Pulse en una miniatura para ir a Google Books.
|
Cargando... Mathematics: The Science of Patterns : The Search for Order in Life, Mind, and the Universe (Scientific American Library)por Keith Devlin
   Ninguno Actualmente no hay Conversaciones sobre este libro.  themulhern | Jun 19, 2020 |  sin reseñas | añadir una reseña
Pertenece a las seriesPertenece a las series editoriales
"With this fascinating volume, Keith Devlin proves that the guiding principles of some of the most mysterious mathematical topics can be made comprehensible. Writing with an elegant lucidity, Devlin shows just why the definition of mathematics as "working with numbers" has been out of date for nearly 2,500 years. And he demonstrates that far from being too abstract to matter, mathematics is instead an essential and uniquely human endeavor, one that helps us understand the universe and ourselves." "In this century alone, there has been a veritable explosion of mathematical activity. A body of knowledge that in 1900 might have filled 80 volumes now would require nearly 100,000. Fields such as algebra and topology have grown tremendously, while complexity theory, dynamical systems theory, and other new areas have developed. And in the last two decades, a common thread running through the many facets of mathematics has been recognized: mathematicians of all kinds now see their work as the study of patterns - real or imagined, visual or mental, arising from the natural world or from within the human mind." "Devlin uses this basic definition as his central theme, revealing the search for patterns that drives the mathematics of counting (natural numbers), reasoning (language and logic), motion (calculus), shape (geometry, tilings), and position (topology, knots, symmetry). Interweaving historical highlights and current developments, and using a minimum of formulas, he lets readers see into the kind of reasoning that allows mathematicians to create and explore arcane subjects. And he makes clear the many ways mathematics informs our perceptions of reality - both the physical, biological, and social worlds without, and the realm of ideas and thoughts within." ""Mathematics, rightly viewed, possesses not only truth, but supreme beauty," the noted philosopher and mathematician Bertrand Russell once wrote. In Mathematics: The Science of Patterns, Keith Devlin makes such a vision accessible, entertaining, and meaningful. It is an insightful, richly illustrated celebration of the simplicity, the precision, the purity, and the elegance of mathematics."--BOOK JACKET.Title Summary field provided by Blackwell North America, Inc. All Rights Reserved No se han encontrado descripciones de biblioteca. |
Debates activosNingunoCubiertas populares
 Google Books — Cargando...GénerosSistema Decimal Melvil (DDC)510.1Natural sciences and mathematics Mathematics General Mathematics Philosophy And PsychologyClasificación de la Biblioteca del CongresoValoraciónPromedio: (4.29)
¿Eres tú?Conviértete en un Autor de LibraryThing. |
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1
Detailed Review:
* Prologue: Gives the means of arriving at the definition, a science of patterns, and also emphasizes the importance of abstraction and notation. Remarks that computers have allowed visualization that was not possible before; Julia sets are older than our ability to easily see what they look like.
* Chapter 1: Counting
Covers the history of the development of the concept of number, as far as can be inferred from archaeological evidence. The Rhind papyrus has a description of how to calculate the volume of the frustum of a pyramid. The numbers used are concrete; but it it s clear that the reader is expected to consider them variables as necessary. A modern day person would use integration to derive a formula for the volume.
Moving on to the Greeks, who were really most interested in geometry. Discusses the Pythagorean theorem and the interpretation of numbers as lengths. The formula for (a + b)^2 and (a -b)^2 is presented as the Greeks would have understood it, geometrically. A proof of the irrationality of root 2 is shown; is that how the Greeks discovered it? A discussion of prime numbers; evidently the fundamental theorem of arithmetic is in Euclid's work, as well as the proof that there is an infinite number of primes.
The primes allow us to leap 1000 or so years forward, and discuss the prime density function, the ratio of the number of primes less than or equal to N to N. Wikipedia discusses the prime _counting_ function: https://en.wikipedia.org/wiki/Prime-counting_function. The proof that the number of primes are infinite is well presented; I've seen other place where the proof fails to properly cover the two possibilities, leaving the reader perplexed and uncertain.
Chebyshev discovers that between every number and its double there is at least one prime. And finally, an upper bound, on the function, 1/ln N, was proved.
The next topic, Finite Arithmetic, is a discussion of abstract algebra. It was then I wanted to find my Abstract Algebra textbook, only to discover that it was almost certainly in my currently inaccessible cubicle.
Chapter 4: Shape
About geometry and algebraic geometry. There is a solid discussion of Euclid and a sidebar on the golden ratio with some unexplained relationships. The regular solids and conic sections are discussed. Properties are described w/out much justification but with fine illustrations. Kepler's wierd obsession with the plantonic solids and the planetary orbits gets a mention. Then algebra is introduced, and there is a perplexing discussion of a general formula for a circle, that yields no insight to me (p.118). Old problems of compass and straightedge construction of various figures are discussed. These all rely on the fact that compass and straightedge constructions only yield a subset of mathematical operations; basic arithmetic and the taking of square roots, and that all the solutions require a calculation that uses more than these. This can't be covered in detail in a book like this, of course. The next step is non-Euclidean geometries: hyperbolic, spherical, Riemannian, projective.