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Mathematics without Apologies: Portrait of a Problematic Vocation
What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers—for the sake of truth, beauty, and practical applications—this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources.
Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party?
Disarmingly candid, relentlessly intelligent, and richly entertaining, Mathematics without Apologies takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.
461 pages, Kindle Edition
First published January 4, 2015
44 people are currently reading
387 people want to read
About the author
Michael Harris
2 books6 followersMichael Howard Harris (born 1954) is an American mathematician. He made notable contributions to the Langlands program, for which he (alongside Richard Taylor) won the 2007 Clay Research Award. In particular, he (jointly with Taylor), proved the local Langlands conjecture for GL(n) over a p-adic local field in (Harris & Taylor 2001), and was part of the team that proved the Sato–Tate conjecture.
Harris attained his doctorate from Harvard University in 1977, under supervision of Barry Mazur. His thesis, entitled "On p-Adic Representations Arising from Descent on Abelian Varieties", was later published in Compositio Mathematica. (From Wikipedia)
https://en.wikipedia.org/wiki/Michael...
Harris attained his doctorate from Harvard University in 1977, under supervision of Barry Mazur. His thesis, entitled "On p-Adic Representations Arising from Descent on Abelian Varieties", was later published in Compositio Mathematica. (From Wikipedia)
https://en.wikipedia.org/wiki/Michael...
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Displaying 1 - 16 of 16 reviews
October 1, 2016
Michael Harris' book is definitely one that mathematicians shouldn't miss. It's also a must-read for people who aren't professional mathematicians but do have a deep appreciation for the subject and have more than a passing interest in understanding what mathematics is "about". It will not, however, win any converts among people who think they "don't like math" and would require serious persuasion to change their minds.
As the title should make clear, the book is intended as a counterpoint to G. H. Hardy's A Mathematician's Apology. Hardy poses the question "[W]hy is it really worthwhile to make a serious study of mathematics? What is the proper justification of a mathematician's life?" Hardy's answer is somewhat defensive and hedged. He rejects a justification based on practical usefulness of mathematics, which he doesn't defend robustly. Instead, he concludes that the justification for the life of any "real" mathematician is to "have added something to knowledge, and have helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any other artists, great or small, who have left some kind of memorial behind them."
Harris, on the other hand, rejects a defensive stance and aims to justify mathematics and the work of mathematicians "without apology". He stands with Hardy in likening mathematics to other fine arts. But he goes further to justify both mathematics and other art forms not in utilitarian or moralistic terms, but in aesthetic terms of the pleasure from creating and appreciating art of high quality. His argument is first set out in Chapter 3 ("Not Merely Good, True, and Beautiful"). And after excursions through diverse mathematical, philosophical, biographical, artistic, and literary topics, the argument concludes in Chapter 10 ("No apology"). Along the way, it's noted that other top tier scientists besides mathematicians have also justified their work in terms of pleasure. There is, for instance, Richard Feynman, one of whose books is The Pleasure Of Finding Things Out: The Best Short Works Of Richard P. Feynman. Although the things studied in physics and mathematics are rather different (though some like Max Tegmark (Our Mathematical Universe: My Quest for the Ultimate Nature of Reality) disagree), there is a distinct pleasure associated with important discoveries in any scientific field. Creative art also involves discovery and "finding things out". Both the creative artist/scientist and others who simply learn to appreciate their work experience the associated pleasure.
Michael Harris' most noteworthy mathematical accomplishment is his work within a vast generalization of number theory (in more classical forms of which Hardy also made his mark many decades ago). This work is part of an extensive series of related conjectures called "The Langlands Program", after Robert Langlands who conceived it and has contributed to its continual expansion. The LP is highly abstruse, technical mathematics, and Harris makes no serious attempt in this book to sketch its outlines. He touches only briefly on some of the more concrete manifestations of the program in connection with what are known as "elliptic curves". A very concise way to describe the LP is in terms of elaborate "correspondences" between seemingly unrelated mathematical objects such as "Galois representations" and "automorphic forms". There's a more complete semi-technical discussion of these correspondences in Edward Frenkel's Love and Math: The Heart of Hidden Reality, and a technical introduction (accessible only to math graduate students and professionals) is An Introduction to the Langlands Program.
