Algorítmica. Historia de la Combinatoria.

Sí señor, así, con mayúsculas, sin complejos: la Cenicienta de las Matemáticas, tantas veces despreciada ya tiene su historia.

Que yo recuerde, Kline en su monumental tratado (Mathematical Thought From Ancient to Modern Times, 1972) ni habla de esta disciplina. Bourbaki, más condescendiente, en sus notas históricas (Élements d´Histoire des  Mathématiques, 1969)  que acompañan a sus famosos seminarios, agrupan el tema que llaman Análisis Combinatorio (y por lo tanto se restringen a la combinatoria enumerativa) con los sistemas de numeración (que realmente, aunque no sin importancia o consecuencias, es algo previo a la actividad Matemática) le dedican 26 líneas. Concretamente citan la formula de selecciones 2 elementos entre n posibles del s. III de nuestra era; a Baskhara y la formula general; a Levi ben Gershon y otras formulas combinatorias; a Cardano y el número de subconjuntos no vacíos de un conjunto de n elementos; Fermat y Pascal con el cálculo de probabilidades; a Pascal de nuevo con su triangulo; y finalmente a Leibniz y de Moivre con respecto a la fórmula general de coeficientes multinomiales.

Si como a mi esto te sabe a poco, el próximo 24 de agosto se va a publicar un libro que contendrá mucho más. Se titula Combinatorics: Ancient & Modern.

Extracto de la presentación en Amazon.

The history of mathematics is a well-studied and vibrant area of research, with books and scholarly articles published on various aspects of the subject. Yet, the history of combinatorics seems to have been largely overlooked. This book goes some way to redress this and serves two main purposes: 1) it constitutes the first book-length survey of the history of combinatorics; and 2) it assembles, for the first time in a single source, researches on the history of combinatorics that would otherwise be inaccessible to the general reader. 

Individual chapters have been contributed by sixteen experts. The book opens with an introduction by Donald E. Knuth to two thousand years of combinatorics. This is followed by seven chapters on early combinatorics, leading from Indian and Chinese writings on permutations to late-Renaissance publications on the arithmetical riangle. The next seven chapters trace the subsequent story, from Euler’s contributions to such wide-ranging topics as partitions, polyhedra, and latin squares to the 20th century advances in combinatorial set theory, enumeration, and graph theory. The book concludes with some combinatorial reflections by the distinguished combinatorialist, Peter J. Cameron. 

This book is not expected to be read from cover to cover, although it can be. Rather, it aims to serve as a valuable resource to a variety of audiences. Combinatorialists with little or no knowledge about the development of their subject will find the historical treatment stimulating. A historian of mathematics will view its assorted surveys as an encouragement for further research in combinatorics. The more general reader will discover an introduction to a fascinating and too little known subject that continues to stimulate and inspire the work of scholars today.

Y puedes ver el índice y hojear el libro en google books.  Hay que reconocer que, vistas las disciplinas que han inspirado en el pasado en las diferentes culturas  las incursiones en la Combinatoria (véase por ejemplo la página 9;  ¿no tendrían otra cosa mejor que hacer que listar permutaciones  ;-)?), se merece la mala fama que tiene (y para más inri, hoy en día se asocian más a las matemáticas recreativas que a cualquier otra cosa).

Sin embargo últimamente esto está cambiando y no sólo por el impacto de las  Computadoras y de las Ciencias Computacionales, que hacen un muy frecuento uso de las  matemáticas discretas.

Para ilustrar esto  un extracto  del último capítulo escrito por  un autor que ha trabajado mucho en temas relacionados con mis propias investigaciones (y que puedes leer completo aquí):

On the importance of discreteness in nature, Steven Pinker has no doubt. He wrote:

“It may not be a coincidence that the two systems in the universe that most impress us with their open-ended complex design – life and mind – are based on discrete combinatorial systems.”

Here, ‘mind’ refers primarily to language, whose combinatorial structure is well described in Pinker’s book. ‘Life’ refers to the genetic code, where DNA molecules can be regarded as words in an alphabet of four letters (the bases adenine, cytosine, guanine and thymine), and three-letter subwords encode amino acids, the building blocks of proteins.

But there are now hints that discreteness plays an even more fundamental role. One of the goals of physics at present is the construction of a theory which could reconcile the two pillars of twentieth-century physics, general relativity and quantum mechanics. In describing string theory, loop quantum gravity, and a variety of other approaches including non-commutative geometry and causal set theory, Smolin [39] argues that all of them involve discreteness at a fundamental level (roughly the Planck scale, which is much too small and fleeting to be directly observed). Indeed, developments such as the holographic principle suggest that the basic currency of the universe may not be space and time, but information, measured in bits. Maybe the ‘theory of everything’ will be combinatorial!

A lo mejor, quién sabe…


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