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Five Golden Rules: Great Theories of 20th-Century Mathematics--and Why They Matter
Praise for Five Golden Rules
"Casti is one of the great science writers of the 1990s. . . . If you'd like to have fun while giving your brain a first-class workout, then check this book out."-Keay Davidson in the San Francisco Examiner.
"Five Golden Rules is caviar for the inquiring reader. . . . There is joy here in watching the unfolding of these intricate and beautiful techniques. Casti's gift is to be able to let the nonmathematical reader share in his understanding of the beauty of a good theory." -Christian Science Monitor.
"Merely knowing about the existence of some of these golden rules may spark new, interesting-maybe revolutionary-ideas in your mind." -Robert Matthews in New Scientist (United Kingdom).
"This book has meat! It is solid fare, food for thought. Five Golden Rules makes math less forbidding and much more interesting." -Ben Bova in the Hartford Courant
"With this groundbreaking work, John Casti shows himself to be a great mathematics writer. Five Golden Rules is a feast of rare new delights all made perfectly comprehensible." -Rudy Rucker, author of The Fourth Dimension.
"With the lucid informality for which he has become known, John Casti has written an engaging and articulate examination of five great mathematical theorems and their myriad applications." -John Allen Paulos, author of A Mathematician Reads the Newspaper.
"Casti is one of the great science writers of the 1990s. . . . If you'd like to have fun while giving your brain a first-class workout, then check this book out."-Keay Davidson in the San Francisco Examiner.
"Five Golden Rules is caviar for the inquiring reader. . . . There is joy here in watching the unfolding of these intricate and beautiful techniques. Casti's gift is to be able to let the nonmathematical reader share in his understanding of the beauty of a good theory." -Christian Science Monitor.
"Merely knowing about the existence of some of these golden rules may spark new, interesting-maybe revolutionary-ideas in your mind." -Robert Matthews in New Scientist (United Kingdom).
"This book has meat! It is solid fare, food for thought. Five Golden Rules makes math less forbidding and much more interesting." -Ben Bova in the Hartford Courant
"With this groundbreaking work, John Casti shows himself to be a great mathematics writer. Five Golden Rules is a feast of rare new delights all made perfectly comprehensible." -Rudy Rucker, author of The Fourth Dimension.
"With the lucid informality for which he has become known, John Casti has written an engaging and articulate examination of five great mathematical theorems and their myriad applications." -John Allen Paulos, author of A Mathematician Reads the Newspaper.
235 pages, Paperback
First published August 25, 1995
5 people are currently reading
163 people want to read
About the author
John L. Casti
60 books22 followersJohn L. Casti (born 1943) is an author, mathematician, and entrepreneur.
As a mathematician and researcher, Casti received his Ph.D. under Richard Bellman at the University of Southern California. He worked at the RAND Corporation in Santa Monica, CA, and served on the faculties of the University of Arizona, New York University and Princeton University, before moving to Vienna in 1973 to become one of the first members of the research staff at the International Institute for Applied Systems Analysis (IIASA) in Laxenburg, Austria. In 1986, he left IIASA to take up a position as a Professor of Operations Research and System Theory at the Technical University of Vienna. He also served as a member of the External Faculty of the Santa Fe Institute in Santa Fe, New Mexico, USA, from 1992-2002, where he worked extensively on the application of biological metaphors to the mathematical modeling of problems in economics, finance and road-traffic networks, as well as on large-scale computer simulations for the study of such networks.
His primary research interests have shifted somewhat in recent years from the natural sciences to the exploration of questions in the social and behavioral realm. One thread has been exploration of the relationship between the social "mood" of a population its biasing effect on actions and behaviors. In this direction, his 2010 book, Mood Matters: From Rising Skirt Lengths to the Collapse of World Powers, published by Copernicus Books, NY, addresses the directions and patterns of social causation and their implications for future trends and collective social events, such as styles in popular culture, the outcome of political processes, and even the rise and fall of civilizations. His most recent book is X-EVENTS: The Collapse of Everything, which addresses the underlying cause of extreme events generated by human inattention, misunderstanding, error, stupidity and/or malevolent intent. The English original edition was published in June 2012 by HarperCollins/Morrow, New York. The book now exists in 15 foreign editions, as well, including German, Japanese, Russian, Dutch, Korean and Portuguese.
As an entrepreneur, Casti formed two companies in Santa Fe and London in 2000, Qforma, Inc. and SimWorld, Ltd, respectively, devoted to the employment of tools and concepts from modern system theory for the solution of problems in business and finance, as well as health care. Qforma merged with SkilaMederi in June 2013. In early 2005 he returned to Vienna where he co-founded The Kenos Circle, a professional society that aims to make use of complexity science in order to gain a deeper insight into the future than that offered by more conventional statistical tools.
