The book focuses on three developments in the field in the late 19th and early 20th centuries: logicism, intuitionism and finitism and their attempts to provide a coherent philosophical foundation for math.
"I hope [the book] will kindle, or rekindle, an interest in these three great attempts to tame mathematics and also lead to a sharp and sophisticated understanding of the subtleties, philosophical and mathematical, that attend them," said George.
The book consists of chapters alternating between the philosophical motivation for a theory in the field and the development of mathematical constructs to support each theory. While it is a study of past systems and does not advance a new theory or advocate one system over another, the book takes a novel approach to the subject that unites the philosophy and the math more fundamentally than other works in the field have previously.
"The book presents these various views, showing to what extent they work and to what extent they have problems," said Velleman. "But it doesn't come to some conclusion that says this one is the answer."
George and Velleman are currently co-teaching Philosophy 50: Philosophy of Mathematics, using the text as a supplement for the lectures, though they intend the book to reach a broader audience than only those new to the subject.
"I hope it will be of interest to students of all ages who are trying to understand some of the issues and details behind the three main projects in the philosophy of mathematics discussed in the book," said George.
"It is an introduction, so intended to be accessible to anyone (who has had some logic), but an opinionated introduction, so ideally of some interest even to those who already know something of the material covered," he added.
The philosophy of mathematics has undergone criticism from those who don't believe math is in need of philosophical contemplation, due to its seemingly intuitive and primary status in the sciences. However, in their book, George and Velleman point to the fact that questions as simple as "What is the number one?" receive different answers from different mathematicians and philosophers, as evidence that the lack of controversy in the field is only indicative of the lack of dialogue.
"Thinking about mathematics allows many philosophical questions about the nature of knowledge, thought and meaning to be raised in a very sharp form," said George. "Far from being esoteric, as philosophical subjects go, the philosophy of mathematics is quite centrally connected to vast and powerful currents in the history of philosophy. Virtually no philosopher worth his salt has failed to take a crack at figuring out what mathematics is about."
Requiring either George's introductory course on logic (Philosophy 13) or Velleman's course on mathematical logic (Math 34), Philosophy 50 has only been offered four times in the past 20 years at the College, due to the scheduling difficulties inherent in a co-taught class.
Many students in this year's class have found that the interrelation between the text and the lectures allows them a deeper understanding of the often subtle and complex subject matter.
"Everything in math builds on previous work and it's important to see how people in the past have attacked certain projects and where that's gotten them," said Michael Stevens '03, a student in Philosophy 50. "The book works just as well as the lectures in making the concepts accessible."
Velleman and George's professional relationship started in the late '80s and the two have collaborated on two articles in the field-"Two conceptions of natural number" and "Leveling the playing field between time and machine."
"[Velleman] is ... very philosophically minded; far more so than many philosophers I know," said George. "It wasn't long before we decided to teach a course together. And it was then we noticed a lacuna, which we eventually set out to remedy."
Velleman was hired as an assistant professor at the College in 1983, after graduating summa cum laude from Dartmouth and earning his M.A. and Ph.D. in mathematics at the University of Wisconsin-Madison, specializing in mathematical logic and set theory, with a minor in philosophy. When George arrived at the College in 1988, after graduating summa cum laude from Columbia University, with an M.A. and Ph.D. in philosophy, specializing in logic, from Harvard University, their collaboration was a natural one.
"[Professor George] is very careful about getting everything exactly right; he's not someone who is satisfied with something that is sort of roughly the right idea," said Velleman. "I guess that's the way I am, too; it's the nature of mathematics ... mathematicians are always after precision. That may have made us compatible: that he's mathematical about his philosophy."
Velleman teaches a course next semester on set theory, which requires several courses in mathematics, in which he will further develop one of the book's early emphases: Zermelo-Frankel Set Theory (ZF). ZF is a system motivated by Gottlob Frege, the founder of logicism, which attempts to describe all of mathematics in the form of logic. "ZF really works-you can derive everything from it-that's the positive, but-partially through working with Professor George-I have become less satisfied with set theory as a foundation for mathematics," said Velleman.
The main challenge to ZF-Intuitionism-challenges some of math's most basic assumptions, including the Platonic perception of infinity. This challenge casts doubt on a great many of math's accepted theorems by questioning whether math-and especially ZF-has given sufficient proof for its axioms.
"[Intuitionists] have this view on how math should be done that's different from all of the math you've ever studied; it's a minority viewpoint not taught in most math departments," said Velleman. "If intuitionists could convince people that they're right, we'd have to change the whole math curriculum."
The closing chapters of the book consider finitism and David Hilbert's efforts (using Hilbert's Program) to reconcile classical mathematics with intuitionism, which most scholars of the philosophy of math consider a proven failure as a result of Godel's incompleteness theorems.
The course follows a similar path as each of the systems and each challenge to each system is elaborated philosophically by George, followed by Velleman's mathematical work, followed by George's raising of doubts, ad infinitum.
"I think the book is very clear; it's distinct of the professors' methods-very precise and lucid in their discussion of the methods and the concepts," said Mihailis Diamantis'04, a student in Philosophy 50.
"I think it's great to have the balance between [math and philosophy]. They play off each other very well, like when Professor George casts doubt on Velleman's seemingly flawless mathematics with philosophical objections [to a premise]," Diamantis added.
George and Velleman re-address philosophy of math questions that have been left unanswered for more than a century. While they don't provide conclusive answers to these questions, they detail provocative suggestions in an attempt to ignite controversy in a mathematical community, including the one at the College, which often dismisses philosophical considerations in favor of purely mathematical work. "I think that [questions in the philosophy of math] are still unresolved, but the attitude of most mathematicians is that they can do their work without them being settled," said Velleman. "We're looking at a lot of very basic questions-Frege wanted to know what the number one is-it's not as if the answer he was going to give was going to change the way people do arithmetic, it's not as if mathematicians who are not logicians can't go forward without these questions being answered."