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Philosophy of Mathematics
A sophisticated, original introduction to the philosophy of mathematics from one of its leading contemporary scholars
Mathematics is one of humanity's most successful yet puzzling endeavors. It is a model of precision and objectivity, but appears distinct from the empirical sciences because it seems to deliver nonexperiential knowledge of a nonphysical reality of numbers, sets, and functions. How can these two aspects of mathematics be reconciled? This concise book provides a systematic yet accessible introduction to the field that is trying to answer that question: the philosophy of mathematics.
Written by Øystein Linnebo, one of the world's leading scholars on the subject, the book introduces all of the classical approaches to the field, including logicism, formalism, intuitionism, empiricism, and structuralism. It also contains accessible introductions to some more specialized issues, such as mathematical intuition, potential infinity, the iterative conception of sets, and the search for new mathematical axioms. The groundbreaking work of German mathematician and philosopher Gottlob Frege, one of the founders of analytic philosophy, figures prominently throughout the book. Other important thinkers whose work is introduced and discussed include Immanuel Kant, John Stuart Mill, David Hilbert, Kurt Gödel, W. V. Quine, Paul Benacerraf, and Hartry H. Field.
Sophisticated but clear and approachable, this is an essential introduction for all students and teachers of philosophy, as well as mathematicians and others who want to understand the foundations of mathematics.
210 pages, Kindle Edition
Published May 30, 2017
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Displaying 1 - 20 of 20 reviews
August 8, 2023
Better Than the Super Bowl
Numbers are words. Numbers exist in the same way that other words exist, namely when they are used in speech or writing. That is to say, they are neither objective nor subjective but inter-subjective. Numbers are names which refer to themselves, and there are specified ways in which numbers may be used with each other. This is their syntax. Like all words, numbers are defined by other words. This is called their semantics. Words are the only things we really know about. We know most about numbers because they are more precisely defined than other words. When we use numbers according to the specified rules, we are able to discover relationships among them, many of which are unexpected. These discoveries constitute the pragmatics of numbers.
This is a summary of my philosophy of mathematics. It is simple, easy to comprehend, comprehensive, and probably wrong. I say this because people far more intelligent than I have agonised over the existence, meaning and practical effect of mathematics without coming to a consensus. I cannot understand why this is so. Linnebo chronicles the last two hundred years or so of philosophical agony about numbers largely through a focus on one man, the German Gottlob Frege. Frege devoted much of his life to proving that arithmetic was entirely rational, that is, that it could be derived from indisputable first principles. He failed.
He failed for technical reasons involving necessary self-contradictions. But some forethought might have suggested that his failure was inevitable for other reasons as well, namely that there is no, and cannot be any, privileged language no matter how well-defined that language is. Language is its own representation. There is nothing beyond or outside of language that can be used to demonstrate its integrity. The logic of any language is contained in its syntax and cannot be gainsaid by reference to any logic applied from outside of itself.* The logics among languages may differ but they cannot be compared in order to determine which is superior. This is the ultimate conclusion of the mathematician Kurt Gödel, who finally killed Frege’s project as a matter of principle.
One implication of my philosophy of mathematics is that by using numbers we submit to them, their syntax, and their semantics in order to experience their pragmatics. No further justification for engaging is]n mathematics, it seems to me, is required. Nor is there any deep mystery, except perhaps psychological, about why we do this. Submission to mathematics, like submission to any language, is the giving over of one’s life to a society, namely that group which jointly participates in it. To use numbers is not to associate with the gods as the ancient Greeks thought. It is to associate with each other, to accept the judgments of each other, and to be recognised for furthering the implicit social project of the language of our association.
It strikes me as more than odd that Linnebo as well as the mathematicians he analyses simply ignore their own social dedication in their language of choice. The primary pragmatic consequence of mathematics is membership. Membership is achieved even in the failure to achieve any other result in mathematics. So, for example, Frege, and Bertrand Russell and even Gödel continued to ‘speak’ mathematics knowing that it is as ‘flawed’ as any other language. It may not be the key to understanding the universe; but it is nevertheless beautiful, intriguing, seductive, and it is most of all a way to spend a harmless afternoon with friends. Certainly better than watching the Super Bowl for those who prefer words to physical violence.
* This applies to the ‘dialects’ within mathematics as well. As Linnebo points out, the logic of Frege was very different from the logic of Kant for example.
