What do you think?
Rate this book
Το μπιζέλι και ο ήλιος: Ένα μαθηματικό παράδοξο: Το θεώρημα Banach-Tarski
Μπορούμε, αλήθεια, να κόψουμε ένα μήλο σε πέντε κομμάτια, να τα ξανα-ενώσουμε και να φτιάξουμε δύο μήλα ολόιδια με το αρχικό;
Τι είναι τα μη μετρήσιμα σύνολα; Τι είναι το αξίωμα της επιλογής; Υπάρχουν πολλά άπειρα και τι είναι οι υπερπεπερασμένοι αριθμοί; Ποια η σχέση του παράδοξου αυτού με τη Θεωρητική Φυσική και την Κοσμολογία; Μπορεί αυτό το θεώρημα να έχει εφαρμογές στην πραγματικότητα;
Το βιβλίο αυτό του Λέοναρντ Γουάπνερ συνδυάζει τις απαντήσεις και μάς παρουσιάζει την απόδειξη του "μαγικού" θεωρήματος των Μπάναχ-Τάρσκι με τρόπο σαφή και απλό. Η αφήγησή του είναι ένα μείγμα Ιστορίας και Φιλοσοφίας των Μαθηματικών που κρατά αμείωτο το ενδιαφέρον ως το τέλος.
Τι είναι τα μη μετρήσιμα σύνολα; Τι είναι το αξίωμα της επιλογής; Υπάρχουν πολλά άπειρα και τι είναι οι υπερπεπερασμένοι αριθμοί; Ποια η σχέση του παράδοξου αυτού με τη Θεωρητική Φυσική και την Κοσμολογία; Μπορεί αυτό το θεώρημα να έχει εφαρμογές στην πραγματικότητα;
Το βιβλίο αυτό του Λέοναρντ Γουάπνερ συνδυάζει τις απαντήσεις και μάς παρουσιάζει την απόδειξη του "μαγικού" θεωρήματος των Μπάναχ-Τάρσκι με τρόπο σαφή και απλό. Η αφήγησή του είναι ένα μείγμα Ιστορίας και Φιλοσοφίας των Μαθηματικών που κρατά αμείωτο το ενδιαφέρον ως το τέλος.
304 pages, Paperback
First published April 1, 2005
25 people are currently reading
1414 people want to read
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 12 of 12 reviews
April 29, 2017
The Banach-Tarski Theorem, is not a easy subject even to mathematicians, yet this book gives a very clear and simplified explanation and proof of this theorem.
The general idea of Banach-Tarski Theorem is that you can make an infinite number of infinities out of just one infinity. So you can split the "mother" infinity in to several infinities of the same size as the mother infinity. Or, in other words, you can rearrange the points of a small object (as small as a pea) to get a bigger object (as big as the sun).
The book is interesting and funny. However, it wasn't very well-written. I've struggled though it, because the language was really weak and the sentences were too long sometimes.
If you have no idea about the Banach-Tarski Theorem, I think this book is a good choice to start with.
The general idea of Banach-Tarski Theorem is that you can make an infinite number of infinities out of just one infinity. So you can split the "mother" infinity in to several infinities of the same size as the mother infinity. Or, in other words, you can rearrange the points of a small object (as small as a pea) to get a bigger object (as big as the sun).
The book is interesting and funny. However, it wasn't very well-written. I've struggled though it, because the language was really weak and the sentences were too long sometimes.
If you have no idea about the Banach-Tarski Theorem, I think this book is a good choice to start with.
April 15, 2008
This book is about the Banach-Tarski Theorem. This was a Mathematical result from the twenties that said there is a way to take apart a solid ball in a finite number of pieces and then twist and turn around the pieces, and then reassemble them into two balls of the same mass and volume. This is all mathematically speaking, of course, so it doesn't mean you can do it in real life.
The proof is presented well, and it provides a lot of background. It is not rigorous, but I definitely now how to do the proof rigorously now.
I liked the author's writing style. I thought it was clear and understandable. It wasn't so slow that I was bored. And I have seen all the mathematics in this book before. This leads me to believe that for non-mathematicians this book may be hard to read. But that is just really hard for me to judge. I think this book would be best for non-mathematician scientists who have lots of mathematical background.
