Proofs and Refutations: The Logic of Mathematical Discovery

Book Preface

Proofs and Refutations is one of the undeniable classics of the philosophy of mathematics. Fifty years have passed since the publication of the articles that make up its central core, but the book has lost neither its freshness nor its provocative vitality. It takes the form of a classroom dialogue in which a group of students and their teacher investigate the problem of whether there is a relation that holds between the number of vertices V, the number of edges E, and the number of faces F of regular polyhedra (e.g. the five Platonic solids). At the outset of the dialogues they have arrived at the formula V – E + F = 2. They conjecture that the result might extend to any polyhedron (Euler’s conjecture), and this is the starting point of a riveting development that carries the reader through the rational reconstruction, as embodied in the class dialogue, of the history of Euler’s conjecture, culminating in Poincaré’s proof. The reconstruction, in strong contrast to a piece of axiomatic mathematics, features contradictions, monsters, counterexamples, conjectures, concept-stretchings, hidden lemmas, proofs, and a wide range of informal moves meant to account for the rationality of the process leading to concept-formation and conjectures/proofs in mathematical practice.

Yet Euler’s conjecture is just a case-study displaying Lakatos’s highly original approach to the philosophy of mathematics. A starker contrast with the formalist foundational approach dominant up to the 1960s (and embodied in philosophies of mathematics of neo-positivist inspiration) can scarcely be imagined. Whereas the latter, inspired by Euclid’s infallibilist dogmatic style, thought of mathematical theories statically as axiomatic systems, Lakatos was after an account of informal mathematics as a
fallible dynamic body of knowledge. Rejecting the positivist distinction between context of discovery and context of justification, he claimed that mathematical practice and its history are not the domain of the irrational but rather display an objectivity and rationality that any philosophy of mathematics worth its name should account for. The tools for addressing the rationality of mathematical growth could not, however, be those of formal logic, whose ‘deductivist style’ could only address issues of the static variety and was thus unable to account for concept-formation and the rational dynamics driving the development of informal mathematics. Rather, Lakatos found inspiration in Polya’s work on mathematical heuristics, Hegel’s dialectic, and Popperian conjecture and refutation. This led to Lakatos’s dialectical methodology, a ‘heuristic style’ that reveals the struggle and the adventure of mathematical creation.

There will always be disagreements as to whether or to what extent Lakatos’s case studies are paradigmatic and can be extended to mathematics as a whole. Scholars will also continue to disagree about the suitability of the dialectical framework for accounting for mathematical growth and the role of mathematical logic in the history and philosophy of mathematics. But the characteristic trait of a classic is its rich and varied legacy. Proofs and Refutations stands up to this test, for it continues to be a source of inspiration to many historians, mathematicians, and philosophers who aspire to develop a philosophy of mathematics that does justice to the static and dynamic complexity of mathematical practice