What do you think?
Rate this book
Routledge Contemporary Introductions to Philosophy
Philosophy of Mathematics
In his long-awaited new edition of Philosophy of Mathematics , James Robert Brown tackles important new as well as enduring questions in the mathematical sciences. Can pictures go beyond being merely suggestive and actually prove anything? Are mathematical results certain? Are experiments of any real value? This clear and engaging book takes a unique approach, encompassing non-standard topics such as the role of visual reasoning, the importance of notation, and the place of computers in mathematics, as well as traditional topics such as formalism, Platonism, and constructivism. The combination of topics and clarity of presentation make it suitable for beginners and experts alike. The revised and updated second edition of Philosophy of Mathematics contains more examples, suggestions for further reading, and expanded material on several topics including a novel approach to the continuum hypothesis.
264 pages, Paperback
First published July 22, 1999
12 people are currently reading
181 people want to read
About the author
James Robert Brown
30 books8 followersDepartment of Philosophy, University of Toronto, Canada
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 8 of 8 reviews
March 18, 2012
I didn't understand a word. Amazing.
November 25, 2007
There's a great discussion of the implications of Wittgenstein's Philosophical Investigations for number theory in here. I really liked this as an introduction to the topic, less esoteric than a lot of the available texts. Since I never got past Algebra II and have never taken an actual philosophy course, this was important to me when I read it.
August 24, 2022
of all the branches of philosophy, the philosophy of mathematics has always been (at least to me) one of the most forbidding. as far as philosophy goes, the problems the philosophy of math poses are superlatively abstract and obscure; it's not for nothing, i suspect, that those philosophers who worked on the philo of math in their lifetimes came out mentally bruised (Frege, Russell), slightly insane (Wittgenstein - though arguably Wittgenstein always was so), or fully demented (Gödel) by the process. and as far as mathematics goes the favourite examples of philosophers are usually incomprehensible to the amateur (largely we have set theory to blame for this. the most philosophically interesting mathematical problems have largely arisen from very, very complex set theory - where the complexity breeds interest).
despite all of this, I've always had something of a buried interest in the philo of math ever since my KI sojourn and the readings we received then ("Numbers Exist", followed immediately by "Numbers Don't Exist"). and it's a pleasure to report that, insofar as it aims to introduce the reader to the philosophy of mathematics, Brown's Philosophy of Mathematics is pretty successful.
Brown takes several unorthodox stands and approaches. he is a Platonist, which should set off alarm bells ringing in the mind of anyone familiar with the philosophy of math. he also prizes non-traditional proof techniques (computers, thought experiments, intuition) much more than do other mathematicians, and it is from such a standpoint that Brown constructs his chapters. these bring the reader through well-worn corners of the philo of math (Gödel's incompleteness theorems, Brouwer's constructivism, the applicability problem) and some more abstruse areas (most notably, Freiling's attempted refutation of the continuum hypothesis - but also Lakatos' revisionist history of mathematical knowledge construction, and different types of notation in knot theory).
all this is generally done with remarkable clarity, which can't always be taken for granted both in philosophy and math. for the first time I understood more or less clearly the continuum hypothesis and Wittgenstein/Kripkenstein's rule-following paradox, both of which are not easy concepts to unpack. (more impressively the book gets you to understand the Brouwer brand of intuitionist logic and the motivation behind constructivism, both of which are sufficiently mindboggling for it to be a minor miracle they can be explained at all.) examples are generally gone through minutely and I found myself having recourse to wikipedia only once to clarify my understanding of an example (in that case it was to get a better grasp on what cardinals and ordinals in set theory are - this isn't unfortunately explained as clearly as could be!). and the breadth is enjoyable and suitable for an introductory tract as this.
yet there are non-negligible debits also that detract rather seriously from the book's success as an introduction. as mentioned above Brown takes no shame in taking a stand and sticking to it all the way. this makes his arguments often interesting, but it means he neglects to explain theories he disagrees with in their full strength. this is the case for formalism, structuralism, and fictionalism - the latter neglect is especially to be regretted because, to my mind, fictionalism is one of the more ontologically convincing accounts of mathematics we have even if it struggles to explain the use of math in science. the merits of these theories are simply not given as much weight as they deserve. conversely, the merits of Brown's arguments for Platonism and intuition are often overstated and one has the sense that they are not as critically examined as they might be if Brown disagreed with them. Brown also commits some painful solecisms (he spells weird as "wierd" and uses "principle" as an adjective - "principal", it seems, is not in his lexicon. though I am sure he would believe it exists in the lexicon of Platonic heaven) and makes some errors in the mathematical examples (most noticeably there is an erroneous transcription of Euler's marvellous proof that zeta(2)=π²/6 - an additional 1/3 term is introduced that disappears without explanation) that can be distracting.
the errors in linguistic and mathematical grammar are mildly lamentable but it is the lack of objectivity that is most problematic for an introduction. it is not destructive - the intrepid, exploratory strengths of the text still stand as does its clarity - but the book would have been so much more powerful if it took fewer easy paths. for anyone wading through the dark forests of the philosophy of mathematics, however, Brown's text is a perfectly adequate light. one must simply note that this light illuminates certain paths that end in a cul-de-sac, and leaves other more promising ones still in the darkness.
despite all of this, I've always had something of a buried interest in the philo of math ever since my KI sojourn and the readings we received then ("Numbers Exist", followed immediately by "Numbers Don't Exist"). and it's a pleasure to report that, insofar as it aims to introduce the reader to the philosophy of mathematics, Brown's Philosophy of Mathematics is pretty successful.
