| Gian-Carlo Rota, the Man | |
| Gianco, my Brother | p. 3 |
| References | p. 13 |
| Remembering Gian-Carlo Rota | p. 17 |
| References | p. 21 |
| Gian-Carlo Rota, the Mathematician | |
| A Glimpse of Vector Invariant Theory. The Points of View of Weyl, Rota, De Concini and Procesi, and Grosshans | p. 25 |
| Introduction | p. 25 |
| The Characteristic Zero Approach | p. 27 |
| Preliminaries | p. 27 |
| Weyl's Theorem (The Reduction Principle) | p. 28 |
| Absolute Invariants for the Special Linear Group S L(d) | p. 29 |
| Absolute Invariants for the Symplectic Group S p2m | p. 29 |
| Absolute Invariants for the Special Orthogonal Group S O(d) | p. 31 |
| Absolute Invariants for the Orthogonal Group O(d) | p. 31 |
| The Straightening Formulae | p. 32 |
| Standard Young Tableaux | p. 32 |
| Bideterminants | p. 33 |
| Pfaffians | p. 33 |
| Gramians | p. 34 |
| The E. Pascal Theorems and the Cancellation Laws | p. 35 |
| The E. Pascal Theorem and the Cancellation Law for Scalar Products | p. 36 |
| The E. Pascal Theorem and the Cancellation Law for Symplectic Products | p. 37 |
| The E. Pascal Theorem and the Cancellation Law for Inner Products | p. 38 |
| The First and the Second Fundamental Theorems | p. 39 |
| The First Fundamental Theorems | p. 39 |
| The Second Fundamental Theorems | p. 40 |
| Grosshans's Theorem | p. 40 |
| References | p. 42 |
| Partitions of a Finite Partially Ordered Set | p. 45 |
| Introduction | p. 45 |
| Background, and Preliminary Results | p. 46 |
| Partitions as Sets of Fibres | p. 50 |
| Partitions as Partially Ordered Sets of Blocks | p. 51 |
| Partitions Induced by Quasiorders | p. 55 |
| Further Work | p. 58 |
| References | p. 59 |
| An Algebra of Pieces of Space - Hermann Grassmann to Gian Carlo Rota | p. 61 |
| Almost Ten Years Later | p. 61 |
| Synthetic Projective Geometry | p. 62 |
| Hermann Grassmann's Algebra | p. 64 |
| Extensors and Vectors | p. 66 |
| Reduced Forms | p. 72 |
| Grassmann-Cayley Algebra, Peano Spaces | p. 73 |
| Whitney Algebra | p. 75 |
| Geometric Product | p. 78 |
| Regressive Product | p. 80 |
| Higher Order Syzygies | p. 86 |
| Balls in Boxes | p. 89 |
| References | p. 90 |
| The Eleventh and Twelveth Problems of Rota's Fubini Lectures: from Cumulants to Free Probability Theory | p. 91 |
| Prologue | p. 91 |
| The Classical Umbral Calculus | p. 92 |
| Generating Functions | p. 94 |
| Sequences of Binomial Type, Bell Umbrae and Poisson Processes | p. 96 |
| Cumulants | p. 99 |
| Singleton Umbra | p. 100 |
| Cumulant Umbra | p. 101 |
| Applications in Statistics | p. 103 |
| U-Statistics | p. 104 |
| Moments of Sampling Distributions | p. 106 |
| Products of Statistics | p. 109 |
| k-statistics | p. 110 |
| Fast Symbolic Computation of k-statistics | p. 112 |
| Sheppard's Corrections | p. 114 |
| Sheffer Sequences | p. 117 |
| Abel Polynomials | p. 120 |
| Quasi-free Cumulant Umbra | p. 122 |
| Boolean Cumulants | p. 123 |
| Free Cumulant Theory | p. 124 |
| References | p. 127 |
| Two Examples of Applied Universal Algebra | p. 131 |
| Prologue | p. 131 |
| The Bohnenblust-Spitzer Identity | p. 132 |
| Baxter Algebras | p. 133 |
| The Standard Baxter Algebra | p. 134 |
| A Precursor Identity | p. 137 |
| Abstract Random Variables | p. 141 |
| References | p. 143 |
| On the Euler Characteristic of Finite Distributive Lattices | p. 145 |
