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Mathematical Fallacies and Paradoxes
From ancient Greek mathematics to 20th-century quantum theory, paradoxes, fallacies and other intellectual inconsistencies have long puzzled and intrigued the mind of man. This stimulating, thought-provoking compilation collects and analyzes the most interesting paradoxes and fallacies from mathematics, logic, physics and language.
While focusing primarily on mathematical issues of the 20th century (notably Godel's theorem of 1931 and decision problems in general), the work takes a look as well at the mind-bending formulations of such brilliant men as Galileo, Leibniz, Georg Cantor and Lewis Carroll ― and describes them in readily accessible detail. Readers will find themselves engrossed in delightful elucidations of methods for misunderstanding the real world by experiment (Aristotle's Circle paradox), being led astray by algebra (De Morgan's paradox), failing to comprehend real events through logic (the Swedish Civil Defense Exercise paradox), mistaking infinity (Euler's paradox), understanding how chance ceases to work in the real world (the Petersburg paradox) and other puzzling problems. Some high school algebra and geometry is assumed; any other math needed is developed in the text. Entertaining and mind-expanding, this volume will appeal to anyone looking for challenging mental exercises.
While focusing primarily on mathematical issues of the 20th century (notably Godel's theorem of 1931 and decision problems in general), the work takes a look as well at the mind-bending formulations of such brilliant men as Galileo, Leibniz, Georg Cantor and Lewis Carroll ― and describes them in readily accessible detail. Readers will find themselves engrossed in delightful elucidations of methods for misunderstanding the real world by experiment (Aristotle's Circle paradox), being led astray by algebra (De Morgan's paradox), failing to comprehend real events through logic (the Swedish Civil Defense Exercise paradox), mistaking infinity (Euler's paradox), understanding how chance ceases to work in the real world (the Petersburg paradox) and other puzzling problems. Some high school algebra and geometry is assumed; any other math needed is developed in the text. Entertaining and mind-expanding, this volume will appeal to anyone looking for challenging mental exercises.
240 pages, Paperback
First published February 1, 1982
14 people are currently reading
149 people want to read
About the author
Bryan Bunch
30 books3 followersBRYAN BUNCH got his B.A. in English (more writing than literature) and taught high-school English and wrote poetry for a year after graduation, then started work in New York publishing as a copyeditor. Shortly after it appeared that the U.S. might lose the space race to the Soviet Union, he returned to his early love of mathematics and began graduate studies, specializing in foundations and logic. This led to changing from reference books to el-high textbooks as a mathematics editor. During 20 years in the textbook field, he also headed the math and science departments, ending his career as editor-in-chief. When he left textbook editing to become a consultant and freelance editor, he left behind with the trade division of his company a proposal for a book, which became published as Mathematical Fallacies and Paradoxes (still in print in a Dover edition). Joining The Hudson Group of writers led to freelance work on reference books. Bryan soon started Scientific Publishing, Inc., devoted to using computers in publishing (the PC was brand new at the time; he bought the first one produced by IBM as soon as it was available) and focusing on current science, science history, and medicine. Bryan ran scientific publishing for 25 years before retiring--to a degree, as he continues to write.
BOOKS BY BRYAN BUNCH
201? Figures among the Stars (YA novel about adventures of young Archimedes and his friends)
2004 The History of Science and Technology, Houghton-Mifflin (with Alexander
Hellemans); also available in leather-bound Easton Press edition and a Kindle edition.
2003 Discover Science Almanac, Hyperion (with Jenny Tesar)
2001 Blackbirch Encyclopedia of Science and Invention, Blackbirch Press (with Jenny Tesar).
2000 Penguin Desk Encyclopedia of Science and Mathematics, Penguin U.S.A. (with Jenny Tesar--named by AAAS Science Books as one of the three best science reference books of the year)
2000 The Kingdom of Infinite Number: A Field Guide, W.H. Freeman (Natural Science Book Club selection; New York Public Library 2001 Books for Teen Age List), also available for Kindle
1998 Satellites and Probes (with Clint Hachett; vol. 12 of Outer Space), Grolier. Highly recommended by Book Report November/December 1998.