For several years, Professor Casti was a Senior Research Scholar at the International Institute for Applied Systems Analysis in Laxenburg, Austria, where he created an initiative for the study on Extreme Events in Human Society. In January 2012 he left IIASA to form a new research institute in Vienna, The X-Center, devoted to the study of human-caused extreme events. The X-Center has now expanded to a network of affiliated X-Centers in Helsinki, Tokyo, Seoul, New York and Singapore. Since early 2013, Dr. Casti has been serving as a Senior Research Fellow at the Center for Complex Systems and Enterprises at the Stevens Institute of Technology in the USA.
As an author, Casti has written more than 120 scientific articles and seven technical monographs and textbooks on mathematical modeling. In addition, he was formerly editor of the journals Applied Mathematics & Computation (Elsevier, New York) and Complexity (Wiley, New York). In 1989 his text/reference works Alternate Realities: Mathematical Models of Nature and Man (Wiley, 1989) was awarded a prize by the Association of American Publishers in a competition among all scholarly books published in mathematics and the natural sciences. In 1992, he also published Reality Rules (Wiley, New York), a t
As a mathematician and researcher, Casti received his Ph.D. under Richard Bellman at the University of Southern California. He worked at the RAND Corporation in Santa Monica, CA, and served on the faculties of the University of Arizona, New York University and Princeton University, before moving to Vienna in 1973 to become one of the first members of the research staff at the International Institute for Applied Systems Analysis (IIASA) in Laxenburg, Austria. In 1986, he left IIASA to take up a position as a Professor of Operations Research and System Theory at the Technical University of Vienna. He also served as a member of the External Faculty of the Santa Fe Institute in Santa Fe, New Mexico, USA, from 1992-2002, where he worked extensively on the application of biological metaphors to the mathematical modeling of problems in economics, finance and road-traffic networks, as well as on large-scale computer simulations for the study of such networks.
His primary research interests have shifted somewhat in recent years from the natural sciences to the exploration of questions in the social and behavioral realm. One thread has been exploration of the relationship between the social "mood" of a population its biasing effect on actions and behaviors. In this direction, his 2010 book, Mood Matters: From Rising Skirt Lengths to the Collapse of World Powers, published by Copernicus Books, NY, addresses the directions and patterns of social causation and their implications for future trends and collective social events, such as styles in popular culture, the outcome of political processes, and even the rise and fall of civilizations. His most recent book is X-EVENTS: The Collapse of Everything, which addresses the underlying cause of extreme events generated by human inattention, misunderstanding, error, stupidity and/or malevolent intent. The English original edition was published in June 2012 by HarperCollins/Morrow, New York. The book now exists in 15 foreign editions, as well, including German, Japanese, Russian, Dutch, Korean and Portuguese.
As an entrepreneur, Casti formed two companies in Santa Fe and London in 2000, Qforma, Inc. and SimWorld, Ltd, respectively, devoted to the employment of tools and concepts from modern system theory for the solution of problems in business and finance, as well as health care. Qforma merged with SkilaMederi in June 2013. In early 2005 he returned to Vienna where he co-founded The Kenos Circle, a professional society that aims to make use of complexity science in order to gain a deeper insight into the future than that offered by more conventional statistical tools.
For several years, Professor Casti was a Senior Research Scholar at the International Institute for Applied Systems Analysis in Laxenburg, Austria, where he created an initiative for the study on Extreme Events in Human Society. In January 2012 he left IIASA to form a new research institute in Vienna, The X-Center, devoted to the study of human-caused extreme events. The X-Center has now expanded to a network of affiliated X-Centers in Helsinki, Tokyo, Seoul, New York and Singapore. Since early 2013, Dr. Casti has been serving as a Senior Research Fellow at the Center for Complex Systems and Enterprises at the Stevens Institute of Technology in the USA.
As an author, Casti has written more than 120 scientific articles and seven technical monographs and textbooks on mathematical modeling. In addition, he was formerly editor of the journals Applied Mathematics & Computation (Elsevier, New York) and Complexity (Wiley, New York). In 1989 his text/reference works Alternate Realities: Mathematical Models of Nature and Man (Wiley, 1989) was awarded a prize by the Association of American Publishers in a competition among all scholarly books published in mathematics and the natural sciences. In 1992, he also published Reality Rules (Wiley, New York), a t
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Displaying 1 - 12 of 12 reviews
December 6, 2012
The Curse of Knowledge strikes again
JDN 2456268 EDT 14:56.
A review of Five Golden Rules by John Casti.
It's a problem that plagues many nonfiction writers. Steven Pinker called it the Curse of Knowledge; Less Wrong refers to it as Inferential Distance. The problem is this: You know what you know, but you don't know what other people don't know. So it's hard to explain things without going over people's heads or seeming condescending.