Some people have found Harris' book frustrating because it seems to lack "structure". In fact, however, it does have structure, but as is often true in advanced mathematics, the structure isn't always readily apparent. As Harris would point out, that is precisely what appeals to mathematicians about their vocation and avocation: its pleasure lies in discovering hidden structures that aren't obvious on the surface. The Langlands Program is an apt metaphor for mathematics itself. Earlier examples of the discovery of rich hidden structures in mathematics abound – from Galois theory to Hilbert spaces.
Many topics are discussed in the book, some only cursorily, some in more depth. Here are just a few of them; mathematical Platonism (Harris, like most mathematicians, is an adherent), category theory, Hindu and Buddhist philosophy, literature (such as Thomas Pynchon's), mathematical tricks, the mind-body problem, economics and finance, mathematical "charisma", life in Paris, mathematics and sex (in reference to Frenkel's Love and Math: The Heart of Hidden Reality) – and much more. Some of this is autobiographical, but there's almost always some connection, anecdotal or otherwise, with the book's main themes. It's a smörgåsbord.
The book definitely isn't light reading. It's also heavily footnoted – an average of about 60 per chapter. (Most are worth reading.) There's also a running series of imagined dialogues – at a very elementary level – on "How to Explain Number Theory at at Dinner Party". So don't expect to finish it in a day or two. A better approach is to take it a chapter at a time, and allow that to sink in before proceeding.
As the title should make clear, the book is intended as a counterpoint to G. H. Hardy's A Mathematician's Apology. Hardy poses the question "[W]hy is it really worthwhile to make a serious study of mathematics? What is the proper justification of a mathematician's life?" Hardy's answer is somewhat defensive and hedged. He rejects a justification based on practical usefulness of mathematics, which he doesn't defend robustly. Instead, he concludes that the justification for the life of any "real" mathematician is to "have added something to knowledge, and have helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any other artists, great or small, who have left some kind of memorial behind them."
Harris, on the other hand, rejects a defensive stance and aims to justify mathematics and the work of mathematicians "without apology". He stands with Hardy in likening mathematics to other fine arts. But he goes further to justify both mathematics and other art forms not in utilitarian or moralistic terms, but in aesthetic terms of the pleasure from creating and appreciating art of high quality. His argument is first set out in Chapter 3 ("Not Merely Good, True, and Beautiful"). And after excursions through diverse mathematical, philosophical, biographical, artistic, and literary topics, the argument concludes in Chapter 10 ("No apology"). Along the way, it's noted that other top tier scientists besides mathematicians have also justified their work in terms of pleasure. There is, for instance, Richard Feynman, one of whose books is The Pleasure Of Finding Things Out: The Best Short Works Of Richard P. Feynman. Although the things studied in physics and mathematics are rather different (though some like Max Tegmark (Our Mathematical Universe: My Quest for the Ultimate Nature of Reality) disagree), there is a distinct pleasure associated with important discoveries in any scientific field. Creative art also involves discovery and "finding things out". Both the creative artist/scientist and others who simply learn to appreciate their work experience the associated pleasure.
Michael Harris' most noteworthy mathematical accomplishment is his work within a vast generalization of number theory (in more classical forms of which Hardy also made his mark many decades ago). This work is part of an extensive series of related conjectures called "The Langlands Program", after Robert Langlands who conceived it and has contributed to its continual expansion. The LP is highly abstruse, technical mathematics, and Harris makes no serious attempt in this book to sketch its outlines. He touches only briefly on some of the more concrete manifestations of the program in connection with what are known as "elliptic curves". A very concise way to describe the LP is in terms of elaborate "correspondences" between seemingly unrelated mathematical objects such as "Galois representations" and "automorphic forms". There's a more complete semi-technical discussion of these correspondences in Edward Frenkel's Love and Math: The Heart of Hidden Reality, and a technical introduction (accessible only to math graduate students and professionals) is An Introduction to the Langlands Program.
