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The Beauty of Doing Mathematics: Three Public Dialogues
If someone told you that mathematics is quite beautiful, you might be surprised. But you should know that some people do mathematics all their lives, and create mathematics, just as a composer creates music. Usually, every time a mathematician solves a problem, this gives rise to many oth ers, new and just as beautiful as the one which was solved. Of course, often these problems are quite difficult, and as in other disciplines can be understood only by those who have studied the subject with some depth, and know the subject well. In 1981, Jean Brette, who is responsible for the Mathematics Section of the Palais de la Decouverte (Science Museum) in Paris, invited me to give a conference at the Palais. I had never given such a conference before, to a non-mathematical public. Here was a could I communicate to such a Saturday afternoon audience what it means to do mathematics, and why one does mathematics? By "mathematics" I mean pure mathematics. This doesn't mean that pure math is better than other types of math, but I and a number of others do pure mathematics, and it's about them that I am now concerned. Math has a bad reputation, stemming from the most elementary levels. The word is in fact used in many different contexts. First, I had to explain briefly these possible contexts, and the one with which I wanted to deal.
144 pages, Paperback
First published January 1, 1984
5 people are currently reading
132 people want to read
About the author
Serge Lang
184 books57 followersSerge Lang was an influential mathematician in the field of number theory. Algebra is his most famous book.
Librarian Note: There is more than one author in the GoodReads database with this name. See this thread for more information.
Librarian Note: There is more than one author in the GoodReads database with this name. See this thread for more information.
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Displaying 1 - 7 of 7 reviews
April 26, 2024
کتاب، متنِ مکتوبِ سه سخنرانیِ تعاملیِ سرژ لانگ برای مخاطب عام است. شور و شوق لانگ و علاقهاش به ریاضی واقعا ستودنی بود. از طرفی چیزی که برایم عجیب بود قدرت همراهی و درک و همفکری آدمهای عادی سخنرانی برای پیشبرد مسائل بود. انگار ریاضی برای آدمهای حاضر در آن سخنرانیها در فرانسه اصلا امری مربوط به سالهای دور مدرسه نبودند و چنان عمیق آن را یاد گرفته بودند که پس از سالها باز هم میتوانستند مطالب نهچندان سادهی لانگ را درک کنند.
August 31, 2022
Serge Lang is a very well known mathematician, if you study math you have probably heard of him... if you study Number Theory specifically, you most certainly have. He wrote a series of text books for Springer-Verlag that are fairly well known in the mathematics community. You can often find them on the bookshelves in a fair amount of offices. Beyond his abilities in mathematics is also known for being an excellent lecturer of the craft. For that reason it comes as no surprised when the Palais de la Decouverte in Paris wanted to offer some lectures in mathematics to a lay crowd Serge Lang was asked to speak.
Lang tasked himself with the challenge of trying to explain and give other people the sense of what a mathematician does, or at least what types of problems they can think about. In the 1980's he gave three talks, one per year. So, it was, naturally, not always the same crowd. However, the audience participation was about the same each time he gave a talk, which was rather fun. This book is the transcript of the lecture and the exchange between Lang and his audience.
Since this was a "Saturday afternoon crowd," as Lang kept reminding us, which I found quite comical eventually, he was faced with the challenge of speaking in front of people that may merely have a passing interest in mathematics and nothing more. They, no doubt, came from varied backgrounds and some may never do much math beyond these talks. Quite a challenge for any professional in their field, but I rather love Lang's strategy as he approached this task. The only way to really, truly, see mathematics for what it really is, is to do mathematics. Mathematics is not really a spectator sport, despite how many people would like it to be just that. So, Lang decided to involve the audience in the mathematics he was doing, calling out for participation. He announces that mathematicians participating is cheating! Mathematicians are seriously bad listeners and a few of them piped up breaking his rules.
Since Lang is a number theorist it came as no surprise that he came up with a talk about prime numbers. One of the things that makes his talks rather fascinating is he tries very hard to start his audience with material they either have seen somewhere or at the very least can figure it out relatively easily. He goes over very well the challenges mathematicians face in trying to find these numbers. He tries not to get overly abstract, for fear of losing his audience, but he does present them with the major formulas and results so far. It is rather difficult to understand once he starts wading into the territory where mathematicians (at the time) have no answer. In my case, I did not study much number theory, so I am an apt pupil for his talk to some degree. I found the sections where he goes for unsolved math a bit difficult, probably because the transcript makes it a bit more challenging, I do wonder if the lecture would have been easier to watch in the moment. Sadly, I probably would have asked too many questions about how things are defined and he would have told me we are getting to specific for the audience to follow.
