The philosophy of mathematics is the philosophical study of the concepts and methods of mathematics. It is concerned with the nature of numbers, geometric objects, and other mathematical concepts; it is concerned with their cognitive origins and with their application to reality. It addresses the validation of methods of mathematical inference. In particular, it deals with the logical problems associated with mathematical infinitude.
Among the sciences, mathematics has a unique relation to philosophy. Since antiquity, philosophers have envied, as the model of logical perfection, because of the clarity of its concepts and the certainty of its conclusions, and have therefore devoted much effort to explaining the nature of mathematics.
This study guide will recommend sources that provide an introduction to the major issues in the philosophy of mathematics, and the historically important views on these issues. Some familiarity with mathematics is a prerequisite for thinking about these issues. The book What is Mathematics?, by Richard Courant and Herbert Robbins, is a brilliant exposition of the topics and methods of modern mathematics. The book is intended for laymen, but none of the essence of the mathematics has been omitted; it is not a simple book, but it is rewarding.
Most philosophers have presented their views about mathematics in works on more general topics. The anthology Philosophy and Mathematics by Robert Baum ,contains selections on mathematics from most major western philosophers, from Plato through Mill. The selections include enough material to provide a context for each philosopher's views on mathematics, and Baum's introductory essays trace the philosophical influences on each thinker.
The most influential views have been those of Plato and Kant, and Baum has a section on each of them. Interested Objectivists may want to supplement Baum's section on Aristotle with a look at Thomas Heath's book, Mathematics in Aristotle. Baum's book also contains some modern essays, of which Max Black's "The Elusiveness of Sets," a criticism of the epistemology of set theorists, is worth reading.
Newton's theory of mechanics, and his invention of the integral and differential calculus in support of it, are among the greatest achievements in history. The central idea of limit is logically subtle (this subtlety is what makes Zeno's Achilles paradox perplexing), and Newton failed to treat limits rigorously. His detractors—most notably Berkeley—made much of this flaw. Cauchy, Weierstrass, and other 19th-century mathematicians developed a rigorous theory of limits, which provided an unassailable foundation for Newton's theory and is a cornerstone of modern mathematical analysis. This epistemological success story is well told in Carl Boyer's The History of the Calculus and Its Conceptual Development.
Another logical gem that is a central feature of modern mathematical analysis is the idea of a well-posed problem, which was introduced by the mathematician Jacques Hadamard. When a new mathematical problem is proposed, the first order of business for mathematicians is to establish that the problem has a solution, that it has only one solution, and that the solution depends in a reasonable way on the data (e.g., if the equation relates voltage to illumination in a light bulb, a tiny increase in voltage should result in only a small increase in illumination).
A problem that has these properties is called "well-posed." When mathematicians establish that a mathematical problem is well-posed, they are ensuring that it is a reasonable question to ask before they try to answer it. Investigators in many other fields would be well advised to adopt such careful epistemological habits. Unfortunately, there is no philosophical introduction to this topic.
The popular current view is that mathematics has passed through a series of logical or epistemological crises that have done it severe damage. For a history of these "crises" (e.g. the invention of non-euclidean geometry and the discovery of the set-theoretic paradoxes), and a thorough survey of the issues in modern mathematical philosophy, see Morris Kline's Mathematics: The Loss of Certainty. Kline was a mathematician; this book accurately reflects the sort of attitude that one encounters among practitioners, and it is well documented with pertinent mathematics.
To determine whether there are flaws in the foundations of a subject, one must first answer the more basic epistemological question of what constitutes a proper foundation. The Objectivist position that all knowledge must be grounded in perception, and grasped and organized conceptually, has played virtually no role in the historical development of the philosophy of mathematics. The primary task of an Objectivist approach is to ground mathematics objectively. An important secondary task is to explain how other epistemological presuppositions have brought about the sense of crisis and doubt that has characterized the field.
Stephan Korner's The Philosophy of Mathematics, an Introductory Essay, is a less historically and mathematically detailed treatment than Kline's, but it is more philosophically sophisticated. Korner dedicates two chapters apiece—one expository and one critical—to each of the three main modern schools of thought on mathematical philosophy: the formalists, the logicists, and the intuitionists. Korner's presentation is clear, concise and unbiased.
