There is a big difference between what mathematicians call a fallacy and what they call a paradox. A fallacy is a flawed proof, such as the “proof” on page fifty-four of the book under review, that all triangles are isosceles. A paradox is an assertion almost impossible to believe but nevertheless true. A good example is the famous twin paradox of relativity. If one twin travels a long distance from the earth, at a fast speed, then returns, she’ll be younger than her stay-at-home sister. The time difference can be arbitrarily large. If the traveling twin goes to a distant galaxy at a velocity near that of light, then returns, thousands of earth years could have gone by. Time travel into the far future (not into the past) is a genuine possibility!
The twin paradox, which incidentally has been empirically confirmed, is not hard to comprehend if one is familiar with the time dilation of relativity theory. Far more mind-blowing is a mathematical result known as the Banach-Tarski paradox after two Polish mathematicians, Stefan Banach and Alfred Tarski.
Tarski is best known for his semantic theory of truth. It eliminates such logical paradoxes as “This statement is false” by forbidding a language to talk about the truth or falsity of statements in the same language. Tarski’s famous example is “Snow is white” is true if and only if snow is white. The sentence inside quotes is in a metalanguage asserting the truth of a statement in the language of physical realism. To talk about true or false in a metalanguage requires a meta-metalanguage, and so on into an infinite hierarchy of languages.
Tarski’s way of defining truth has the great merit of applying both to the language of science and to languages of logic and mathematics. It played a key role in the fading of efforts by followers of Dewey and William James to define truth as the passing of tests for truth.
The original version of the Banach-Tarski (BT) paradox shows how a solid sphere can be sliced into a finite number of point sets that can be shifted about to make two balls identical in shape and size to the original! Raphael Robinson, an American mathematician, reduced the required number of point sets to five—four, each with an infinity of points, and a single point from the original sphere’s center.
Other forms of the BT paradox are even more counterintuitive. Each of the two magically created balls can be dissected into point sets that can be rearranged to make four balls, and this magnification can be continued to produce as many balls as one pleases. In a still crazier version, a tiny sphere can be cut into point sets that recombine to make a sphere of any size—hence the book’s title, The Pea and the Sun. Even worse, the two objects may be of any size or shape. As the author puts it, a mosquito can be transformed into an elephant!
It must be said at once that such miraculous changes can only be done in theory. There is no way to slice an apple into parts that will make two apples or an elephant, no way, as Wapner writes, to multiply gold or enlarge loaves and fishes to feed starving millions. The BT paradoxes occur only within an abstract system of mathematics which deals with what are called “transfinite sets”—sets first recognized by the great German mathematician Georg Cantor.
Leonard Wapner, professor of mathematics at El Camino College in Torrance, California, makes the BT paradox the centerpiece of a marvelous book (his first, by the way). Chapters proving the paradox are tough going for mathematically uninformed readers, but don’t let that put you off if you are such a person. Wapner has surrounded his central theme with a wealth of easily understood topics, much of it recreational, related in some way to the BT paradox.
The book opens with a crystal-clear introduction to Cantor’s infinite ladder of transfinite numbers. Cantor called the lowest number aleph-zero. It counts all rational numbers—the natural numbers 1, 2, 3, … and their integral fractions. Like all transfinite numbers, the set has the bizarre property that its points can be put into one-to-one correspondence with an infinite portion of itself. Even Galileo knew this curious property. He noticed, perhaps with surprise, that the counting numbers could be paired one to one with a subset such as the square numbers.
Of course, the counting numbers can be paired with even “smaller” infinite sets such as the prime numbers. Although in one sense there are far more counting numbers than squares, there is another sense in which their numbers are the same. This strange property underlies a story, well retold by Wapner, about tasks facing the manager of Hotel Infinity.
The hotel’s endless numbers of rooms are all occupied when ten travelers arrive, each demanding a room. No problem. The manager simply shifts everyone from a room n to room n+10. This leaves the first ten rooms vacant. Next week an infinity of travelers arrives, each wanting a room. Again, no problem. The clever manager moves each guest to a room with a number twice that of his former room. This opens all odd-numbered rooms.
Cantor was able to show that aleph-one, the next higher number to aleph-zero, counted sets that could not be put into one-to-one correspondence with the rational numbers. It was, therefore, a higher infinity. Cantor did this by an ingenious “diagonal” technique.