The second talk was on Diophantine equations. This was a more fascinating talk to read about, probably because I found the math more interesting. Once again, towards the end when he tries to wade into the unsolvable territory, his need to stay away from technicalities sort of mars the readers understanding of what he is driving at. However, up until that happens, it is a wonderful read. One of the things that I appreciated in these talks was that he tried to share why he found mathematics beautiful. Why he found joy in working on and he tried to share that joy through interesting questions. He tried to start with smaller ideas that grow into big problems to show the audience why he would bother to spend time thinking about such things.
The first two talks were only an hour long, followed by a brief question and answer period. In the second talk chunk of this was cut out because he had a conversation with an audience member about the state of mathematics education and I truly wish they had left that exchange in. He, like me, had commented about looking at a recent high school text and being infuriated by it. I tend to feel a similar way about the majority of text books that are out there. Many of them try to display facts or try to make math "ultra concrete" with applications question, or "make math fun," but all these books sorely miss the point. Many of them barely bother to even list the rules we are working with or give insight or even a slight inkling to the structure of mathematics. And even if they present the rules, they scarcely ever tie it all together at the end showing us the broader picture for what we have built, a sorely missed opportunity.
The last talk, which I think he had planned to be the last, because he was running out of lay topics that turn into unsolved problems that don't rely on intense technical work, was about geometry. Now, this is not his field, so I went in with a little bit of trepidation. He wanted to talk about a recent result that had been worked on in the year prior to his talk. Now, geometry is something I have a bit more experience with, however, I have not run into some of the things he talks about here. This is a major deficiency in my education at this time, which I hope to rectify. His talk started off strong as he tried to give an audience of the idea of dimension and what it means for a set to be open or closed and how this relates to something like a sphere with a boundary versus a sphere without a boundary and what that "means". He did pretty good discussing equivalences of geometric objects and cutting and deforming. The book actually does a wonderful job of including diagrams. They took two breaks in this long sequence of a lecture and each time more people left, but the third our there were still about a hundred people left out of the 250+ starting. Now, this third hour is where he fell victim to trying to discuss very complex matters that are very hard to understand, in my opinion, if you try to avoid technicalities. I was probably left with a fair amount of questions after his discussion of the hyperbolic metric, because there was much more I wanted to know about it and he probably would not have been able to say anything without writing down and deriving certain formulas.
He tried to give a decent sense of what topology and differential geometry are trying to talk about at times. His big goal was to talk about a conjecture presented by Thurston and I still don't really know what it was about or why it was very important. So, I think in his attempt to present this very complex idea, it was too difficult to truly say simply. Now, I do know Thurston and I know Hubbard has written a book built on his ideas into a subject called Teichmuller Theory. Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. This material is hardly for the faint of heart and you no doubt need an extremely strong understanding of things like Differential Geometry to even begin reading such material.
In the end, I found this transcript fun to read. Lang's enthusiasm for the subject oozes off the page and he even went through and edited the transcript to include any footnotes to add further details he felt would be useful. If you are enthusiastic about mathematics in general, or if you study maths and would just like to engage a good speaker (albeit not through the spoken word), then I think you will find some merit in this little book. I wish they had preserved some Lang lectures in video form, and I don't know if any exist, but I think this would have been even more powerful to watch happen in real time. Serge Lang, I'm sure, is missed in the mathematical community, but with books like this his voice can live on and I'm glad Springer-Verlag put these books together.
Lang tasked himself with the challenge of trying to explain and give other people the sense of what a mathematician does, or at least what types of problems they can think about. In the 1980's he gave three talks, one per year. So, it was, naturally, not always the same crowd. However, the audience participation was about the same each time he gave a talk, which was rather fun. This book is the transcript of the lecture and the exchange between Lang and his audience.
Since this was a "Saturday afternoon crowd," as Lang kept reminding us, which I found quite comical eventually, he was faced with the challenge of speaking in front of people that may merely have a passing interest in mathematics and nothing more. They, no doubt, came from varied backgrounds and some may never do much math beyond these talks. Quite a challenge for any professional in their field, but I rather love Lang's strategy as he approached this task. The only way to really, truly, see mathematics for what it really is, is to do mathematics. Mathematics is not really a spectator sport, despite how many people would like it to be just that. So, Lang decided to involve the audience in the mathematics he was doing, calling out for participation. He announces that mathematicians participating is cheating! Mathematicians are seriously bad listeners and a few of them piped up breaking his rules.