| Introduction | p. 145 |
| Posets and Distributive Lattices | p. 146 |
| Euler Characteristic | p. 147 |
| Uniform Distributions of the Characteristic | p. 151 |
| X-uniform Lattices | p. 151 |
| Order-preserving Euler Characteristic | p. 152 |
| Rank X-uniform Lattices | p. 153 |
| Pseudo-planar Distributive Lattices | p. 154 |
| Graphs and Hypergraphs | p. 158 |
| Dual Göet;del Lattices | p. 160 |
| Tree maps | p. 162 |
| References | p. 165 |
| Rota, Probability, Algebra and Logic | p. 167 |
| MV-algebraic States and De Finetti Coherence Criterion | p. 167 |
| MV-algebraic and C*-algebraic States After Elliott Classification | p. 171 |
| MV-algebraic ¿-states and Carathéodory Probability Theory | p. 176 |
| References | p. 180 |
| A Symbolic Treatment of Abel Polynomials | p. 183 |
| Introduction | p. 183 |
| Abel polynomials, Lagrange inversion formula and sequences of binomial type | p. 185 |
| Generalized Abel polynomials | p. 187 |
| Cumulants and convolutions | p. 191 |
| References | p. 195 |
| Gian-Carlo Rota, the Philosopher | |
| Ethics in Thought. Gian-Carlo Rota and Philosophy | p. 199 |
| Telling/Inventing the Truth | p. 199 |
| The Winding Streets, and the Straight and Wide Avenue of Precision | p. 202 |
| The Food of Philosophical Thought and the Medicine of Axiomatics | p. 204 |
| The Ethics of Knowledge and the Genesis of its Disciplines | p. 206 |
| References | p. 208 |
| Indiscrete Variations on Gian-Carlo Rota's Themes | p. 211 |
| Introduction | p. 211 |
| The Existence of Mathematical Objects | p. 211 |
| Irrelevance of the Existence of Mathematical Objects | p. 211 |
| Mathematical Objects as Hypotheses | p. 213 |
| Hypotheses vs. Fictions | p. 215 |
| Existence and Identity | p. 215 |
| The Inexhaustibility of Mathematical Objects | p. 216 |
| Definition in Mathematics | p. 217 |
| Definition, Description and Analysis of Concepts | p. 217 |
| Definition in Mathematics and in Philosophy | p. 218 |
| The Alleged Circularity of Definitions and Theorems | p. 219 |
| The Notion of Proof | p. 219 |
| Proof as the Opening up of Possibilities | p. 220 |
| Axiomatic Presentation and Göet;del's Incompleteness Theorems | p. 221 |
| Are there Definitive Proofs? | p. 221 |
| Proof and Evidence | p. 222 |
| The Relation of Philosophy of Mathematics to Mathematics | p. 224 |
| The Descriptive Character of the Philosophy of Mathematics | p. 224 |
| The Need for a Drastic Overhaul of Logic | p. 224 |
| Rota's Place in the Philosophy of Mathematics | p. 225 |
| References | p. 227 |
| On the Courage Needed to Do Phenomenology. Rota and Analytic Philosophy | p. 229 |
| Twilight of an Idol | p. 230 |
| Three Senses of Phenomenology | p. 234 |
| Coda | p. 240 |
| References | p. 240 |
| Rota's Philosophical Insights | p. 241 |
| Introduction | p. 241 |
| What Kind of Reductionism? | p. 241 |
| Rota's Objectivism and Husserl's Naturalism | p. 244 |
| Beyond Classical Logic | p. 245 |
| Concluding Remarks | p. 248 |
| References | p. 249 |
| "A Minority View". Gian-Carlo Rota's Phenomenological Realism | p. 251 |
| Considerations on the Problem of Realism | p. 252 |
| Phenomenological Realism | p. 253 |
| Fringe Phenomena and State-of-Mind | p. 254 |
| Psychologistic Misunderstandings | p. 256 |
| Descriptions, Not Prescriptions | p. 258 |
| References | p. 259 |
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