The logicist school, whose central figures are Bertrand Russell and Gottlob Frege, had as its purpose to "reduce mathematics to logic." Russell's Introduction to Mathematical Philosophy is a nontechnical introduction to the logicist program. The logicist conception of logic is radically different from the Objectivist, or more generally, the Aristotelian conception of logic; and it is a view of logic presupposed in most modern mathematical philosophy. Russell's Introduction is an exceptionally clear exposition of this conception of logic and its application to mathematics. It is valuable as a guide to the premises that an objective approach to the foundations of mathematics will have to challenge.
The works of Henry Veatch, notably Intentional Logic, criticize Russell's conception of logic from an Aristotelian perspective. Veatch argues from a tenet with which Objectivism agrees—that consciousness is intentional, that it is always of or about a world that exists and has identity independently of consciousness.
The formalist school was founded by the mathematician David Hilbert. Formalists seek to express mathematics as strictly formal logical systems, and to study them as such, without concern for their meaning. (This is in contrast to the logicists, who seek to establish the meaning of mathematical notions by defining them in terms of concepts of logic.) Their primary motivation was to justify the mathematics of infinite sets, which had been developed by Georg Cantor in the late 19th century. The formalists hoped to express the mathematics of infinite sets in such a system, and to establish the consistency of that system by finite methods. If they succeeded in this, they thought, they would have justified the use of infinite sets without having to address the thorny question of just what such sets are.
The formalist approach is explained and illustrated in Godel's Proof by Ernest Nagel and James Newman. This short book is a masterpiece in making sophisticated material accessible to non-experts. The book starts with an exposition of formalism, and concludes with a very readable outline of the proof of Kurt Godel's incompleteness theorem. This theorem showed, on the formalists' own terms, that their program was untenable.
The intuitionists, whose leader was the mathematician L.E.J. Brouwer, are best known for their conservatism regarding mathematical infinitude. They are opposed to the application of the law of excluded middle to statements involving mathematical infinitudes, as in a proof that takes the following form: either there is a number with the property P or there is not; if not, a consequence follows that is known to be false; therefore there exists a number with the property P. Such proofs do not tell us what the number in question is, or why it has the property. Constructive proofs, by contrast, do provide this information, and intuitionists require constructive proofs of mathematical theorems.
The intuitionists find their philosophical roots in Kant. Yet their caution regarding the infinite should appeal to Objectivists. Their position on the law of excluded middle may be interpreted as a demand that a statement be established as meaningful before the laws of logic are applied to it, a demand that Objectivism certainly endorses. Their insistence on constructive proofs may be seen as a means of specifying what is meant by the existence of a number.
Unfortunately, intuitionists are not always clear about the meaning and philosophical foundations of their positions; they attend to mathematical details at the expense of philosophical exposition. There is no introduction like Russell's or Nagel and Newman's. There are several pieces by intuitionists— Brouwer, Heyting and Dummett— in the collection Philosophy of Mathematics, Selected Readings, edited by Paul Benacerraf and Hilary Putnam. The introduction to this volume also contains a clear discussion of intuitionist principles.
A proper understanding of abstraction is a prerequisite for explaining mathematical concepts. Historical theories of mathematical concepts have tended to embody the worst aspects of historical theories of universals; Platonic realism, Kantian idealism, and extreme nominalism dominate the subject.
Ayn Rand's identification of the nature of universals and her analysis of the process of abstraction have much to contribute to the philosophy of mathematics. There is, however, no Objectivist literature on this topic. An indication of an Objectivist approach to the subject is given in the essay "The Cognitive Basis of Arithmetic" by David Ross. Comments by Ayn Rand on various mathematical topics are contained in the appendix to the 1990 edition of Introduction to Objectivist Epistemology.
Objectivism recognizes a deeper connection between mathematics and philosophy than advocates of other philosophies have imagined. According to Ayn Rand 's theory, the process of concept-formation involves the grasp of quantitative relationships among units and the omission of their specific measurements. It thus places mathematics at the core of human knowledge as a crucial element of the process of abstraction. This is a radical, new view of the role of mathematics in philosophy. As Leonard Peikoff has put it in Objectivism: The Philosophy of Ayn Rand:
Thus, an area that an Objectivist philosophy of mathematics must address is the meaning and structure of measurement in the measurement omission theory; this subfield of the philosophy of mathematics might be called the mathematics of philosophy. For the Objectivist view, see Rand's discussions in Introduction to Objectivist Epistemology, Peikoff's Objectivism: The Philosophy of Ayn Rand, and David Kelley's "A Theory of Abstraction.