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Combinators: A Centennial View

by Stephen Wolfram (Author)
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Combinators have inspired ideas about computation ever since they were first invented in 1920, and in this innovative book, Stephen Wolfram provides a modern view of combinators and their significance. Informed by his work on the computational universe of possible programs and on computational language design, Wolfram explains new and existing ideas about combinators with unique clarity and stunning visualizations, as well as provides insights on their historical connections and the curious story of Moses Schönfinkel, inventor of combinators. Though invented well before Turing machines, combinators have often been viewed as an inaccessibly abstract approach to computation. This book brings them to life as never before in a thought-provoking and broadly accessible exposition of interest across mathematics and computer science, as well as to those concerned with the foundations of formal and computational thinking, and with the history of ideas.

Contents

Preface

Combinators: A Centennial View:
Ultimate Symbolic Abstraction • Computing with Combinators • A Hundred Years Later... • Combinators in the Wild: Some Zoology • Visualizing Combinators • Updating Schemes and Multiway Systems • The Question of Evaluation Order • The World of the S Combinator • Causal Graphs and the Physicalization of Combinators • Combinator Expressions as Dynamical Systems • Equality and Theorem Proving for Combinators • Lemmas and the Structure of Combinator Space • Empirical Computation Theory with Combinators • The Future of Combinators • Historical & Other Notes

Combinators and the Story of Computation
The Abstract Representation of Things • What Is Mathematics―and Logic―Made Of? • Combinators Arrive • What Is Their Mathematics? • Gödel's Theorem and Computability • Lambda Calculus • Practical Computation • Combinators in Culture • Designing Symbolic Language • Combinators in the Computational Universe • Combinators All the Way Down?

Where Did Combinators Come From? Hunting the Story of Moses Schönfinkel
 December 7, 1920 • Who Was Moses Schönfinkel? • The Beginning of the Story • Going to College in Odessa • Göttingen, Center of the Mathematical Universe • Problems Are Brewing • The 1924 Paper • The "1927" Paper • To Moscow and Beyond... • Other Schönfinkels... • Haskell Curry • Schönfinkel Rediscovered • What Should We Make of Schönfinkel?

A Little Closer to Finding What Became of Moses Schönfinkel, Inventor of Combinators

1920, 2020 and a $20,000 Prize: Announcing the S Combinator Challenge Hiding in Plain Sight for a Century? • The Basic Setup • The Operation of the S Combinator Challenge

Excerpts from A New Kind of Science (2002)

A Bibliography of Combinators Foundational Documents • Books • Surveys & Summaries • Combinators as Symbolic Expressions • Combinators as Mathematical Constructs • Combinator Computation • Extensions & Applications • Confusing Issues

Index

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  1. ISBN-10
    1579550436
  2. ISBN-13
    978-1579550431
  3. Publisher
    Wolfram Media, Inc.
  4. Publication date
    June 17, 2021
  5. Language
    English
  6. Dimensions
    7.5 x 1 x 9.75 inches
  7. Print length
    362 pages
  8. See all details
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From the Publisher

Combinators: A Centennial View by Stephen Wolfram

"The book has a lot of illustrations, high printing quality and colors—a lot of colors! . . . A good example of how a modern book about programming or math should look."

— Verified Purchaser

From the Preface

"I'd had it in my calendar for more than twenty years: December 7, 2020—the centenary of combinators. But how should it be celebrated? I'd first learned about combinators more than forty years ago—and a few times I'd explicitly used them. But they—and their origins—had always maintained a certain air of mystery for me.

So a few months ago I decided that the way I wanted to celebrate their centenary was finally to do my best to understand how combinators work, and how they came to be invented. This book is the result."

Contents

Preface

Combinators: A Centennial View: Ultimate Symbolic Abstraction · Computing with Combinators · A Hundred Years Later... · Combinators in the Wild: Some Zoology · Visualizing Combinators · Updating Schemes and Multiway Systems · The Question of Evaluation Order · The World of the S Combinator · Causal Graphs and the Physicalization of Combinators · Combinator Expressions as Dynamical Systems · Equality and Theorem Proving for Combinators · Lemmas and the Structure of Combinator Space · Empirical Computation Theory with Combinators · The Future of Combinators

Combinators and the Story of Computation: The Abstract Representation of Things · What Is Mathematics—and Logic—Made Of? · Combinators Arrive · What Is Their Mathematics? · Gödel's Theorem and Computability · Lambda Calculus · Practical Computation · Combinators in Culture · Designing Symbolic Language · Combinators in the Computational Universe · Combinators All the Way Down?

Where Did Combinators Come From? Hunting the Story of Moses Schönfinkel: December 7, 1920 · Who Was Moses Schönfinkel? · The Beginning of the Story · Going to College in Odessa · Göttingen, Center of the Mathematical Universe · Problems Are Brewing · The 1924 Paper · The "1927" Paper · To Moscow and Beyond... · Other Schönfinkels... · Haskell Curry · Schönfinkel Rediscovered · What Should We Make of Schönfinkel?

A Little Closer to Finding What Became of Moses Schönfinkel, Inventor of Combinators

1920, 2020 and a $20,000 Prize: Announcing the S Combinator Challenge

Excerpts from A New Kind of Science (2002)

A Bibliography of Combinators

Index

Author Stephen Wolfram, creator of Mathematica, Wolfram Language and Wolfram|Alpha

About the Author

Stephen Wolfram is an award-winning scientist and bestselling author, and the creator of some of the world's most respected software systems—Mathematica, Wolfram|Alpha and Wolfram Language. For more than 35 years, he has been CEO of the global technology company Wolfram Research, as well as responsible for a series of groundbreaking advances in basic science, including the recent Wolfram Physics Project.