Numbers are words. Numbers exist in the same way that other words exist, namely when they are used in speech or writing. That is to say, they are neither objective nor subjective but inter-subjective. Numbers are names which refer to themselves, and there are specified ways in which numbers may be used with each other. This is their syntax. Like all words, numbers are defined by other words. This is called their semantics. Words are the only things we really know about. We know most about numbers because they are more precisely defined than other words. When we use numbers according to the specified rules, we are able to discover relationships among them, many of which are unexpected. These discoveries constitute the pragmatics of numbers.
This is a summary of my philosophy of mathematics. It is simple, easy to comprehend, comprehensive, and probably wrong. I say this because people far more intelligent than I have agonised over the existence, meaning and practical effect of mathematics without coming to a consensus. I cannot understand why this is so. Linnebo chronicles the last two hundred years or so of philosophical agony about numbers largely through a focus on one man, the German Gottlob Frege. Frege devoted much of his life to proving that arithmetic was entirely rational, that is, that it could be derived from indisputable first principles. He failed.
He failed for technical reasons involving necessary self-contradictions. But some forethought might have suggested that his failure was inevitable for other reasons as well, namely that there is no, and cannot be any, privileged language no matter how well-defined that language is. Language is its own representation. There is nothing beyond or outside of language that can be used to demonstrate its integrity. The logic of any language is contained in its syntax and cannot be gainsaid by reference to any logic applied from outside of itself.* The logics among languages may differ but they cannot be compared in order to determine which is superior. This is the ultimate conclusion of the mathematician Kurt Gödel, who finally killed Frege’s project as a matter of principle.
One implication of my philosophy of mathematics is that by using numbers we submit to them, their syntax, and their semantics in order to experience their pragmatics. No further justification for engaging is]n mathematics, it seems to me, is required. Nor is there any deep mystery, except perhaps psychological, about why we do this. Submission to mathematics, like submission to any language, is the giving over of one’s life to a society, namely that group which jointly participates in it. To use numbers is not to associate with the gods as the ancient Greeks thought. It is to associate with each other, to accept the judgments of each other, and to be recognised for furthering the implicit social project of the language of our association.
It strikes me as more than odd that Linnebo as well as the mathematicians he analyses simply ignore their own social dedication in their language of choice. The primary pragmatic consequence of mathematics is membership. Membership is achieved even in the failure to achieve any other result in mathematics. So, for example, Frege, and Bertrand Russell and even Gödel continued to ‘speak’ mathematics knowing that it is as ‘flawed’ as any other language. It may not be the key to understanding the universe; but it is nevertheless beautiful, intriguing, seductive, and it is most of all a way to spend a harmless afternoon with friends. Certainly better than watching the Super Bowl for those who prefer words to physical violence.
* This applies to the ‘dialects’ within mathematics as well. As Linnebo points out, the logic of Frege was very different from the logic of Kant for example.
July 23, 2022
I might flippantly describe this as a decent introduction to the philosophy of mathematics for people who already have a strong background in mathematics and the philosophy thereof. Each chapter is very short, and the book owes some of its brevity to assumptions that the reader is well acquainted with related subjects, notably formal logic notation. Some things are not given sufficient or sufficiently clear explanation for introductory purposes, like the difference between one theory and some subtly different theory. It is informative, but you will likely need to refer to supplementary sources from time to time.
The chapter structure is a little odd - Linnebo will discuss a particular school of thought in one chapter, then spend a few chapters on competing ideas before returning to explore more recent developments along similar lines.
Linnebo, of course, has his own allegiances within the philosophy of math. To his credit, he is clear about what they are - he is a Fregean, and emphatically rejects empiricism - and tries to affect neutrality when summarizing the arguments of his opponents. The extent to which he succeeds is debatable. I am not an empiricist, but I should note that I know one who thinks empiricism was sold short.
August 28, 2021
This book was very interesting and I did enjoy a lot of the book, but many of the concepts also went over my head. I feel like this will be a good book to return to in the future. I would definitely recommend this book to anyone who is interested in a harder read and is willing to stop quite often to stop and process the book.
May 5, 2020
This book radiates intelligence.
It has the clearest language of anything I've encountered in philosophy, and it manages to cover an extraordinary range of viewpoints in the short space between its covers. But it's only an appetiser, and the reader who wants to grasp the range of its contents must look further.
