Bøger af Craig Smorynski

The problems stimulate, the stories of their successful solutions inspire, and their applications are wide.The literature abounds with books dedicated to mathematical problems - collections of problems, hints on how to solve them, and even histories of the paths to the solutions of some famous ones.

"The binomial theorem is usually quite rightly considered as one of the mostimportant theorems in the whole of analysis." Thus wrote Bernard Bolzanoin 1816 in introducing the first correct proof of Newton's generalisation of acentury and a half earlier of a result familiar to us all from elementary algebra. Bolzano's appraisal may surprise the modern reader familiar only with the finite algebraic version of the Binomial Theorem involving positive integralexponents, and may also appear incongruous to one familiar with Newton'sseries for rational exponents. Yet his statement was a sound judgment back inthe day. Here the story of the Binomial Theorem is presented in all its glory,from the early days in India, the Moslem world, and China as an essentialtool for root extraction, through Newton's generalisation and its central rolein infinite series expansions in the 17th and 18th centuries, and to its rigorous foundation in the 19th.The exposition is well-organised and fairly complete with all the necessarydetails, yet still readable and understandable for those with a limited mathematical background, say at the Calculus level or just below that.The present book, with its many citations from the literature, will be ofinterest to anyone concerned with the history or foundations of mathematics.

Mathematics originates with intuition. But intuition alone can only go so far and formalism develops to handle the more difficult problems. Formalism, however, has its inherent dangers. There are three types of formalism. Type Iformalism, exemplified in the work of Euler, is basically heuristic reasoning, the use of familiar reasoning in areas where the reasoning might not or ought not apply. The results include startling successes, and also theorems admitting exceptions. Type II formalism, associated with names like Bolzano, Cauchy, and Weierstrass, attempts to clarify the situation by means of precise definitions of the terms used. Type III formalism, the axiomatic method, leaves the fundamental concepts undefined, but offers precise rules for their use. Such precision deserts intuition and one pays the price. Most dramatically, the formal definitions of Type II formalism allow for the construction of monsters -bizarre counterexamples that exhibit behaviour inconsistent with existing intuition. The initially repellant nature of these "monsters" leads to dissatisfaction that is only dispelled by their growing familiarity and applicability. The present book covers the history of formalism in mathematics from Euclid through the 20th century. It should be of interest to advancedmathematics students, anyone who teaches mathematics, and anyone generally interested in the foundation of mathematics.

This book introduces elementary probability through its history, eschewing the usual drill in favour of a discussion of the problems that shaped the field's development. Numerous excerpts from the literature, both from the pioneers in the field and its commentators, some given new English translations, pepper the exposition. First, for the reader without a background in the Calculus, itoffers a brief intuitive explanation of some of the concepts behind the notation occasionally used in the text, and, for those with a stronger background, it gives more detailed presentations of some of the more technical results discussed in the text.Special features include two appendices on the graphing calculator and on mathematical topics. The former begins with a short course on the use of the calculator to raise the reader up from the beginner to a more advanced level, and then finishes with some simulations of probabilistic experimentson the the calculator. The mathematical appendix likewise serves a dual purpose.The book should be accessible to anyone taking or about to take a course in the Calculus, and certainly is accessible to anyone who has already had such acourse. It should be of special interest to teachers, statisticians, or anyone who uses probability or is interested in the history of mathematics or sciencein general.

It discusses the evolution of the theorem and the concepts behind it, how the theorem relates to other fundamental results in calculus, and modern re-evaluations of its role in the standard calculus course. The mean value theorem is one of the central results of calculus.