A few of Stephen Wolfram's other books:

  • What Is ChatGPT Doing ... and Why Does It Work?
  • A New Kind of Science
  • Metamathematics: Foundations & Physicalization
  • The Second Law: Resolving the Mystery of the Second Law of Thermodynamics
  • Idea Makers: Personal Perspectives on the Lives & Ideas of Some Notable People
  • An Elementary Introduction to the Wolfram Language, Third Edition

Combinators: A Centennial View Metamathematics: Foundations & Physicalization A New Kind of Science The Second Law: Resolving the Mystery of the Second Law of Thermodynamics A Project to Find the Fundamental Theory of Physics Twenty Years of A New Kind of Science
Combinators: A Centennial View Metamathematics: Foundations & Physicalization A New Kind of Science The Second Law: Resolving the Mystery of the Second Law of Thermodynamics A Project to Find the Fundamental Theory of Physics Twenty Years of A New Kind of Science
Customer Reviews
4.1 out of 5 stars
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4.4 out of 5 stars
46
4.3 out of 5 stars
661
4.2 out of 5 stars
43
4.5 out of 5 stars
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3.6 out of 5 stars
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Price $32.87 $41.54 $25.93 $45.04 $34.36 $30.10
Page Count 362 456 1280 584 778 384
Publication Year 2021 2022 2002 2023 2020 2022

Editorial Reviews

From the Back Cover

"MAT018000 "

About the Author

Stephen Wolfram is a distinguished scientist and best-selling author, and the creator of some of the world's most respected software systems, including Mathematica, Wolfram|Alpha and the Wolfram Language. For more than 30 years he has been the CEO of the global technology company Wolfram Research.

Product details

  • Publisher ‏ : ‎ Wolfram Media, Inc. (June 17, 2021)
  • Language ‏ : ‎ English
  • Hardcover ‏ : ‎ 362 pages
  • ISBN-10 ‏ : ‎ 1579550436
  • ISBN-13 ‏ : ‎ 978-1579550431
  • Item Weight ‏ : ‎ 1.92 pounds
  • Dimensions ‏ : ‎ 7.5 x 1 x 9.75 inches

About the author

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Stephen Wolfram

Stephen Wolfram

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Stephen Wolfram has had a unique trajectory in science, technology and business. Widely known for his discoveries in basic science and his groundbreaking 2002 book "A New Kind of Science," he has spent more than three decades building what is now Wolfram Language: the knowledge-based computer language that powers Mathematica and Wolfram|Alpha and has contributed to countless inventions and discoveries, as well as to the education of several generations of students.

Wolfram was born in London and educated at Eton, Oxford and Caltech, earning his PhD in physics in 1979 at the age of 20. After a brief but distinguished academic career, he founded Wolfram Research in 1987 and as CEO has built it into one of the world’s most respected and innovative software companies, whose products are relied on by millions of people around the world.

Customer reviews

4.1 out of 5 stars
66 global ratings

Top reviews from the United States

Constantine V.
5.0 out of 5 stars Very approachable book on the quite a hard subject
Reviewed in the United States on October 4, 2021
Verified Purchase
The book is split in several distinct parts, each can be read separately, but connected by a common theme of Combinators. The "main" first part is the analysis of all possible S and SK combinator words up to the length 16, with interesting observations about time/space complexity, halting, execution order, opportunities for performance optimization, etc. After that historical excurse into discovery of combinators and an amazing rich bio of Moses Schonfinkel.

The book has a lot of illustrations, high printing quality and colors - a lot of colors ! I think this is a good example of how a modern book about programming or math should look.

I have "New Kind of Science", but was not able to finish it. This book actually reads much better and easier.

p.s.: Author quite often mentions Mathematica and Wolfram language, as well as his personal achievements or opinions about things. While this might annoy some readers, I felt like this added more flavor to why the subject of the book is important to author.
19 people found this helpful
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Nahatz
5.0 out of 5 stars Very readable.
Reviewed in the United States on August 2, 2021
Verified Purchase
Somewhat prone to self promotion.
Otherwise a very good read.
21 people found this helpful
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Donald Bosley
3.0 out of 5 stars Read up on combinators, before you read the book on combinators.
Reviewed in the United States on August 21, 2022
Verified Purchase
Although it's a beautifully printed book, and well organized with some interesting history, it was overall underwhelming. If you don't already know about combinators, which might be why you bought the book about combinators, you'll probably find getting through even the early material difficult or impossible.
9 people found this helpful
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John Narofsky
1.0 out of 5 stars Impenetrable self-promotion
Reviewed in the United States on May 2, 2024
Verified Purchase
I bought this book to learn about combinators. I apparently need to buy a different book than one that claims to describe and teach about combinators.
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John LeClair
3.0 out of 5 stars Dear Author, enable Amazon "Look-Inside" previewing in future
Reviewed in the United States on January 10, 2022
or I won't even consider purchasing. I have been burnt too many times without viewing chapters, index and a few pages in the middle of the book. The book looks interesting, but I have not read or purchased it.
19 people found this helpful
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Top reviews from other countries

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Baddel.Does
5.0 out of 5 stars Wolfram für Schönfinkel
Reviewed in Germany on July 13, 2021
Verified Purchase
Stephen Wolfram hat etwas übrig für verkannte Genies, Moses Schönfinkel ist einer davon. Ok, vielleicht einer von hundert, meine Frau sagt von tausend Menschen versteht überhaupt das Problem, dem sich Schönfinkel gewidmet hat, aber trotzdem ist er ein Genie.
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