The book was praised as accessible and well-written and the reaction to its contemporary focus was largely positive, although some academic reviewers felt that it should have covered the historical debates over logicism, formalism and intuitionism in more detail.
[1][2] Colyvan described his intention for the book as being a textbook that "[gets] beyond the first half of the twentieth century and [explores] the issues capturing the attention of contemporary philosophers of mathematics".
[1][3] Chapter 1 briefly covers what Colyvan calls the "big isms" which dominated early 20th century philosophy of mathematics: logicism, formalism and intuitionism.
[9] It gives an introduction to the fictionalism of Hartry Field and his nominalisation program, which Colyvan calls the hard road to nominalism.
[1] These include the fictionalism of Jody Azzouni and the metaphorical account of mathematical language propounded by Stephen Yablo.
[7] He presents the examples of the mathematical explanations of the life cycles of periodical cicadas, why hive-bee honeycomb has a hexagonal structure, the distribution of asteroids across the solar system, and Lorentz contraction to support his argument.
[2][5] To illustrate the unreasonable effectiveness of mathematics, Colyvan writes about how James Clerk Maxwell formulated the Maxwell–Ampère law as an analogue of Newtonian gravitational theory, but that it produced completely novel predictions that ended up being confirmed.
[8] Colyvan presents the case study of mathematical models in population ecology to illustrate this mapping account.
"[2] In a review in Teaching Philosophy, Carl Wagner said that "Colyvan has a real talent for conveying the excitement of these ongoing debates, and encouraging readers to develop their own views on these issues".
He argued that "[f]or too long now, the field has been frozen in the age of formalism, logicism and intuitionism" and that with regards to its goal of presenting more contemporary material, the book was "a splendid success".
He said that the book's "knack for jumping right to the heart of the issue" meant that it was "[n]ever overwhelming" and concluded that it was "a pleasure to teach from and [that he could] report that students having their first exposure to topics in the philosophy of mathematics have found it to be both accessible and stimulating.
"[4] Zach Weber said of the book in the Australasian Journal of Philosophy that "Colyvan has condensed his own body of research into a highly accessible textbook.
He also said that the book "covers a surprisingly wide range of topics" which he felt increased its utility in creating courses with different focuses.
[1] Friedman-Biglin felt that "students will find this book an excellent place to begin studying philosophy of mathematics, and it could easily serve as the basis for an interesting course for undergraduates.
"[9] Weber felt that the book's "conversational style and brisk pacing" made it "clearly designed for a lively undergraduate course".
"[7] Soto similarly recommended the book for mathematicians and scientists as well as philosophers, saying that it provided an "insightful guide to debates that encompass their areas".
Irvine felt that "If there is a weakness with the book, it is that the traditional debates over logicism, formalism, and intuitionism are covered in less than half a dozen pages, leaving readers wondering what all the fuss was about."
"[1] In contrast to these comments, Wagner called Colyvan's coverage of the "big isms" in chapter 1 "a masterly piece of compressed exposition".
[7] Pettigrew commented on the lack of coverage of category theory, reverse mathematics, and automated reasoning and computer-aided proofs but went on to say "no textbook can cover all topics, and one might feel that these belong more naturally to a more advanced course in the subject.
However, he felt that concerns about the lack of mathematical details in some areas do not "carry much weight" as "the main line of argument in Colyvan's book might have been obscured by including too many formalisms".