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A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age
A New York Times Book Review Editors' Choice
"Wilkinson has accomplished something more moving and original, braiding his stumbling attempts to get better at math with his deepening awareness that there’s an entire universe of understanding that will, in some fundamental sense, forever lie outside his reach." ―Jennifer Szalai, The New York Times
"There is almost no writer I admire as much as I do Alec Wilkinson. His work has enduring brilliance and humanity.” ―Susan Orlean, author of The Library Book
A spirited, metaphysical exploration into math's deepest mysteries and conundrums at the crux of middle age.
Decades after struggling to understand math as a boy, Alec Wilkinson decides to embark on a journey to learn it as a middle-aged man. What begins as a personal challenge―and it's challenging―soon transforms into something greater than a belabored effort to learn math. Despite his incompetence, Wilkinson encounters a universe of unexpected mysteries in his pursuit of mathematical knowledge and quickly becomes fascinated; soon, his exercise in personal growth (and torture) morphs into an intellectually expansive exploration.
In A Divine Language , Wilkinson, a contributor to The New Yorker for over forty years, journeys into the heart of the divine aspect of mathematics―its mysteries, challenges, and revelations―since antiquity. As he submits himself to the lure of deep mathematics, he takes the reader through his investigations into the subject’s big questions―number theory and the creation of numbers, the debate over math’s human or otherworldly origins, problems and equations that remain unsolved after centuries, the conundrum of prime numbers. Writing with warm humor and sharp observation as he traverses practical math’s endless frustrations and rewards, Wilkinson provides an awe-inspiring account of an adventure from a land of strange sights. Part memoir, part metaphysical travel book, and part journey in self-improvement, A Divine Language is one man’s second attempt at understanding the numbers in front of him, and the world beyond.
"Wilkinson has accomplished something more moving and original, braiding his stumbling attempts to get better at math with his deepening awareness that there’s an entire universe of understanding that will, in some fundamental sense, forever lie outside his reach." ―Jennifer Szalai, The New York Times
"There is almost no writer I admire as much as I do Alec Wilkinson. His work has enduring brilliance and humanity.” ―Susan Orlean, author of The Library Book
A spirited, metaphysical exploration into math's deepest mysteries and conundrums at the crux of middle age.
Decades after struggling to understand math as a boy, Alec Wilkinson decides to embark on a journey to learn it as a middle-aged man. What begins as a personal challenge―and it's challenging―soon transforms into something greater than a belabored effort to learn math. Despite his incompetence, Wilkinson encounters a universe of unexpected mysteries in his pursuit of mathematical knowledge and quickly becomes fascinated; soon, his exercise in personal growth (and torture) morphs into an intellectually expansive exploration.
In A Divine Language , Wilkinson, a contributor to The New Yorker for over forty years, journeys into the heart of the divine aspect of mathematics―its mysteries, challenges, and revelations―since antiquity. As he submits himself to the lure of deep mathematics, he takes the reader through his investigations into the subject’s big questions―number theory and the creation of numbers, the debate over math’s human or otherworldly origins, problems and equations that remain unsolved after centuries, the conundrum of prime numbers. Writing with warm humor and sharp observation as he traverses practical math’s endless frustrations and rewards, Wilkinson provides an awe-inspiring account of an adventure from a land of strange sights. Part memoir, part metaphysical travel book, and part journey in self-improvement, A Divine Language is one man’s second attempt at understanding the numbers in front of him, and the world beyond.
304 pages, Hardcover
Published July 12, 2022
235 people are currently reading
1095 people want to read
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Displaying 1 - 30 of 74 reviews
August 12, 2022
Disappointed.
The reviews, as usual, make this book sound more interesting than it is.
Very little of the book deals with his attempt to learn math. He is easily distracted, and it shows. It seems like he spent more time researching math, and its relationship to the Divine, than to studying. Many, many quotes he jotted down in his researches, and he had to share all of them with us. Some pages seems to be nothing more than recitation of quotes.
The discussion of the poker player - what is the relationship of this to Alec's learning math in old age? The whole segement seemed tossed in last minute.
Would have been a good, long, magazine article, not book length.
There were a few, good, sections, with some, brief, humor.
The reviews, as usual, make this book sound more interesting than it is.
Very little of the book deals with his attempt to learn math. He is easily distracted, and it shows. It seems like he spent more time researching math, and its relationship to the Divine, than to studying. Many, many quotes he jotted down in his researches, and he had to share all of them with us. Some pages seems to be nothing more than recitation of quotes.
The discussion of the poker player - what is the relationship of this to Alec's learning math in old age? The whole segement seemed tossed in last minute.
Would have been a good, long, magazine article, not book length.
There were a few, good, sections, with some, brief, humor.
September 6, 2022
As someone reasonably adept at math, I picked up A Divine Language because I wanted to have a better understanding of what some of the difficulties are for people who struggle with the subject. I wish I could say that I learned more from this book than I did, especially when it comes to understanding the underlying thought patterns that make math difficult for so many people.
For example, on the cancellation of units: "I am baffled by the absurd idea that in ... 2 x 105 m^3 x 1.1 kg/m^3 I can shed the cubic meters, since they cancel each other, but cubic meters are things and not numbers." I was astonished by his bafflement: even the author must have implicitly done such cancellations in his everyday life — e.g., 2 hours x 60 miles/hour gives you a number with units of miles — so why does he regard the generalized case of this procedure as "absurd"? But there's little more said here beyond cubic meters being things and not numbers. I was hoping for a further exploration since this is what I came to the book for: understanding the barriers that others are encountering.
One takeaway for me from reading about the author's experience in attempting to learn algebra, geometry, and calculus is discovering how deeply compartmentalized someone's knowledge can be even within one subject area. For example, the author clearly understands calculus at a conceptual level even if the mechanics and the procedures present him with great difficulty. However, with trigonometry it was quite the opposite: "Figuring the ratios of circles according to their radians, I felt as I did the work, flawlessly as it turned out for a change, that I understood nothing of the concepts I was enacting."
While I appreciate the author's work here — making a concerted effort "at the edge of old age" to grapple with a subject that you have struggled with your entire life takes dedication, and to write a book about that process in which you honestly discuss your own limitations and shortcomings takes more openness and self-acceptance than many of us are capable of — I wish there were fewer pages devoted to mathematical philosophy and history (which are indeed interesting but are areas I'm already somewhat familiar with) and more pages focusing on the difficulties with specific mathematical procedures or methods of thinking and how he worked his way through them or around them. Nonetheless, reading this book did help me better understand the nature of "math anxiety" and the need to remain cognizant of the fact that what can be obvious to one person can be extremely non-intuitive and challenging to another.
My rating: two stars ("it was ok").
For example, on the cancellation of units: "I am baffled by the absurd idea that in ... 2 x 105 m^3 x 1.1 kg/m^3 I can shed the cubic meters, since they cancel each other, but cubic meters are things and not numbers." I was astonished by his bafflement: even the author must have implicitly done such cancellations in his everyday life — e.g., 2 hours x 60 miles/hour gives you a number with units of miles — so why does he regard the generalized case of this procedure as "absurd"? But there's little more said here beyond cubic meters being things and not numbers. I was hoping for a further exploration since this is what I came to the book for: understanding the barriers that others are encountering.
One takeaway for me from reading about the author's experience in attempting to learn algebra, geometry, and calculus is discovering how deeply compartmentalized someone's knowledge can be even within one subject area. For example, the author clearly understands calculus at a conceptual level even if the mechanics and the procedures present him with great difficulty. However, with trigonometry it was quite the opposite: "Figuring the ratios of circles according to their radians, I felt as I did the work, flawlessly as it turned out for a change, that I understood nothing of the concepts I was enacting."
While I appreciate the author's work here — making a concerted effort "at the edge of old age" to grapple with a subject that you have struggled with your entire life takes dedication, and to write a book about that process in which you honestly discuss your own limitations and shortcomings takes more openness and self-acceptance than many of us are capable of — I wish there were fewer pages devoted to mathematical philosophy and history (which are indeed interesting but are areas I'm already somewhat familiar with) and more pages focusing on the difficulties with specific mathematical procedures or methods of thinking and how he worked his way through them or around them. Nonetheless, reading this book did help me better understand the nature of "math anxiety" and the need to remain cognizant of the fact that what can be obvious to one person can be extremely non-intuitive and challenging to another.
My rating: two stars ("it was ok").
January 23, 2023
When I first read a review of A Divine Language by Alec Wilkinson, I was pretty certain that I’d enjoy reading it. My intuition proved correct. Like Wilkinson, I was not very good at math in school. Not as bad as Wilkinson claims to have been (he reports he had to cheat on tests to get by), but I never felt at ease in math; I never felt—or only rarely—that I got it. Looking back at myself, I’ve always chosen to think that my lack of aptitude in math was a result of a failure to apply myself (a euphemism for laziness). Other subjects, such as history, literature, and science, came much easier to me. (However, when it came to science, I could have done without the tedious labs and experiments. I wanted to know about science, not do science.) So when I completed a course in trigonometry in high school, I ended my formal math education. But having gotten away from formal requirements for learning about math, I’ve been haunted by a sense that there is something there that I’ve missed, some way of knowing and understanding that deeply enhances my understanding of the world. Alec Wilkinson, it seems, had similar thoughts.
A part of Wilkinson’s book consists of his travails as he attempts to master this form of knowledge that eluded him in his youth. It should be easier now that he’s older and wiser—the math that he’s learning is really kid’s stuff. Not so. He just needs to apply himself more diligently. Not so simple. He reads and thinks too literally (pun intended). It seems so. Of course, if this is all that his book consisted of, it would merely prove to be a tale of woe and frustration, albeit with an ultimate claim of triumph (spoiler alert: he gains some competence in calculus). But Wilkinson, an established author and New Yorker contributor, wouldn’t publish a book that’s simply a journal about his quest for mathematical competence. So it’s also an inquiry into the world of math.
