Meta-calculus


Book Description

This book describes systems of calculus, called meta-calculi, that arose from the problem of measuring stock-price performance when taking all intermediate prices into consideration. The meta-calculi provide mathematical tools for use in science, engineering, and mathematics. They appear to have potential for use as alternatives to the classical calculus of Newton and Leibniz. It may well be that they can be used to define new concepts, to yield new or simpler laws, or to formulate or solve problems.




Bigeometric Calculus


Book Description

This book contains a detailed account of the bigeometric calculus, a non-Newtonian calculus in which the power functions play the role that the linear functions play in the classical calculus of Newton and Leibniz. This nonlinear system provides mathematical tools for use in science, engineering, and mathematics. It appears to have considerable potential for use as an alternative to the classical calculus. It may well be that the bigeometric calculus can be used to define new concepts, to yield new or simpler laws, or to formulate or solve problems.







Pattern Calculus


Book Description

Over time, basic research tends to lead to specialization – increasingly narrow t- ics are addressed by increasingly focussed communities, publishing in increasingly con ned workshops and conferences, discussing increasingly incremental contri- tions. Already the community of programming languages is split into various s- communities addressing different aspects and paradigms (functional, imperative, relational, and object-oriented). Only a few people manage to maintain a broader view, and even fewer step back in order to gain an understanding about the basic principles, their interrelation, and their impact in a larger context. The pattern calculus is the result of a profound re-examination of a 50-year - velopment. It attempts to provide a unifying approach, bridging the gaps between different programming styles and paradigms according to a new slogan – compu- tion is pattern matching. It is the contribution of this book to systematically and elegantly present and evaluate the power of pattern matching as the guiding paradigm of programming. Patterns are dynamically generated, discovered, passed, applied, and automatically adapted, based on pattern matching and rewriting technology, which allows one to elegantly relate things as disparate as functions and data structures. Of course, pattern matching is not new. It underlies term rewriting – it is, for example, inc- porated in, typically functional, programming languages, like Standard ML – but it has never been pursued as the basis of a unifying framework for programming.







Formal Logic


Book Description

"Logic", one of the central words in Western intellectual history, compre hends in its meaning such diverse things as the Aristotelian syllogistic, the scholastic art of disputation, the transcendental logic of the Kantian critique, the dialectical logic of Hegel, and the mathematical logic of the Principia Mathematica of Whitehead and Russell. The term "Formal Logic", following Kant is generally used to distinguish formal logical reasonings, precisely as formal, from the remaining universal truths based on reason. (Cf. SCHOLZ, 1931). A text-book example of a formal-logical inference which from "Some men are philosophers" and "All philosophers are wise" concludes that "Some men are wise" is called formal, because the validity of this inference depends only on the form ofthe given sentences -in particular it does not depend on the truth or falsity of these sentences. (On the dependence of logic on natural language, English, for example, compare Section 1 and 8). The form of a sentence like "Some men are philosophers", is that which remains preserved when the given predicates, here "men" and "philosophers" are replaced by arbitrary ones. The form itself can thus be represented by replacing the given predicates by variables. Variables are signs devoid of meaning, which may serve merely to indicate the place where meaningful constants (here the predicates) are to be inserted. As variables we shall use - as did Aristotle - letters, say P, Q and R, as variables for predicates.




The First Nonlinear System of Differential and Integral Calculus


Book Description

The book contains a detailed account of the first non-Newtonian calculus. In this system, the exponential functions play the role that the linear functions play in the classical calculus of Newton and Leibniz. This nonlinear system provides mathematical tools for use in science, engineering, and mathematics. It appears to have considerable potential for use as an alternative to the classical calculus. It may well be that this non-Newtonian calculus can be used to define new concepts, to yield new or simpler laws, or to formulate or solve problems.




The World of Mathematics


Book Description




Godel's Proof


Book Description

In 1931 Kurt Gödel published his paper, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." Gödel's paper challenged certain basic assumptions underlying much research in mathematics and logic. However, few scholars were unable to understand Gödel's ideas. Ernest Nagel and James Newman provide a readable and accessible explanation of the main ideas and broad implications of Gödel's discovery.




Typed Lambda Calculi and Applications


Book Description

This book constitutes the refereed proceedings of the 4th International Conference on Typed Lambda Calculi and Applications, TLCA'99, held in L'Aquila, Italy in April 1999. The 25 revised full papers presented were carefully reviewed and selected from a total of 50 submissions. Also included are two invited demonstrations. The volume reports research results on various aspects of typed lambda calculi. Among the topics addressed are noncommutative logics, type theory, algebraic data types, logical calculi, abstract data types, and subtyping.