1998 Family Encyclopedia of Disease, W. H. Freeman (Editor).
1997 Mathematical Fallacies and Paradoxes, Dover Publications (paperbound)
1996 The Globe Junior High School Science Series, New Revised Edition, Globe Book Company (Senior Author).
Handbook of Current Science and Technology (Gale Research).
1994 Handbook of Current Health and Medicine (Gale Research).
1993 The Timetables of Technology (with Alexander Hellemans), Simon and Schuster; also available as a CD ROM in 1997.
1992 The Henry Holt Handbook of Current Science and Technology, Henry Holt and Company; listed by Library Journal as one of the best science books of 1992 and by the American Librarians Association among the best reference sources of 1992.
1991 The Timetables of Science, new, updated edition (with Alexander Hellemans), Touchstone Books (Library of Science Book Club, Quality Paperback Book Club)
1990 Excel in Graphing Level H (with Margaret Hill), Modern Curriculum Press.
1989 Reality's Mirror: Exploring the Mathematics of Symmetry, John Wiley; listed by Library Journal as one of the best science books of 1989; also published in Japan in Japanese.
1988 The Timetables of Science (with Alexander Hellemans), Simon and Schuster; also published in England, in Germany (in German), in Japan (in Japanese), and Romania (in Rumanian).
1986 The Globe Junior High School Science Series, Globe Book Company (Senior Author).
1985 Harper & Row Elementary Mathematics, Grades K 8, Macmillan McGraw (Co author).
1984 The Science Almanac, Doubleday
1984 A Practical Herb Garden (with recipes), TAB Books.
1983 Algebra One, McDougal, Littell (Co author). Fun with Math, World Book Childcraft.
1982 Mathematical Fallacies and Paradoxes, Van Nostrand Reinhold
BOOKS BY BRYAN BUNCH
201? Figures among the Stars (YA novel about adventures of young Archimedes and his friends)
2004 The History of Science and Technology, Houghton-Mifflin (with Alexander
Hellemans); also available in leather-bound Easton Press edition and a Kindle edition.
2003 Discover Science Almanac, Hyperion (with Jenny Tesar)
2001 Blackbirch Encyclopedia of Science and Invention, Blackbirch Press (with Jenny Tesar).
2000 Penguin Desk Encyclopedia of Science and Mathematics, Penguin U.S.A. (with Jenny Tesar--named by AAAS Science Books as one of the three best science reference books of the year)
2000 The Kingdom of Infinite Number: A Field Guide, W.H. Freeman (Natural Science Book Club selection; New York Public Library 2001 Books for Teen Age List), also available for Kindle
1998 Satellites and Probes (with Clint Hachett; vol. 12 of Outer Space), Grolier. Highly recommended by Book Report November/December 1998.
1998 Family Encyclopedia of Disease, W. H. Freeman (Editor).
1997 Mathematical Fallacies and Paradoxes, Dover Publications (paperbound)
1996 The Globe Junior High School Science Series, New Revised Edition, Globe Book Company (Senior Author).
Handbook of Current Science and Technology (Gale Research).
1994 Handbook of Current Health and Medicine (Gale Research).
1993 The Timetables of Technology (with Alexander Hellemans), Simon and Schuster; also available as a CD ROM in 1997.
1992 The Henry Holt Handbook of Current Science and Technology, Henry Holt and Company; listed by Library Journal as one of the best science books of 1992 and by the American Librarians Association among the best reference sources of 1992.
1991 The Timetables of Science, new, updated edition (with Alexander Hellemans), Touchstone Books (Library of Science Book Club, Quality Paperback Book Club)
1990 Excel in Graphing Level H (with Margaret Hill), Modern Curriculum Press.
1989 Reality's Mirror: Exploring the Mathematics of Symmetry, John Wiley; listed by Library Journal as one of the best science books of 1989; also published in Japan in Japanese.
