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Conceptual Mathematics: A First Introduction to Categories [Englisch] [Gebundene Ausgabe]

F. William Lawvere , Stephen H. Schanuel
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Kurzbeschreibung

30. Juli 2009
In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics introduces this tool for the learning, development, and use of mathematics, to beginning students and also to practising mathematical scientists. This book provides a skeleton key that makes explicit some concepts and procedures that are common to all branches of pure and applied mathematics. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories. This second edition provides links with more advanced topics of possible study. In the new appendices and annotated bibliography the reader will find concise introductions to adjoint functors and geometrical structures, as well as sketches of relevant historical developments.


Produktinformation

  • Gebundene Ausgabe: 408 Seiten
  • Verlag: Cambridge University Press; Auflage: 2 (30. Juli 2009)
  • Sprache: Englisch
  • ISBN-10: 0521894859
  • ISBN-13: 978-0521894852
  • Größe und/oder Gewicht: 2,8 x 16,8 x 24 cm
  • Durchschnittliche Kundenbewertung: 5.0 von 5 Sternen  Alle Rezensionen anzeigen (1 Kundenrezension)
  • Amazon Bestseller-Rang: Nr. 2.413.461 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)
  • Komplettes Inhaltsverzeichnis ansehen

Produktbeschreibungen

Pressestimmen

‘This text, written by two experts in Category Theory and tried out carefully in courses at SUNY Buffalo, provides a simple and effective first course on conceptual mathematics.’ American Mathematical Monthly

‘… every mathematician should know the basic ideas and techniques explained in this book …’ Monatshefte für Mathematik

‘Conceptual Mathematics provides an excellent introductory account to categories for those who are starting from scratch. It treats material which will appear simple and familiar to many philosophers, but in an unfamiliar way.’ Studies in History and Philosophy of Modern Physics

‘Category Theory slices across the artificial boundaries dividing algebra, arithmetic, calculus, geometry, logic, topology. If you have students you wish to introduce to the subject, I suggest this delightfully elementary book . Lawvere is one of the greatest visionaries of mathematics in the last half of the twentieth century. He characteristically digs down beneath the foundations of a concept in order to simplify its understanding. Schanuel has published research in diverse areas of Algebra, Topology, and Number Theory and is known as a great teacher. I have recommended this book to motivated high school students. I certainly suggest it for undergraduates. I even suggest it for the mathematician who needs a refresher on modern concepts.’ National Association of Mathematicians Newsletter

‘Conceptual Mathematics is the first book to serve both as a skeleton key to mathematics for the general reader or beginning student and as an introduction to categories for computer scientists, logicians, physicists, linguists, etc … The fundamental ideas are illuminated in an engaging way.’ L’Enseignment Mathématique

Über das Produkt

Conceptual Mathematics introduces the concept of category to beginning students and practising mathematical scientists based on a leisurely introduction to the important categories of directed graphs and discrete dynamical systems. The expanded second edition approaches more advanced topics via historical sketches and a concise introduction to adjoint functors.

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5 von 5 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Ein ursprüngliches Buch 26. April 2011
Von Hans Beck
Format:Taschenbuch
Dieses Buch ist ungewöhnlich. Es ist keines der Mathe-Bücher, die von der ersten Seite weg den Leser mit einer Unzahl von Definitionen belästigen, und den Leser mit der Frage allein lassen: "Warum das alles?", "Wozu braucht man das ?". Stattdessen geht das Buch tief in die Grundlagen des mathematischen Verständnissen von Abbildungen und von Kategorien. Es erzählt von dem Sinn dieser Dinge und führt so zu einem tieferen Verständnis von Kategorien. Viele Fragen (Aufgaben möchte man sie fast nicht nennen) laden zum Mitdenken ein. Natürlich hat diese Gründlichkeit einen Preis: es werden viele Konstrukte der Kategorientheorie nicht angesprochen. Allerdings besteht eine grosse Chance, nach der Lektüre dieses Buches die "üblichen" Mathematikbücher zu Kategorientheorie besser zu verstehen. Also ein sehr empfehlenswertes Buch für Interessierte !
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5.0 von 5 Sternen A bird's eye view of the mathematical landscape 18. August 2010
Von Michael - Veröffentlicht auf Amazon.com
Format:Taschenbuch
Over the last two years I have revisited different sections of this book and gain new profound insights with every read. With some dedication and time, this book will surely enrich your life! What this book offers is the truth. The concepts presented in this book are the underlying unifying ideas which make up mathematics itself in an even more general and profound sense than Set Theory (in fact, one of the authors has rigorously shown that set theory is a very special case of what is presented in this book). We can encounter categories not only at the microscopic level (where we define the fundamental ideas that allow us to construct mathematical concepts from the ground up), but at the macroscopic level as well (where complex constructions in distant fields become analogous to the microscopic building blocks). With these ideas we can show that multiplication and addition are actually more appropriately opposites of one another than addition and subtraction or multiplication and division. This book is the key to beginning a journey to discovering the true nature of mathematics. To continue (or supplement) your journey, also pick up a copy of Sets for Mathematics By F. William Lawvere and Robert Rosebrugh. With time and practice (attempt the exercises from both books!!!) you will be greatly rewarded. As a student of Mathematics, this has paid off in ways I never thought possible and continues to provide insight to nearly everything I learn in school and on my own.