The author also discusses the different schools of mathematical thought and goes into the details of the different mathematical philosophies. I think he did a good job at this.
Towards the end of the book the author goes into a rather lengthy discussion of how quantum physicists try to incorporate this theorem into their work. It was interesting but seemed a little lame and speculative to me. The author concludes by saying that the advanced mathematical tools that we have, end up heavily influencing physicists.
The proof is presented well, and it provides a lot of background. It is not rigorous, but I definitely now how to do the proof rigorously now.
I liked the author's writing style. I thought it was clear and understandable. It wasn't so slow that I was bored. And I have seen all the mathematics in this book before. This leads me to believe that for non-mathematicians this book may be hard to read. But that is just really hard for me to judge. I think this book would be best for non-mathematician scientists who have lots of mathematical background.
The author also discusses the different schools of mathematical thought and goes into the details of the different mathematical philosophies. I think he did a good job at this.
Towards the end of the book the author goes into a rather lengthy discussion of how quantum physicists try to incorporate this theorem into their work. It was interesting but seemed a little lame and speculative to me. The author concludes by saying that the advanced mathematical tools that we have, end up heavily influencing physicists.
April 18, 2020
The Banach-Tarski theorem states that a ball (i.e. a solid sphere) can be split into countably many pieces and re-assembled into another ball of different radius. Like the author, as a maths student I was unsettled by this result, as were many others: according to an obituary of Tarski, an Illinois citizen once demanded that the state legislature outlaw its teaching.
The paradoxical nature of the theorem lies in our wanting to infer from mathematics to the physical world. While this is valid for many mathematical results, here it is not, because the axioms used to construct the theorem allow physically unrealistic sets to exist (in a Platonic sense). In particular, the “measure” of a three-dimensional set (such as a ball) corresponds to the volume, but some sets are non-measurable, and this is the kind of set into which the theorem divides the ball. In other words, the pieces of the original ball don’t have defined volumes.
The culprit is the axiom of choice. Without it, there are no unmeasurable sets. (This is shown, albeit with a caveat, by something called the Solovay model.) Bertrand Russell’s explanation of the axiom of choice may be the best: from an infinite series of pairs of shoes, you can specify a rule to choose one from each pair; but how can you do that for an infinite series of pairs of socks? The axiom of choice simply asserts that there always is such a rule.
Paradox (etymologically “beyond belief”) can be a tricky concept to pin down. The author distinguishes three types: 1) statements which offend common sense but are true, 2) those which appear true but are self-contradictory, i.e. fallacies, and 3) those which lead to contradictory conclusions, known as antinomies. The Banach-Tarski theorem is an example of the first type: the reasoning is correct, and any objection has to be with the premises (axioms). Russell’s paradox, and the related Burali-Forti paradox, are of the third kind: they assert sets whose existence leads to contradictions. This is avoided by modern set theories which constrains the kind of set that can be defined. In fact, the main set theory in current use was jointly specified by Zermelo, the same person who made the axiom of choice explicit.
The grand plan of the book is to take the reader right through from first principles to a proof of the Banach-Tarski theorem. This is ambitious but seems to be achieved. Although my main interest lay in the history and interpretation of the theorem. I didn’t need to be convinced that the proof was valid, nor was I really interested in exactly how it was proved. (I already knew that, for example, that the set of real numbers can be 1-1 related with certain of its subsets.)
The publishing is let down by an idiosyncratic mix of fonts, clip art style illustrations and slack editing, e.g. MC Escher drawings, and the lyrics to “Hotel Infinity”, a pastiche of the Eagles song. I didn’t see any mathematical errors, although some of the material seems unnecessary, e.g. matrix formulations of symmetries, and relativistic time dilation.
The paradoxical nature of the theorem lies in our wanting to infer from mathematics to the physical world. While this is valid for many mathematical results, here it is not, because the axioms used to construct the theorem allow physically unrealistic sets to exist (in a Platonic sense). In particular, the “measure” of a three-dimensional set (such as a ball) corresponds to the volume, but some sets are non-measurable, and this is the kind of set into which the theorem divides the ball. In other words, the pieces of the original ball don’t have defined volumes.