1988 The Timetables of Science (with Alexander Hellemans), Simon and Schuster; also published in England, in Germany (in German), in Japan (in Japanese), and Romania (in Rumanian).
1986 The Globe Junior High School Science Series, Globe Book Company (Senior Author).
1985 Harper & Row Elementary Mathematics, Grades K 8, Macmillan McGraw (Co author).
1984 The Science Almanac, Doubleday
1984 A Practical Herb Garden (with recipes), TAB Books.
1983 Algebra One, McDougal, Littell (Co author). Fun with Math, World Book Childcraft.
1982 Mathematical Fallacies and Paradoxes, Van Nostrand Reinhold
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Displaying 1 - 6 of 6 reviews
February 8, 2021
Really dense at times (at least to my brain), but fun to read and think about. Intro to the book says the math is no more than high school level, and in notation that’s true (majority is sequences and series/elementary logic), but without familiarity of higher-level (college) discrete and continuous math some concepts will be hard to get through. A decent amount of similarity to GEB, but way shorter and more math-y.
April 1, 2021
I read this on a plug from Grant Sanderson (of 3Blue1Brown fame). It didn't really live up to my hopes and generally lacked consistency. There are certainly some interesting takeaways and highlights in the later chapters — relating to Gödel Incompleteness, General Relativity, and Set Theory. However, the parts that felt most interesting to me are the ones that draw heavily from Douglas Hofstadter's work. I think I would be better served reading those sections of GEB again instead of having Bunch interpret them. But this version is certainly far more concise (and so might make for a decent primer for some readers).
The first 4 or 5 chapters felt overly simplistic and featured excessive arithmetic working. I enjoyed much of the following chapters, though:
6. The Limits of Thought
7. Misunderstanding Space and Time
8. Moving Against Infinity
Most readers with a decent background in theoretical CS or mathematics would probably be best served just skipping forward to chapter 5 or 6 right away.
Selected highlights from the book:
"What to do with a paradox ? If you are sure that no contradiction results, incorporate the paradox into mathematics and declare it a paradox no longer."
"the real numbers cannot be put into a one-to-one correspondence with the natural numbers. There are two kinds of infinity after all!"
"the Cantor paradox. Cantor had shown that for any set whatsoever, the set of subsets of the set contains more members than the set itself. What about the set of all sets? Since the set of all sets includes all possible sets, each of its subsets must be members of it. So there cannot be more subsets than there are members of the set of all sets."
"Axiomatic set theory explicitly eliminates the known paradoxes, but cannot be shown to be consistent. Therefore, other paradoxes can occur at any time."
"you can show that if you can compute the Gödel number of a particular proof by following the rules for computing Gödel numbers from the axioms, then that statement can be proved. What Gödel did, however, was to compute the Gödel number of a statement that says “This statement cannot be proved.” [...] there exists, for any axiomatic system that is strong enough to derive the natural numbers, a statement such that neither its truth nor its falsity can be proved, unless the system is inconsistent. The last sentence is Gödel’s theorem—technically, Gödel’s incompleteness theorem, since he is responsible for several other important results of modern mathematics."
"Instead of very restrictive methods, such as the straightedge and the compass, mathematicians began to show that problems existed that could not be solved by very general methods. In fact, several mathematicians described such general methods that they came to believe that these methods were equivalent to all possible methods. This belief, known as the Church-Turing thesis (after Alonzo Church and Alan Turing), can be more specifically stated as “The only method of calculating a number or set of numbers in a finite number of steps is to use the class of methods that have been identified already.” [...] most mathematicians believe that the Church-Turing thesis is true (although it cannot be proved to be true)."