And I agree with Wilkinson that math isn’t just a challenge like walking so many steps a day, learning a foreign language, or how to play chess (all rewarding endeavors); but math represents something more. It’s like the obelisk in Kubrick’s 2001: even a dumb ape like me knows that this object, this subject, represents much more than what’s presented to the naked eye and greater than what I can ever hope to fully grasp. There’s something “inside it” that can provide a leap forward in knowledge and understanding. And Wilkinson, although he uses different words, intuits that same mysterious something. Happily, he takes his readers down the trodden paths as well as his experiences hacking through the jungles of mathematical concepts and formulas. Thus, we get insights from contemporary mathematicians (like a helpful cousin who’s a University of Chicago mathematician and her mathematician-husband) and others who submit to his interviews. He also delves some into some history of math and some philosophical issues implicit in the subject. For instance, is math, more or less following Plato, math as the divine language? Or is math simply a tool developed by humans to address certain needs? If you think it’s merely a human invention, as I’m wont to do, you have to explain why the tool fits so neatly with what appears to be the underlying structure of the universe. There isn’t, so far as I can tell from this book or my other reading, any definitive answer to these and related issues about mathematics. But like many fields of endeavor, the quest can be as instructive and enlightening as any answers claimed.
So did reading this book encourage me to go learn more math? No, I think not. My best excuse? Wilkinson reports that he’s on “the edge of old age.” I contend that I’m over the edge. I’m within shouting distance of beginning my eighth decade on this earth. But, still, it’s a temptation. There is a fascinating if perplexing world—or perhaps a whole other world—out there that we call mathematics. Who knows (unless one tries) what doors we old folks might yet open?
5/5 stars for diverse & intriguing subjects (including persons and the nature of math itself), & for its clarity of exposition.
A part of Wilkinson’s book consists of his travails as he attempts to master this form of knowledge that eluded him in his youth. It should be easier now that he’s older and wiser—the math that he’s learning is really kid’s stuff. Not so. He just needs to apply himself more diligently. Not so simple. He reads and thinks too literally (pun intended). It seems so. Of course, if this is all that his book consisted of, it would merely prove to be a tale of woe and frustration, albeit with an ultimate claim of triumph (spoiler alert: he gains some competence in calculus). But Wilkinson, an established author and New Yorker contributor, wouldn’t publish a book that’s simply a journal about his quest for mathematical competence. So it’s also an inquiry into the world of math.
And I agree with Wilkinson that math isn’t just a challenge like walking so many steps a day, learning a foreign language, or how to play chess (all rewarding endeavors); but math represents something more. It’s like the obelisk in Kubrick’s 2001: even a dumb ape like me knows that this object, this subject, represents much more than what’s presented to the naked eye and greater than what I can ever hope to fully grasp. There’s something “inside it” that can provide a leap forward in knowledge and understanding. And Wilkinson, although he uses different words, intuits that same mysterious something. Happily, he takes his readers down the trodden paths as well as his experiences hacking through the jungles of mathematical concepts and formulas. Thus, we get insights from contemporary mathematicians (like a helpful cousin who’s a University of Chicago mathematician and her mathematician-husband) and others who submit to his interviews. He also delves some into some history of math and some philosophical issues implicit in the subject. For instance, is math, more or less following Plato, math as the divine language? Or is math simply a tool developed by humans to address certain needs? If you think it’s merely a human invention, as I’m wont to do, you have to explain why the tool fits so neatly with what appears to be the underlying structure of the universe. There isn’t, so far as I can tell from this book or my other reading, any definitive answer to these and related issues about mathematics. But like many fields of endeavor, the quest can be as instructive and enlightening as any answers claimed.
So did reading this book encourage me to go learn more math? No, I think not. My best excuse? Wilkinson reports that he’s on “the edge of old age.” I contend that I’m over the edge. I’m within shouting distance of beginning my eighth decade on this earth. But, still, it’s a temptation. There is a fascinating if perplexing world—or perhaps a whole other world—out there that we call mathematics. Who knows (unless one tries) what doors we old folks might yet open?
5/5 stars for diverse & intriguing subjects (including persons and the nature of math itself), & for its clarity of exposition.
August 9, 2022
“Unless one has exercised one’s mind seriously at the gymnasium of mathematics one is incapable of precise thought, which amounts to saying that one is good for nothing.” Simone Weil quoted in Wilkinson (2022, p. 4)
If true, this is a terrible thought for me because I was a serial failure in mathematics throughout my schooling. I picked this book up because I’m a senior citizen who’s been afraid of math since Grade Four, when my teacher was Miss Rogers, whose one and only teaching method was humiliation. On and off since then, I’ve thought about trying to teach myself what I missed, but I’ve never done it.
You’re certainly led to think that this book is about a senior citizen learning math, but it is only so in a small part. It’s a book that might have made a good, short New Yorker article but it was self-indulgent to make it into a book. At this length, it is unbearably digressive and repetitious. I really dislike authors who read too much tangential material while doing their research and then feel compelled to include it all in their book. Not recommended.
If true, this is a terrible thought for me because I was a serial failure in mathematics throughout my schooling. I picked this book up because I’m a senior citizen who’s been afraid of math since Grade Four, when my teacher was Miss Rogers, whose one and only teaching method was humiliation. On and off since then, I’ve thought about trying to teach myself what I missed, but I’ve never done it.
You’re certainly led to think that this book is about a senior citizen learning math, but it is only so in a small part. It’s a book that might have made a good, short New Yorker article but it was self-indulgent to make it into a book. At this length, it is unbearably digressive and repetitious. I really dislike authors who read too much tangential material while doing their research and then feel compelled to include it all in their book. Not recommended.
November 4, 2022
Hmmm.
I was very curious about why a person, who had math anxiety as a kid, would try to learn mathematics when he turned 65. And, then write a book about it.
My curiosity is sated.
I was very curious about why a person, who had math anxiety as a kid, would try to learn mathematics when he turned 65. And, then write a book about it.
My curiosity is sated.
November 11, 2024
You think that Doestoevsky's characters suffer inordinately but then you meet Amie, this guy's mathematician niece who's expected to get him to grasp the barest of math ideas, and discover there must exist subtler layers of pain than you ever thought possible and that you yourself are capable of a whole new dimension of empathy (sympathy, rather).
The bland pablum of trite truisms is often rightfully bashed, but a book such as this reminds you why
directives like "Write What You Know" have needfully come into existence. Wilkinson can certainly turn out a competent phrase every now and then, so I hardly doubt that when he sets his talents on a topic actually worth reading about, the end result might be passable. I don't feel particularly inclined to submit this proposition to empirical testing, though.
That is not to say that there is nothing of value in this 278-pages long narrative (278 is so large a number, in fact, that you'd need to use all of your fingers in both hands almost twenty-eight times in a row to be able to actually count it. It's also both an even number and not a prime, which I hope is not too intimidating of a fact). Those things of value are mostly third-party quotations and allusions to other, better works. It doesn't take a mathematical prodigy, however, to realize that from a cost-efficiency standpoint, waddling through the bog of Wilkinson's navel-gazing and self-glorifying incompetence is simply not worth it.
To its credit, I must admit that the book's build is sturdy, the jacket has a pleasant handfeel and the cover is very pretty, all of which make "A Divine Language" both an optimal piece of coffee-table décor and the perfect gift for people who like books but hate reading them.
I did not very much like this book.
Q.E.D.
The bland pablum of trite truisms is often rightfully bashed, but a book such as this reminds you why
directives like "Write What You Know" have needfully come into existence. Wilkinson can certainly turn out a competent phrase every now and then, so I hardly doubt that when he sets his talents on a topic actually worth reading about, the end result might be passable. I don't feel particularly inclined to submit this proposition to empirical testing, though.
That is not to say that there is nothing of value in this 278-pages long narrative (278 is so large a number, in fact, that you'd need to use all of your fingers in both hands almost twenty-eight times in a row to be able to actually count it. It's also both an even number and not a prime, which I hope is not too intimidating of a fact). Those things of value are mostly third-party quotations and allusions to other, better works. It doesn't take a mathematical prodigy, however, to realize that from a cost-efficiency standpoint, waddling through the bog of Wilkinson's navel-gazing and self-glorifying incompetence is simply not worth it.
To its credit, I must admit that the book's build is sturdy, the jacket has a pleasant handfeel and the cover is very pretty, all of which make "A Divine Language" both an optimal piece of coffee-table décor and the perfect gift for people who like books but hate reading them.
I did not very much like this book.
Q.E.D.
September 24, 2022
I ordered this book after reading a review in the FT . I thought , surely if this man can learn algebra , trig, and calculus at the edge of old age , I can too .
After all , my knowledge and problem solving skills, not to mention self confidence and tenacity , are vastly greater than in my teenage years .
After flipping through the book , I will borrow a quote .
“Looking at this subject matter decades later is like meeting an old acquaintance after many years and saying , now I remember why I didn’t like him .”
Book has been donated to the library .
After all , my knowledge and problem solving skills, not to mention self confidence and tenacity , are vastly greater than in my teenage years .
After flipping through the book , I will borrow a quote .
“Looking at this subject matter decades later is like meeting an old acquaintance after many years and saying , now I remember why I didn’t like him .”
Book has been donated to the library .
August 9, 2022
Super intrigued by this book. It begs so many questions about the nature of mathematics and learning and teaching in general!