Aleph-one counts all the real numbers—the rationals plus the irrationals such as pi and the square root of 2. The number of points on a line is aleph-one. So is the number of points on a square, or a cube, or any solid object of higher dimensions. Aleph-two counts all the subsets of aleph-one. Aleph-three counts the subsets of aleph-two. Each aleph can be expressed by 2 raised to the power of its preceding aleph. The ladder of alephs is infinite, although beyond aleph-three the alephs have almost no mathematical uses.
A tantalizing question arises. Is it possible that there are transfinite numbers between two alephs, especially between aleph-one and aleph-two? Cantor tried without success to prove there are no such in-betweens. As Wapner informs us, the question was not laid to rest until Stanford University’s Paul Cohen proved that the problem is Gödel-undecidable within standard set theory. One may assume without contradiction that no such numbers exist, or one can assume an infinity of in-between numbers.
Cohen also startled his colleagues with another unexpected result. The BT paradox requires for its proof a notorious axiom called the “axiom of choice.” It asserts that from any number of sets you can always select exactly one member of each set to create a new set. This obviously can be done with a finite or an infinite number of finite sets, but when each set is also infinite, severe difficulties arise.
Bertrand Russell, Wapner tells us, explained this by considering shoes and socks. From an infinite set of pairs of shoes you can take, say, a right shoe from each pair to form another infinite set. But if each set consists of an infinity of identical socks, there is no clear rule about how to select exactly one sock from each set. The axiom allows you to do this even though you can’t specify exactly how to do it.
The BT paradox is one of many point-set paradoxes that cannot be proved without the axiom of choice. Cohen was able to show that a consistent set theory may include or exclude the axiom of choice. It remains a mysterious “existence” axiom independent of all the other axioms of standard set theory.
Although Cantor’s alephs no longer worry today’s mathematicians, Wapner cites several mathematicians of the past who regarded Cantor’s alephs as mystical nonsense. Henri Poincaré called them a “malady, a perverse illness from which one day mathematics would be cured.” Wapner also quotes Hermann Weyl’s description of Cantor’s ladder as “fog on fog.” Leopold Kronecker, Cantor’s former Berlin teacher, branded Cantor a “charlatan” and “corrupter of youth.”
Cantor, Wapner reveals, was a devout Protestant who believed that his work was inspired by God. Outside of mathematics he held wild opinions, spending his later years trying to convince the world that Francis Bacon wrote all of Shakespeare’s plays. After several nervous breakdowns, he ended his days in a mental facility.
Several chapters in Wapner’s book cover entertaining topics that in some way resemble the BT paradox. One chapter is titled “Baby BTs.” A variety of geometrical paradoxes are presented in which a polygon is cut into a small number of parts that can be reassembled to make another polygon with a different area! Paul Curry, an amateur Manhattan magician, was the first to show how pieces of a polygon could be rearranged to form a seemingly identical polygon with a hole!
Wapner reproduces puzzle-maker Sam Loyd’s bewildering paradox of a vanishing Chinese warrior. Fourteen warriors are around the rim of a disc. After a small rotation of the disc, one warrior vanishes. A linear version of the paradox, also reproduced by Wapner, involves fifteen leprechauns in a row. By switching two rectangular pieces, a leprechaun totally disappears. Which one vanished? And where did the little fellow go? In another chapter Wapner displays a number of beautiful dissections in which a polygon is cut into parts which reform to make a polygon of different shape. He outlines a lovely proof that any given polygon can be dissected into a finite number of pieces that will form any desired different polygon of the same area. There is a companion proof, described by Wapner, that similar transformations cannot be done with certain solid objects.
An old counterfeiting method is explained. It allows a thief to slice each of nine currency bills into two parts; then the parts can be rearranged to make ten bills. It’s almost a BT paradox! Bill numbers on left and right sides, and top and bottom, are there precisely to foil this technique for multiplying currency.
