platonic rationalism that requires rational intuition, called 'constituted platonism', is proposed.
## Mathematics and Reality

Mary Leng defends a philosophical account of the nature of mathematics which views it as a kind of fiction (albeit an extremely useful fiction). On this view, the claims of our ordinary mathematical theories are more closely analogous to utterances made in the context of storytelling than to utterances whose aim is to assert literal truths.
## Plato's Ghost

Plato's Ghost is the first book to examine the development of mathematics from 1880 to 1920 as a modernist transformation similar to those in art, literature, and music. Jeremy Gray traces the growth of mathematical modernism from its roots in problem solving and theory to its interactions with physics, philosophy, theology, psychology, and ideas about real and artificial languages. He shows how mathematics was popularized, and explains how mathematical modernism not only gave expression to the work of mathematicians and the professional image they sought to create for themselves, but how modernism also introduced deeper and ultimately unanswerable questions. Plato's Ghost evokes Yeats's lament that any claim to worldly perfection inevitably is proven wrong by the philosopher's ghost; Gray demonstrates how modernist mathematicians believed they had advanced further than anyone before them, only to make more profound mistakes. He tells for the first time the story of these ambitious and brilliant mathematicians, including Richard Dedekind, Henri Lebesgue, Henri Poincaré, and many others. He describes the lively debates surrounding novel objects, definitions, and proofs in mathematics arising from the use of naïve set theory and the revived axiomatic method--debates that spilled over into contemporary arguments in philosophy and the sciences and drove an upsurge of popular writing on mathematics. And he looks at mathematics after World War I, including the foundational crisis and mathematical Platonism. Plato's Ghost is essential reading for mathematicians and historians, and will appeal to anyone interested in the development of modern mathematics.
## Alcinous: The Handbook of Platonism

John Dillon presents an English translation of Alcinous' Handbook of Platonism, accompanied by an introduction and a philosophical commentary which reveal the intellectual background to the ideas in the work. The Handbook purports to be an introduction to the doctrines of Plato, but in fact gives us an excellent survey of Platonist thought in the second century AD. - ;Clarendon Later Ancient Philosophers This series, which is modelled on the familiar Clarendon Aristotle and Clarendon Plato Series, is designed to encourage philosophers and students of philosophy to explore the fertile terrain of later ancient philosophy. The texts range in date from the first century BC to the fifth century AD, and they cover all the parts and all the schools of philosophy. Each volume contains a substantial introduction, an English translation, and a critical commentary on the philosophical claims and arguments of the text. The accurate and faithful translations are highly readable and accompanied by notes on textual problems that affect the philosophical interpretation. No knowledge of Greek or Latin is assumed. The Handbook of Platonism, or Didaskalikos, attributed to Alcinous (long identified with the Middle Platonist Albinus, but on inadequate grounds), is a central text of later Platonism. In Byzantine times, in the Italian Renaissance, and even up to 1800, it was regarded as an ideal introduction to Plato's thought. In fact it is far from being this, but it is an excellent source for our understanding of Platonism in the second century AD. Neglected after a more accurate view of Plato's thought established itself in the nineteenth century, the Handbook is only now coming to be properly appreciated for what it is. It presents a survey of Platonist doctrine, divided into the topics of Logic, Physics, and Ethics, and pervaded with Aristotelian and Stoic doctrines, all of which are claimed for Plato. John Dillon presents an English translation of this work, accompanied by an introduction and a philosophical commentary in which he disentangles the various strands of influence on the text, elucidates the complex scholastic tradition that lies behind it, and thus reveals the sources and subsequent influence of the ideas expounded. -
## An Aristotelian Realist Philosophy of Mathematics

Mathematics is as much a science of the real world as biology is. It is the science of the world's quantitative aspects (such as ratio) and structural or patterned aspects (such as symmetry). The book develops a complete philosophy of mathematics that contrasts with the usual Platonist and nominalist options.
## Logicism, Intuitionism, and Formalism

This anthology reviews the programmes in the foundations of mathematics from the classical period and assesses their possible relevance for contemporary philosophy of mathematics. A special section is concerned with constructive mathematics.
## Constructibility and Mathematical Existence

This book is concerned with `the problem of existence in mathematics'. It develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. It explores the philosophical implications of such an approach through an examination of the writings of Field, Burgess, Maddy, Kitcher, and others.
## Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries)

A portrait of the eminent twentieth-century mathematician discusses his theorem of incompleteness, relationships with such contemporaries as Albert Einstein, and untimely death as a result of mental instability and self-starvation.

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