For much of the first half, I found myself unsettled by the premise: what does it really mean to "learn mathematics"? Learning how to factor a polynomial or take the derivative of a function is one thing, but surely those procedural skills are part of a much broader landscape. I was a little taken aback by how quickly Wilkinson jumped into that big picture: high school geometry and the cardinalities of infinity seem like vastly different topics to me, and seeing them juxtaposed made me wonder what, if anything, learning about those two topics have in common. Admittedly, at times I felt like the idea of "learning math" was over-simplified. There are so many different skills required in being a math student that it's hard to really combine them into one. Yet, this tension kept me engrossed and I could not help but read on.
Throughout the book, I often wrestled with the question of whether high school math is really something that ought to be learned independently. As a teacher, one of the biggest challenges I have (and one of the things that makes my job so fun) is helping learners see the big picture, finding those moments to highlight how an ostensibly tedious and random procedure is able to stand in place for a much more complicated and nuanced idea. This balance between symbolic manipulation and abstraction is one that is so very challenging to strike, such that when Wilkinson named this tension himself in the closing chapter, I was so impressed. At the end of the day, I don't really care if my students learn the properties of logarithms. What I do care about is the opportunity I have to help frame their notion of what learning actually feels like and looks like, helping them realize what their minds are capable of largely on their own. How much of that comes naturally?
Additionally, I couldn't help but think of the nature of assessment: I loved the little bit about Wilkinson quickly abandoning the state standardized algebra test. That is one very specific metric for measuring algebraic content knowledge, but at the end of the day, what did he really want to measure? His knowledge of algebra? His appreciation of math? His willingness to be embraced by and appreciate mathematics? As teachers, what do we want to measure when we assess our students?
At times, I felt reluctant to give book 5 stars. There were many times when the math teacher in me felt very antsy about a slightly misguided opinion of algebra or calculus or an arguably reductive conclusion about the nature of mathematics. At the end of the day, however, it was important to remember that that was not really the goal of this book.
I am appreciative of Wilkinson's honesty and vulnerability through this journey, and there were countless moments where I was in awe of his ability to step back and reflect on his experience as a learner. I honestly could not put this book down, and I'm so intrigued to know what other people, particularly non-math teachers thought! Either way, one I'm going to be thinking about for a long time to come.
For much of the first half, I found myself unsettled by the premise: what does it really mean to "learn mathematics"? Learning how to factor a polynomial or take the derivative of a function is one thing, but surely those procedural skills are part of a much broader landscape. I was a little taken aback by how quickly Wilkinson jumped into that big picture: high school geometry and the cardinalities of infinity seem like vastly different topics to me, and seeing them juxtaposed made me wonder what, if anything, learning about those two topics have in common. Admittedly, at times I felt like the idea of "learning math" was over-simplified. There are so many different skills required in being a math student that it's hard to really combine them into one. Yet, this tension kept me engrossed and I could not help but read on.
Throughout the book, I often wrestled with the question of whether high school math is really something that ought to be learned independently. As a teacher, one of the biggest challenges I have (and one of the things that makes my job so fun) is helping learners see the big picture, finding those moments to highlight how an ostensibly tedious and random procedure is able to stand in place for a much more complicated and nuanced idea. This balance between symbolic manipulation and abstraction is one that is so very challenging to strike, such that when Wilkinson named this tension himself in the closing chapter, I was so impressed. At the end of the day, I don't really care if my students learn the properties of logarithms. What I do care about is the opportunity I have to help frame their notion of what learning actually feels like and looks like, helping them realize what their minds are capable of largely on their own. How much of that comes naturally?
Additionally, I couldn't help but think of the nature of assessment: I loved the little bit about Wilkinson quickly abandoning the state standardized algebra test. That is one very specific metric for measuring algebraic content knowledge, but at the end of the day, what did he really want to measure? His knowledge of algebra? His appreciation of math? His willingness to be embraced by and appreciate mathematics? As teachers, what do we want to measure when we assess our students?
At times, I felt reluctant to give book 5 stars. There were many times when the math teacher in me felt very antsy about a slightly misguided opinion of algebra or calculus or an arguably reductive conclusion about the nature of mathematics. At the end of the day, however, it was important to remember that that was not really the goal of this book.
I am appreciative of Wilkinson's honesty and vulnerability through this journey, and there were countless moments where I was in awe of his ability to step back and reflect on his experience as a learner. I honestly could not put this book down, and I'm so intrigued to know what other people, particularly non-math teachers thought! Either way, one I'm going to be thinking about for a long time to come.
December 18, 2022
"Everyone needs to make his or her own mistakes, and one of mine might have been to study math."
I loved this book. A 65 year-old writer sets out to learn mostly high school mathematics, the algebra and geometry and calculus that eluded him in the past, and he writes about his emotional experiences during his year of self study. On the way he includes lots of snippets of conversations he has with his mathematician niece, interviews prodigious mathematicians and poker players, and quotes from the dozens of books he read on the history and philosophy of mathematics.
This is the kind of nontechnical math book I've fantasized about for a long time, but if you tried to describe this sort of project to an actual math person, they would be both unable to imagine it and completely disinterested. (I know because I have.)
A lot of this memoir is about fixing the mistakes we feel we made in childhood. It's also about how the education system implants a kind of trauma in us: "Even in an empty room, at the edge of old age, I feel the pressure of the classroom, and the requirement from my childhood to be the bright boy."
Like me, Wilkinson had math anxiety as a student, and he didn't know how to tackle topics that were difficult for him. In his 60s, he learned metacognition and tried to find new ways to overcome old bad habits. I don't agree with all of the conclusions he reached about education and learning, but it was enjoyable to read about someone pondering the same subjects that I have pondered in my post-school years.
This felt like a series of wonderful dinner table conversations, touching on a variety of things I've always romanticized in my head, from Euclid to Newton to Einstein, mysticism and God and the infinity of human imagination. It was wonderful to have my fancies indulged, to not just get a finger wagged at me by someone wondering why I'm interested in the narrative of math more than the nitty gritty technical bits.
Very enjoyable.
P.S. To the 1-star reviews complaining that Wilkinson didn't get good enough at math or whatever: I'm sorry, are you his school teacher???
I loved this book. A 65 year-old writer sets out to learn mostly high school mathematics, the algebra and geometry and calculus that eluded him in the past, and he writes about his emotional experiences during his year of self study. On the way he includes lots of snippets of conversations he has with his mathematician niece, interviews prodigious mathematicians and poker players, and quotes from the dozens of books he read on the history and philosophy of mathematics.
This is the kind of nontechnical math book I've fantasized about for a long time, but if you tried to describe this sort of project to an actual math person, they would be both unable to imagine it and completely disinterested. (I know because I have.)
A lot of this memoir is about fixing the mistakes we feel we made in childhood. It's also about how the education system implants a kind of trauma in us: "Even in an empty room, at the edge of old age, I feel the pressure of the classroom, and the requirement from my childhood to be the bright boy."
Like me, Wilkinson had math anxiety as a student, and he didn't know how to tackle topics that were difficult for him. In his 60s, he learned metacognition and tried to find new ways to overcome old bad habits. I don't agree with all of the conclusions he reached about education and learning, but it was enjoyable to read about someone pondering the same subjects that I have pondered in my post-school years.
This felt like a series of wonderful dinner table conversations, touching on a variety of things I've always romanticized in my head, from Euclid to Newton to Einstein, mysticism and God and the infinity of human imagination. It was wonderful to have my fancies indulged, to not just get a finger wagged at me by someone wondering why I'm interested in the narrative of math more than the nitty gritty technical bits.
Very enjoyable.
P.S. To the 1-star reviews complaining that Wilkinson didn't get good enough at math or whatever: I'm sorry, are you his school teacher???
July 30, 2022
Wilkinson begins his book by admitting that he passed high school math by cheating (copying off other students on exam days). He didn't resort to this because of laziness or lack of motivation but because of a genuine "disconnect" with math. He wondered if it were true that people differ markedly in their capacity to learn math. So he decided to devote himself for an entire year to try to master the concepts and skills that had seemed so out of his intellectual reach in adolescence. The book recounts his efforts and experiences along with a very engaging description of many of the logical problems of pure mathematics and how they developed throughout history. So, there are two reasons to be interested in this book: (1) hearing his personal experiences and difficulties with algebra, geometry, and calculus; and (2) being introduced to some of the interesting problems in number theory and pure mathematics.
When I heard of this book, I immediately got it and read it. Why? Because, me too, at the edge of old age, I took up studying math again. Like Wilkinson, I was always somewhat disappointed that I hadn't done better in math when in college. So, I wanted to see what I could do. Unlike Wilkinson, I didn't use textbooks and study alone. Instead I took advantage of the many math courses available on MOOC (Massive Open Online Course) platforms. (I would recommend particularly those developed at MIT and offered on edx.org). I've been continuing this for the past 6 years or so.
Like Wilkinson, I think I would have to accept the idea that there are some cognitive differences among people with regard to math skills. Whether this is a general intellectual ability or a group of specific intellectual abilities, it seems that there are limits to what levels of understanding one can achieve. But I have found it a lot of fun to keep pushing myself and exploring.
When I heard of this book, I immediately got it and read it. Why? Because, me too, at the edge of old age, I took up studying math again. Like Wilkinson, I was always somewhat disappointed that I hadn't done better in math when in college. So, I wanted to see what I could do. Unlike Wilkinson, I didn't use textbooks and study alone. Instead I took advantage of the many math courses available on MOOC (Massive Open Online Course) platforms. (I would recommend particularly those developed at MIT and offered on edx.org). I've been continuing this for the past 6 years or so.
Like Wilkinson, I think I would have to accept the idea that there are some cognitive differences among people with regard to math skills. Whether this is a general intellectual ability or a group of specific intellectual abilities, it seems that there are limits to what levels of understanding one can achieve. But I have found it a lot of fun to keep pushing myself and exploring.