After a detailed proof of the BT paradoxes, Wapner turns his attention to some speculations about the future of mathematics. He considers the implications of Moore’s Law, which predicts that every eighteen months computer power will double. The incredible speed of today’s supercomputers has led to what Wapner calls “experimental mathematics.” The computer has become a tool, like a telescope or a microscope, for testing and even suggesting conjectures. Occasionally a computer proof will require a printout so vast it can be checked only by another computer. Some computer programs will not validate a proof, but establish a result with only a very high degree of probability. The resemblance to science is obvious. Future computer “proofs” may end, Wapner jokingly writes, not with Q.E.D., but with “You can bet on it—trust me!”
Tomorrow’s computers, Wapner believes, will be enormously faster. They may use light instead of electricity to twiddle symbols. They may exploit the properties of DNA. Looming large on the horizon are quantum computers capable of speeds Wapner calls “astronomical.”
Wapner ends his book with a profound question which science is nowhere close to answering: Will Cantor’s alephs and the BT paradoxes ever find applications in the physical world? To my astonishment, Wapner introduces three scientists who are seriously speculating on just such applications. Two American physicists, Roger S. Jones and Bruno Augenstein, conjecture that the BT paradox may actually play a role in the behavior of hadrons! How can a muon be exactly like an electron except that it is larger, heavier, and short-lived? Has an electron been magnified by something akin to BT magnification?
The third scientist is the astrophysicist M. S. El Naschie, at the University of Cairo. Wapner lists two of his papers in his seven-page bibliography. Naschie wonders if a BT magnification may have produced the Big Bang. If the universe ever stops expanding, and goes the other way toward a Big Crunch, perhaps BT compression will be at work!
In considering such fantastic speculations it is good to realize that transfinite sets seem to “exist” only in the Platonic world of pure mathematics. Only in that world can a mosquito’s infinity of points be put into one-to-one correspondence with the points of an elephant. What is so amazing is that in Plato’s realm a mosquito can be cut into a finite number of parts that will reassemble to make an elephant. In the real world, of course, no material object has an infinity of points. Indeed, it has no points at all. Points exist only in formal mathematical systems. Matter is made of molecules, in turn made of atoms, in turn made of particles which could be vibrating loops of string. Almost all of a material object is empty space.
The “Go-Go Principle” permits anything to occur that is not logically forbidden. Wapner reminds us that both relativity and quantum mechanics bristle with paradoxes that are extreme violations of common sense but which are known to be true. Perhaps someday scientists may discover that Nature knows all about transfinite sets and BT paradoxes. On some level, far below quantum mechanics and possible superstrings, or in dimensions high above those we know, Einstein’s Old One may be juggling Cantor’s alephs in ways we cannot yet—perhaps never can—comprehend.
The twin paradox, which incidentally has been empirically confirmed, is not hard to comprehend if one is familiar with the time dilation of relativity theory. Far more mind-blowing is a mathematical result known as the Banach-Tarski paradox after two Polish mathematicians, Stefan Banach and Alfred Tarski.
Tarski is best known for his semantic theory of truth. It eliminates such logical paradoxes as “This statement is false” by forbidding a language to talk about the truth or falsity of statements in the same language. Tarski’s famous example is “Snow is white” is true if and only if snow is white. The sentence inside quotes is in a metalanguage asserting the truth of a statement in the language of physical realism. To talk about true or false in a metalanguage requires a meta-metalanguage, and so on into an infinite hierarchy of languages.
Tarski’s way of defining truth has the great merit of applying both to the language of science and to languages of logic and mathematics. It played a key role in the fading of efforts by followers of Dewey and William James to define truth as the passing of tests for truth.
The original version of the Banach-Tarski (BT) paradox shows how a solid sphere can be sliced into a finite number of point sets that can be shifted about to make two balls identical in shape and size to the original! Raphael Robinson, an American mathematician, reduced the required number of point sets to five—four, each with an infinity of points, and a single point from the original sphere’s center.
Other forms of the BT paradox are even more counterintuitive. Each of the two magically created balls can be dissected into point sets that can be rearranged to make four balls, and this magnification can be continued to produce as many balls as one pleases. In a still crazier version, a tiny sphere can be cut into point sets that recombine to make a sphere of any size—hence the book’s title, The Pea and the Sun. Even worse, the two objects may be of any size or shape. As the author puts it, a mosquito can be transformed into an elephant!