Some people have found Harris' book frustrating because it seems to lack "structure". In fact, however, it does have structure, but as is often true in advanced mathematics, the structure isn't always readily apparent. As Harris would point out, that is precisely what appeals to mathematicians about their vocation and avocation: its pleasure lies in discovering hidden structures that aren't obvious on the surface. The Langlands Program is an apt metaphor for mathematics itself. Earlier examples of the discovery of rich hidden structures in mathematics abound – from Galois theory to Hilbert spaces.
Many topics are discussed in the book, some only cursorily, some in more depth. Here are just a few of them; mathematical Platonism (Harris, like most mathematicians, is an adherent), category theory, Hindu and Buddhist philosophy, literature (such as Thomas Pynchon's), mathematical tricks, the mind-body problem, economics and finance, mathematical "charisma", life in Paris, mathematics and sex (in reference to Frenkel's Love and Math: The Heart of Hidden Reality) – and much more. Some of this is autobiographical, but there's almost always some connection, anecdotal or otherwise, with the book's main themes. It's a smörgåsbord.
The book definitely isn't light reading. It's also heavily footnoted – an average of about 60 per chapter. (Most are worth reading.) There's also a running series of imagined dialogues – at a very elementary level – on "How to Explain Number Theory at at Dinner Party". So don't expect to finish it in a day or two. A better approach is to take it a chapter at a time, and allow that to sink in before proceeding.
August 27, 2017
I was an ideal target reader for this book, which means the size of my bullseye was larger than most... and it still missed.
September 5, 2022
A long, rambling book that is unsure of its focus, audience, or intent.
Pros:
1.) Written by a good author.
Michael Harris is passionate and fully immersed in the field of mathematics. He's been at the heart of academic professional mathematics for over the majority of his life. If you look up "Professional Academic Mathematician" in the dictionary, no doubt you'd read "see Harris, Michael (1954)". If anyone could articulate what he does, why he does it, the problems of the profession, and what it's like to be a modern academic mathematician, it'd be him. By "academic", I mean as opposed to a mathematician in the employ of a financial firm, or doing correlation studies for firms pushing foods, drugs, and their relative safety/effectivity, or computer scientist finding the best algorithm to minimize network traffic or predict desires of the user.
2.) Individual sections are great.
There are chapters here that, individually, would make great essays that college students could do an Adler-style Socratic Seminar on. There are parts of the chapter "Not Merely Good, True, and Beautiful" that are awesome. There are sections on mathematical history that are decent. There are parts of "How to Explain Number Theory at a Dinner Party" that are good (though why "How to Explain Number Theory at a Dinner Party" is spread across 4 disparate sections across the book is a mystery to me). This book has many "diamonds in the rough", made even rougher because they appear to have been split into pieces for no discernable reason, and to only detrimental effect. More on this later.
3.) Diagrams, pictures, and equations included are helpful.
The diagrams, when included, certainly help the book. They are sparse, but when used they are used well.
Cons:
1.) Audience seems schizophrenic.
To whom is this book addressed? It seems addressed to many different people at once. Sometimes it appears to be addressed to the common man who wants to learn about math history. The history is boiled down, and the audience spoon-fed stories. Other times, it seems aimed to the amateur mathematician who seeks learn about the roots of mathematics and where it points. The math here is harder, with more equations and diagrams than verbal stories. Yet other times, it seems addressed to professional academic mathematicians, where names and problems are dropped with zero explanation.
2.) Focus seems schizophrenic.
The focus vacillates wildly without apparent reason, transition to or from, or forewarning.
Is this book about number theory? Is it about mathematical history? Is it auto-biography of Harris' academic career? Is it about to what extent professional mathematicians should romanticize their profession and enjoy movies featuring mathematicians? A musing of whether mathematicians are to blame for modern problems like gerrymandering and the financial crisis? A musing on the mind-body problem? The intersections between math and romantic love?
The answer to these questions is... "yes". All those things. Not at once, but in sections. It's 3 or 4 books put into a blender, but not blended enough to make a homogenous whole. A bit like Frankenstein's monster, we're left to make sense of a messy amalgamation. The "Notes" section from from page 327-395, if that's any indication of the rambling nature of this book.