The culprit is the axiom of choice. Without it, there are no unmeasurable sets. (This is shown, albeit with a caveat, by something called the Solovay model.) Bertrand Russell’s explanation of the axiom of choice may be the best: from an infinite series of pairs of shoes, you can specify a rule to choose one from each pair; but how can you do that for an infinite series of pairs of socks? The axiom of choice simply asserts that there always is such a rule.
Paradox (etymologically “beyond belief”) can be a tricky concept to pin down. The author distinguishes three types: 1) statements which offend common sense but are true, 2) those which appear true but are self-contradictory, i.e. fallacies, and 3) those which lead to contradictory conclusions, known as antinomies. The Banach-Tarski theorem is an example of the first type: the reasoning is correct, and any objection has to be with the premises (axioms). Russell’s paradox, and the related Burali-Forti paradox, are of the third kind: they assert sets whose existence leads to contradictions. This is avoided by modern set theories which constrains the kind of set that can be defined. In fact, the main set theory in current use was jointly specified by Zermelo, the same person who made the axiom of choice explicit.
The grand plan of the book is to take the reader right through from first principles to a proof of the Banach-Tarski theorem. This is ambitious but seems to be achieved. Although my main interest lay in the history and interpretation of the theorem. I didn’t need to be convinced that the proof was valid, nor was I really interested in exactly how it was proved. (I already knew that, for example, that the set of real numbers can be 1-1 related with certain of its subsets.)
The publishing is let down by an idiosyncratic mix of fonts, clip art style illustrations and slack editing, e.g. MC Escher drawings, and the lyrics to “Hotel Infinity”, a pastiche of the Eagles song. I didn’t see any mathematical errors, although some of the material seems unnecessary, e.g. matrix formulations of symmetries, and relativistic time dilation.
November 6, 2007
3.5 stars. fascinating subject, a lot of fun to read, and educational for me at the time, but sloppily written, fully of errors, and not brilliant.
September 28, 2020
It's math in words and numbers! This is a very fun book, it goes through some deep mathematical ideas without getting too stuck up with the jargon. Additionally, he is full of graphs to create intuition. I loved the philosophy bites in the beginning, about what it takes to be a platonist. Anyway, great book all around! It's mostly structured to get you to the Banach Tarski Paradox, so all the chapters teach intuitively the pre-requisites to understanding it. I would buy this for a smart teenager who was into mathematics.
July 21, 2023
Just an amazing book that reveals very interesting paradoxes that changed my point of view.
July 7, 2014
The clearest exposition of the Banach-Tarski I've ever seen. Come to think of it, the only exposition of the Banach-Tarski paradox I've ever seen. Not so hard mathematics that prove you can disassemble a pea into 5 or more pieces and reassemble them into a sphere the size of the sun.
August 21, 2024
A semi-rigorous explanation of the paradoxical theorem that a pea can be cut into a finite number of pieces, and then, once those pieces are rotated, can be re-assembled into the sun. The main theorems and its prerequisites are presented and the author does his very best to explain them. His explanations are much more insightful at the beginning than at the end. It is almost as he was slightly rushed to finish the book as he got closer to the finish line --- which I guess is true of just about any writer.
As for the theorem: Yes, it is mathematically true, but there are of course, two gigantic caveats.
1) The kind of mathematical congruence the theorem concerns itself with is not the kind of congruence high school students learn from the venerable Euclid (paper and scissors congruence). Rather, the proof is about point-wise congruence, in which many shenanigans about completed infinite sets are exploited that makes the theorem less paradoxical.
2) Ultimately, the theorem relies on "bringing" enough "points from infinity" to make the theorem work. Which is to say, ultimately the theorem relies on completed infinities for the proof.
2a) There is a great misconception in Mathematics that the axiom of choice creates opportunities to prove incongruous or paradoxical results. In fact, the paradoxes really come from completed infinite sets --- often paired with the use of the axiom of choice in order to make use of the infinite sets. A criticism that I have not seen fully addressed is the mismatch between the proof --- which is only allowed to take into account COMPLETING infinities, and the result of the proof, which is allowed to be expressed into completed infinities.
