From Combinatorics to Philosophy (eBook)

From Combinatorics to Philosophy 2
Foreword 5
List of talks 10
Contents 12
Part I Gian-Carlo Rota, the Man 17
Gianco, my Brother 18
Ester Gasperoni Rota 18
References 28
Remembering Gian-Carlo Rota 31
Joseph J. Kohn 31
References 35
Part II Gian-Carlo Rota, the Mathematician 36
A Glimpse of Vector Invariant Theory. The Points of View of Weyl, Rota, De Concini and Procesi, and Grosshans 37
Andrea Brini 37
Introduction 37
The Characteristic Zero Approach 39
Preliminaries 39
Weyl's Theorem (The Reduction Principle) 40
Absolute Invariants for the Special Linear Group SL(d) 41
Absolute Invariants for the Symplectic Group Sp2m 41
Absolute Invariants for the Special Orthogonal Group SO(d) 43
Absolute Invariants for the Orthogonal Group O(d) 43
The Straightening Formulae 44
Standard Young Tableaux 44
Bideterminants 45
Pfaffians 45
Gramians 46
The E. Pascal Theorems and the Cancellation Laws 47
The E. Pascal Theorem and the Cancellation Law for Scalar Products 48
The E. Pascal Theorem and the Cancellation Law for Symplectic Products 49
The E. Pascal Theorem and the Cancellation Law for Inner Products 50
The First and the Second Fundamental Theorems 51
The First Fundamental Theorems 51
The Second Fundamental Theorems 52
Grosshans's Theorem 52
References 54
Partitions of a Finite Partially Ordered Set 56
Pietro Codara 56
Introduction 56
Background, and Preliminary Results 57
Partitions as Sets of Fibres 61
Partitions as Partially Ordered Sets of Blocks 62
Partitions Induced by Quasiorders 66
Further Work 69
References 70
An Algebra of Pieces of Space --- Hermann Grassmann to Gian Carlo Rota 1
Henry Crapo 1
Synthetic Projective Geometry 72
Hermann Grassmann's Algebra 74
Extensors and Vectors 76
Reduced Forms 82
Grassmann-Cayley Algebra, Peano Spaces 83
Whitney Algebra 85
Geometric Product 88
Regressive Product 90
Higher Order Syzygies 96
Balls in Boxes 99
References 100
The Eleventh and Twelveth Problems of Rota's Fubini Lectures: from Cumulants to Free Probability Theory 101
Elvira Di Nardo and Domenico Senato 101
Prologue 101
The Classical Umbral Calculus 102
Generating Functions 104
Sequences of Binomial Type, Bell Umbrae and Poisson Processes 106
Cumulants 109
Singleton Umbra 110
Cumulant Umbra 111
Applications in Statistics 113
U-Statistics 114
Moments of Sampling Distributions 116
Products of Statistics 119
k-statistics 120
Fast Symbolic Computation of k-statistics 122
Sheppard's Corrections 124
Sheffer Sequences 127
Abel Polynomials 130
Quasi-free Cumulant Umbra 132
Boolean Cumulants 133
Free Cumulant Theory 134
References 137
Two Examples of Applied Universal Algebra 141
Joseph P. S. Kung 141
Prologue 141
The Bohnenblust--Spitzer Identity 142
Baxter Algebras 143
The Standard Baxter Algebra 144
A Precursor Identity 147
Abstract Random Variables 151
References 153
On the Euler Characteristic of Finite Distributive Lattices 154
Emanuele Munarini 154
Introduction 154
Posets and Distributive Lattices 155
Euler Characteristic 156
Uniform Distributions of the Characteristic 160
-uniform Lattices 160
Order-preserving Euler Characteristic 161
Rank -uniform Lattices 162
Pseudo-planar Distributive Lattices 163
Graphs and Hypergraphs 167
Dual Gödel Lattices 169
Tree maps 171
References 174
Rota, Probability, Algebra and Logic 176
Daniele Mundici 176
MV-algebraic States and De Finetti Coherence Criterion 176
MV-algebraic and C*-algebraic States After Elliott Classification 180
MV-algebraic -states and Carathéodory Probability Theory 185
References 189
A Symbolic Treatment of Abel Polynomials 191
Pasquale Petrullo 191
Introduction 191
Abel polynomials, Lagrange inversion formula and sequences of binomial type 193
Generalized Abel polynomials 195
Cumulants and convolutions 199
References 203
Part III Gian-Carlo Rota, the Philosopher 205
Ethics in Thought. Gian-Carlo Rota and Philosophy 206
Francesca Bonicalzi 206
Telling/Inventing the Truth 206
The Winding Streets, and the Straight and Wide Avenue of Precision 209
The Food of Philosophical Thought and the Medicine of Axiomatics 211
The Ethics of Knowledge and the Genesis of its Disciplines 213
References 213
Indiscrete Variations on Gian-Carlo Rota's Themes 217
Carlo Cellucci 217
Introduction 217
The Existence of Mathematical Objects 217
Irrelevance of the Existence of Mathematical Objects 217
Mathematical Objects as Hypotheses 219
Hypotheses vs. Fictions 221
Existence and Identity 221
The Inexhaustibility of Mathematical Objects 222
Definition in Mathematics 223
Definition, Description and Analysis of Concepts 223
Definition in Mathematics and in Philosophy 224
The Alleged Circularity of Definitions and Theorems 225
The Notion of Proof 225
Proof as the Opening up of Possibilities 226
Axiomatic Presentation and Gödel's Incompleteness Theorems 227
Are there Definitive Proofs? 227
Proof and Evidence 228
The Relation of Philosophy of Mathematics to Mathematics 230
The Descriptive Character of the Philosophy of Mathematics 230
The Need for a Drastic Overhaul of Logic 230
Rota's Place in the Philosophy of Mathematics 231
References 233
On the Courage Needed to Do Phenomenology. Rota and Analytic Philosophy 235
Albino Lanciani and Claudio Majolino 235
Twilight of an Idol 236
Three Senses of Phenomenology 240
Coda 246
References 246
Rota's Philosophical Insights 247
Massimo Mugnai 247
Introduction 247
What Kind of Reductionism? 247
Rota's Objectivism and Husserl's Naturalism 250
Beyond Classical Logic 251
Concluding Remarks 254
References 254
``A Minority View". Gian-Carlo Rota's Phenomenological Realism 256
Fabrizio Palombi 256
Considerations on the Problem of Realism 257
Phenomenological Realism 258
Fringe Phenomena and State-of-Mind 259
Psychologistic Misunderstandings 261
Descriptions, Not Prescriptions 263
References 264