3.) Comes across as elitist.
I'm sure this wasn't the author's intent (he refers to himself as a "bottom-feeder" at one point), but it still comes across in spades. There are a few reasons for this. For one, the writing takes the form of a self-narrative, so expect plenty of "Once when I...". The book is, inherently, about Michael Harris; the focus is HIS thoughts, HIS history, and HIS musings. Chapter 2 is called "How I Acquired Charisma". He speaks about how he was in enrolled is a summer program in high school after being enrolled as a two-year experimental course on vector geometry. How he visited nearly every college mathematics department. How marveled teachers drove him to see lectures. One sentence reads "The way a graduate student, barely past 20, chooses a thesis adviser, and, in so doing, nearly always determines a permanent career orientation, has always fascinated me." How many times did he encounter 20-yr-olds doing graduate work? Does he realize this is rare? Apparently not; he has seen it often, and it fascinates him. It comes across as out of touch, if not preening. I kept on writing "As one does..." sarcastically in the margin.
A second reason is that there isn't a topic that this books sticks to longer than 10 pages. This lack of focus makes it seem as though Harris is talking to himself and we are a captive audience along for the ride. Other contributing factors include quotations with no translations (most are translated, but many are not). It's as if he author is stating "If you're not a trilingual polyglot, why even crack open this book?"
The last reason is the title itself. Harris points out that he is a mathematician that gives little to no regard to the good, bad, beautiful, or pragmatic of his field. If asked "What good do you do", contra Hardy, he seems to say "I'm barely going to address the topic, and seem offended that you'd even ask". If you're trying to find someone who's going to defend the math department from budget cuts, this wouldn't be your guy. It all comes across as elitist, serving to validate the inhumane, out-of-touch view of mathematicians than many seem to have, a view Harris seems to both share and embrace (yet still seems confused as to why it is off-putting to others).
4.) Despite the subtitle, you won't learn much about the vocation.
Reading this book is reading about Harris and his view on thigs. I didn't learn much about the vocation of mathematics. His experience didn't seem emblematic of others. Questions remain unanswered. Why is the vocation "problematic"? How do mathematicians in applied fields view those in theoretical? How mathematicians in disparate fields feel about each other? Is there competition between them, and if so, is it good-spirited or nasty? I came to this book to learn about how mathematicians feel about their own career choice and vocation. I left unsatisfied.
Pros:
1.) Written by a good author.
Michael Harris is passionate and fully immersed in the field of mathematics. He's been at the heart of academic professional mathematics for over the majority of his life. If you look up "Professional Academic Mathematician" in the dictionary, no doubt you'd read "see Harris, Michael (1954)". If anyone could articulate what he does, why he does it, the problems of the profession, and what it's like to be a modern academic mathematician, it'd be him. By "academic", I mean as opposed to a mathematician in the employ of a financial firm, or doing correlation studies for firms pushing foods, drugs, and their relative safety/effectivity, or computer scientist finding the best algorithm to minimize network traffic or predict desires of the user.
2.) Individual sections are great.
There are chapters here that, individually, would make great essays that college students could do an Adler-style Socratic Seminar on. There are parts of the chapter "Not Merely Good, True, and Beautiful" that are awesome. There are sections on mathematical history that are decent. There are parts of "How to Explain Number Theory at a Dinner Party" that are good (though why "How to Explain Number Theory at a Dinner Party" is spread across 4 disparate sections across the book is a mystery to me). This book has many "diamonds in the rough", made even rougher because they appear to have been split into pieces for no discernable reason, and to only detrimental effect. More on this later.
3.) Diagrams, pictures, and equations included are helpful.
The diagrams, when included, certainly help the book. They are sparse, but when used they are used well.
Cons:
1.) Audience seems schizophrenic.
To whom is this book addressed? It seems addressed to many different people at once. Sometimes it appears to be addressed to the common man who wants to learn about math history. The history is boiled down, and the audience spoon-fed stories. Other times, it seems aimed to the amateur mathematician who seeks learn about the roots of mathematics and where it points. The math here is harder, with more equations and diagrams than verbal stories. Yet other times, it seems addressed to professional academic mathematicians, where names and problems are dropped with zero explanation.