It has the clearest language of anything I've encountered in philosophy, and it manages to cover an extraordinary range of viewpoints in the short space between its covers. But it's only an appetiser, and the reader who wants to grasp the range of its contents must look further.
May 1, 2019
Good summary but completely ignores the writings of Wittgenstein and pragmatism arguing that mathematics is an invented wordgame though this argument can probably be seen through the sections on nominalism and formalism.
May 23, 2023
could have been more thorough in explicating views of intuitionists in math...
October 27, 2018
That book is exactly what it says it is: an excellent overview of the philosophy of mathematics with short chapters on all the main currents. That is going to be very hard to beat in just under 200 pages.
It is the latest and, in my opinion, the best volume in the "Princeton foundations of contemporary philosophy" series. While some of the books in the series were very good (the ones on physics and biology come to mind), a few were mediocre and undigestible (epistemology!).
The writing is clear, concise, never boring, and gives a prominent role to Frege and Godel.
It is the latest and, in my opinion, the best volume in the "Princeton foundations of contemporary philosophy" series. While some of the books in the series were very good (the ones on physics and biology come to mind), a few were mediocre and undigestible (epistemology!).
The writing is clear, concise, never boring, and gives a prominent role to Frege and Godel.
Want to read
February 10, 2020Math ch. 1-7
October 10, 2025
An introduction to the thing, for people who already know it.
Linnebo provides a general overview of the philosophy of mathematics, emphasizing positions on realism on object, formalization with regards to the Fregean approach, and mathematical structure. It will surprise few that Linnebo is a structuralist himself, which animates what areas he chose to focus on. And while I share the structuralist philosophy (in an in re sort close to Aristotelian Realism mediated by Naturalism a la the later Penelope Maddy) I still think these choices are rather blatant and a more nuanced introduction would have covered more that would justify a Platonist, Nominalist, Fictionalist, Intuitionist, whatever-ist views that the Structuralist school is weaker on.
Linnebo provides a general overview of the philosophy of mathematics, emphasizing positions on realism on object, formalization with regards to the Fregean approach, and mathematical structure. It will surprise few that Linnebo is a structuralist himself, which animates what areas he chose to focus on. And while I share the structuralist philosophy (in an in re sort close to Aristotelian Realism mediated by Naturalism a la the later Penelope Maddy) I still think these choices are rather blatant and a more nuanced introduction would have covered more that would justify a Platonist, Nominalist, Fictionalist, Intuitionist, whatever-ist views that the Structuralist school is weaker on.
January 20, 2021
Short and succinct
Great intro to phil of maths. Some parts were challenging for me despite my phil background, yet it was intriguing and well written. I think one has to have a love for maths to really get into phil of maths, so it may not be for you if you're not a fan of maths. The author was particularly good at focusing on the unique aspects of maths.
Great intro to phil of maths. Some parts were challenging for me despite my phil background, yet it was intriguing and well written. I think one has to have a love for maths to really get into phil of maths, so it may not be for you if you're not a fan of maths. The author was particularly good at focusing on the unique aspects of maths.
November 6, 2018
3.5/5.0
September 13, 2022
Pretty hardcore introduction, but I understood this in couple of days only, so it was pretty good. Anyways I have strong mathematics & philosophy background, so maybe this was easier because of that.
June 21, 2023
So lit
August 27, 2023
The very best introduction to the philosophy of mathematics that I read.
February 25, 2024
Great introduction to the subject.
May 10, 2022
Linnebo does a decent task of orienting one to the various mental models that have shaped mathematics' development over the years! Totally a resourceful read.
April 6, 2022
None of the books in this series that I have seen are introductory. They are for graduate students specializing in the field.
Linnebo spends little time on infinity. He is mostly concerned with an account of mathematics. Mathematics is an odd discipline. It seems to give knowledge of the world, but it is not an empirical science. He covers logicism, formalism, intuitionism and structuralism, which I have not seen before.
Linnebo spends little time on infinity. He is mostly concerned with an account of mathematics. Mathematics is an odd discipline. It seems to give knowledge of the world, but it is not an empirical science. He covers logicism, formalism, intuitionism and structuralism, which I have not seen before.
Read
June 18, 2020The book has been insightful as it let me better understand the links between Mathematics and Philosophy which are foundations for the Natural Sciences
September 23, 2023
3 years of groundwork comes to a nought.
February 1, 2025
So basically numbers don’t exist and all mathematical objects are a consequence of mathematical structures.
Displaying 1 - 20 of 20 reviews