It must be said at once that such miraculous changes can only be done in theory. There is no way to slice an apple into parts that will make two apples or an elephant, no way, as Wapner writes, to multiply gold or enlarge loaves and fishes to feed starving millions. The BT paradoxes occur only within an abstract system of mathematics which deals with what are called “transfinite sets”—sets first recognized by the great German mathematician Georg Cantor.
Leonard Wapner, professor of mathematics at El Camino College in Torrance, California, makes the BT paradox the centerpiece of a marvelous book (his first, by the way). Chapters proving the paradox are tough going for mathematically uninformed readers, but don’t let that put you off if you are such a person. Wapner has surrounded his central theme with a wealth of easily understood topics, much of it recreational, related in some way to the BT paradox.
The book opens with a crystal-clear introduction to Cantor’s infinite ladder of transfinite numbers. Cantor called the lowest number aleph-zero. It counts all rational numbers—the natural numbers 1, 2, 3, … and their integral fractions. Like all transfinite numbers, the set has the bizarre property that its points can be put into one-to-one correspondence with an infinite portion of itself. Even Galileo knew this curious property. He noticed, perhaps with surprise, that the counting numbers could be paired one to one with a subset such as the square numbers.
Of course, the counting numbers can be paired with even “smaller” infinite sets such as the prime numbers. Although in one sense there are far more counting numbers than squares, there is another sense in which their numbers are the same. This strange property underlies a story, well retold by Wapner, about tasks facing the manager of Hotel Infinity.
The hotel’s endless numbers of rooms are all occupied when ten travelers arrive, each demanding a room. No problem. The manager simply shifts everyone from a room n to room n+10. This leaves the first ten rooms vacant. Next week an infinity of travelers arrives, each wanting a room. Again, no problem. The clever manager moves each guest to a room with a number twice that of his former room. This opens all odd-numbered rooms.
Cantor was able to show that aleph-one, the next higher number to aleph-zero, counted sets that could not be put into one-to-one correspondence with the rational numbers. It was, therefore, a higher infinity. Cantor did this by an ingenious “diagonal” technique.
Aleph-one counts all the real numbers—the rationals plus the irrationals such as pi and the square root of 2. The number of points on a line is aleph-one. So is the number of points on a square, or a cube, or any solid object of higher dimensions. Aleph-two counts all the subsets of aleph-one. Aleph-three counts the subsets of aleph-two. Each aleph can be expressed by 2 raised to the power of its preceding aleph. The ladder of alephs is infinite, although beyond aleph-three the alephs have almost no mathematical uses.
A tantalizing question arises. Is it possible that there are transfinite numbers between two alephs, especially between aleph-one and aleph-two? Cantor tried without success to prove there are no such in-betweens. As Wapner informs us, the question was not laid to rest until Stanford University’s Paul Cohen proved that the problem is Gödel-undecidable within standard set theory. One may assume without contradiction that no such numbers exist, or one can assume an infinity of in-between numbers.
Cohen also startled his colleagues with another unexpected result. The BT paradox requires for its proof a notorious axiom called the “axiom of choice.” It asserts that from any number of sets you can always select exactly one member of each set to create a new set. This obviously can be done with a finite or an infinite number of finite sets, but when each set is also infinite, severe difficulties arise.
Bertrand Russell, Wapner tells us, explained this by considering shoes and socks. From an infinite set of pairs of shoes you can take, say, a right shoe from each pair to form another infinite set. But if each set consists of an infinity of identical socks, there is no clear rule about how to select exactly one sock from each set. The axiom allows you to do this even though you can’t specify exactly how to do it.
The BT paradox is one of many point-set paradoxes that cannot be proved without the axiom of choice. Cohen was able to show that a consistent set theory may include or exclude the axiom of choice. It remains a mysterious “existence” axiom independent of all the other axioms of standard set theory.
Although Cantor’s alephs no longer worry today’s mathematicians, Wapner cites several mathematicians of the past who regarded Cantor’s alephs as mystical nonsense. Henri Poincaré called them a “malady, a perverse illness from which one day mathematics would be cured.” Wapner also quotes Hermann Weyl’s description of Cantor’s ladder as “fog on fog.” Leopold Kronecker, Cantor’s former Berlin teacher, branded Cantor a “charlatan” and “corrupter of youth.”
Cantor, Wapner reveals, was a devout Protestant who believed that his work was inspired by God. Outside of mathematics he held wild opinions, spending his later years trying to convince the world that Francis Bacon wrote all of Shakespeare’s plays. After several nervous breakdowns, he ended his days in a mental facility.