"What Banach and Tarski showed is that: If you have two geometric figures in a Euclidean space that consist of an ordinary outside surface and some points inside, then there is a finite decomposition that shows that they are equivalent. (The same situation exists for the surface of a sphere.) When this result is reported to nonmathematicians, it is often stated something like this: Banach and Tarski proved that it was possible to disassemble a small sphere, such as a pea or a grapefruit, and reassemble it with no interior gaps into a sphere the size of the earth or the sun. Indeed, that is one of the consequences of the theorem."
"according to Albert Einstein’s theories of relativity, you do not live in Euclidean three-space. You live in Minkowski four-space. [...] Although Minkowski space resembles Euclidean space in many ways, the concept of length is quite different."
"The Dichotomy paradox concerns a traveler who is to walk a certain distance. First, he or she must walk half the distance, then half of what remains, then half of what remains, and so forth. It is clear that there will always be half of the remaining distance to go to any point during the walk. Thus, the person can never complete the walk."
"The mathematician’s view depends again on the reality of the continuum. But what physicists have found out about the real world casts great doubt on that point of view. In 1900, Max Planck discovered—somewhat to his regret—that energy is not continuous. Instead of being available along the continuum (in any amount one could calculate), energy is only available in small packets—quanta."
"Zeno recognized early on that the mathematical way of looking at the world and the scientific way of looking at the world produced contradictory results. As mathematics has grown independently (to some degree) over the centuries, it has been necessary again and again to change the rules slightly so that mathematical paradoxes will become mere fallacies. But from the beginning, from Zeno’s time, it was clear that mathematics does not correspond exactly to the real world. Of course, this does not mean that mathematics is in any way not useful in describing or discovering the real world. It certainly is. But what is discovered, as in the case of quanta, may not fit with mathematics. If you assume that an arrow behaves like a collection of mathematical points, you can use mathematics to describe its motion. If you concentrate on the arrow being a finite collection of small packets of energy, none of which can be located at a particular mathematical point, then you are up the creek. This situation remains an unresolved paradox."
The first 4 or 5 chapters felt overly simplistic and featured excessive arithmetic working. I enjoyed much of the following chapters, though:
6. The Limits of Thought
7. Misunderstanding Space and Time
8. Moving Against Infinity
Most readers with a decent background in theoretical CS or mathematics would probably be best served just skipping forward to chapter 5 or 6 right away.
Selected highlights from the book:
"What to do with a paradox ? If you are sure that no contradiction results, incorporate the paradox into mathematics and declare it a paradox no longer."
"the real numbers cannot be put into a one-to-one correspondence with the natural numbers. There are two kinds of infinity after all!"
"the Cantor paradox. Cantor had shown that for any set whatsoever, the set of subsets of the set contains more members than the set itself. What about the set of all sets? Since the set of all sets includes all possible sets, each of its subsets must be members of it. So there cannot be more subsets than there are members of the set of all sets."
"Axiomatic set theory explicitly eliminates the known paradoxes, but cannot be shown to be consistent. Therefore, other paradoxes can occur at any time."
"you can show that if you can compute the Gödel number of a particular proof by following the rules for computing Gödel numbers from the axioms, then that statement can be proved. What Gödel did, however, was to compute the Gödel number of a statement that says “This statement cannot be proved.” [...] there exists, for any axiomatic system that is strong enough to derive the natural numbers, a statement such that neither its truth nor its falsity can be proved, unless the system is inconsistent. The last sentence is Gödel’s theorem—technically, Gödel’s incompleteness theorem, since he is responsible for several other important results of modern mathematics."
"Instead of very restrictive methods, such as the straightedge and the compass, mathematicians began to show that problems existed that could not be solved by very general methods. In fact, several mathematicians described such general methods that they came to believe that these methods were equivalent to all possible methods. This belief, known as the Church-Turing thesis (after Alonzo Church and Alan Turing), can be more specifically stated as “The only method of calculating a number or set of numbers in a finite number of steps is to use the class of methods that have been identified already.” [...] most mathematicians believe that the Church-Turing thesis is true (although it cannot be proved to be true)."