November 14, 2022
I found so many gems, saw myself in many ways, and was treated to a new (surprisingly accesible) way of thinking about math. My copy is annotated and dog-eared.
December 7, 2022
I loved this book, but it won’t be for everyone. I needed to read it slowly, a few pages a day, and I sometimes skipped sections. It’s the experience of a man who, having flailed his way about as far as algebra II, decides at sixty-three years of age to go back and study mathematics with the goal of understanding calculus. He doesn’t achieve his goal, but his insights along the way, along with a few nuggets of history, fascinated me. Because I studied applied mathematics, I missed out on the wonders and magic of theoretical (“pure”) math. Now that learning math doesn’t seem like an existential threat to me (I’m no longer being graded!), I could relax and enjoy the wonder. A reading friend who is a physicist and a down-to-earth applied mathematician didn’t like the book. He wanted some diagrams and found the author too long winded. I admit that there were the sections I skimmed, and also I also tired of the author’s whining about having felt duped as a boy in math class, but the little nuggets of wonder meant so much to me. Maybe any book describing pure mathematics for a non mathematician would have done the same, but this is the one that opened the world to me, so I had fun with it.
April 8, 2025
I feel I have so much in common with this author and wish I could speak to him. Yet I don't understand how he made it a year through his math journey complaining about calculations and admiring the philosophy of math without realizing that he could pivot into those areas that he loves. He concludes feeling like he failed without realizing that his version of success was right in front of him the whole time. Also he had some pretty fundamental misunderstandings about a lot of concepts & history that somehow made it through this book's editor.
September 5, 2022
Mathematics done for its own sake is a social participatory sport. You learn by doing it, making mistakes, and interacting with others. Mistakes, and interacting with others about misunderstandings, is essential to gain understanding. It is not about you, it is about the math. Struggling alone and staring at a book while hiding your failures is a tough go.
The author uses, and refuses to change from, the tough go approach. Before he begins his efforts the author's mathematically competent niece tells him he is going to overthink learning math. This book is the result of that overthinking.
The book gives a reasonably written, thoroughly sourced history of math and the philosophy of math. There are sections on Cantor's work on set theory and the infinite (circa 1900), and Yitang Zhang's bounded gap between primes theorem (2014). This is math of the highest order, and I was happy to read the accounts. There are also long sections on Plato's thoughts, and the religious thoughts of Kepler and Newton. None of this, not even the math part, is not much related to the task of gaining a fundamental understanding of algebra and calculus.
The book is also a memoir. To my mind, the memoir and the overview mixed and muddled each other as opposed to complementing each other.
The author relates that he "learned" division in the 7th grade, but had forgotten it by the start of the 8th. Very embarrassing. His response was to start copy classmates' answers and he never recovered. He may have learned a procedure for doing division in base 10, but gives the impression that he had no clue on what it meant to divide. If you understand that division is repeated subtraction, and you understand subtraction, you can sort out the problem even if you forget the long division procedure.
Bertrand Russell is quoted on procedure vs understanding being a problem with how math is frequently taught. The author does not comment on the connection between Russell's thoughts and his division experience, or his attempted learning.
A friend told the author he was repeating the procedure vs understanding mistake with his efforts in algebra and beyond. The friend offered help giving him a broader more conceptual view in an effort to gain a better understanding. The author refuses the offer, which left me stunned.
Imagine you are struggling trying to learn conversational French reading everything from a book. A friend offers you a free extended stay among friendly native French speakers. You refuse because you might embarrass yourself with a French grammar mistake. It was a slog to see him repeatedly defeating himself in this manner.
If anyone would like to take their own stab at learning or relearning math, I suggest starting with Exploding Dots on the Global Math Project: https://globalmathproject.org/explodi...
This will give you a good foundation in the basics of arithmetic. You will find some insights even though we all "know" arithmetic.
Follow up with anything you can find by James Tanton. Other names to be aware of are Jo Boaler, and Po-shen Lo. There is a wealth of material available for math enthusiasts of all levels on the web. YouTube and Twitter are good places to look. All of it learning math, as opposed to learning the history or philosophy of math. As mentioned in another review, free edX classes are frequently quite good.
Don't forget that mistakes are an essential part of the process, and actively interacting with others about misunderstandings is both faster and much more productive than struggling alone.
The author uses, and refuses to change from, the tough go approach. Before he begins his efforts the author's mathematically competent niece tells him he is going to overthink learning math. This book is the result of that overthinking.
The book gives a reasonably written, thoroughly sourced history of math and the philosophy of math. There are sections on Cantor's work on set theory and the infinite (circa 1900), and Yitang Zhang's bounded gap between primes theorem (2014). This is math of the highest order, and I was happy to read the accounts. There are also long sections on Plato's thoughts, and the religious thoughts of Kepler and Newton. None of this, not even the math part, is not much related to the task of gaining a fundamental understanding of algebra and calculus.
The book is also a memoir. To my mind, the memoir and the overview mixed and muddled each other as opposed to complementing each other.
The author relates that he "learned" division in the 7th grade, but had forgotten it by the start of the 8th. Very embarrassing. His response was to start copy classmates' answers and he never recovered. He may have learned a procedure for doing division in base 10, but gives the impression that he had no clue on what it meant to divide. If you understand that division is repeated subtraction, and you understand subtraction, you can sort out the problem even if you forget the long division procedure.
Bertrand Russell is quoted on procedure vs understanding being a problem with how math is frequently taught. The author does not comment on the connection between Russell's thoughts and his division experience, or his attempted learning.
A friend told the author he was repeating the procedure vs understanding mistake with his efforts in algebra and beyond. The friend offered help giving him a broader more conceptual view in an effort to gain a better understanding. The author refuses the offer, which left me stunned.
Imagine you are struggling trying to learn conversational French reading everything from a book. A friend offers you a free extended stay among friendly native French speakers. You refuse because you might embarrass yourself with a French grammar mistake. It was a slog to see him repeatedly defeating himself in this manner.
If anyone would like to take their own stab at learning or relearning math, I suggest starting with Exploding Dots on the Global Math Project: https://globalmathproject.org/explodi...
This will give you a good foundation in the basics of arithmetic. You will find some insights even though we all "know" arithmetic.
Follow up with anything you can find by James Tanton. Other names to be aware of are Jo Boaler, and Po-shen Lo. There is a wealth of material available for math enthusiasts of all levels on the web. YouTube and Twitter are good places to look. All of it learning math, as opposed to learning the history or philosophy of math. As mentioned in another review, free edX classes are frequently quite good.
Don't forget that mistakes are an essential part of the process, and actively interacting with others about misunderstandings is both faster and much more productive than struggling alone.
Read
August 20, 2024At the age of 65 Wilkinson decided to go back and learn high school the right way. He admits that he did not understand algebra and geometry in high school, and he never took calculus. He confesses that to pass Algebra 2, " I copied a term paper and almost got caught." I can sympathize. I enjoyed geometry but by the end of algebra I was right at the edge of my math skills. I also never took Calculus.
Wilkinson is a very good New Yorker writer. He gives this project the full New Yorker treatment. He not only tells us his story, he also interviews big time mathematicians, and reads a pile of the literature about the philosophy and history of math. He delves into the physiology and psychology of learning math.
He has a niece, Amie Wilkinson, who is a professor of mathematics at the University of Chicago. She agrees to act as his tutor. This is a long way from picking up "Algebra for Dummies" and teaching yourself. (Although Wilkinson does consult that tome.)
Wilkinson does not go into minute detail about learning his lessons. He does pick examples of problems that stump him and uses them to explore what kind of mental blocks he has that slow him down. One that keeps coming up is the necessity to doubt everything. Amie tells him repeatedly that he first needs to learn the procedures. He has to learn how to follow the recipe. Asking why it is this way and not that way will just slow him down and confuse him. He admits that he keeps thinking that the theory or the book is wrong because what they are doing doesn't make sense.
He also struggles with word problems, another common roadblock. He thinks it is because he can't help seeing them as stories rather than as just scaffolding for a math problem.
Writing about calculus is difficult, as is doing calculus. That section is the least satisfying, probably because by that point Wilkinson does seem to have just given up and focused on following the recipe.
Some of the digressions are fascinating. Yitang Zhang was a part time calculus teacher at the University of New Hampshire. He had published one professional paper. In 2010 he dedicated himself to solving the "bounded gap" problem in numbers theory. It is a famous unproven conjecture about the frequency of prime numbers. He worked night and day on the problem and in 2013 he published a solution. The solution was immediately recognized as a massive breakthrough in the whole field of number theory. It inspired a huge rush of discoveries using Zhang's break through.
Wilkinson met Zhang and points out that Zhang is an extreme example of the typical mathematician personality. He is monomaniacally focused. He has a phenomenal memory and his work is more important to him then almost anything else. The unusual thing about him is that he was 58 years old when made his major discovery. That is unheard of. The accepted wisdom is that even great mathematicians never make major discoveries after the age of 30.
Some of the digressions are not as interesting. Philosophy of mathematics is a dry topic which rests mostly on theory and speculation. Wilkinson is fascinated by the various theories about whether numbers represent an objective reality or are simply concepts humans invented.
The ending is honest. "Mathematics did not embrace me. It tolerated me." but "I am pleased for what learning of a pure type has done for me." I am happy for him, but he didn't convince me to take the plunge back into those deep waters.
Wilkinson is a very good New Yorker writer. He gives this project the full New Yorker treatment. He not only tells us his story, he also interviews big time mathematicians, and reads a pile of the literature about the philosophy and history of math. He delves into the physiology and psychology of learning math.
