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Philosophy of Mathematics: Selected Readings
The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.
612 pages, Paperback
First published September 1, 1983
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About the author
Paul Benacerraf
9 books6 followersPaul Joseph Salomon Benacerraf was a French-born American philosopher working in the field of the philosophy of mathematics who taught at Princeton University his entire career, from 1960 until his retirement in 2007. Benacerraf was appointed Stuart Professor of Philosophy in 1974, and retired as the James S. McDonnell Distinguished University Professor of Philosophy.
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Displaying 1 - 11 of 11 reviews
July 28, 2013
This is something of an introduction to the philosophy of math for me. I have read some papers and one book on the subject, but nothing covering this broad of an area. The collection is packed full of the papers that other papers cite. I do like going to the sources and this is a extremely illuminating anthology of primary sources. You not only get the feel of the early to mid 20th century philosophy of math, which always interested me, but you get a whole lot of paradigmatic arguments of major schools of thought in the field.
The only criticism would be a mild one. Given the size of the book, it seems impossible, but it would be exceedingly helpful as a textbook if the editors (as esteemed as they seem to be) were to give small introductions or contextualization to each paper.
As is, though, it provides a great breadth of helpful papers and arguments in the field. It also sharply points out some major distinctions in philosophy, let alone phil of math (e.g. realism/anti-realism, epistemology stances, etc). I found some papers to be difficult but clear enough in context. The order of the papers (within the larger sections) were helpful in suggesting a dialogue between the authors.
Overall, wonderful and I will keep this book on hand for rereading. It isn't exactly a book you can read once.
The only criticism would be a mild one. Given the size of the book, it seems impossible, but it would be exceedingly helpful as a textbook if the editors (as esteemed as they seem to be) were to give small introductions or contextualization to each paper.
As is, though, it provides a great breadth of helpful papers and arguments in the field. It also sharply points out some major distinctions in philosophy, let alone phil of math (e.g. realism/anti-realism, epistemology stances, etc). I found some papers to be difficult but clear enough in context. The order of the papers (within the larger sections) were helpful in suggesting a dialogue between the authors.
Overall, wonderful and I will keep this book on hand for rereading. It isn't exactly a book you can read once.
November 24, 2015
This is a fairly abstract book to read. There are many essays in this book that question how we should characterize our relationship with numbers, or how numbers are or are not in the world. Nearly nothing is taken for granted; at its most abstract even the relationship between mathematics and logic is questioned. Ultimately the various kinds of consistencies that are abstractly drawn are compared for structural "differences" within the relationships. Most of these are digitally produced by comparison of different domains as rational "thresholds" defined in the same manner that Alan Badiou utilizes to match mathematics and ontology. Even within math we can find kernels which serve as "ontological differences" that are irreducible.
There are a variety of different approaches here. In a big way, this volume presents a smattering of approaches and best serves as an introductory text. Doubtless much of the information presented is beyond most people's comprehension, so readership is small. But it is rewarding to see that even among analytic philosophers there remains a variety of approaches which are irreducible to one another. The multiplicity of approaches is how a particular field remains vital and renewable. There are also some surprising mentions of Kantian antinomies (alongside Skolem's Paradox and Godel's Incompleteness Theorem), which was a delightful connection, given that Kant presented philosophy as logic, an approach that this volume does well to expand on.
I rather enjoyed reading this text. There are many connections that can be made (and aren't in this text) between various conceptions and how they functionally differentiate different modalities of making sense (consistency). I would like to see a post-structural approach to mathematics, one that extends these various conceptions to apply them in imaginative and intuitive ways. After all, the foundation of philosophy, mathematics and logic relies on raw taxonomy, as John Stuart Mill's A System of Logic: Ratiocination and Induction provides expansionist evidence for. While traditionally these areas are separated by content, they remain joined at their roots, as modalities of pure categorization, and consistent extension. There is much to be gleamed from these set of relationships, as new connections allow for a more general grasp of complexity as concepts and differences can be (un)translatable given different logical domains.
