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Lectures on the Philosophy of Mathematics
An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice.
In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.
In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.
352 pages, Paperback
Published March 9, 2021
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508 people want to read
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Displaying 1 - 12 of 12 reviews
July 22, 2025
I would not teach a first - or second - or nth - phil of math class without this text at my elbow.
So far, I've used it with (1) philosophy grad students and (2) high-school students who test at least four grades above age level and want to spend their summer learning logic eight hours a day. (No undergrads yet.)
With the younger students, there's one small effect I'm rueful about, and it's not, of course, that they sometimes encounter background material they're unfamiliar with. That happens whenever one reads something interesting, and good readers are intrigued, not dissuaded, by it. The one qualification, ironic rather than harmful, is that they don't fully appreciate what Hamkins has achieved with the exposition, because they haven't seen these things expounded poorly and at great length, as in the course of a typical undergrad and grad education in mathematics. But rather than trying to fix that by providing them with less elegant treatments, I'm leaving that to their future education.
To the publisher: If the gods are merciful, the second edition will spare us the School of Athens kitsch and, more importantly, the @#(%*&!#&$B high-gloss pages that make it impossible for students to take good notes without being initiated into the mysteries of blotting paper.
So far, I've used it with (1) philosophy grad students and (2) high-school students who test at least four grades above age level and want to spend their summer learning logic eight hours a day. (No undergrads yet.)
With the younger students, there's one small effect I'm rueful about, and it's not, of course, that they sometimes encounter background material they're unfamiliar with. That happens whenever one reads something interesting, and good readers are intrigued, not dissuaded, by it. The one qualification, ironic rather than harmful, is that they don't fully appreciate what Hamkins has achieved with the exposition, because they haven't seen these things expounded poorly and at great length, as in the course of a typical undergrad and grad education in mathematics. But rather than trying to fix that by providing them with less elegant treatments, I'm leaving that to their future education.
To the publisher: If the gods are merciful, the second edition will spare us the School of Athens kitsch and, more importantly, the @#(%*&!#&$B high-gloss pages that make it impossible for students to take good notes without being initiated into the mysteries of blotting paper.
August 16, 2024
Very extensive review of the main discoveries and their implications in current mathematics: from incompleteness to constructivsm.
You need a strong background to read this. As many of the mathematical concepts are not slowly/formally developed, but deeply explored. Perfect for mathematics students that want to understand the big implications of what they are studying.
It also made me laugh a couple of times :)
You need a strong background to read this. As many of the mathematical concepts are not slowly/formally developed, but deeply explored. Perfect for mathematics students that want to understand the big implications of what they are studying.
It also made me laugh a couple of times :)
March 27, 2025
Absolutely cracking read. Hamkins starts off by asking what a number is; and then takes on a journey through geometry, logic, computability, and set theory. Hamkins is a professional mathematician, which gives the book a relative pragmatic feeling: what do the concepts mean to a working research mathematician?
The book may require a reasonable level of mathematical sophistication on the reader's part.
To my surprise, it had jokes. In homage to Russell, the dedication is to all authors who do not dedicate their book to themselves
The book may require a reasonable level of mathematical sophistication on the reader's part.
To my surprise, it had jokes. In homage to Russell, the dedication is to all authors who do not dedicate their book to themselves
March 10, 2022
Not just for wonks, actually a digestible read for a more general, albeit committed, readership.
How does mathematics underlie the development of philosophical thought and indeed scholarship more broadly?
An eye-opening read!!
How does mathematics underlie the development of philosophical thought and indeed scholarship more broadly?
An eye-opening read!!
March 25, 2024
Tough read. I'd say I understood about 70% of it, but am planning to go back to read through the more difficult sections again (the chapters on Proof and Incompleteness in particular). I would recommend to anyone who yearns for philosophically-motivated explanations for mathematics.
I wish some explanations were longer. The author seemingly expects the reader to have a bunch of background knowledge for the topics he covers. This was frustrating, though I got used to it, and I had a good time interacting with ChatGPT to go over some proofs and concepts.
Some things I found novel and insightful...
- We seem to adopt a "strategic reading" of delta-epsilon assertions (in analysis proofs). Or rather, we tend to view these proofs as a sort of game (and I believe Terry Tao has discussed this approach to proving these types of theorems, though would have to look). The author speculates: "Perhaps because human evolution took place in a challenging environment of essentially game-theoretic human choices, with consequences for strategic failures, we seem to have an innate capacity for the strategic reasoning underlying these complex, alternating-quantifier mathematical assertions."
- One could imagine two different visions of computational power: a hierarchical vision, where more problems become decidable as the machines get more powerful, or a threshold vision, where machines only have to get sufficiently powerful to solve all the potentially decidable problems. The threshold vision happens to be correct..
