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Mathematics | Finite Versus Infinite - Contributions to an Eternal Dilemma

Finite Versus Infinite

Contributions to an Eternal Dilemma

Calude, Cristian S., Paun, Gheorghe

Softcover reprint of the original 1st ed. 2000, X, 374 p.

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The finite - infinite interplay is central in human thinking, from ancient philosophers and mathematicians (Zeno, Pythagoras), to modern mathe­ matics (Cantor, Hilbert) and computer science (Turing, Godel). Recent developments in mathematics and computer science suggest a) radically new answers to classical questions (e. g. , does infinity exist?, where does infinity come from?, how to reconcile the finiteness of the human brain with the infinity of ideas it produces?), b) new questions of debate (e. g. , what is the role played by randomness?, are computers capable of handling the infinity through unconventional media of computation?, how can one approximate efficiently the finite by the infinite and, conversely, the infinite by finite?). Distinguished authors from around the world, many of them architects of the mathematics and computer science for the new century, contribute to the volume. Papers are as varied as Professor Marcus' activity, to whom this volume is dedicated. They range from real analysis to DNA com­ puting, from linguistics to logic, from combinatorics on words to symbolic dynamics, from automata theory to geography, and so on, plus an incursion into the old history of conceptions about infinity and a list of philosophical "open problems". They are mainly mathematical and theoretical computer science texts, but not all of them are purely mathematical.

Content Level » Research

Keywords » Algebra - Automata Theory - Cantor - Finite - Mathematical Logic - Neural Computing - Problem-solving - combinatorics - complexity - databases - function - logic - mathematics

Related subjects » Artificial Intelligence - Mathematics - Theoretical Computer Science

Table of contents 

Smoothing Data: When Finite and Infinite Dimensional Help Each Other (Umberto Amato, Dan Tudor Vuza).- Rearrangements of Real Functions derived from the Combinatorics of Young Tableaux (Hari Bercovici).- A Possible Distance for Sequences of Equal and Unequal Length (Massimo Borelli, Andrea Sgarro).- Solving Finitely Refutable Mathematical Problems (Cristian Calude, Helmut Juergensen, Shane Legg).- State Complexity of Regular Languages: Finite versus Infinite (Cezar Campeanu, Kai Salmoaa, Sheng Yu).- A Century of Controversy over the Foundations of Mathematics (Gregory Chaitin).- Continuity, Discontinuity, and Negotiation of Meaning in Distributed Virtual Environments (Patrick Coppock).- Finite versus Infinite in Singularity Dynamics (Florin Diacu).- An Interesting Serendipitous Real Number (John Ewing, Ciprian Foias).- Algebraic Representations of Regular Array Languages (Rudolf Freund, Alexandru Mateescu, Arto Salomaa).- Rough Set Processing of Vague Information using Fuzzy Similarity Relations (Salvatore Greco, Benedetto Matarazzo, Roman Slowinski).- Relativised Code Concepts and Multi-Tube DNA Dictionaries (Tom Head).- Uniform Scattered Factors (Lucian Llie, Ion Petre, Grzegorz Rozenberg).- Pregroups: A New Algebraic Approach to Sentence Structure (Jim Lambek).- Splicing Normalization and Regularity (Vincenzo Manca).- DNA Pattern Multigrammars (Brian Mayoh).- Finite and Infinite in Geography (Georges Nicolas).- Ultrafilters, Dictators and Gods (Piergiorgio Odifreddi).- Mathematical Structures for the Chomsky Grammar of English (Luis Peris-Vine).- On the Behaviour of Semiautomata (Sergiu Rudeanu).- Infinity: An Outline of Conceptions in Mythology, Cosmology and Natural Philosophy (Jouko Seppanen).- Finite versus Infinite Neural Computation (Hava Sigeleman).- On Information-Theoretical Aspects of Relational Databases (Dan Simovici, Szymon Jaroszewicz).- A Version of Omega for which ZFC cannot Predict a Single Bit (Robert Solovay).- On the Power of Reading the WholeInfinite Input Tape (Ludwig Staiger).- Finite and Infinite in Logic (Victor Vianu).

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