He has a niece, Amie Wilkinson, who is a professor of mathematics at the University of Chicago. She agrees to act as his tutor. This is a long way from picking up "Algebra for Dummies" and teaching yourself. (Although Wilkinson does consult that tome.)
Wilkinson does not go into minute detail about learning his lessons. He does pick examples of problems that stump him and uses them to explore what kind of mental blocks he has that slow him down. One that keeps coming up is the necessity to doubt everything. Amie tells him repeatedly that he first needs to learn the procedures. He has to learn how to follow the recipe. Asking why it is this way and not that way will just slow him down and confuse him. He admits that he keeps thinking that the theory or the book is wrong because what they are doing doesn't make sense.
He also struggles with word problems, another common roadblock. He thinks it is because he can't help seeing them as stories rather than as just scaffolding for a math problem.
Writing about calculus is difficult, as is doing calculus. That section is the least satisfying, probably because by that point Wilkinson does seem to have just given up and focused on following the recipe.
Some of the digressions are fascinating. Yitang Zhang was a part time calculus teacher at the University of New Hampshire. He had published one professional paper. In 2010 he dedicated himself to solving the "bounded gap" problem in numbers theory. It is a famous unproven conjecture about the frequency of prime numbers. He worked night and day on the problem and in 2013 he published a solution. The solution was immediately recognized as a massive breakthrough in the whole field of number theory. It inspired a huge rush of discoveries using Zhang's break through.
Wilkinson met Zhang and points out that Zhang is an extreme example of the typical mathematician personality. He is monomaniacally focused. He has a phenomenal memory and his work is more important to him then almost anything else. The unusual thing about him is that he was 58 years old when made his major discovery. That is unheard of. The accepted wisdom is that even great mathematicians never make major discoveries after the age of 30.
Some of the digressions are not as interesting. Philosophy of mathematics is a dry topic which rests mostly on theory and speculation. Wilkinson is fascinated by the various theories about whether numbers represent an objective reality or are simply concepts humans invented.
The ending is honest. "Mathematics did not embrace me. It tolerated me." but "I am pleased for what learning of a pure type has done for me." I am happy for him, but he didn't convince me to take the plunge back into those deep waters.
March 13, 2023
This one was a disappointment and was almost a DNF, but I was far enough into it that I decided to keep going even though I wasn’t enjoying it very much. The author repeats himself repetitively, constantly complaining about how much difficulty he has understanding math and then going off on some wild tangent about philosophy or history or literature or some discussion he had with a person who does understand math. Some of those tangents were interesting, but others didn’t add much and/or didn’t feel very relevant to the topic at hand. What I was hoping for from the author was some insight into what exactly made it difficult for him to understand how to do math and what compelled him to make the choice to study it as he entered old age rather than other subjects with which he may have had a hard time when he was younger. What I got was a series of complaints about math, followed by quotes from books and essays other people have written about math, followed by more complaints, followed by more quotes, followed by digressions about poker or poetry, followed by more complaints, etc. This would have been a better book if it had been half the length and better organized.
Part of the reason I chose this book was because, when my husband retired, he did something similar to the author’s project, i.e. studying various topics he had neglected in school or hadn’t had a chance to study. He chose to use the Great Courses to help him along and one of the courses he chose was Mastering the Fundamentals of Mathematics by James Sellers (which my husband says is excellent). His choice of that course puzzled me because, though mathematics was easy for me when I was in school and, though knowing how to do math is a necessary skill for my job, I have never felt particularly drawn to the subject. My husband regretted sloughing it off when he was young and wanted to correct a gap in his knowledge base, as mathematics is foundational for other subjects that interest him such as astronomy and ballistics. In my husband’s case, his lack of math knowledge was the result of willful neglect of the subject when he was a young hooligan. In the case of the author of this book, I have to agree with the author’s niece that his main problem with math is that he tends to overthink things.
Part of the reason I chose this book was because, when my husband retired, he did something similar to the author’s project, i.e. studying various topics he had neglected in school or hadn’t had a chance to study. He chose to use the Great Courses to help him along and one of the courses he chose was Mastering the Fundamentals of Mathematics by James Sellers (which my husband says is excellent). His choice of that course puzzled me because, though mathematics was easy for me when I was in school and, though knowing how to do math is a necessary skill for my job, I have never felt particularly drawn to the subject. My husband regretted sloughing it off when he was young and wanted to correct a gap in his knowledge base, as mathematics is foundational for other subjects that interest him such as astronomy and ballistics. In my husband’s case, his lack of math knowledge was the result of willful neglect of the subject when he was a young hooligan. In the case of the author of this book, I have to agree with the author’s niece that his main problem with math is that he tends to overthink things.
August 15, 2024
On page 97 the text claims (incorrectly) that a dodecahedron has 20 sides. This is an editorial mistake so basic and glaring that I question the seriousness of the whole work.
This memoir is of the genre "post middle-aged male writer muses about things he's read as they relate to some project he's doing." I was reminded a little bit of Alberto Manguel's The Library at Night and William Styron's Darkness Visible. The genre as a whole is sort of inherently tedious for obvious reasons. A Divine Language is on the heavier side on the tedium scale.
The main trouble I have with this book is that he spends an awful lot of time telling us what a hard time he's had learning mathematics; he spends an awful lot of time telling us the history/philosophy/quirks of mathematics (stories that you'll find in most every prose survey of math); but he never seems to get around to /showing/ us what he's learned.
I'm really eager to read a book about re-learning math, but I would rather have him explain what problems he's having and how he went about overcoming them. I want to see the struggle, not just hear about it. Instead he prefers to focus on whatever comment from his niece was most unhelpful, before veering into vague descriptions of wrestling some monster. Speak concretely, dude! There's just a lot that feels inconsistent and half-baked here.
This memoir is of the genre "post middle-aged male writer muses about things he's read as they relate to some project he's doing." I was reminded a little bit of Alberto Manguel's The Library at Night and William Styron's Darkness Visible. The genre as a whole is sort of inherently tedious for obvious reasons. A Divine Language is on the heavier side on the tedium scale.
The main trouble I have with this book is that he spends an awful lot of time telling us what a hard time he's had learning mathematics; he spends an awful lot of time telling us the history/philosophy/quirks of mathematics (stories that you'll find in most every prose survey of math); but he never seems to get around to /showing/ us what he's learned.
I'm really eager to read a book about re-learning math, but I would rather have him explain what problems he's having and how he went about overcoming them. I want to see the struggle, not just hear about it. Instead he prefers to focus on whatever comment from his niece was most unhelpful, before veering into vague descriptions of wrestling some monster. Speak concretely, dude! There's just a lot that feels inconsistent and half-baked here.
September 11, 2022
Its good if overlong intro starts honestly "the following account and its many digressions is about what happens when an untrained mind tries to train itself" but as feared its technical content was limited to criticism and how to learn. The title would be less misleading if 3 words are omitted: algebra, geometry, calculus. Little of those subjects appear in this book.
Instead the author talks about the feeling he gets learning X without actually describing X. X could be any mysterious subject: math, love, astrology, music... He cites too many psychological and philosophical references, plus a few mathematicians when they talk about psychology, trying to explain his attempt to learn.
Seems that he is too suspicious of math being legitimate. He seems to agree with a fellow NYer writer who asks if he's "for or against" math? He jumps the divide between pure math and applied math, to add a meta-math layer maybe trying to explain why math exists.
Disappointing. I took an incomplete in the course.
Instead the author talks about the feeling he gets learning X without actually describing X. X could be any mysterious subject: math, love, astrology, music... He cites too many psychological and philosophical references, plus a few mathematicians when they talk about psychology, trying to explain his attempt to learn.
Seems that he is too suspicious of math being legitimate. He seems to agree with a fellow NYer writer who asks if he's "for or against" math? He jumps the divide between pure math and applied math, to add a meta-math layer maybe trying to explain why math exists.
Disappointing. I took an incomplete in the course.
May 3, 2025
I feel like I must be doing the equivalent of drunk shopping with Libby; when the service notified me that this was available I was surprised, as I didn't recall ever hearing about, let alone placing a hold on, this book. Nevertheless, it looked interesting and I got it and started reading. I was also astonished that so many people were queued up behind me to read this book. By its title alone it doesn't look like it should be popular. But evidently it is relatively in-demand.
Wilkinson struggled with math(s) as a boy and decided to tackle it again in his late 60s. In this book he describes his journey and many mathematical tangents. Many of the tangents into the philosophy of mathematics, especially Platonism, and reflections on the divine.
The good - this made me reflect a lot on my own mathematical journey. I didn't struggle with maths as a child. It didn't occur to me that I was good at maths until a primary school teacher asked me to explain negative numbers to a classmate. I drew her a number line and showed her how negative numbers were on the left, you reached them by subtracting, and she just didn't get it. I had no way to explain it to her further because it was obvious in my mind and you can't really explain something that's obvious. (One reason why I think mathematics teachers ought to be the ones who struggled.)
I hit my own wall a lot later than Wilkinson did - I did an undergrad degree in mathematics, but had no desire to go further. I think I have a relatively good relationship with maths to this day, but I don't exercise my mathematical muscle much and I know it's atrophied. In fact, reading this book may prompt me to take it up again. I've been thinking of doing some reading in category theory, possibly with a couple of Eugenia Cheng's books.
So that's great.
The bad - first of all, Wilkinson's attitude. I felt so bad for his poor niece Amie (a professional mathematician) whom he badgered with harangues about the irrationality of mathematics when it was just his own inability to see.