- The hyperreal numbers isn't unique, i.e. there are multiple structures that satisfy the axioms for hyperreal numbers (and they are not isomorphic). But the differences between these structures happens to not be relevant to the proofs in nonstandard analysis.
If you find any of these topics stimulating, then you should read the book.
I also plan on watching the lectures: https://youtube.com/playlist?list=PLg...
I wish some explanations were longer. The author seemingly expects the reader to have a bunch of background knowledge for the topics he covers. This was frustrating, though I got used to it, and I had a good time interacting with ChatGPT to go over some proofs and concepts.
Some things I found novel and insightful...
- We seem to adopt a "strategic reading" of delta-epsilon assertions (in analysis proofs). Or rather, we tend to view these proofs as a sort of game (and I believe Terry Tao has discussed this approach to proving these types of theorems, though would have to look). The author speculates: "Perhaps because human evolution took place in a challenging environment of essentially game-theoretic human choices, with consequences for strategic failures, we seem to have an innate capacity for the strategic reasoning underlying these complex, alternating-quantifier mathematical assertions."
- One could imagine two different visions of computational power: a hierarchical vision, where more problems become decidable as the machines get more powerful, or a threshold vision, where machines only have to get sufficiently powerful to solve all the potentially decidable problems. The threshold vision happens to be correct..
- The hyperreal numbers isn't unique, i.e. there are multiple structures that satisfy the axioms for hyperreal numbers (and they are not isomorphic). But the differences between these structures happens to not be relevant to the proofs in nonstandard analysis.
If you find any of these topics stimulating, then you should read the book.
I also plan on watching the lectures: https://youtube.com/playlist?list=PLg...
June 28, 2023
Mathematics is such a rich subject standing just below philosophies mantle. Mathematicians have marvelled at the elegance of their models. They have wondered how mathematical objects so aptly corresponds to the behaviour and dynamics of the natural world. Some have said that mathematics is the language of the universe. Others have claimed that it is merely a functional class of objects derived from and for convention and practical utility. These deep questions concerning the ontology of mathematics is discussed, researched, and debated in the philosophy of the mathematics. Joel David Hamkins a legendary set theorist and philosopher of mathematics decides to take on the subject in his book Lectures on The Philosophy of Mathematics. He introduces the subject by partitioning it into different topics each given their own chapter. Each chapter starts with an abstract, which gives the reader an overview of the chapters content. The chapters end with exercises for the mathematically initiated and motivated. This book stands above all others in terms of the depth, breadth, and brilliance. One particular feature of the book that stands out to me is the preservation of mathematical rigor. It would be fair to say that Hamkins is writing a mathematicians philosophy of mathematics introductory text. His chapters on set theory and incompleteness are truly fascinating. They’re I was driven into the weeds of Transfinite hierarchies of sets whose sizes defy imagination. I was truly impressed by how the author was able to dispel content that was both complex and demanding in terms of perquisites in around 200 pages. Therefore, my final thoughts are that this is a fine and splendid introduction to the philosophy of mathematics from a set theorist at the height of his intellectual powers.
February 26, 2023
I went out looking for an introduction to Set Theory and the Philosophy of Math. Hamkins delivered.
As someone who enjoys math as a passtime, I went into the first few chapters with confidence and ease. As the subject matter became more dense, this book goes from introductory to specialist fairly promptly.
I wouldn’t say this is unattainable to the casual reader, but I would say that unless you are a committed individual or an academic, this book probably isn’t for you.
Very well structured and explained, I will revisit this again in the future.
As someone who enjoys math as a passtime, I went into the first few chapters with confidence and ease. As the subject matter became more dense, this book goes from introductory to specialist fairly promptly.
I wouldn’t say this is unattainable to the casual reader, but I would say that unless you are a committed individual or an academic, this book probably isn’t for you.
Very well structured and explained, I will revisit this again in the future.
September 3, 2023
Beautiful book for the mathematically inclined and curious. This is the kind of foundational content that one yearns to understand when first starting a maths undergrad, but which seldom forms a significant part of a mathematics curriculum.
Joel is an excellent writer and explainer, and we had a lively conversation topics in this book on the Paradigm podcast here https://youtu.be/563qSYUByak
It's also worth checking out Joel's Oxford University lectures that follow this book, which he's kindly published on Youtube.
Joel is an excellent writer and explainer, and we had a lively conversation topics in this book on the Paradigm podcast here https://youtu.be/563qSYUByak
It's also worth checking out Joel's Oxford University lectures that follow this book, which he's kindly published on Youtube.
August 15, 2023
Very good, but uneven in the background knowledge required of the reader
Want to read
October 14, 2023
Read
June 15, 2024Uma das coisas mais legais que eu já li.
Displaying 1 - 12 of 12 reviews