Five Golden Rules is supposed to be a book about cutting-edge mathematics for people who don't know a lot of mathematics. It is in fact a book about cutting-edge mathematics for people who do know a lot of mathematics. I got quite a bit out of it, but I've studied abstract algebra and real analysis. Also, some of the topology still confused me, perhaps because I've never formally studied topology. How do you cut a hole in a surface and then stitch the hole closed with a Moebius strip? I'm still confused by higher-dimensional non-orientable surfaces. Also, I still don't quite get the deeper philosophical implications of Godel's incompleteness theorems. I've heard everything from "It undermines rationality itself" to "It's basically trivial". I assume the truth is somewhere in between? (I actually lean more towards the "basically trivial" side of things; a lot of paradoxes really seem like they are more statements about language than they are about truth. "It is raining in Bangladesh but Patrick Julius doesn't know that." "Patrick Julius cannot coherently assert this sentence." Both of these sentences could very well be true, but I can't assert them if they are. Is this a problem?) Casti is quite noncommital on what he thinks Godel implies.
The basic format of the book centers around five seminal branches of 20th century mathematics: game theory, topology, computer science, singularity theory, and linear optimization. If you already have the basic knowledge of each field, you can get a lot out of the way Casti ties everything together with the passion of a real working mathematician. The joy he feels from exploring mathematics can be felt through the words.
But if you don't at least know calculus, this book is going to make very little sense to you. He tries to make it non-mathematical, but fails really quite miserably. It's a much more pleasant read than your average math textbook, but it requires a comparable level of background knowledge.
It's unfortunate really; I'd love to have a book that explains these deep mathematical concepts to people who don't know a lot of math. Unfortunately, Five Golden Rules isn't that book. Instead, it's a useful synthesis and a pleasant read for those of us who already have the necessary background.
JDN 2456268 EDT 14:56.
A review of Five Golden Rules by John Casti.
It's a problem that plagues many nonfiction writers. Steven Pinker called it the Curse of Knowledge; Less Wrong refers to it as Inferential Distance. The problem is this: You know what you know, but you don't know what other people don't know. So it's hard to explain things without going over people's heads or seeming condescending.
Five Golden Rules is supposed to be a book about cutting-edge mathematics for people who don't know a lot of mathematics. It is in fact a book about cutting-edge mathematics for people who do know a lot of mathematics. I got quite a bit out of it, but I've studied abstract algebra and real analysis. Also, some of the topology still confused me, perhaps because I've never formally studied topology. How do you cut a hole in a surface and then stitch the hole closed with a Moebius strip? I'm still confused by higher-dimensional non-orientable surfaces. Also, I still don't quite get the deeper philosophical implications of Godel's incompleteness theorems. I've heard everything from "It undermines rationality itself" to "It's basically trivial". I assume the truth is somewhere in between? (I actually lean more towards the "basically trivial" side of things; a lot of paradoxes really seem like they are more statements about language than they are about truth. "It is raining in Bangladesh but Patrick Julius doesn't know that." "Patrick Julius cannot coherently assert this sentence." Both of these sentences could very well be true, but I can't assert them if they are. Is this a problem?) Casti is quite noncommital on what he thinks Godel implies.
The basic format of the book centers around five seminal branches of 20th century mathematics: game theory, topology, computer science, singularity theory, and linear optimization. If you already have the basic knowledge of each field, you can get a lot out of the way Casti ties everything together with the passion of a real working mathematician. The joy he feels from exploring mathematics can be felt through the words.
But if you don't at least know calculus, this book is going to make very little sense to you. He tries to make it non-mathematical, but fails really quite miserably. It's a much more pleasant read than your average math textbook, but it requires a comparable level of background knowledge.
It's unfortunate really; I'd love to have a book that explains these deep mathematical concepts to people who don't know a lot of math. Unfortunately, Five Golden Rules isn't that book. Instead, it's a useful synthesis and a pleasant read for those of us who already have the necessary background.
September 13, 2022
This was never going to be an 'easy' read. As one much better review put it "Five Golden Rules is supposed to be a book about cutting-edge mathematics for people who don't know a lot of mathematics. It is in fact a book about cutting-edge mathematics for people who do know a lot of mathematics". I did not take the time to immerse myself in too many of the equations but tried to follow the written logic and basic ideas where it interested me. The ideas were very well-presented and there was quite a bit that was interesting, especially where I was more familiar with some aspects. The five theories are: Game Theory (Minimax); Brower Fixed Point theorem (Topology); Morse's Theorem, Catastrophe theory; The Halting Theorem--Computability, Turing, Godel and optimization, Linear/Dynamic Programming, Simplex Method. You get read about great ideas and minds and how these ideas were developed and that is always rewarding even if essential details remain obscure! And at least I have now heard of a Möbius strip and that a coffee cup can be transformed into a donut!
