Damiani / D'Antona / Marra From Combinatorics to Philosophy

The Legacy of G.-C. Rota
1. Auflage 2009
ISBN: 978-0-387-88753-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 262 Seiten

From Combinatorics to Philosophy: The Legacy of G. -C. Rota provides an assessment of G. -C. Rota's legacy to current international research issues in mathematics, philosophy and computer science. This volume includes chapters by leading researchers, as well as a number of invited research papers. Rota's legacy connects European and Italian research communities to the USA by providing inspiration to several generations of researchers in combinatorics, philosophy and computer science. From Combinatorics to Philosophy: The Legacy of G. -C. Rota is of valuable interest to research institutions and university libraries worldwide. This book is also designed for advanced-level students in mathematics, computer science, and philosophy.

Autoren/Hrsg.

Damiani, Ernesto

D'Antona, Ottavio

Marra, Vincenzo

Palombi, Fabrizio

Inhaltsverzeichnis

1;From Combinatorics to Philosophy;2
2;Foreword;5
3;List of talks;10
4;Contents;12
5;Part I Gian-Carlo Rota, the Man;17
5.1;Gianco, my Brother;18
5.1.1;Ester Gasperoni Rota;18
5.1.1.1;References;28
5.2;Remembering Gian-Carlo Rota;31
5.2.1;Joseph J. Kohn;31
5.2.1.1;References;35
6;Part II Gian-Carlo Rota, the Mathematician;36
6.1;A Glimpse of Vector Invariant Theory. The Points of View of Weyl, Rota, De Concini and Procesi, and Grosshans;37
6.1.1;Andrea Brini;37
6.1.1.1;Introduction;37
6.1.1.2;The Characteristic Zero Approach;39
6.1.1.2.1;Preliminaries;39
6.1.1.2.2;Weyl's Theorem (The Reduction Principle);40
6.1.1.2.3;Absolute Invariants for the Special Linear Group SL(d);41
6.1.1.2.4; Absolute Invariants for the Symplectic Group Sp2m;41
6.1.1.2.5;Absolute Invariants for the Special Orthogonal Group SO(d);43
6.1.1.2.6;Absolute Invariants for the Orthogonal Group O(d);43
6.1.1.3;The Straightening Formulae;44
6.1.1.3.1;Standard Young Tableaux;44
6.1.1.3.2;Bideterminants;45
6.1.1.3.3;Pfaffians;45
6.1.1.3.4;Gramians;46
6.1.1.4;The E. Pascal Theorems and the Cancellation Laws;47
6.1.1.4.1;The E. Pascal Theorem and the Cancellation Law for Scalar Products;48
6.1.1.4.2;The E. Pascal Theorem and the Cancellation Law for Symplectic Products;49
6.1.1.4.3;The E. Pascal Theorem and the Cancellation Law for Inner Products;50
6.1.1.5;The First and the Second Fundamental Theorems;51
6.1.1.5.1;The First Fundamental Theorems;51
6.1.1.5.2;The Second Fundamental Theorems;52
6.1.1.6;Grosshans's Theorem;52
6.1.1.7;References;54
6.1.2;Partitions of a Finite Partially Ordered Set;56
6.1.2.1;Pietro Codara;56
6.1.2.1.1;Introduction;56
6.1.2.1.2;Background, and Preliminary Results;57
6.1.2.1.3;Partitions as Sets of Fibres;61
6.1.2.1.4;Partitions as Partially Ordered Sets of Blocks;62
6.1.2.1.5;Partitions Induced by Quasiorders;66
6.1.2.1.6;Further Work;69
6.1.2.1.7;References;70
6.1.3;An Algebra of Pieces of Space --- Hermann Grassmann to Gian Carlo Rota ;1
6.1.3.1;Henry Crapo;1
6.1.3.1.1;Synthetic Projective Geometry;72
6.1.3.1.2;Hermann Grassmann's Algebra;74
6.1.3.1.3;Extensors and Vectors;76
6.1.3.1.4;Reduced Forms;82
6.1.3.1.5;Grassmann-Cayley Algebra, Peano Spaces;83
6.1.3.1.6;Whitney Algebra;85
6.1.3.1.7;Geometric Product;88
6.1.3.1.8;Regressive Product;90
6.1.3.1.9;Higher Order Syzygies;96
6.1.3.1.10;Balls in Boxes;99
6.1.3.1.11;References;100
6.1.4;The Eleventh and Twelveth Problems of Rota's Fubini Lectures: from Cumulants to Free Probability Theory;101
6.1.4.1;Elvira Di Nardo and Domenico Senato;101
6.1.4.1.1;Prologue;101
6.1.4.1.2;The Classical Umbral Calculus;102
6.1.4.1.2.1;Generating Functions;104
6.1.4.1.3;Sequences of Binomial Type, Bell Umbrae and Poisson Processes;106
6.1.4.1.4;Cumulants;109
6.1.4.1.4.1;Singleton Umbra;110
6.1.4.1.4.2;Cumulant Umbra;111
6.1.4.1.5;Applications in Statistics;113
6.1.4.1.5.1;U-Statistics;114
6.1.4.1.5.2;Moments of Sampling Distributions;116
6.1.4.1.5.3;Products of Statistics;119
6.1.4.1.5.4;k-statistics;120
6.1.4.1.5.5;Fast Symbolic Computation of k-statistics;122
6.1.4.1.5.6;Sheppard's Corrections;124