2.) Focus seems schizophrenic.
The focus vacillates wildly without apparent reason, transition to or from, or forewarning.
Is this book about number theory? Is it about mathematical history? Is it auto-biography of Harris' academic career? Is it about to what extent professional mathematicians should romanticize their profession and enjoy movies featuring mathematicians? A musing of whether mathematicians are to blame for modern problems like gerrymandering and the financial crisis? A musing on the mind-body problem? The intersections between math and romantic love?
The answer to these questions is... "yes". All those things. Not at once, but in sections. It's 3 or 4 books put into a blender, but not blended enough to make a homogenous whole. A bit like Frankenstein's monster, we're left to make sense of a messy amalgamation. The "Notes" section from from page 327-395, if that's any indication of the rambling nature of this book.
3.) Comes across as elitist.
I'm sure this wasn't the author's intent (he refers to himself as a "bottom-feeder" at one point), but it still comes across in spades. There are a few reasons for this. For one, the writing takes the form of a self-narrative, so expect plenty of "Once when I...". The book is, inherently, about Michael Harris; the focus is HIS thoughts, HIS history, and HIS musings. Chapter 2 is called "How I Acquired Charisma". He speaks about how he was in enrolled is a summer program in high school after being enrolled as a two-year experimental course on vector geometry. How he visited nearly every college mathematics department. How marveled teachers drove him to see lectures. One sentence reads "The way a graduate student, barely past 20, chooses a thesis adviser, and, in so doing, nearly always determines a permanent career orientation, has always fascinated me." How many times did he encounter 20-yr-olds doing graduate work? Does he realize this is rare? Apparently not; he has seen it often, and it fascinates him. It comes across as out of touch, if not preening. I kept on writing "As one does..." sarcastically in the margin.
A second reason is that there isn't a topic that this books sticks to longer than 10 pages. This lack of focus makes it seem as though Harris is talking to himself and we are a captive audience along for the ride. Other contributing factors include quotations with no translations (most are translated, but many are not). It's as if he author is stating "If you're not a trilingual polyglot, why even crack open this book?"
The last reason is the title itself. Harris points out that he is a mathematician that gives little to no regard to the good, bad, beautiful, or pragmatic of his field. If asked "What good do you do", contra Hardy, he seems to say "I'm barely going to address the topic, and seem offended that you'd even ask". If you're trying to find someone who's going to defend the math department from budget cuts, this wouldn't be your guy. It all comes across as elitist, serving to validate the inhumane, out-of-touch view of mathematicians than many seem to have, a view Harris seems to both share and embrace (yet still seems confused as to why it is off-putting to others).
4.) Despite the subtitle, you won't learn much about the vocation.
Reading this book is reading about Harris and his view on thigs. I didn't learn much about the vocation of mathematics. His experience didn't seem emblematic of others. Questions remain unanswered. Why is the vocation "problematic"? How do mathematicians in applied fields view those in theoretical? How mathematicians in disparate fields feel about each other? Is there competition between them, and if so, is it good-spirited or nasty? I came to this book to learn about how mathematicians feel about their own career choice and vocation. I left unsatisfied.