Several chapters in Wapner’s book cover entertaining topics that in some way resemble the BT paradox. One chapter is titled “Baby BTs.” A variety of geometrical paradoxes are presented in which a polygon is cut into a small number of parts that can be reassembled to make another polygon with a different area! Paul Curry, an amateur Manhattan magician, was the first to show how pieces of a polygon could be rearranged to form a seemingly identical polygon with a hole!
Wapner reproduces puzzle-maker Sam Loyd’s bewildering paradox of a vanishing Chinese warrior. Fourteen warriors are around the rim of a disc. After a small rotation of the disc, one warrior vanishes. A linear version of the paradox, also reproduced by Wapner, involves fifteen leprechauns in a row. By switching two rectangular pieces, a leprechaun totally disappears. Which one vanished? And where did the little fellow go? In another chapter Wapner displays a number of beautiful dissections in which a polygon is cut into parts which reform to make a polygon of different shape. He outlines a lovely proof that any given polygon can be dissected into a finite number of pieces that will form any desired different polygon of the same area. There is a companion proof, described by Wapner, that similar transformations cannot be done with certain solid objects.
An old counterfeiting method is explained. It allows a thief to slice each of nine currency bills into two parts; then the parts can be rearranged to make ten bills. It’s almost a BT paradox! Bill numbers on left and right sides, and top and bottom, are there precisely to foil this technique for multiplying currency.
After a detailed proof of the BT paradoxes, Wapner turns his attention to some speculations about the future of mathematics. He considers the implications of Moore’s Law, which predicts that every eighteen months computer power will double. The incredible speed of today’s supercomputers has led to what Wapner calls “experimental mathematics.” The computer has become a tool, like a telescope or a microscope, for testing and even suggesting conjectures. Occasionally a computer proof will require a printout so vast it can be checked only by another computer. Some computer programs will not validate a proof, but establish a result with only a very high degree of probability. The resemblance to science is obvious. Future computer “proofs” may end, Wapner jokingly writes, not with Q.E.D., but with “You can bet on it—trust me!”
Tomorrow’s computers, Wapner believes, will be enormously faster. They may use light instead of electricity to twiddle symbols. They may exploit the properties of DNA. Looming large on the horizon are quantum computers capable of speeds Wapner calls “astronomical.”
Wapner ends his book with a profound question which science is nowhere close to answering: Will Cantor’s alephs and the BT paradoxes ever find applications in the physical world? To my astonishment, Wapner introduces three scientists who are seriously speculating on just such applications. Two American physicists, Roger S. Jones and Bruno Augenstein, conjecture that the BT paradox may actually play a role in the behavior of hadrons! How can a muon be exactly like an electron except that it is larger, heavier, and short-lived? Has an electron been magnified by something akin to BT magnification?
The third scientist is the astrophysicist M. S. El Naschie, at the University of Cairo. Wapner lists two of his papers in his seven-page bibliography. Naschie wonders if a BT magnification may have produced the Big Bang. If the universe ever stops expanding, and goes the other way toward a Big Crunch, perhaps BT compression will be at work!
In considering such fantastic speculations it is good to realize that transfinite sets seem to “exist” only in the Platonic world of pure mathematics. Only in that world can a mosquito’s infinity of points be put into one-to-one correspondence with the points of an elephant. What is so amazing is that in Plato’s realm a mosquito can be cut into a finite number of parts that will reassemble to make an elephant. In the real world, of course, no material object has an infinity of points. Indeed, it has no points at all. Points exist only in formal mathematical systems. Matter is made of molecules, in turn made of atoms, in turn made of particles which could be vibrating loops of string. Almost all of a material object is empty space.
The “Go-Go Principle” permits anything to occur that is not logically forbidden. Wapner reminds us that both relativity and quantum mechanics bristle with paradoxes that are extreme violations of common sense but which are known to be true. Perhaps someday scientists may discover that Nature knows all about transfinite sets and BT paradoxes. On some level, far below quantum mechanics and possible superstrings, or in dimensions high above those we know, Einstein’s Old One may be juggling Cantor’s alephs in ways we cannot yet—perhaps never can—comprehend.
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