2b) The Hilbert Hotel is a great example of this. It is argued that an infinite hotel will always have room for one more guest, even if the hotel is full. The guest is accommodated as follows: he is placed in room 1; the guest in room 1 is shifted to room 2; the guest in room 2 is shifted to room 3; etc. Then it is argued that for any guest in any finite room, that particular guest will eventually have a room to spend the night ---framing the proof using finite elements. Therefore, it is concluded, since this is true for any guest in any finite room, it is true for the entire infinite set of rooms: all the infinite guests will not be left out in the cold night, but will have a room to sleep in --- making a conclusion about the completed infinite set. And somehow the propagating wave of room changes that the arrival of the new guest precipitated gets "shifted" out of sight by the ever retreating horizon to infinity.
3) And this is how the theorem ultimately gets proved. The pea is cleverly partitioned into a number of non-measurable sets, and then any missing points that are needed to enlarge the pea into the sun get "shifted from infinity" (the reverse of the Hilbert Hotel).
The more I think about it, the more I am convinced that completed infinities should not be allowed in mathematics --- though I am happy to include COMPLETING infinities. But this is suicide for a professional mathematician, I think. I think that the whole of analysis might have to go, as it seems to require the completed real number line, and I don't think that there are professional mathematicians who would go for this. I should look more into constructive mathematics.
As for the theorem: Yes, it is mathematically true, but there are of course, two gigantic caveats.
1) The kind of mathematical congruence the theorem concerns itself with is not the kind of congruence high school students learn from the venerable Euclid (paper and scissors congruence). Rather, the proof is about point-wise congruence, in which many shenanigans about completed infinite sets are exploited that makes the theorem less paradoxical.
2) Ultimately, the theorem relies on "bringing" enough "points from infinity" to make the theorem work. Which is to say, ultimately the theorem relies on completed infinities for the proof.
2a) There is a great misconception in Mathematics that the axiom of choice creates opportunities to prove incongruous or paradoxical results. In fact, the paradoxes really come from completed infinite sets --- often paired with the use of the axiom of choice in order to make use of the infinite sets. A criticism that I have not seen fully addressed is the mismatch between the proof --- which is only allowed to take into account COMPLETING infinities, and the result of the proof, which is allowed to be expressed into completed infinities.
2b) The Hilbert Hotel is a great example of this. It is argued that an infinite hotel will always have room for one more guest, even if the hotel is full. The guest is accommodated as follows: he is placed in room 1; the guest in room 1 is shifted to room 2; the guest in room 2 is shifted to room 3; etc. Then it is argued that for any guest in any finite room, that particular guest will eventually have a room to spend the night ---framing the proof using finite elements. Therefore, it is concluded, since this is true for any guest in any finite room, it is true for the entire infinite set of rooms: all the infinite guests will not be left out in the cold night, but will have a room to sleep in --- making a conclusion about the completed infinite set. And somehow the propagating wave of room changes that the arrival of the new guest precipitated gets "shifted" out of sight by the ever retreating horizon to infinity.
3) And this is how the theorem ultimately gets proved. The pea is cleverly partitioned into a number of non-measurable sets, and then any missing points that are needed to enlarge the pea into the sun get "shifted from infinity" (the reverse of the Hilbert Hotel).
The more I think about it, the more I am convinced that completed infinities should not be allowed in mathematics --- though I am happy to include COMPLETING infinities. But this is suicide for a professional mathematician, I think. I think that the whole of analysis might have to go, as it seems to require the completed real number line, and I don't think that there are professional mathematicians who would go for this. I should look more into constructive mathematics.
September 30, 2023
Very delightful book that lets one understand how the Banach-Tarski paradox is proved. The author does a good job of motivating and explaining the paradox. I found chapters 1 through 6 to be the most engaging. The ending chapters' speculations are just that, speculations, but at least show some of the ideas in the air when the book was written.
If you're interested in the Banach-Tarski paradox, then I think this is a great introduction.
If you're interested in the Banach-Tarski paradox, then I think this is a great introduction.
October 5, 2021
nice at the beginning, a bit chaotic at the end. Rather unreadable without adequate mathematical background.
Read
February 1, 2022This is a great popularization. The middle part of it is hard slogging for non-mathematicians, but it's as close as most people will ever get to understanding the Banach-Tarski Paradox.
April 22, 2025
I fw this book- the banach tarski paradox is really interesting and this book explains it in a beautiful way.
Not for the faint of heart
xoxo
karan
Not for the faint of heart
xoxo
karan
Displaying 1 - 12 of 12 reviews