September 11, 2017
Mathematics can be made simple, even obvious; and beautiful, and even useful. Just read Ian Stewart’s 17 Equations That Changed the World. But there are other more provocative views. You just need to read Michael Harris’ mathematics without apologies. Harris is certainly not as easy to read as Stewart. But it is as (maybe more) enriching. His Chapter 3 for example is entitled Not Merely Good, True and Beautiful. In this world of increasing pressure to justify the usefulness of science, the author fights back. “There is now a massive literature on the pressures facing university laboratories. These books mostly ignore mathematics, where stakes are not so high and opportunities for commercial applications are scarce, especially in the pure mathematics.” [Page 55] But even Truth seems to be at stake.“If one really thinks deeply about the possbility that the foundations of mathematics are inconsistent, this is extremely unsettling for any rational mind” [Voevodsky quoted on page 58] and a few lines before “Bombieri recalled the concerns about the consistency, reliability, and truthfulness of mathematics that surfaced during the Foundations Crisis and alluded to the ambiguous status of computer proofs and too-long proofs.” Finally Harris mentions some confusion about Beauty quoting Villani: “The artistic aspect of our discipline is [so] evident” that we don’t see how anyone could miss it.. immediatley adding that “what generally makes a mathematician progress is the desire to produce something beautiful.” Harris then quotes an art expert advising museum-goers to “let go of [their] preconceived notions that art has to be beautiful”. [Page 63] Harris adds that “the utility of practical applications, the guarantee of absolute certainty and the vision of mathematics as an art form – the good, the true and the beautiful, for short – have the advantage of being ready to hand with convenient associations, though we should keep in mind that what you are willing to see as good depends on your perspective, and on the other hand the true and beautiful can themselves be understood as goods.” [Pages 63-4] The short answer to the “why” question is going to be that mathematicains engage in mathematics because it gives us pleasure. [Page 68]
After the claim in his Chapter 3 that mathematics was “Not Merely Good, True and Beautiful”, Harris goes on with provocative and thoughful arguments about the relations that mathematics have with Money (Chapter 4 – Megaloprepeia), with the Body (Chapter 6 – Further Investigations of the Mind-Body problem), with Foundations (Chapter 7 – The Habit of Clinging to an Ultimate Ground) and even with tricks (Chapter 8 – The Science of Tricks), Harris finally comes back to Apologies after a personal chapter about inspiration and work (Chapter 9 – A Mathematical Dream and Its Interpretation). The author made me discover, shame on me, that “apology” does not mean only praise, but also excuse or defense. Difficulty and confusion of the vocabulary, indeed a recurrent theme of Harris’ book. Let me be quite clear again. I did not understand everything and I imagined Harris could have created a new index. This new Index could be 0 for Maths Giants or Supergiants, humans who could be awarded the Fields Medal, the Abel Prize or equivalent, 1 for those who can understand (everything) that has been written in mathematics by those with 0 Index; then 2, for those who can understand (everything) that has been written in mathematics by those with 1 Index, etc… I do not know where the index would stop and perhaps it already exists… I woudl like to believe that I was at the Index 3 but not sure! But then I made my discovery about “apology”, I put myself down at Index 5… Harris goes even stronger than Hardy with his “No Apologies” even if he quotes him: Irony has not spoken its last word on the flight from utility [of science], even when utility is understood, with Hardy, as that which “tends to accentuate the existing inequalities in the distribution of wealth”. [Page 296] I think harris has written a very useful book about mathematics. I add another example on the nature of mathematical beauty: “there is a very high degree of unexpectedness, combined with inevitability and economy” [Page 307] When looking for more information about Harris, I found his web page which begins with the quote "La libertad es como un número primo." Roberto Bolaño, Los Detectives Salvajes. When I discovered Bolaño a few years ago, it was such a shock that I read everything I could find. Again without understanding everything. But if you read Harris’ chapter 9, you will undestand that “not understanding everything” may not be that important, compared to the impact that (apparent) confusion may create…
