An Introduction to the Philosophy of Mathematics is a 2012 textbook on the philosophy of mathematics by Mark Colyvan. It has a focus on issues in contemporary philosophy, such as the mathematical realism–anti-realism debate and the philosophical significance of mathematical practice, and largely skips over historical debates. It covers a range of topics in contemporary philosophy of mathematics including various forms of mathematical realism, the Quine–Putnam indispensability argument, mathematical fictionalism, mathematical explanation, the "unreasonable effectiveness of mathematics", paraconsistent mathematics, and the role of mathematical notation in the progress of mathematics. 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. Other aspects of the book that received praise were its coverage of mathematical explanation, its appeal to mathematicians and other non-philosophers, and its discussion questions and further readings, whilst its epilogue and short length received a more mixed reception.
Overview An Introduction to the Philosophy of Mathematics is a textbook on the philosophy of mathematics focusing on the issue of mathematical realism, i.e. the question of whether or not there are mathematical objects, and mathematical explanation. 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". As a result, the book focuses less on historical debates in the philosophy of mathematics than other similar textbooks and more on contemporary issues, including the philosophy of mathematical practice.
Summary The book has eight chapters and an epilogue with each chapter ending with a list of discussion questions and further readings. Chapter 1 briefly covers what Colyvan calls the "big isms" which dominated early 20th century philosophy of mathematics: logicism, formalism and intuitionism. It then turns to the philosophical issues raised by Paul Benacerraf in his papers "What is Mathematical Truth?" (1965) and "What Numbers Could Not Be" (1971).
Chapter 2 concerns the limits of mathematics and relevant constraining mathematical theorems. It discusses the Löwenheim–Skolem theorem and its connection with Cantor's theorem, including a proof of Cantor's theorem and an explanation of why the two theorems are not contradictory. It also discusses Gödel's incompleteness theorems and Gödel and Cohen's work on the independence of the continuum hypothesis. These results are used to motivate the debate between mathematical realism and anti-realism.
Following Hilary Putnam, Colyvan distinguishes between realism about mathematical truths and realism about mathematical objects in chapter 3. He argues that "‘it seems a very quick path from objective truth to objects" and so focuses subsequent discussion on realism about mathematical objects. The chapter goes on to distinguish between various types of realism, including full-blooded platonism, structuralism and the physicalist realism of writers such as Penelope Maddy, and to cover naturalism and the Quine–Putnam indispensability argument. Objections to the indispensability argument from Maddy, Hartry Field and Elliott Sober are also presented.
Chapter 4 focuses on mathematical anti-realism (aka nominalism), specifically mathematical fictionalism. It gives an introduction to the fictionalism of Hartry Field and his nominalisation program, which Colyvan calls the hard road to nominalism. Colyvan also covers so-called easy roads to nominalism; such views are "easy" because they do not attempt to remove mathematical entities from our best scientific theories as Field's nominalisation project attempts to. These include the fictionalism of Jody Azzouni and the metaphorical account of mathematical language propounded by Stephen Yablo. In the chapter, Colyvan objects to Yablo's views, claiming that mathematics appears in scientific explanations and that where metaphorical language is used in explanations, it is being used as a shorthand for a non-metaphorical explanation or else it must be interpreted literally.
Following on from this discussion, chapter 5 concerns mathematical explanation and the "explanatory turn" in the realism–anti-realism debate in which the indispensability argument was reframed in terms of the explanatory power of mathematics. It begins with a consideration of different theories of explanation, resulting in Colyvan advocating for a unification account. This is the view that explanations work by bringing multiple different phenomena under the same theoretical framework. He then distinguishes between intra-mathematical explanations, mathematical explanations of mathematical facts, and extra-mathematical explanations, mathematical explanations of non-mathematical, empirical facts. He uses the unification account of explanation to attempt to explain the difference between mathematical proofs that are explanatory and those that are not, citing proofs of Euclid's theorem, Rolle's theorem and the formula for the sum of the first n natural numbers as examples of non-explanatory proofs. He then moves on to extra-mathematical explanations, arguing that mathematics is more than just a descriptive tool and provides genuine explanations of empirical facts. 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.
Chapter 6 is about the applicability of mathematics and its "unreasonable effectiveness" when applied within science. 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. He attempts to explain this unreasonable effectiveness by providing a mapping account of mathematical application. According to this account, mathematical models act as maps of physical systems by abstracting away from particular details to more structural features of the system. In this way, the abstract structures of mathematics can be used to represent physical systems via similarity relations. Colyvan presents the case study of mathematical models in population ecology to illustrate this mapping account.
Chapter 7 explores issues surrounding paraconsistent mathematics and logic. Colyvan argues that mathematical theory can be inconsistent whilst still being useful, pointing to naive set theory and early infinitesimal calculus as examples of mathematical theories that were later proved to be inconsistent but were fruitfully worked on by mathematicians. He also proposes that the logic used by mathematicians must be some kind of contradiction-tolerant or paraconsistent logic rather than classical logic to account for this fact. He provides the Logic of Paradox (LP) as an example of such a paraconsistent logic which does not lead to the principle of explosion by using modified truth tables and a third truth value i which he suggests should be referred to as "true and false".
Chapter 8 is on the philosophical significance of mathematical notation. Colyvan argues that mathematical progress can sometimes be attributed to changes in notation. The chapter includes a number of examples to support this idea. For example, Colyvan says that the shift from Roman numerals to Arabic numerals could have prompted mathematical progress because Arabic numerals, unlike Roman numerals, have recursion built in. Another example provided is the relevance of mathematical notation in the proof of the impossibility of squaring the circle which is used to illustrate the idea that the same procedure being represented in different ways can reveal non-obvious connections within mathematics. The chapter also considers the importance of definition in mathematics using the example of the evolving definition of the term polyhedron.
The epilogue is titled "Desert Island Theorems" and contains a list of 20 important theorems and 5 open problems which Colyvan believes all philosophers of mathematics should know. There is also a two-page list of "interesting numbers". Short discussions on the philosophical importance and impact of each of these theorems, problems and numbers is also included after each item.
Reception The contemporary focus of the book was met with praise. Noah Friedman-Biglin, reviewing the book in Metascience, felt this feature of the book was "distinctive" and praised the coverage of mathematical explanation which he called "a topic which is attracting the interest of many professional philosophers of mathematics now". Richard Pettigrew, reviewing the book in The Bulletin of Symbolic Logic, felt that the book "really begins" at chapter 5 where it moves from a "fairly standard, if admirably clear, presentation of well-worn material" to "an exciting exploration of nascent topics on which there is still relatively little literature." He said that the book's use of examples whilst exploring these topics instead of fully formed arguments was a feature of the book that "will provoke [students] to formulate their own philosophical hypotheses and arguments more readily than more traditional textbooks." He concluded that "while some of the book might lack a little of the detail and rigour I'd like future students of the topic to value, Colyvan has written the first textbook that initiates the student into the current period in the philosophy of mathematics." 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 described chapter 5 as "[standing] out even from the other uniformly excellent chapters of this book" and gave specific praise to its coverage of extra-mathematical explanations for being "particularly interesting". Jean-Pierre Marquis characterised the book in Mathematical Reviews as "a warm breeze after a cold winter in the rarefied atmosphere of the philosophy of mathematics." 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". The contemporary focus of the book, as well as its use of actual mathematics, were also identified as interesting aspects of the book by Mark Hunacek, who reviewed the book for the Mathematical Association of America.