"What Banach and Tarski showed is that: If you have two geometric figures in a Euclidean space that consist of an ordinary outside surface and some points inside, then there is a finite decomposition that shows that they are equivalent. (The same situation exists for the surface of a sphere.) When this result is reported to nonmathematicians, it is often stated something like this: Banach and Tarski proved that it was possible to disassemble a small sphere, such as a pea or a grapefruit, and reassemble it with no interior gaps into a sphere the size of the earth or the sun. Indeed, that is one of the consequences of the theorem."
"according to Albert Einstein’s theories of relativity, you do not live in Euclidean three-space. You live in Minkowski four-space. [...] Although Minkowski space resembles Euclidean space in many ways, the concept of length is quite different."
"The Dichotomy paradox concerns a traveler who is to walk a certain distance. First, he or she must walk half the distance, then half of what remains, then half of what remains, and so forth. It is clear that there will always be half of the remaining distance to go to any point during the walk. Thus, the person can never complete the walk."
"The mathematician’s view depends again on the reality of the continuum. But what physicists have found out about the real world casts great doubt on that point of view. In 1900, Max Planck discovered—somewhat to his regret—that energy is not continuous. Instead of being available along the continuum (in any amount one could calculate), energy is only available in small packets—quanta."
"Zeno recognized early on that the mathematical way of looking at the world and the scientific way of looking at the world produced contradictory results. As mathematics has grown independently (to some degree) over the centuries, it has been necessary again and again to change the rules slightly so that mathematical paradoxes will become mere fallacies. But from the beginning, from Zeno’s time, it was clear that mathematics does not correspond exactly to the real world. Of course, this does not mean that mathematics is in any way not useful in describing or discovering the real world. It certainly is. But what is discovered, as in the case of quanta, may not fit with mathematics. If you assume that an arrow behaves like a collection of mathematical points, you can use mathematics to describe its motion. If you concentrate on the arrow being a finite collection of small packets of energy, none of which can be located at a particular mathematical point, then you are up the creek. This situation remains an unresolved paradox."
July 16, 2019
Some very good bits, but inconsistent as far as variety of examples and many classic problems are missing.
The book would have been better structured around the more unique examples and their history rather than presented as a puzzler.
The book would have been better structured around the more unique examples and their history rather than presented as a puzzler.
April 3, 2013
Perfect for those without much mathematical background but determined to change that. Typos here and there, but the manner in which the author covers his material is refreshing -- informative, comprehensive, accurate, but does not risk scaring the layman/philosopher away.
As for it's "pop"-ness, the style is quite honest with its own material, relative to other pop-mathematics books I've skimmed or seen. There is little over-hyping (though mathematics needs none of that) of the various paradoxes it discusses, and Bunch does get down and dirty with a bit of proper logic to argue its points. Where it is out of its depth, Bunch is not afraid to inform the reader, rather than giving the reader a false sense of deep mathematical understanding.
There is a list of selected readings at the end of the book, itself also quite good.
As for it's "pop"-ness, the style is quite honest with its own material, relative to other pop-mathematics books I've skimmed or seen. There is little over-hyping (though mathematics needs none of that) of the various paradoxes it discusses, and Bunch does get down and dirty with a bit of proper logic to argue its points. Where it is out of its depth, Bunch is not afraid to inform the reader, rather than giving the reader a false sense of deep mathematical understanding.
There is a list of selected readings at the end of the book, itself also quite good.
May 22, 2010
The beginning pages are a little too elementary, though they set good groundwork for the paradoxes to come. Once the book gets into the problems of infinity and logic vs. mathematical logic, then we got some good stuff happening that makes for intriguing reading.
October 31, 2015
Some interesting ideas, a lot of stuff I've covered before. A decent read.
Displaying 1 - 6 of 6 reviews