Secondly, there's nothing here about trying to learn better (besides by developing a better attitude, which he does towards the end). It seems like he just took three regular textbooks in algebra/geometry/calculus and struggled through them. If it were me, I would have maybe started with some mathematical games. He struggled with algebra, why not try Dragonbox Algebra? Instead of complaining about abstruse textbooks, why not try the many pedagogical efforts to make maths more digestible, such as John Mighton's JUMP books? (I don't actually know if JUMP covers the subjects Wilkinson wanted to cover; I'm just giving an example of an effort I know about.) Thirdly, he does admit that he could have hired a tutor and it would have gone better, but he had barred it as part of his ridiculous rules about his project, which was just annoying.
Thirdly, it feels like he spends as much time wandering away from mathematics as he does working at it. I recognise that this would have been a boring book if he had not, but I have to admit that whenever I saw him start discussing the philosophy of mathematics I felt like he should have spent the time just working harder at the maths. I don't really think Platonism vs anti-Platonism is decidable; I'm happy to just say "yes the universe is mathematical" and carry on. (Interestingly, Amie seems to feel the opposite: that mathematics is constructed in the human mind. But she does seem equally uninterested in the topic.) I will concede that some of the asides were interesting, mostly interviews with working mathematicians and how they felt when lighting upon a discovery.
Last notes on the mathematics-and-the-divine stuff. The only money I've ever really directly made from mathematics was when I was flying back from a semester in Budapest doing basically nothing but mathematics. My seatmate on the plane, upon hearing this, mused on why mathematicians seem to have the highest rate of religiosity among scientists, and I flippantly said, "Maybe because they think about infinities everyday?" This remark impressed him enough that when I said separately that I really wanted some British coins (the ones that together form the coat of arms had just come out), he emptied his wallet looking for a full set, despite my protestations. (I definitely wouldn't say this is the only thing I ever got out of maths: a husband was the best prize, for sure.)
For my part, I'm decidedly not religious, but I think I've had two experiences in my life that feel to me like what other people, religious people, describe as "feeling the spirit". So I guess they're the closest I've come to touching the divine. The second one was about the Law of Large Numbers - a professor had remarked that it was like the universe was designed for us to be able to learn, and it set off this tingly feeling in me that I can still remember to this day. I got that from mathematics, too, and I'm only grateful for having persisted with it as long as I did, and I have no regrets giving up on it when I did. I guess I'm pretty content with my mathematical journey, although - as I have noted above - maybe I will continue someday, on a hobbyist level.
Wilkinson struggled with math(s) as a boy and decided to tackle it again in his late 60s. In this book he describes his journey and many mathematical tangents. Many of the tangents into the philosophy of mathematics, especially Platonism, and reflections on the divine.
The good - this made me reflect a lot on my own mathematical journey. I didn't struggle with maths as a child. It didn't occur to me that I was good at maths until a primary school teacher asked me to explain negative numbers to a classmate. I drew her a number line and showed her how negative numbers were on the left, you reached them by subtracting, and she just didn't get it. I had no way to explain it to her further because it was obvious in my mind and you can't really explain something that's obvious. (One reason why I think mathematics teachers ought to be the ones who struggled.)
I hit my own wall a lot later than Wilkinson did - I did an undergrad degree in mathematics, but had no desire to go further. I think I have a relatively good relationship with maths to this day, but I don't exercise my mathematical muscle much and I know it's atrophied. In fact, reading this book may prompt me to take it up again. I've been thinking of doing some reading in category theory, possibly with a couple of Eugenia Cheng's books.
So that's great.
The bad - first of all, Wilkinson's attitude. I felt so bad for his poor niece Amie (a professional mathematician) whom he badgered with harangues about the irrationality of mathematics when it was just his own inability to see.
Secondly, there's nothing here about trying to learn better (besides by developing a better attitude, which he does towards the end). It seems like he just took three regular textbooks in algebra/geometry/calculus and struggled through them. If it were me, I would have maybe started with some mathematical games. He struggled with algebra, why not try Dragonbox Algebra? Instead of complaining about abstruse textbooks, why not try the many pedagogical efforts to make maths more digestible, such as John Mighton's JUMP books? (I don't actually know if JUMP covers the subjects Wilkinson wanted to cover; I'm just giving an example of an effort I know about.) Thirdly, he does admit that he could have hired a tutor and it would have gone better, but he had barred it as part of his ridiculous rules about his project, which was just annoying.
Thirdly, it feels like he spends as much time wandering away from mathematics as he does working at it. I recognise that this would have been a boring book if he had not, but I have to admit that whenever I saw him start discussing the philosophy of mathematics I felt like he should have spent the time just working harder at the maths. I don't really think Platonism vs anti-Platonism is decidable; I'm happy to just say "yes the universe is mathematical" and carry on. (Interestingly, Amie seems to feel the opposite: that mathematics is constructed in the human mind. But she does seem equally uninterested in the topic.) I will concede that some of the asides were interesting, mostly interviews with working mathematicians and how they felt when lighting upon a discovery.
Last notes on the mathematics-and-the-divine stuff. The only money I've ever really directly made from mathematics was when I was flying back from a semester in Budapest doing basically nothing but mathematics. My seatmate on the plane, upon hearing this, mused on why mathematicians seem to have the highest rate of religiosity among scientists, and I flippantly said, "Maybe because they think about infinities everyday?" This remark impressed him enough that when I said separately that I really wanted some British coins (the ones that together form the coat of arms had just come out), he emptied his wallet looking for a full set, despite my protestations. (I definitely wouldn't say this is the only thing I ever got out of maths: a husband was the best prize, for sure.)
For my part, I'm decidedly not religious, but I think I've had two experiences in my life that feel to me like what other people, religious people, describe as "feeling the spirit". So I guess they're the closest I've come to touching the divine. The second one was about the Law of Large Numbers - a professor had remarked that it was like the universe was designed for us to be able to learn, and it set off this tingly feeling in me that I can still remember to this day. I got that from mathematics, too, and I'm only grateful for having persisted with it as long as I did, and I have no regrets giving up on it when I did. I guess I'm pretty content with my mathematical journey, although - as I have noted above - maybe I will continue someday, on a hobbyist level.
March 11, 2024
The actual title for this book at times appeared to be: “Accomplished writer with ill feelings towards his earlier encounter with post-arithmetic math seeks to even the score by taking on Algebra, Geometry, and Calculus at a much later point in life”. What could happen?
There is much to like in this book. I sympathize with the author and have tried at various points to reintroduce my self to these threshold math experiences. I even found myself in a situation earlier in my career where if I did not learn some calculus based statistics then I would have to rethink my career aspirations. I decided to “knuckle down” and learn the math/stats and never regretted it.
At various points throughout the book, the author is effective and even eloquent in passing on his triumphs in math to the reader. He is an effective communicator and I was left wondering why he did not just sign up to teach a high school algebra class. Take on such a class and be forced to prep it and I guarantee that the requisite math will be learned. I also enjoyed Mr. Wilkinson bringing up flurries of math info along the way but also with sufficient cites for following through on his suggestions.
What didn’t I like about the book? Hmmm…? Mr. Wilkinson has structured important parts of his book (the first part) around the idea that he has been let down by math teachers, math textbooks, and even by mathematics itself. Sigh. I would have thought that with all his experience as a writer, Wilkinson would know that if literally millions of teenagers can learn algebra to some extent while you the narrator flounder, then perhaps - just perhaps - the problem lies with him rather than with the algebra establishment conspiring against him. The idea that the behavior of assembled masses may be informative, especially if they succeed to some extent, would tell Wilkinson from the start that the learning problem may well be all about him. He seems to pick this up later in the book and - surprise! surprise! The book gets much better when he does.
He also has a member of his extended family who teaches advanced math at the University of Chicago whom he called when he is frustrated. I hope his interactions with her are more invention than reality - you know “based on a true story”. It appears that he treats he very poorly and that to her credit, she takes issue with the continual “algebra ate my homework” rant. She is far too polite.
In between the rants and the review of basic concepts and examples, the book actually drops a few interesting point down onto the table without even realizing it. For example, early in the book, a point was raised about doing algebra with “word problems.” In the course of mentioning this, the math professor wonders why the author is so interested in word problems, adding that these problems are not seen as very interesting at all by math professors.
Why teach students by word problems when these are not of interest to serious mathaticians? Perpahs because most of the student learning algebra are not being trained to go on and become math Ph.D.s but rather to go out into the world and actually use the math for some tasks in the world. If so, the use of word problems would possibly help students try to apply the math to real situations. It is very clear that math capability via word problems is of interest to career recruiters, especially for firms inn high tech innovative industries, where students are hired for potential careers rather than particular individuals jobs. This suggest that while the mathematicians can and should teach to core math ideas, there also needs to be investments by firms to ensure that students learn how to apply the math in particular situations. That is something that can be written about without so much whining.
Warts and all, this was an interesting and worthwhile book to read.
There is much to like in this book. I sympathize with the author and have tried at various points to reintroduce my self to these threshold math experiences. I even found myself in a situation earlier in my career where if I did not learn some calculus based statistics then I would have to rethink my career aspirations. I decided to “knuckle down” and learn the math/stats and never regretted it.
At various points throughout the book, the author is effective and even eloquent in passing on his triumphs in math to the reader. He is an effective communicator and I was left wondering why he did not just sign up to teach a high school algebra class. Take on such a class and be forced to prep it and I guarantee that the requisite math will be learned. I also enjoyed Mr. Wilkinson bringing up flurries of math info along the way but also with sufficient cites for following through on his suggestions.