After the claim in his Chapter 3 that mathematics was “Not Merely Good, True and Beautiful”, Harris goes on with provocative and thoughful arguments about the relations that mathematics have with Money (Chapter 4 – Megaloprepeia), with the Body (Chapter 6 – Further Investigations of the Mind-Body problem), with Foundations (Chapter 7 – The Habit of Clinging to an Ultimate Ground) and even with tricks (Chapter 8 – The Science of Tricks), Harris finally comes back to Apologies after a personal chapter about inspiration and work (Chapter 9 – A Mathematical Dream and Its Interpretation). The author made me discover, shame on me, that “apology” does not mean only praise, but also excuse or defense. Difficulty and confusion of the vocabulary, indeed a recurrent theme of Harris’ book. Let me be quite clear again. I did not understand everything and I imagined Harris could have created a new index. This new Index could be 0 for Maths Giants or Supergiants, humans who could be awarded the Fields Medal, the Abel Prize or equivalent, 1 for those who can understand (everything) that has been written in mathematics by those with 0 Index; then 2, for those who can understand (everything) that has been written in mathematics by those with 1 Index, etc… I do not know where the index would stop and perhaps it already exists… I woudl like to believe that I was at the Index 3 but not sure! But then I made my discovery about “apology”, I put myself down at Index 5… Harris goes even stronger than Hardy with his “No Apologies” even if he quotes him: Irony has not spoken its last word on the flight from utility [of science], even when utility is understood, with Hardy, as that which “tends to accentuate the existing inequalities in the distribution of wealth”. [Page 296] I think harris has written a very useful book about mathematics. I add another example on the nature of mathematical beauty: “there is a very high degree of unexpectedness, combined with inevitability and economy” [Page 307] When looking for more information about Harris, I found his web page which begins with the quote "La libertad es como un número primo." Roberto Bolaño, Los Detectives Salvajes. When I discovered Bolaño a few years ago, it was such a shock that I read everything I could find. Again without understanding everything. But if you read Harris’ chapter 9, you will undestand that “not understanding everything” may not be that important, compared to the impact that (apparent) confusion may create…
April 12, 2020
Deep and uneven. Highly discursive, with a lot of insight into the mindset and motivations of the pure mathematician and a lot of... other stuff. I get the sense that no matter what you’re looking for, 50% of the book will be fascinating and the other 50% will have you rolling your eyes.
September 14, 2017
Overall a good book, but it "wanders" somewhat. Also could have used a better edit to shorten/simplify some very long sentences.
April 26, 2020
The title is obviously a reference to Hardy's "Mathematician's Apolog". Harris set out to answer the question of what mathematicians do and why bother when surrounded by people "who declare an odd sort of pride that they are mathematically illiterate". The answer really lies in three different angles: 1. fruitfulness for practical applications; 2. unique capacity to demonstrate truth not subject to doubt; and 3. aesthetics. All this is in the intro. I'm starting to get excited. Unfortunately, the rest of the book is ... weird. Harris made a big deal how difficult it is to write a book about math. The rest of the book is perhaps trying to convince himself of his own axiom. The book reads at times like a stream of consciousness notebook with too much full-paragraph quotations. At other times, it reads like bad philosophy of the occult.
Harris gets an A for discussing an interesting subject; he gets a B- for putting in the effort with an occasional interesting pointer or quote; he'll have to get an F in putting together unfiltered notes and pass them as a book. 2.1 stars rounded up to 3 for making me appreciate Hardy's book better now.
Harris gets an A for discussing an interesting subject; he gets a B- for putting in the effort with an occasional interesting pointer or quote; he'll have to get an F in putting together unfiltered notes and pass them as a book. 2.1 stars rounded up to 3 for making me appreciate Hardy's book better now.
May 30, 2025
You can watch a video review of it here:
https://youtu.be/g8zYNL-Hw9I
https://youtu.be/g8zYNL-Hw9I
May 14, 2017
There are many interesting citations in this book. Nothing to hurry, slow down while reading to identify the connections across a variety of literature. The conversation between the performing artist and the number theorist are very compelling.
Definitely, may be, will read it twice.
Definitely, may be, will read it twice.
January 31, 2016
The title is obviously a reference to the classic book by G. H. Hardy, “A Mathematician's Apology” and Harris acknowledges that fact. Hardy’s book is one that all math students should read, for it contains a great deal of wisdom regarding what mathematicians do and how they fade over time.
This book goes far deeper into what mathematics is and what it is that mathematicians do. There is a lot of philosophy, an occasional splash of silliness, a great deal of the history of mathematics and many mentions of some of the quirks of famous mathematicians. Harris also throws in many autobiographical references.
The mathematics itself is all over the spectrum, from extremely abstract and complex ideas to simple concepts such as a small complete graph. One person that is repeatedly mentioned is Alexander Grothendieck, an incredibly prolific mathematician that was also higher on the scale of eccentricity than most math people.
Although there is the occasional mention of very advanced mathematics, this book is generally a work of popular mathematics and nearly all can be understood by nearly all. Even though it is understandable, it is sometimes a slog to read through it as it occasionally descends to the level of tedium. If I were to assign this as a reading in a math class it would examined one chapter at a time. That is due to the major problem of the book, the lack of continuity.