Since Lang is a number theorist it came as no surprise that he came up with a talk about prime numbers. One of the things that makes his talks rather fascinating is he tries very hard to start his audience with material they either have seen somewhere or at the very least can figure it out relatively easily. He goes over very well the challenges mathematicians face in trying to find these numbers. He tries not to get overly abstract, for fear of losing his audience, but he does present them with the major formulas and results so far. It is rather difficult to understand once he starts wading into the territory where mathematicians (at the time) have no answer. In my case, I did not study much number theory, so I am an apt pupil for his talk to some degree. I found the sections where he goes for unsolved math a bit difficult, probably because the transcript makes it a bit more challenging, I do wonder if the lecture would have been easier to watch in the moment. Sadly, I probably would have asked too many questions about how things are defined and he would have told me we are getting to specific for the audience to follow.
The second talk was on Diophantine equations. This was a more fascinating talk to read about, probably because I found the math more interesting. Once again, towards the end when he tries to wade into the unsolvable territory, his need to stay away from technicalities sort of mars the readers understanding of what he is driving at. However, up until that happens, it is a wonderful read. One of the things that I appreciated in these talks was that he tried to share why he found mathematics beautiful. Why he found joy in working on and he tried to share that joy through interesting questions. He tried to start with smaller ideas that grow into big problems to show the audience why he would bother to spend time thinking about such things.
The first two talks were only an hour long, followed by a brief question and answer period. In the second talk chunk of this was cut out because he had a conversation with an audience member about the state of mathematics education and I truly wish they had left that exchange in. He, like me, had commented about looking at a recent high school text and being infuriated by it. I tend to feel a similar way about the majority of text books that are out there. Many of them try to display facts or try to make math "ultra concrete" with applications question, or "make math fun," but all these books sorely miss the point. Many of them barely bother to even list the rules we are working with or give insight or even a slight inkling to the structure of mathematics. And even if they present the rules, they scarcely ever tie it all together at the end showing us the broader picture for what we have built, a sorely missed opportunity.
The last talk, which I think he had planned to be the last, because he was running out of lay topics that turn into unsolved problems that don't rely on intense technical work, was about geometry. Now, this is not his field, so I went in with a little bit of trepidation. He wanted to talk about a recent result that had been worked on in the year prior to his talk. Now, geometry is something I have a bit more experience with, however, I have not run into some of the things he talks about here. This is a major deficiency in my education at this time, which I hope to rectify. His talk started off strong as he tried to give an audience of the idea of dimension and what it means for a set to be open or closed and how this relates to something like a sphere with a boundary versus a sphere without a boundary and what that "means". He did pretty good discussing equivalences of geometric objects and cutting and deforming. The book actually does a wonderful job of including diagrams. They took two breaks in this long sequence of a lecture and each time more people left, but the third our there were still about a hundred people left out of the 250+ starting. Now, this third hour is where he fell victim to trying to discuss very complex matters that are very hard to understand, in my opinion, if you try to avoid technicalities. I was probably left with a fair amount of questions after his discussion of the hyperbolic metric, because there was much more I wanted to know about it and he probably would not have been able to say anything without writing down and deriving certain formulas.
He tried to give a decent sense of what topology and differential geometry are trying to talk about at times. His big goal was to talk about a conjecture presented by Thurston and I still don't really know what it was about or why it was very important. So, I think in his attempt to present this very complex idea, it was too difficult to truly say simply. Now, I do know Thurston and I know Hubbard has written a book built on his ideas into a subject called Teichmuller Theory. Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. This material is hardly for the faint of heart and you no doubt need an extremely strong understanding of things like Differential Geometry to even begin reading such material.
In the end, I found this transcript fun to read. Lang's enthusiasm for the subject oozes off the page and he even went through and edited the transcript to include any footnotes to add further details he felt would be useful. If you are enthusiastic about mathematics in general, or if you study maths and would just like to engage a good speaker (albeit not through the spoken word), then I think you will find some merit in this little book. I wish they had preserved some Lang lectures in video form, and I don't know if any exist, but I think this would have been even more powerful to watch happen in real time. Serge Lang, I'm sure, is missed in the mathematical community, but with books like this his voice can live on and I'm glad Springer-Verlag put these books together.
August 22, 2024
I read of the first two lectures, and personally they're quite hard to follow because so much is unexplained, so I think he had quite the attentive audience, especially on the discussion of Diophantine equations.
Still though, it's always interesting to learn what some mathematician thinks of things about his field.
Reading about people talk about math makes me feel so inspired sometimes, I'm glad I picked this up when I was kind of down.
Still though, it's always interesting to learn what some mathematician thinks of things about his field.
Reading about people talk about math makes me feel so inspired sometimes, I'm glad I picked this up when I was kind of down.
