''Proofs and Refutations: The Logic of Mathematical Discovery'' is a 1976 book by philosopher
Imre Lakatos
Imre Lakatos (, ; hu, Lakatos Imre ; 9 November 1922 – 2 February 1974) was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its "methodology of proofs and refutations" in its pr ...
expounding his view of the progress of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. The book is written as a series of
Socratic dialogue
Socratic dialogue ( grc, Σωκρατικὸς λόγος) is a genre of literary prose developed in Greece at the turn of the fourth century BC. The earliest ones are preserved in the works of Plato and Xenophon and all involve Socrates as the p ...
s involving a group of students who debate the proof of the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
defined for the
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on th ...
. A central theme is that
definition
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitio ...
s are not carved in stone, but often have to be patched up in the light of later insights, in particular failed
proofs. This gives mathematics a somewhat experimental flavour. At the end of the Introduction, Lakatos explains that his purpose is to challenge
formalism in mathematics, and to show that
informal mathematics
Informal mathematics, also called naïve mathematics, has historically been the predominant form of mathematics at most times and in most cultures, and is the subject of modern ethno-cultural studies of mathematics. The philosopher Imre Lakatos in ...
grows by a logic of "proofs and refutations".
Background
The 1976 book ''Proofs and Refutations'' is based on the first three chapters of his 1961 four-chapter doctoral thesis ''Essays in the Logic of Mathematical Discovery''. But its first chapter is Lakatos's own revision of its chapter 1 that was first published as ''Proofs and Refutations'' in four parts in 1963–4 in the ''
British Journal for the Philosophy of Science
''British Journal for the Philosophy of Science'' (''BJPS'') is a peer-reviewed, academic journal of philosophy, owned by the British Society for the Philosophy of Science (BSPS) and published by University of Chicago Press. The journal publishes ...
''.
Synopsis
Many important logical ideas are explained in the book. For example, the difference between a
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
to a
lemma (a so-called 'local counterexample') and a counterexample to the specific conjecture under attack (a 'global counterexample' to the Euler characteristic, in this case) is discussed.
Lakatos argues for a different kind of textbook, one that uses heuristic style. To the critics that say such a textbook would be too long, he replies: 'The answer to this pedestrian argument is: let us try.'
The book includes two appendices. In the first, Lakatos gives examples of the heuristic process in mathematical discovery. In the second, he contrasts the deductivist and heuristic approaches and provides heuristic analysis of some 'proof generated' concepts, including
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
,
bounded variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a conti ...
, and the
Carathéodory definition of a measurable set.
The pupils in the book are named after letters of the Greek alphabet.
Method
Though the book is written as a narrative, it aims to develop an actual method of investigation based upon "proofs and refutations". In Appendix I, Lakatos summarizes this method by the following list of stages:
# Primitive conjecture.
# Proof (a rough thought-experiment or argument, decomposing the primitive conjecture into subconjectures).
# "Global" counterexamples (counterexamples to the primitive conjecture) emerge.
# Proof re-examined: the "guilty lemma" to which the global counter-example is a "local" counterexample is spotted. This guilty lemma may have previously remained "hidden" or may have been misidentified. Now it is made explicit, and built into the primitive conjecture as a condition. The theorem - the improved conjecture - supersedes the primitive conjecture with the new proof-generated concept as its paramount new feature.
He goes on and gives further stages that might sometimes take place:
- Proofs of other theorems are examined to see if the newly found lemma or the new proof-generated concept occurs in them: this concept may be found lying at cross-roads of different proofs, and thus emerge as of basic importance.
- The hitherto accepted consequences of the original and now refuted conjecture are checked.
- Counterexamples are turned into new examples - new fields of inquiry open up.
Publication history
The 1976 book has been translated into more than 15 languages worldwide, including Chinese, Korean, Serbo-Croat and Turkish, and went into its second Chinese edition in 2007.
Impact on teaching
A number of mathematics teachers have implemented Lakatos' method of proofs and refutations in the classroom, when teaching other mathematical topics. The method has been applied to the analysis and presentation of problem solving in mechanics
by high school to college level students.
The
Mathematical Association of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...
has included this book on a list of books that they consider to be "essential for undergraduate mathematics libraries".
Notes
References
* & {{ISBN, 978-0-521-29038-8. John Worrall & Elie Zahar were the editors of this posthumous book.
*Gábor Kutrovátz
''Imre Lakatos’s Philosophy of Mathematics'' Eötvös Loránd University, 2005.
1976 non-fiction books
Science studies
Philosophy of science literature
Mathematics books
Dialogues