June 28, 2008
This book does a good job of explaining some of the main accomplishments of Mathematics of this last century. The ideas are well developed and explained with many examples - hypothetical and real. (I have read only Chptrs 1,2,5)
December 3, 2023
This book was a dense but informative window into some profound results from mathematical theory. I appreciated the author's active effort in trying to connect the heavy theory to the impact and real life applications of the theories. While I appreciated the pictures and step by step approach used to describe the key results it was very difficult to understand and there were many passages that I had to read multiple times. This was certainly not the type of book where the reader could be distracted or disengaged. However, I felt the knowledge that was earned was worth the effort and made it feel like more of an accomplishment when I was able to internalize what the author was saying and predict the resulting inferences or ways that the math could be used or applied. It was also a throwback to mathematics classes with the backhanded compliment of "a perceptive reader will notice," which I found to be a challenge to be and do better while reading the book. Another saving grace for this book was the decision by the author to limit the length and scope to five key results. Overall, a challenging and enriching read that was incomprehensible at parts but with key results that will change the way I approach problems.
Read
December 29, 2025Once again, a math book that tries to explain some deep topics to a very wide audience, resulting in a mildly unsatisfying level of exposition: not quite easy enough for the lay-person, but not remotely rigorous enough for someone with more mathematical knowledge. The chapters would be good to skim for broad strokes about why a certain theorem/subfield is interesting before moving on to some of the references provided in the "further reading" sections.
Read
February 25, 2010It's not that I didn't enjoy this book, it's just that I can think of very little to say about it. I think I fall in a no-man's-land between the mathematically inclined who would find it dull and the lay reader who would consider the math to be so far over their head that it's not worth trying to understand directly, so an indirect approach, describing the applications of these six major mathematical concepts rather than their inherent logical elegance, would be indicated.
That said, the section I found most interesting was the one on the concept I was most familiar with, the halting problem; I'd encountered it in its barebones incarnation, following trivially from the invention of the universal Turing machine, back in my theory of computation classes, but before that, I'd seen it in Gödel Escher Bach as containing a deep analogy with Cantor's diagonalization and as the final reification of the Incompleteness Theorem. Seeing a topic I knew that well explained as to a novice was like learning it anew again, and I think I wasn't aware of the limits of a lay explanation because I already knew the full version.
The other topics--the minimax theorem, Brouwer's fixed-point theorem, Morse's theorem and the simplex method--are, I'm quite sure, fascinating topics, and I was prompted to go read a bit more on my own, but the presentaiton in the book just doesn't square with my own style of learning and my level of understanding.
That said, the section I found most interesting was the one on the concept I was most familiar with, the halting problem; I'd encountered it in its barebones incarnation, following trivially from the invention of the universal Turing machine, back in my theory of computation classes, but before that, I'd seen it in Gödel Escher Bach as containing a deep analogy with Cantor's diagonalization and as the final reification of the Incompleteness Theorem. Seeing a topic I knew that well explained as to a novice was like learning it anew again, and I think I wasn't aware of the limits of a lay explanation because I already knew the full version.
The other topics--the minimax theorem, Brouwer's fixed-point theorem, Morse's theorem and the simplex method--are, I'm quite sure, fascinating topics, and I was prompted to go read a bit more on my own, but the presentaiton in the book just doesn't square with my own style of learning and my level of understanding.
December 17, 2014
Casti presents five important advance in 20th c. mathematics very clearly with interesting examples. I found the book quite engaging. If you are not very comfortable with mathematics, the book is a bit of a challenge but well worth the challege. I look forward to reading his second volume of important mathematics. Nice read to discover the importance of mathematics and get a feel for what these ideas are. The book also contains some important historical and personality notes.
September 3, 2009
Despite the claims, this book is possibly not an easy read. It does help that I am somewhat familiar with 3 of the 5 golden rules and to a large extent, he did a great job simplifying some of the more complex notions. Highly recommended for those who had exposure in those topics, e.g., Game Theory, Turing machine, etc., and is feeling a little nostalgic, like me.
October 29, 2007
interesting overview of:
* Game Theory (minmax)
* Brower Fixed point theorem (Topology)
* Morse's Singularity Theory, Catastrophe theory
* The halting theorem Computability, Turing, Godel
* Optimization, Linear/Dynamic Programming, Simplex Method
* Game Theory (minmax)
* Brower Fixed point theorem (Topology)
* Morse's Singularity Theory, Catastrophe theory
* The halting theorem Computability, Turing, Godel
* Optimization, Linear/Dynamic Programming, Simplex Method
December 26, 2014
Interesting presentation of 5 recently-developed brahces of math.
July 28, 2014
Took me forever. I understood a bit of the first chapter and it all went downhill from there...
November 26, 2013
A little too dense for what I was looking for
Displaying 1 - 12 of 12 reviews