From Combinatorics to Philosophy - The Legacy of G.-C. Rota

From Combinatorics to Philosophy

Foreword

List of talks

Contents

Part I Gian-Carlo Rota, the Man

Gianco, my Brother

Ester Gasperoni Rota

References

Remembering Gian-Carlo Rota

Joseph J. Kohn

References

Part II Gian-Carlo Rota, the Mathematician

A Glimpse of Vector Invariant Theory. The Points of View of Weyl, Rota, De Concini and Procesi, and Grosshans

Andrea Brini

Introduction

The Characteristic Zero Approach

Preliminaries

Weyl's Theorem (The Reduction Principle)

Absolute Invariants for the Special Linear Group SL(d)

Absolute Invariants for the Symplectic Group Sp2m

Absolute Invariants for the Special Orthogonal Group SO(d)

Absolute Invariants for the Orthogonal Group O(d)

The Straightening Formulae

Standard Young Tableaux

Bideterminants

Pfaffians

Gramians

The E. Pascal Theorems and the Cancellation Laws

The E. Pascal Theorem and the Cancellation Law for Scalar Products

The E. Pascal Theorem and the Cancellation Law for Symplectic Products

The E. Pascal Theorem and the Cancellation Law for Inner Products

The First and the Second Fundamental Theorems

The First Fundamental Theorems

The Second Fundamental Theorems

Grosshans's Theorem

References

Partitions of a Finite Partially Ordered Set

Pietro Codara

Introduction

Background, and Preliminary Results

Partitions as Sets of Fibres

Partitions as Partially Ordered Sets of Blocks

Partitions Induced by Quasiorders

Further Work

References

An Algebra of Pieces of Space --- Hermann Grassmann to Gian Carlo Rota

Henry Crapo

Synthetic Projective Geometry

Hermann Grassmann's Algebra

Extensors and Vectors

Reduced Forms

Grassmann-Cayley Algebra, Peano Spaces

Whitney Algebra

Geometric Product

Regressive Product

Higher Order Syzygies

Balls in Boxes

References

The Eleventh and Twelveth Problems of Rota's Fubini Lectures: from Cumulants to Free Probability Theory

Elvira Di Nardo and Domenico Senato

Prologue

The Classical Umbral Calculus

Generating Functions

Sequences of Binomial Type, Bell Umbrae and Poisson Processes

Cumulants

Singleton Umbra

Cumulant Umbra

Applications in Statistics

U-Statistics

Moments of Sampling Distributions

Products of Statistics

k-statistics

Fast Symbolic Computation of k-statistics

Sheppard's Corrections

Sheffer Sequences

Abel Polynomials

Quasi-free Cumulant Umbra

Boolean Cumulants

Free Cumulant Theory

References

Two Examples of Applied Universal Algebra

Joseph P. S. Kung

Prologue

The Bohnenblust--Spitzer Identity

Baxter Algebras

The Standard Baxter Algebra

A Precursor Identity

Abstract Random Variables

References

On the Euler Characteristic of Finite Distributive Lattices

Emanuele Munarini

Introduction

Posets and Distributive Lattices

Euler Characteristic

Uniform Distributions of the Characteristic

-uniform Lattices

Order-preserving Euler Characteristic

Rank -uniform Lattices

Pseudo-planar Distributive Lattices

Graphs and Hypergraphs

Dual Gödel Lattices

Tree maps

References

Rota, Probability, Algebra and Logic

Daniele Mundici

MV-algebraic States and De Finetti Coherence Criterion

MV-algebraic and C*-algebraic States After Elliott Classification

MV-algebraic -states and Carathéodory Probability Theory

References

A Symbolic Treatment of Abel Polynomials

Pasquale Petrullo

Introduction

Abel polynomials, Lagrange inversion formula and sequences of binomial type

Generalized Abel polynomials

Cumulants and convolutions

References

Part III Gian-Carlo Rota, the Philosopher

Ethics in Thought. Gian-Carlo Rota and Philosophy

Francesca Bonicalzi

Telling/Inventing the Truth

The Winding Streets, and the Straight and Wide Avenue of Precision

The Food of Philosophical Thought and the Medicine of Axiomatics

The Ethics of Knowledge and the Genesis of its Disciplines

References

Indiscrete Variations on Gian-Carlo Rota's Themes

Carlo Cellucci

Introduction

The Existence of Mathematical Objects

Irrelevance of the Existence of Mathematical Objects

Mathematical Objects as Hypotheses

Hypotheses vs. Fictions

Existence and Identity

The Inexhaustibility of Mathematical Objects

Definition in Mathematics

Definition, Description and Analysis of Concepts

Definition in Mathematics and in Philosophy

The Alleged Circularity of Definitions and Theorems

The Notion of Proof

Proof as the Opening up of Possibilities

Axiomatic Presentation and Gödel's Incompleteness Theorems

Are there Definitive Proofs?

Proof and Evidence

The Relation of Philosophy of Mathematics to Mathematics

The Descriptive Character of the Philosophy of Mathematics

The Need for a Drastic Overhaul of Logic

Rota's Place in the Philosophy of Mathematics

References

On the Courage Needed to Do Phenomenology. Rota and Analytic Philosophy

Albino Lanciani and Claudio Majolino

Twilight of an Idol

Three Senses of Phenomenology

Coda

References

Rota's Philosophical Insights

Massimo Mugnai

Introduction

What Kind of Reductionism?

Rota's Objectivism and Husserl's Naturalism

Beyond Classical Logic

Concluding Remarks

References

``A Minority View". Gian-Carlo Rota's Phenomenological Realism

Fabrizio Palombi

Considerations on the Problem of Realism

Phenomenological Realism

Fringe Phenomena and State-of-Mind

Psychologistic Misunderstandings

Descriptions, Not Prescriptions

References