This book was made available for free for review purposes
This book goes far deeper into what mathematics is and what it is that mathematicians do. There is a lot of philosophy, an occasional splash of silliness, a great deal of the history of mathematics and many mentions of some of the quirks of famous mathematicians. Harris also throws in many autobiographical references.
The mathematics itself is all over the spectrum, from extremely abstract and complex ideas to simple concepts such as a small complete graph. One person that is repeatedly mentioned is Alexander Grothendieck, an incredibly prolific mathematician that was also higher on the scale of eccentricity than most math people.
Although there is the occasional mention of very advanced mathematics, this book is generally a work of popular mathematics and nearly all can be understood by nearly all. Even though it is understandable, it is sometimes a slog to read through it as it occasionally descends to the level of tedium. If I were to assign this as a reading in a math class it would examined one chapter at a time. That is due to the major problem of the book, the lack of continuity.
This book was made available for free for review purposes
May 9, 2015
"When this book was nearly done and my colleagues started asking me what it is about, I found it simplest to answer that it's about how hard it is to write a book about mathematics."
That's the first sentence of this mathematically based tome. While I found it usually intriguing, it also came across as selectively exclusive for those with a strong math (and math philosophy) background. Often times I could just barely grasp the concept described. This is why I didn't rate this book higher. I wanted to, but it seemed to be inflating my opinion of it.
All that being said, I still enjoyed this book. As a mathematician (student) myself, I too struggle with trying to explain to others not only why we do what we do, but also why we should be paid to do it.
Sadly, this book mostly just reflects on how other mathematicians disagree among themselves on the ultimate purpose of mathematical research. But if nothing else, Harris's almost too-dry wit allows those of us in on the joke to smirk once in a while.
That's the first sentence of this mathematically based tome. While I found it usually intriguing, it also came across as selectively exclusive for those with a strong math (and math philosophy) background. Often times I could just barely grasp the concept described. This is why I didn't rate this book higher. I wanted to, but it seemed to be inflating my opinion of it.
All that being said, I still enjoyed this book. As a mathematician (student) myself, I too struggle with trying to explain to others not only why we do what we do, but also why we should be paid to do it.
Sadly, this book mostly just reflects on how other mathematicians disagree among themselves on the ultimate purpose of mathematical research. But if nothing else, Harris's almost too-dry wit allows those of us in on the joke to smirk once in a while.
July 12, 2016
By its own admission this is very much an "assemblage" drawing on a wide variety of historical, philosophical, sociological etc. resources. To be praised for its well-informed questioning of the received picture of the mathematical life. MWA opens the reader onto the experience of working mathematicians in the 21st century, navigating the demands of the powers that be, academic politics, romanticism, finance mathematics, dreams – only ever taking up the clichés of truth, beauty, madness in order to interrogate their validity. The style and range wont be for everyone, there is no overarching coherence or method, but surely a unique perspective.
July 13, 2016
A quixotic, erudite, gossipy (of the academic variety), tongue-in-cheek, intellectual free-associating book grappling with the motivations and culture of research mathematics written by a top-flight mathematician. Using a whirlwind of literature, mythology (both scientific and religious), philosophy (eastern and western), art, pop-culture references and a good deal of high-level mathematical ideas to give an irresistible insider's view of mathematical motivations that will require some effort on the reader's part to keep all the threads together.
January 20, 2017
Unfortunately this book hasn't clicked with me. The chapters on the Number theory haven't revealed anything new or interesting. The scope has been covered in high school syllabus. Apart from that a few anecdotes about famous mathematicians and a lot about the "philosophy of mathematics" or as what mathematicians think about their work. Not my cup of tea.
March 9, 2015
Parts of this book are interesting, others not so much. A word of warning though. In order to understand what he's getting at, it helps to have some familiarity with mathematics, not necessarily with the content, but with the way mathematicians approach the subject.
March 31, 2015
There were some interesting ideas presented, but overall the book seemed to ramble from one thing to another. Too many ideas and arguments for one volume, it seems.
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