There are a variety of different approaches here. In a big way, this volume presents a smattering of approaches and best serves as an introductory text. Doubtless much of the information presented is beyond most people's comprehension, so readership is small. But it is rewarding to see that even among analytic philosophers there remains a variety of approaches which are irreducible to one another. The multiplicity of approaches is how a particular field remains vital and renewable. There are also some surprising mentions of Kantian antinomies (alongside Skolem's Paradox and Godel's Incompleteness Theorem), which was a delightful connection, given that Kant presented philosophy as logic, an approach that this volume does well to expand on.
I rather enjoyed reading this text. There are many connections that can be made (and aren't in this text) between various conceptions and how they functionally differentiate different modalities of making sense (consistency). I would like to see a post-structural approach to mathematics, one that extends these various conceptions to apply them in imaginative and intuitive ways. After all, the foundation of philosophy, mathematics and logic relies on raw taxonomy, as John Stuart Mill's A System of Logic: Ratiocination and Induction provides expansionist evidence for. While traditionally these areas are separated by content, they remain joined at their roots, as modalities of pure categorization, and consistent extension. There is much to be gleamed from these set of relationships, as new connections allow for a more general grasp of complexity as concepts and differences can be (un)translatable given different logical domains.
February 25, 2009
A must-have book, if you're a student of the philosophy of mathematics. This was THE anthology of important essays when it came out, and remained so for a lot of years. As an historical map of the phil-math terrain, this book is essential.
April 18, 2025
okay 4 starts because this bad boy was THICK. both content and the physical copy were DENSE. I read this for a class my last semester of my undergraduate philosophy program. It was great to read while also conducting copious amounts of research on the side to decipher what I was reading, so it's definitely a little (a lot) inaccessible at times. that said, it basically contains all the original texts of the philosophy of mathematics most from the early twentieth century. I read this alongside Gödel, Esher, Bach by Hofsdadter (which was a lot more accessible in my opinion) which is much more contemporary and explanatory (actually described a lot of the terms or assumption of this book).
Coming out of reading this, though, I am certainly a much more competent reader. my confidence in struggling through difficult text is much higher.
Coming out of reading this, though, I am certainly a much more competent reader. my confidence in struggling through difficult text is much higher.
August 27, 2021
Philosophy of math in the wake of 19th-century rigor, Cantor's infinity, the logicist program of Frege and Russell, and the foundational crisis of Mathematics in the form of Godel. Intuitionists, Logicists, Realists (Platonists), Structuralists all argue their sides on the ontology and nature of mathematics with conundrums of whether it is real or are they a fiction and how to handle infinity. Classic stuff from this book on the topic from the 1980s.
January 13, 2025
pretty good overview of philosophy of mathematics (particularly the last 150 years or so), though i wouldnt mind some contexualization for some of the essays at least. though as a collection of essays it makes a good reference i suppose. way more references to wittgenstein than i expected is also a plus
Want to read
February 11, 2020Math anthology
May 28, 2024
this the last time i meddle with a subject without first teaching myself the fundamentals
November 30, 2024
Get the second edition. I got the first, and although it does contain some great essays that the second lacks, the logical notation is so outdated it’s almost unreadable at times.
August 20, 2007
This was a collection of essays organized around a few of the major themes in early 20th century mathematical philosophy, such as formalism vs. intuitionism, the nature of mathematical truth, etc. The essays presented a variety of viewpoints on each topic, and gave a good sense of the dialogue and debate that was taking place in the field at the time. A lot of the great 20th century mathematical philosophers are represented here, including Russell, Frege, Goedel, and Hilbert. The essays go fairly deep into the topics they cover, and demand a solid background in mathematical philosophy.
I think most people would be better served by a modern book written on this subject, as the essays offer little background information, don't always use consistent terminology, and have a lot of overlap. However, if you prefer to read primary sources, this is a good collection to start with.
I think most people would be better served by a modern book written on this subject, as the essays offer little background information, don't always use consistent terminology, and have a lot of overlap. However, if you prefer to read primary sources, this is a good collection to start with.
January 9, 2008
An important book in the field
Displaying 1 - 11 of 11 reviews