A startling demonstration presented in this book is that Cantor's Diagonal Argument in generalized form not only proves that there are infinite different levels of infinity, but also Godel's Incompleteness Theorem! Also contained is a convincingly appropriate abstraction of the characteristic function of any subobject with respect to any object it is contained in (in any sufficiently rich category). In other words, mappings in the context of a chosen category with domain X and a particular codomain Omega can correspond exactly with all objects contained within X. The latest Edition elaborates on this notion of parthood as well as introduces adjoint functors.
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5.0 von 5 Sternen For High School students and Professional Scientists 29. Mai 2010
Von Bonvibre Prosim - Veröffentlicht auf Amazon.com
Format:Taschenbuch
Not long ago, I spoke with a professor at strong HBCU department. Her Ph.D. was nearly twenty years ago, but I shocked her with the following statement, "Most of our beginning graduate students [even those in Applied Mathematics] are entering with the basic knowledge and language of Category Theory. These days one might find Chemists, Computer Scientists, Engineers, Linguists and Physicists expressing concepts and asking questions in the language of Category Theory because it slices across the artificial boundaries dividing algebra, arithmetic, calculus, geometry, logic, topology. If you have students you wish to introduce to the subject, I suggest a delightfully elementary book called Conceptual Mathematics by F. William Lawvere and Stephen H. Schanuel" [Cambridge University Press 1997].
From the introduction: "Our goal in this book is to explore the consequences of a new and fundamental insight about the nature of mathematics which has led to better methods for understanding and usual mathematical concepts. While the insight and methods are simple ... they will require some effort to master, but you will be rewarded with a clarity of understanding that will be helpful in unraveling the mathematical aspect of any subject matter."
Who are the authors? Lawvere is one of the greatest visionaries of mathematics in the last half of the twentieth century. He characteristically digs down beneath the foundations of a concept in order to simplify its understanding. Though Schanuel has published research in diverse areas of Algebra, Topology, and Number Theory, he is known as a great teacher. The book is an edited transcript of a course taught by Lawvere and Schanuel to American undergraduate math students. The book was actually chosen as one of the items in the Library of Science Book Club. The concepts of Category Theory in Conceptual Mathematics are presented in the same way Lawvere and Schanuel implemented it, in a real classroom setting, addressing common questions of students (yes these are real people) at crucial points in the book.
The book comes with thirty-three Sessions instead of Chapters. Some Sessions can be understood in a single class or hour. Others may take longer. There are also numerous Examples, Problems, and five Tests of the student's understanding.
The title of Session 1 is "Galileo and the flight of a bird" and motivates the notion map. The sixth part of Session 5 is called "Stacking in a Chinese restaurant" and helps motivate sections and retractions. Session 10 motivates the Brouwer Fixed Point Theorem. Less you think this is all Abstract Mathematical nonsense, Session 15 is called "Objectification of properties in dynamical systems." The title of Session 20 is "Points of an object."
I have recommended Lawvere and Schanuel to motivated high school students. I certainly suggest this clearly written "Conceptual Mathematics" for undergraduates. I even suggest it for the mathematician who needs a refresher on modern concepts.

This a re-print of a review I wrote for the quarterly of the National Association of Mathematics.
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5.0 von 5 Sternen My favorite maths book! 25. November 2010
Von King Yin Yan - Veröffentlicht auf Amazon.com
Format:Taschenbuch
It has flaws, but is still one of the greatest maths book I've read. Aimed at high-school level and up, but towards the end it gets a bit complicated, so I doubt if a high school kid can fully understand it without consulting other books. But, most of the book is really easy to read, and the authors' effort to write such a book is admirable.

Lawvere is one of the developers of topos theory, where he found an axiomatization of the category of sets.

The last 2 sections are an introduction to topoi and logic. One key fact seems to be missing which caused me some perplexing: In the category of subobjects, 2 subobjects A and B has A > B if A includes B. Thus, the relation ">" creates a partial order amongst the subobjects. If A > B and B < A, then A = B, thus inducing an equivalence class, denoted by [A]. This is the reason why the subobject classifier has internal structure (different "shades" of truth values).

Also, the relation of topology to logic is analogous to the relation of classical propositional logic to the Boolean algebra of sets, with the sets replaced by open sets in topological space.

I've only read the 1st edition. The 2nd edition's first part is the same as the 1st edition, with additional advanced topics at the very end.
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5.0 von 5 Sternen ||||| A Beautiful Work -- once in a blue moon sort of thing ! 7. Juli 2011
Von Farogh Dovlatashahi - Veröffentlicht auf Amazon.com
Format:Taschenbuch
Such an excellent work as one is given to saying asto all productions of Lawvere's. This book, on the face of it, seems easy, even elementary. But there is, as Lawvere has said, an awful lot here. A book is elegant if it achieves to say a great deal with ease and a sense of depth of coverage.

The path to Cateogories and Toposes is via two book: Cat for working mathematicians and Sheaves in Geometry and Logic both by Mac Lane. But these are anything but easy or elementary.

There is a problem with mathematical texts of a pedagogic kind, one that this book avoids: their writers often confuse teaching with forma exposition. They don't "talk" to one but go off at their own formal tangents.
6 von 6 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Intuitive and insightful 22. Februar 2011
Von Dmitri - Veröffentlicht auf Amazon.com
Format:Taschenbuch|Verifizierter Kauf
This book is a simple introduction to category theory. It is strictly about categories, examples are easy to understand and relevant to beginners. You will not see references to vector spaces, matrices, algebras or topologies, but you will understand what category theory is about.
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