What didn’t I like about the book? Hmmm…? Mr. Wilkinson has structured important parts of his book (the first part) around the idea that he has been let down by math teachers, math textbooks, and even by mathematics itself. Sigh. I would have thought that with all his experience as a writer, Wilkinson would know that if literally millions of teenagers can learn algebra to some extent while you the narrator flounder, then perhaps - just perhaps - the problem lies with him rather than with the algebra establishment conspiring against him. The idea that the behavior of assembled masses may be informative, especially if they succeed to some extent, would tell Wilkinson from the start that the learning problem may well be all about him. He seems to pick this up later in the book and - surprise! surprise! The book gets much better when he does.
He also has a member of his extended family who teaches advanced math at the University of Chicago whom he called when he is frustrated. I hope his interactions with her are more invention than reality - you know “based on a true story”. It appears that he treats he very poorly and that to her credit, she takes issue with the continual “algebra ate my homework” rant. She is far too polite.
In between the rants and the review of basic concepts and examples, the book actually drops a few interesting point down onto the table without even realizing it. For example, early in the book, a point was raised about doing algebra with “word problems.” In the course of mentioning this, the math professor wonders why the author is so interested in word problems, adding that these problems are not seen as very interesting at all by math professors.
Why teach students by word problems when these are not of interest to serious mathaticians? Perpahs because most of the student learning algebra are not being trained to go on and become math Ph.D.s but rather to go out into the world and actually use the math for some tasks in the world. If so, the use of word problems would possibly help students try to apply the math to real situations. It is very clear that math capability via word problems is of interest to career recruiters, especially for firms inn high tech innovative industries, where students are hired for potential careers rather than particular individuals jobs. This suggest that while the mathematicians can and should teach to core math ideas, there also needs to be investments by firms to ensure that students learn how to apply the math in particular situations. That is something that can be written about without so much whining.
Warts and all, this was an interesting and worthwhile book to read.
February 2, 2023
Wilkinson’s book, A Divine Language (2022), about learning algebra, geometry, and calculus in his sixties, is a compelling read about the struggle to master mathematical topics beyond basic arithmetic. The author reflected on his inability to master algebra, geometry, and calculus as an elementary through high school student.
An apt description for adding and subtracting versus using “x” and “y” to solve equations was offered: “A problem in arithmetic was vertical, one number beneath another, and a problem in algebra, an equation, was horizontal.” It was easy to learn multiplication and long division in elementary school too, but much harder to grasp algebra in middle school.
“Mathematics is severe and faultless,” notes Wilkinson. “It became the language of science because of its precision. [Einstein’s] theory of relativity can be written in prose, but e = mc2 is more succinct.”
Mathematics can be difficult to comprehend. “In On Proof and Progress in Mathematics, William Thurston writes, “The transfer of understanding from one person to another is not automatic. It is hard and tricky.”
Wilkinson’s insists one must learn to separate his or ego from always being right, otherwise frustration and dismay sets in, “I had to disengage myself from believing that my identity was attached to my being right. That I was able to, barely, was a surprise to me.”
If one can master mathematics, he or she can more easily delve into the field of physics, if they so desire. In 1939, physicist Paul Dirac said, “Pure mathematics and physics are becoming ever more closely connected, though their methods remain different. One may describe the situation by saying the mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen.”
Even as a sixty-year-old, Wilkinson observes, “In math I am failing against a pitiless standard. There is no ambiguity, whereas failure in the eyes of others, as a writer, say, is a matter of other people’s judgment, of taste and bias and opinion, all of which are culturally conditioned and can be questioned. There are very few absolute standards to life, however much a person sometimes wish that there were.” He adds, “Everyone needs to make his or her own mistakes, and some of mine might have been to study math. We have things to tell each other in the form of advice and instruction, but very little of it is ironclad or will mean the same to one person as another. Experience is individual. And making a mistake is maybe not the same thing as failing.”
He posits that even failing to grasp all of mathematics’ concepts, learning about oneself can be achieved. Wilkinson, in a reflective mood, writes, “If I had challenged myself, instead of giving up, I might have more to show for it. There is only so much, though, that one can reasonably expect of one’s childhood self.”
If anything is learned from his musings, one should keep on learning and striving. Enjoy the read, A Divine Language.
An apt description for adding and subtracting versus using “x” and “y” to solve equations was offered: “A problem in arithmetic was vertical, one number beneath another, and a problem in algebra, an equation, was horizontal.” It was easy to learn multiplication and long division in elementary school too, but much harder to grasp algebra in middle school.
“Mathematics is severe and faultless,” notes Wilkinson. “It became the language of science because of its precision. [Einstein’s] theory of relativity can be written in prose, but e = mc2 is more succinct.”
Mathematics can be difficult to comprehend. “In On Proof and Progress in Mathematics, William Thurston writes, “The transfer of understanding from one person to another is not automatic. It is hard and tricky.”
Wilkinson’s insists one must learn to separate his or ego from always being right, otherwise frustration and dismay sets in, “I had to disengage myself from believing that my identity was attached to my being right. That I was able to, barely, was a surprise to me.”
If one can master mathematics, he or she can more easily delve into the field of physics, if they so desire. In 1939, physicist Paul Dirac said, “Pure mathematics and physics are becoming ever more closely connected, though their methods remain different. One may describe the situation by saying the mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen.”
Even as a sixty-year-old, Wilkinson observes, “In math I am failing against a pitiless standard. There is no ambiguity, whereas failure in the eyes of others, as a writer, say, is a matter of other people’s judgment, of taste and bias and opinion, all of which are culturally conditioned and can be questioned. There are very few absolute standards to life, however much a person sometimes wish that there were.” He adds, “Everyone needs to make his or her own mistakes, and some of mine might have been to study math. We have things to tell each other in the form of advice and instruction, but very little of it is ironclad or will mean the same to one person as another. Experience is individual. And making a mistake is maybe not the same thing as failing.”
He posits that even failing to grasp all of mathematics’ concepts, learning about oneself can be achieved. Wilkinson, in a reflective mood, writes, “If I had challenged myself, instead of giving up, I might have more to show for it. There is only so much, though, that one can reasonably expect of one’s childhood self.”
If anything is learned from his musings, one should keep on learning and striving. Enjoy the read, A Divine Language.
February 19, 2023
From the title, I was expecting a book about learning math, the challenges, and a dive into ways of teaching it to those who've been mangled by our math educational system.
The author was one of those who didn't "get math" in school and he seems to have sadly been wounded by it. Unfortunately he embarks on his journey without the help of a learning coach. He describes his frustrations and realizes some of his behavior in re-tackling maths is counterproductive (e.g. using a faulty approach even though he knows its wrong, purely out of obstinacy).
It would have been cool to see him work with a coach, to see the author figure out where the blockages are, and how to get around them. (The kind of coach like Eloise Ristad, a musical educator who wrote "A Soprano on Her Head", except one who does math).
There is one tantalizing example that Wilkinson offers. In one chapter he talks about reading a math textbook: "This book has at the beginning the sentence 'A variable is a letter used to represent one or more numbers.' I would find it easier if the sentence were written, “In algebra, when you don’t know a number, you use a letter, called a variable.” In the phrasing of the textbook explanation, questions arise for me: How do I use a variable? In what circumstances? Does one letter represent more than one number, or do I need additional letters? In my version I am presented with a rule and am prepared to understand a following rule: when more than one number is unknown, you will use more than one letter, one for each unknown."
Spot on! In this excerpt the author shows a tremendous amount of insight into the nature of mathematical definition. His rephrasing of the textbook is excellent, showing a deep understanding. This paragraph convinces me that he could have become a mathematical adept with the right coach.
I would have enjoyed seeing more of his thought patterns. I get the impression at the end of the book that he isn't satisfied at the result of his journey. His sidetrack into mathematical philosophy (Platonism etc) seems like a retreat from the battlefield. :)
There are also sidetracks into the lives of eccentric math prodigies - Godel et al. This makes for interesting reading but again it's not relevant to learning math. (Plus it reaffirms the stereotype of the mathematical egghead).
On the plus side, Wilkinson is a talented writer. The prose is smooth and he's done his research. Some things I learned: Ramanujan's ugliest formula :), Anne Carson's lovely poem - Essay on What I Think About Most. (Wilkinson uses only the first few lines of the poem which talks about the shame of error, which is sad because the poem is about how metaphor is tied up with error, and what is math but a form of metaphor?)
Anyhow I hope Wilkinson doesn't shelve his math books. His journey may not be over....
The author was one of those who didn't "get math" in school and he seems to have sadly been wounded by it. Unfortunately he embarks on his journey without the help of a learning coach. He describes his frustrations and realizes some of his behavior in re-tackling maths is counterproductive (e.g. using a faulty approach even though he knows its wrong, purely out of obstinacy).
It would have been cool to see him work with a coach, to see the author figure out where the blockages are, and how to get around them. (The kind of coach like Eloise Ristad, a musical educator who wrote "A Soprano on Her Head", except one who does math).
There is one tantalizing example that Wilkinson offers. In one chapter he talks about reading a math textbook: "This book has at the beginning the sentence 'A variable is a letter used to represent one or more numbers.' I would find it easier if the sentence were written, “In algebra, when you don’t know a number, you use a letter, called a variable.” In the phrasing of the textbook explanation, questions arise for me: How do I use a variable? In what circumstances? Does one letter represent more than one number, or do I need additional letters? In my version I am presented with a rule and am prepared to understand a following rule: when more than one number is unknown, you will use more than one letter, one for each unknown."
Spot on! In this excerpt the author shows a tremendous amount of insight into the nature of mathematical definition. His rephrasing of the textbook is excellent, showing a deep understanding. This paragraph convinces me that he could have become a mathematical adept with the right coach.
I would have enjoyed seeing more of his thought patterns. I get the impression at the end of the book that he isn't satisfied at the result of his journey. His sidetrack into mathematical philosophy (Platonism etc) seems like a retreat from the battlefield. :)
There are also sidetracks into the lives of eccentric math prodigies - Godel et al. This makes for interesting reading but again it's not relevant to learning math. (Plus it reaffirms the stereotype of the mathematical egghead).
