The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, ...and surreal numbers were inspired by the game of Go. Yet even with such groundbreaking findings and a wealth of popular-level books exploring puzzles and brainteasers, research in recreational mathematics has often been neglected.The Mathematics of Various Entertaining Subjectsbrings together authors from a variety of specialties to present fascinating problems and solutions in recreational mathematics.
Contributors to the book show how sophisticated mathematics can help construct mazes that look like famous people, how the analysis of crossword puzzles has much in common with understanding epidemics, and how the theory of electrical circuits is useful in understanding the classic Towers of Hanoi puzzle. The card game SET is related to the theory of error-correcting codes, and simple tic-tac-toe takes on a new life when played on an affine plane. Inspirations for the book's wealth of problems include board games, card tricks, fake coins, flexagons, pencil puzzles, poker, and so much more.
Looking at a plethora of eclectic games and puzzles,The Mathematics of Various Entertaining Subjectsis sure to entertain, challenge, and inspire academic mathematicians and avid math enthusiasts alike.
Is there really a God, and if so, what is God actually like? Is there an afterlife, and if so, is there such a thing as eternal punishment for unrepentant sinners, as many orthodox Christians and ...Muslims believe? And is it really true that our unconscious minds are connected to a higher spiritual reality, and if so, could this higher spiritual reality be the very same thing that religionists call God? In his latest book, Raymond M. Smullyan invites the reader to explore some beautiful and some horrible ideas related to religious and mystical thought. In Part One, Smullyan uses the writings on religion by fellow polymath Martin Gardner as the starting point for some inspired ideas about religion and belief. Part Two focuses on the doctrine of Hell and its justification, with Smullyan presenting powerful arguments on both sides of the controversy. If God asked you to vote on the retention or abolition of Hell, he asks, how would you vote? Smullyan has posed this question to many believers and received some surprising answers. In the last part of his treasurable triptych, Smullyan takes up the beautiful and inspiring ideas of Richard Bucke and Edward Carpenter on Cosmic Consciousness. Readers will delight in Smullyan's observations on religion and in his clear-eyed presentation of many new and startling ideas about this most wonderful product of human consciousness.
Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently ...undecidable. His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.
Truth and Provability Smullyan, Raymond M.
The Mathematical intelligencer,
03/2013, Letnik:
35, Številka:
1
Journal Article
Recenzirano
As is well known, in the first quarter of the twentieth century, there were two mathematical systems, Principia Mathematica, and Zermelo-Fraenkel Set Theory, which were so powerful that it was ...generally assumed that all mathematical questions could he decided--that is, either proved or disproved--in each of the systems. However, in 1933, the logician Kurt Godel startled the mathematical world by proving that this was not the case--that in each of these systems, as well as in a variety of related systems, there must be sentences that, though true, could not be proved within the systems. This celebrated result is known as Godel's Incompleteness Theorem. Closely related to this result is a theorem of the logician Alfred Tarski, roughly to the effect that in these systems truth of sentences of the system is not definable in the system. Tarski's theorem provides a proof of Godel's theorem, which in many ways is simpler than Gadels original proof. Here, Smullyan provides the essential ideas behind the proofs of the Godel and Tarski theorems.
This book serves both as a completely self-contained introduction and as an exposition of new results in the field of recursive function theory and its application to formal systems.
This book serves both as a completely self-contained introduction and as an exposition of new results in the field of recursive function theory and its application to formal systems.
In this chapter we lay the groundwork for the remainder of this study. The results of #A are needed for Chapter III; Section #B is needed for Chapter IV. The results of #C, though not actually needed ...for our further development of recursive function theory, are of some independent interest and are useful in certain metamathematical applications.
(a) Common Representability. Let (E1) and (E2) be 2 elementary formal systems over a common alphabet K. We shall call them Independent iff they contain no common predicates. By (E1) U (E2) we mean the E.F.S. over K whose axioms are those of
RECURSIVE FUNCTION THEORY Smullyan, Raymond M
Theory of Formal Systems. (AM-47),
03/2016, Letnik:
47
Book Chapter
In this chapter we present a connected development of recursive function theory from the viewpoint of elementary formal systems. Sections #A and #B are completely independent. The results of #A are ...fundamental for all of Chapter V. The reader desirous of quickly seeing the applications of Chapter III to mathematical logic (as developed in the supplement) might wish to read #B before #A.
In Chapter II we showed that certain operations (e.g., union, intersection, existential quantification, finite quantification, explicit transformations) preserve recursive enumerability. One of the principal aims of this sections is to define what it means for an operation