May 1, 2025
If you studied math or physics at the University, I can bet you prepared your linear algebra exam on Lang book too, an exceptional one!
The book of this review is at a totally different level. It is a transcript of seminars and lessons given by Serge to people not having any math expertise but just curiosity, or to students of first high school years. Lang shows how it is possible to engage people with math in a really interesting way throwing away all standard boring teaching approaches!
I'm fully with him! The teaching of math and science at school is designed poorly, and so it appears boring and the performances are low too. Teachers focus on a bunch of rules and formulas without giving a big picture of why, of how results appeared.
The beauty of science and math goes with the historical paths, failures and lives of the people who created them.
Enjoy the book!
The book of this review is at a totally different level. It is a transcript of seminars and lessons given by Serge to people not having any math expertise but just curiosity, or to students of first high school years. Lang shows how it is possible to engage people with math in a really interesting way throwing away all standard boring teaching approaches!
I'm fully with him! The teaching of math and science at school is designed poorly, and so it appears boring and the performances are low too. Teachers focus on a bunch of rules and formulas without giving a big picture of why, of how results appeared.
The beauty of science and math goes with the historical paths, failures and lives of the people who created them.
Enjoy the book!
September 27, 2023
Niente di che.
il titolo è accattivante, in realtà è una raccolta di lezioni tenute da Lang, prima con i pubblico non professionista e poi con delle scuole superiori.
I capitoli più interessanti e godibili sono quest'ultimi
Si ripetono le stesse domande a sfinimento "lei è un matematico?" e risposte piccate di Lang, dialoghi che non portano niente di utile al lettore.
LANG Lei è un matematico? SIGNORE Un po'.
LANG E già troppo! Vorrei che i matematici non intervenissero. Cens i matematici ne conoscono la risposta, ma questa conferenza non è per loro.
Dunque, si scava un buco, e si ottiene un oggetto del genere.
UNA PERSONA Una bottiglia di Klein.
LANG Qualcuno di voi ne sa anche troppo.
BAMBINA Una piramide?
LANG No, è equivalente alla sfera."
SIGNORA Una scatola senza il coperchio?
LANG Sì, ma avrà un bordo. Voglio una superficie senza bordo. Quella di prima non ce l'ha, e nemmeno la sfera. Voglio una superficie compatta.
STUDENTE UNIVERSITARIO Si possono fare due buchi, a mo' di occhiali.
LANG Ecco, è proprio quel che volevo sentirvi dire. Ma tu sei un matematico? Oh no, no! Non intervenire. È naturale per te che sei un matematico. Ma non stai al gioco. Ecco per ché vi chiedo di non intervenire. Voglio che la gente pensi con la pro pria testa.
il titolo è accattivante, in realtà è una raccolta di lezioni tenute da Lang, prima con i pubblico non professionista e poi con delle scuole superiori.
I capitoli più interessanti e godibili sono quest'ultimi
Si ripetono le stesse domande a sfinimento "lei è un matematico?" e risposte piccate di Lang, dialoghi che non portano niente di utile al lettore.
LANG Lei è un matematico? SIGNORE Un po'.
LANG E già troppo! Vorrei che i matematici non intervenissero. Cens i matematici ne conoscono la risposta, ma questa conferenza non è per loro.
Dunque, si scava un buco, e si ottiene un oggetto del genere.
UNA PERSONA Una bottiglia di Klein.
LANG Qualcuno di voi ne sa anche troppo.
BAMBINA Una piramide?
LANG No, è equivalente alla sfera."
SIGNORA Una scatola senza il coperchio?
LANG Sì, ma avrà un bordo. Voglio una superficie senza bordo. Quella di prima non ce l'ha, e nemmeno la sfera. Voglio una superficie compatta.
STUDENTE UNIVERSITARIO Si possono fare due buchi, a mo' di occhiali.
LANG Ecco, è proprio quel che volevo sentirvi dire. Ma tu sei un matematico? Oh no, no! Non intervenire. È naturale per te che sei un matematico. Ma non stai al gioco. Ecco per ché vi chiedo di non intervenire. Voglio che la gente pensi con la pro pria testa.
April 7, 2023
i first read a sample of this book one year ago when i was really struggling with feeling dumb as hell in my physics class and it honestly made me feel better about it. i enjoyed the dialogue and the last chapter the most, but the book was a bit hard to follow. i think it would have definitely translated better seeing the dialogue in person. either way, i will definitely be revisiting this in the future.
October 18, 2014
Fun read. Especially the last chapter about geometries.
Displaying 1 - 7 of 7 reviews