Brown takes several unorthodox stands and approaches. he is a Platonist, which should set off alarm bells ringing in the mind of anyone familiar with the philosophy of math. he also prizes non-traditional proof techniques (computers, thought experiments, intuition) much more than do other mathematicians, and it is from such a standpoint that Brown constructs his chapters. these bring the reader through well-worn corners of the philo of math (Gödel's incompleteness theorems, Brouwer's constructivism, the applicability problem) and some more abstruse areas (most notably, Freiling's attempted refutation of the continuum hypothesis - but also Lakatos' revisionist history of mathematical knowledge construction, and different types of notation in knot theory).
all this is generally done with remarkable clarity, which can't always be taken for granted both in philosophy and math. for the first time I understood more or less clearly the continuum hypothesis and Wittgenstein/Kripkenstein's rule-following paradox, both of which are not easy concepts to unpack. (more impressively the book gets you to understand the Brouwer brand of intuitionist logic and the motivation behind constructivism, both of which are sufficiently mindboggling for it to be a minor miracle they can be explained at all.) examples are generally gone through minutely and I found myself having recourse to wikipedia only once to clarify my understanding of an example (in that case it was to get a better grasp on what cardinals and ordinals in set theory are - this isn't unfortunately explained as clearly as could be!). and the breadth is enjoyable and suitable for an introductory tract as this.
yet there are non-negligible debits also that detract rather seriously from the book's success as an introduction. as mentioned above Brown takes no shame in taking a stand and sticking to it all the way. this makes his arguments often interesting, but it means he neglects to explain theories he disagrees with in their full strength. this is the case for formalism, structuralism, and fictionalism - the latter neglect is especially to be regretted because, to my mind, fictionalism is one of the more ontologically convincing accounts of mathematics we have even if it struggles to explain the use of math in science. the merits of these theories are simply not given as much weight as they deserve. conversely, the merits of Brown's arguments for Platonism and intuition are often overstated and one has the sense that they are not as critically examined as they might be if Brown disagreed with them. Brown also commits some painful solecisms (he spells weird as "wierd" and uses "principle" as an adjective - "principal", it seems, is not in his lexicon. though I am sure he would believe it exists in the lexicon of Platonic heaven) and makes some errors in the mathematical examples (most noticeably there is an erroneous transcription of Euler's marvellous proof that zeta(2)=π²/6 - an additional 1/3 term is introduced that disappears without explanation) that can be distracting.
the errors in linguistic and mathematical grammar are mildly lamentable but it is the lack of objectivity that is most problematic for an introduction. it is not destructive - the intrepid, exploratory strengths of the text still stand as does its clarity - but the book would have been so much more powerful if it took fewer easy paths. for anyone wading through the dark forests of the philosophy of mathematics, however, Brown's text is a perfectly adequate light. one must simply note that this light illuminates certain paths that end in a cul-de-sac, and leaves other more promising ones still in the darkness.
January 4, 2019
As an introduction to various positions in philosophy of mathematics, it seems fairly good. Brown's Platonistic bias is very strong, and he tends to not give other positions a fair presentation. In addition, he includes several factual errors indicative of an embarrassing lack of thorough research. Brown seems to forget that this is an introduction to various positions, and he constantly argues for the superiority of the Platonist position - but he does so superficially, bringing up opposing views and dismissing them without sufficient (or at times, any) reason.
July 23, 2012
FULLY AGREE WITH THE FOLLOWING QUOTE:
A philosopher who has nothing to do with geometry is only half a
philosopher, and a mathematician with no element of philosophy in
him is only half a mathematician. These disciplines have estranged
themselves from one another to the detriment of both.
Frege
however, this book is a very dull and undisciplined treatment of the potential fusion between two key elements of the human experience
A philosopher who has nothing to do with geometry is only half a
philosopher, and a mathematician with no element of philosophy in
him is only half a mathematician. These disciplines have estranged
themselves from one another to the detriment of both.
Frege
however, this book is a very dull and undisciplined treatment of the potential fusion between two key elements of the human experience
December 21, 2019
An accessible introduction to the subject that would make a nice jumping off point for the mathematics major having an interest in philosophy, but lacking a background in the relevant areas. I imagine this book is somewhat unique in its consideration of visual reasoning and the use of diagrams as learning aids & possible modes of proof. As another reviewer mentioned, Brown is not shy about his Platonist sympathies, but I found the authorial bias to be quite refreshing for a book of this nature.
April 14, 2018
Fantastic book. Fascinating questions are raised and addressed, in an easy to read and convincing style. If the topic interests you, this is a good place to start.
April 10, 2021
Great book - wrong title.
Displaying 1 - 8 of 8 reviews