6.1.4.1.6;Sheffer Sequences;127
6.1.4.1.6.1;Abel Polynomials;130
6.1.4.1.6.2;Quasi-free Cumulant Umbra;132
6.1.4.1.6.3;Boolean Cumulants;133
6.1.4.1.7;Free Cumulant Theory;134
6.1.4.1.8;References;137
6.1.5;Two Examples of Applied Universal Algebra;141
6.1.5.1;Joseph P. S. Kung;141
6.1.5.1.1;Prologue;141
6.1.5.1.2;The Bohnenblust--Spitzer Identity;142
6.1.5.1.3;Baxter Algebras;143
6.1.5.1.4;The Standard Baxter Algebra;144
6.1.5.1.5;A Precursor Identity;147
6.1.5.1.6;Abstract Random Variables;151
6.1.5.1.7;References;153
6.1.6;On the Euler Characteristic of Finite Distributive Lattices;154
6.1.6.1;Emanuele Munarini;154
6.1.6.1.1;Introduction;154
6.1.6.1.2;Posets and Distributive Lattices;155
6.1.6.1.3;Euler Characteristic;156
6.1.6.1.4;Uniform Distributions of the Characteristic;160
6.1.6.1.4.1;-uniform Lattices;160
6.1.6.1.4.2;Order-preserving Euler Characteristic;161
6.1.6.1.4.3;Rank -uniform Lattices;162
6.1.6.1.5;Pseudo-planar Distributive Lattices;163
6.1.6.1.6;Graphs and Hypergraphs;167
6.1.6.1.7;Dual Gödel Lattices;169
6.1.6.1.8;Tree maps;171
6.1.6.1.9;References;174
6.1.7;Rota, Probability, Algebra and Logic;176
6.1.7.1;Daniele Mundici;176
6.1.7.1.1;MV-algebraic States and De Finetti Coherence Criterion;176
6.1.7.1.2;MV-algebraic and C*-algebraic States After Elliott Classification;180
6.1.7.1.3;MV-algebraic -states and Carathéodory Probability Theory;185
6.1.7.1.4;References;189
6.1.8;A Symbolic Treatment of Abel Polynomials;191
6.1.8.1;Pasquale Petrullo;191
6.1.8.1.1;Introduction;191
6.1.8.1.2;Abel polynomials, Lagrange inversion formula and sequences of binomial type;193
6.1.8.1.3;Generalized Abel polynomials;195
6.1.8.1.4;Cumulants and convolutions;199
6.1.8.1.5;References;203
7;Part III Gian-Carlo Rota, the Philosopher;205
7.1;Ethics in Thought. Gian-Carlo Rota and Philosophy;206
7.1.1;Francesca Bonicalzi;206
7.1.1.1;Telling/Inventing the Truth;206
7.1.1.2;The Winding Streets, and the Straight and Wide Avenue of Precision;209
7.1.1.3;The Food of Philosophical Thought and the Medicine of Axiomatics;211
7.1.1.4;The Ethics of Knowledge and the Genesis of its Disciplines;213
7.1.1.5;References;213
7.2;Indiscrete Variations on Gian-Carlo Rota's Themes;217
7.2.1;Carlo Cellucci;217
7.2.1.1;Introduction;217
7.2.1.2;The Existence of Mathematical Objects;217
7.2.1.2.1;Irrelevance of the Existence of Mathematical Objects;217
7.2.1.2.2;Mathematical Objects as Hypotheses;219
7.2.1.2.3;Hypotheses vs. Fictions;221
7.2.1.2.4;Existence and Identity;221
7.2.1.2.5;The Inexhaustibility of Mathematical Objects;222
7.2.1.3;Definition in Mathematics;223
7.2.1.3.1;Definition, Description and Analysis of Concepts;223
7.2.1.3.2;Definition in Mathematics and in Philosophy;224
7.2.1.3.3;The Alleged Circularity of Definitions and Theorems;225
7.2.1.4;The Notion of Proof;225
7.2.1.4.1;Proof as the Opening up of Possibilities;226
7.2.1.4.2;Axiomatic Presentation and Gödel's Incompleteness Theorems;227
7.2.1.4.3;Are there Definitive Proofs?;227
7.2.1.4.4;Proof and Evidence;228
7.2.1.5;The Relation of Philosophy of Mathematics to Mathematics;230
7.2.1.5.1;The Descriptive Character of the Philosophy of Mathematics;230
7.2.1.5.2;The Need for a Drastic Overhaul of Logic;230
7.2.1.6;Rota's Place in the Philosophy of Mathematics;231
7.2.1.7;References;233
7.3;On the Courage Needed to Do Phenomenology. Rota and Analytic Philosophy;235
7.3.1;Albino Lanciani and Claudio Majolino;235
7.3.1.1;Twilight of an Idol;236
7.3.1.2;Three Senses of Phenomenology;240
7.3.1.3;Coda;246
7.3.1.4;References;246
7.4;Rota's Philosophical Insights;247
7.4.1;Massimo Mugnai;247
7.4.1.1;Introduction;247
7.4.1.2;What Kind of Reductionism?;247
7.4.1.3;Rota's Objectivism and Husserl's Naturalism;250
7.4.1.4;Beyond Classical Logic;251
7.4.1.5;Concluding Remarks;254
7.4.1.6;References;254
7.5;``A Minority View". Gian-Carlo Rota's Phenomenological Realism;256
7.5.1;Fabrizio Palombi;256
7.5.1.1;Considerations on the Problem of Realism;257
7.5.1.2;Phenomenological Realism;258
7.5.1.3;Fringe Phenomena and State-of-Mind;259
7.5.1.4;Psychologistic Misunderstandings;261
7.5.1.5;Descriptions, Not Prescriptions;263
7.5.1.6;References;264