Ebook: Finite Versus Infinite: Contributions to an Eternal Dilemma

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.

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The finite-infinite interplay is central in the human thinking, from ancient philosophers and mathematicians (Zenon, Pythagoras), to modern mathematics (Cantor, Hilbert) and computer science (Turing, Goedel). Recent developments in mathematics and computer science suggest radically new answers to classical questions such as:
Does infinity exist?
Where does infinity come from?
How can we reconcile the finiteness of the human brain with the infinity of ideas it produces?
Well-known authors from around the world, many of them architects of the mathematics and computer science for the new century, contribute to the volume. While mathematical in spirit, contributions have many connections with computer science, cognitive science, linguistics, philosophy, physics, biology and semiotics.

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The finite-infinite interplay is central in the human thinking, from ancient philosophers and mathematicians (Zenon, Pythagoras), to modern mathematics (Cantor, Hilbert) and computer science (Turing, Goedel). Recent developments in mathematics and computer science suggest radically new answers to classical questions such as:
Does infinity exist?
Where does infinity come from?
How can we reconcile the finiteness of the human brain with the infinity of ideas it produces?
Well-known authors from around the world, many of them architects of the mathematics and computer science for the new century, contribute to the volume. While mathematical in spirit, contributions have many connections with computer science, cognitive science, linguistics, philosophy, physics, biology and semiotics.

Content:
Front Matter....Pages i-x
Smoothing Data: When Finite and Infinite Dimensional Help Each Other....Pages 1-16
Rearrangements of Real Functions Derived from the Combinatorics of Young Tableaux....Pages 17-26
A Possibilistic Distance for Sequences of Equal and Unequal Length....Pages 27-38
Solving Problems with Finite Test Sets....Pages 39-52
State Complexity of Regular Languages: Finite versus Infinite....Pages 53-73
A Century of Controversy over the Foundations of Mathematics....Pages 75-100
Finite Versus Infinite in Singularity Dynamics....Pages 101-117
An Interesting Serendipitous Real Number....Pages 119-126
Algebraic Representations of Regular Array Languages....Pages 127-148
Rough Set Processing of Vague Information Using Fuzzy Similarity Relations....Pages 149-173
Relativized Code Concepts and Multi-Tube DNA Dictionaries....Pages 175-186
Uniformly Scattered Factors....Pages 187-198
Splicing Normalization and Regularity....Pages 199-215
Infinitely Many Infinities....Pages 217-230
Finite and Infinite in Geography....Pages 231-237
Ultrafilters, Dictators, and Gods....Pages 239-245
On the Behaviour of Semiautomata....Pages 247-256
Infinity — An Outline of Conceptions in Mythology, Cosmology and Natural Philosophy....Pages 257-283
Finite Versus Infinite Neural Computation....Pages 285-299
On Information-Theoretical Aspects of Relational Databases....Pages 301-322
A Version of ? for which ZFC Cannot Predict a Single Bit....Pages 323-334
On the Power of Reading the Whole Infinite Input Tape....Pages 335-348
Finite and Infinite in Logic....Pages 349-371
Back Matter....Pages 373-374

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The finite-infinite interplay is central in the human thinking, from ancient philosophers and mathematicians (Zenon, Pythagoras), to modern mathematics (Cantor, Hilbert) and computer science (Turing, Goedel). Recent developments in mathematics and computer science suggest radically new answers to classical questions such as:
Does infinity exist?
Where does infinity come from?
How can we reconcile the finiteness of the human brain with the infinity of ideas it produces?
Well-known authors from around the world, many of them architects of the mathematics and computer science for the new century, contribute to the volume. While mathematical in spirit, contributions have many connections with computer science, cognitive science, linguistics, philosophy, physics, biology and semiotics.

Content:
Front Matter....Pages i-x
Smoothing Data: When Finite and Infinite Dimensional Help Each Other....Pages 1-16
Rearrangements of Real Functions Derived from the Combinatorics of Young Tableaux....Pages 17-26
A Possibilistic Distance for Sequences of Equal and Unequal Length....Pages 27-38
Solving Problems with Finite Test Sets....Pages 39-52
State Complexity of Regular Languages: Finite versus Infinite....Pages 53-73
A Century of Controversy over the Foundations of Mathematics....Pages 75-100
Finite Versus Infinite in Singularity Dynamics....Pages 101-117
An Interesting Serendipitous Real Number....Pages 119-126
Algebraic Representations of Regular Array Languages....Pages 127-148
Rough Set Processing of Vague Information Using Fuzzy Similarity Relations....Pages 149-173
Relativized Code Concepts and Multi-Tube DNA Dictionaries....Pages 175-186
Uniformly Scattered Factors....Pages 187-198
Splicing Normalization and Regularity....Pages 199-215
Infinitely Many Infinities....Pages 217-230
Finite and Infinite in Geography....Pages 231-237
Ultrafilters, Dictators, and Gods....Pages 239-245
On the Behaviour of Semiautomata....Pages 247-256
Infinity — An Outline of Conceptions in Mythology, Cosmology and Natural Philosophy....Pages 257-283
Finite Versus Infinite Neural Computation....Pages 285-299
On Information-Theoretical Aspects of Relational Databases....Pages 301-322
A Version of ? for which ZFC Cannot Predict a Single Bit....Pages 323-334
On the Power of Reading the Whole Infinite Input Tape....Pages 335-348
Finite and Infinite in Logic....Pages 349-371
Back Matter....Pages 373-374
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