Online research in philosophy

Bertrand Russell’s 1906 article ‘The Theory of Implication’ contains an algebraic weak completeness proof for classical propositional logic. Russell did not present it as such. We give an exposition of the proof and investigate Russell’s view of what he was about, whether he could have appreciated the proof for what it is, and why there is no parallel of the proof in Principia Mathematica.

The authors present the main ideas of the computer-assisted proof of Mischaikow and Mrozek that chaos is really present in the Lorenz equations. Methodological consequences of this proof are examined. It is shown that numerical calculations can constitute an essential part of mathematical proof not only in the discrete mathematics but also in the mathematics of continua.

The standard approach to what I call “proof-theoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of proof-theoretic semantics, this paper investigates in detail various notions of proof-theoretic validity and offers certain improvements of the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and validity concepts used in normalization theory. It is argued that these two sorts of concepts must be kept strictly apart.

This paper explores some of the benefits informal logic may have for the analysis of mathematical inference. It shows how Stephen Toulmin’s pioneering treatment of defeasible argumentation may be extended to cover the more complex structure of mathematical proof. Several common proof techniques are represented, including induction, proof by cases, and proof by contradiction. Affinities between the resulting system and Imre Lakatos’s discussion of mathematical proof are then explored.

'Numbers and Proofs' presents a gentle introduction to the notion of proof to give the reader an understanding of how to decipher others' proofs as well as construct their own. Useful methods of proof are illustrated in the context of studying problems concerning mainly numbers (real, rational, complex and integers). An indispensable guide to all students of mathematics. Each proof is preceded by a discussion which is intended to show the reader the kind of thoughts they might have before any attempt proof is made. Established proofs which the student is in a better position to follow then follow. Presented in the author's entertaining and informal style, and written to reflect the changing profile of students entering universities, this book will prove essential reading for all seeking an introduction to the notion of proof as well as giving a definitive guide to the more common forms. Stressing the importance of backing up "truths" found through experimentation, with logically sound and watertight arguments, it provides an ideal bridge to more complex undergraduate maths.

Certain anti-realisms about mathematics are distinguished by their taking proof rather than truth as the central concept in the account of the meaning of mathematical statements. This notion of proof which is meaning determining or canonical must be distinguished from a notion of demonstration as more generally conceived. This paper raises a set of objections to Dummett's characterisation of the notion via the notion of a normalised natural deduction proof. The main complaint is that Dummett's use of normalised natural deduction proofs relies on formalisation playing a role for which it is unfit. Instead I offer an alternative account which does not rely on formalisation and go on to examine the relation of proof to canonical proof, arguing that rather than requiring an explicit characterisation of canonical proofs we need to be more aware of the complexities of that relation.

A textbook on proof in mathematics, inspired by an Aristotelian point of view on mathematics and proof. It explains how to prove statements in mathematics, from evident premises. It deals with the proof of "all" statements, "some" statements, multiple quantifiers and mathematical induction.

Proof, Logic and Formalization addresses the various problems associated with finding a philosophically satisfying account of mathematical proof. It brings together many of the most notable figures currently writing on this issue in an attempt to explain why it is that mathematical proof is given prominence over other forms of mathematical justification. The difficulties that arise in accounts of proof range from the rightful role of logical inference and formalization to questions concerning the place of experience in proof and the possibility of eliminating impredictive reasoning from proof. Students and lecturers of philosophy, philosophy of logic, and philosophy of mathematics will find this to be essential reading. A companion volume entitled Proof and Logic in Mathematics is also available from Routledge.

Preface -- Introduction -- There is only one reality -- The ultimate perspective and the ultimate drama -- Proof #1: Science -- Proof #2: History -- Proof #3: Prophecy -- Proof #4: Supernatural -- Proof #5: Psychology -- Proof #6: Sociology -- Proof #7: Inerrancy -- Proof #8: Micro-science -- Proof #9: Logic -- Proof #10: The only provably -- Inerrant, complete system -- Why proof is important -- Personal iplications of proof.