On the plus side, Wilkinson is a talented writer. The prose is smooth and he's done his research. Some things I learned: Ramanujan's ugliest formula :), Anne Carson's lovely poem - Essay on What I Think About Most. (Wilkinson uses only the first few lines of the poem which talks about the shame of error, which is sad because the poem is about how metaphor is tied up with error, and what is math but a form of metaphor?)
Anyhow I hope Wilkinson doesn't shelve his math books. His journey may not be over....
April 27, 2024
Anyone thinking about reading A Divine Language should know that it is not some variation on Math for Dummies, and it is not Math for Old Folks. The book is more like a memoir of the author's experience of realizing fairly late in life that he never mastered the sort of mathematics that is typically taught in grade school and high school (which is true of more people than it is not) and always felt guilty about it. So he tried to remedy that shortcoming with a crash self-study program. He dove in at the beginning algebra level and continued until he got through fundamental calculus. In the end, he concluded that he had succeeded in some small ways and failed in others, but the experience had taught him a lot—as experience is wont to do.
He had the advantage of an advisor in the form of a relative, Amie (a cousin or niece as I recall) who is a mathematician (and so is her husband) whom he was free to call and rant to pretty much any time he was feeling frustrated. She led him by the hand through many difficulties. Most beneficially, she taught him to deconstruct problems and solve them in parts or steps, a valuable lesson that can be applied to many pursuits.
I relate strongly to author Alec Wilkinson's experience, and maybe other readers will, too. I've never been afraid of math, nor do I think of myself as math illiterate. In high school, I took an academic path toward engineering but ended up in music. However, the most advanced math that I learned was college algebra and trigonometry, which I liked, but I never got to study calculus. Several times I've set out to learn it on my own in my adult life (and I'm now 80 years old), e.g. by plowing through Khan Academy (which I regard as an excellent online teaching resource), but I'm otherwise quite busy, so I've never been able to get enough momentum to get very far—which would necessarily involve reviewing some things I knew at one time but need to be reminded of until they are intuitive.
The value in reading A Divine Language is in coming to appreciate (or reappreciate) how deeply mathematics is embedded in everything we experience in life. Wilkinson also reminds the reader that math, unlike science, is about absolute truth, and whereas scientists often disagree, mathematicians rarely do.
He had the advantage of an advisor in the form of a relative, Amie (a cousin or niece as I recall) who is a mathematician (and so is her husband) whom he was free to call and rant to pretty much any time he was feeling frustrated. She led him by the hand through many difficulties. Most beneficially, she taught him to deconstruct problems and solve them in parts or steps, a valuable lesson that can be applied to many pursuits.
I relate strongly to author Alec Wilkinson's experience, and maybe other readers will, too. I've never been afraid of math, nor do I think of myself as math illiterate. In high school, I took an academic path toward engineering but ended up in music. However, the most advanced math that I learned was college algebra and trigonometry, which I liked, but I never got to study calculus. Several times I've set out to learn it on my own in my adult life (and I'm now 80 years old), e.g. by plowing through Khan Academy (which I regard as an excellent online teaching resource), but I'm otherwise quite busy, so I've never been able to get enough momentum to get very far—which would necessarily involve reviewing some things I knew at one time but need to be reminded of until they are intuitive.
The value in reading A Divine Language is in coming to appreciate (or reappreciate) how deeply mathematics is embedded in everything we experience in life. Wilkinson also reminds the reader that math, unlike science, is about absolute truth, and whereas scientists often disagree, mathematicians rarely do.
October 5, 2022
At times, when problems got difficult, and I failed one after another, I felt as if I’d been left at a senior center and given a textbook of problems to pass my day, because the attendants had been told I enjoyed math.
That's just one of the many passages where I laughed out loud while also wincing, because ouch. I guess this is the knowing laughter Brené Brown talks about?
As a fellow math-phobic, I found this book absolutely delightful and inspiring. I could relate to the blackout frustration in dealing with math problems, while also getting oddly... curious about trying to learn math for reals now at an older age. And I think I'm falling into the same pit that Wilkinson did: as we get older, we presume that through maturity and experience middle and high school topics will be a frigging breeze to learn, if we revisit them now. But instead, we find that our brains can't handle something that adolescents are expected routinely to power through. Math can't be negotiated with or flubbed through by bullshittery and guesswork. After reading Wilkinson's experience with the frustrations and the mind-blowing moments of dealing with math, I'm tempted to compare math to tap dance, among other dances. I used to find dancing easy - waltzes, tango, random folk dances, lindy hop... but then tap dance kicked my ass because any minor mistake in tempo or choreography would be instantly AUDIBLE. Like math, it didn't care that you tried your best.
What a digression. Anyway. This book is funny, and scary, too: if I try to learn math now, will I just prove that I'm a dolt if I don't get it? Will I think less of myself if I fail? Can't I continue to live my days in blissful ignorance that if I only cared enough I could totally learn...? Wilkinson's account is a wonderful testament to simultaneously both neuroplasticity and the limits of our learning.
That's just one of the many passages where I laughed out loud while also wincing, because ouch. I guess this is the knowing laughter Brené Brown talks about?
As a fellow math-phobic, I found this book absolutely delightful and inspiring. I could relate to the blackout frustration in dealing with math problems, while also getting oddly... curious about trying to learn math for reals now at an older age. And I think I'm falling into the same pit that Wilkinson did: as we get older, we presume that through maturity and experience middle and high school topics will be a frigging breeze to learn, if we revisit them now. But instead, we find that our brains can't handle something that adolescents are expected routinely to power through. Math can't be negotiated with or flubbed through by bullshittery and guesswork. After reading Wilkinson's experience with the frustrations and the mind-blowing moments of dealing with math, I'm tempted to compare math to tap dance, among other dances. I used to find dancing easy - waltzes, tango, random folk dances, lindy hop... but then tap dance kicked my ass because any minor mistake in tempo or choreography would be instantly AUDIBLE. Like math, it didn't care that you tried your best.
What a digression. Anyway. This book is funny, and scary, too: if I try to learn math now, will I just prove that I'm a dolt if I don't get it? Will I think less of myself if I fail? Can't I continue to live my days in blissful ignorance that if I only cared enough I could totally learn...? Wilkinson's account is a wonderful testament to simultaneously both neuroplasticity and the limits of our learning.
July 30, 2023
Interesting premise. The author is an accomplished author, but has sort of regretted that he had done so poorly in High School math. So he sets aside a year to try to teach himself Algebra, Geometry, and Calculus.
He leans reasonably heavily for tutorial advice on his niece, a U of Chicago mathematician. (Full disclosure: I know her, and that’s more or less how I came to read this book.)
There’s a lot about philosophy of math, which I don’t find that intersting (and the author was not able to make it interesting to me). There are a few stories about particular mathematicians, and how they came across what they came across, and how they feel about it. Those were interesting.
I don’t think he did a great job of explaining his difficulties; indeed, he did try to explain some how he doesn’t quite understand what his difficulties are.
There were some points where he was talking about some mathematical concept, and it didn’t quite come out right. That is, it wasn’t the way I’m used to hearing those parts of math described. Again, that might have been part of the point---he’s not a mathematician, and you might expect his way of seeing things to be a little different. (Although there was one conjecture that he just stated wrong---what he stated is not what the conjecture is. His statement was trivially true. It was clear (to me, at least) what he meant. Still, I was surprised the editors didn’t catch that.)
He leans reasonably heavily for tutorial advice on his niece, a U of Chicago mathematician. (Full disclosure: I know her, and that’s more or less how I came to read this book.)
There’s a lot about philosophy of math, which I don’t find that intersting (and the author was not able to make it interesting to me). There are a few stories about particular mathematicians, and how they came across what they came across, and how they feel about it. Those were interesting.
I don’t think he did a great job of explaining his difficulties; indeed, he did try to explain some how he doesn’t quite understand what his difficulties are.
There were some points where he was talking about some mathematical concept, and it didn’t quite come out right. That is, it wasn’t the way I’m used to hearing those parts of math described. Again, that might have been part of the point---he’s not a mathematician, and you might expect his way of seeing things to be a little different. (Although there was one conjecture that he just stated wrong---what he stated is not what the conjecture is. His statement was trivially true. It was clear (to me, at least) what he meant. Still, I was surprised the editors didn’t catch that.)
August 2, 2022
For the most part, this is a beautiful exploration of the "limits" of what any one person can know in a lifetime. I would imagine those explorations apply as well to literature and other sciences as well as to math. Who hasn't wished at one time or another, as a non-specialist, to have read all of Shakespeare or all of the Bible or all of Austen or Tolstoy or Turgenev, not just to have read but to remember and have the literature woven into one's consciousness? To wish to know mathematics, as Wilkinson wants thoroughly to know mathematics, seems at times self-defeating and self-punishing. Even if one learns "adolescent" mathematics (high school geometry, algebra, and calculus) in public school, does one expect to retain that knowledge without constant reinforcement over a lifetime? What Wilkinson expects of himself as both young student and mature seeker seems unrealistic to me, as unrealistic as what he considers the edge of old age to be. Some of the most enlightening moments in the book are concerned with the question of whether mathematics is invented or discovered, a product of the cleverness of humans or a product of an other- or self-designed universe.
February 6, 2023
I loved this book - it is one of the best non-fiction titles that I have read in some time. Also, I feel so grateful to author Alec Wilkinson. The author invests a significant amount of time, despite being at an age where most of his peers are beginning to really enjoy life, to re-learn mathematics (algebra, geometry and calculus) with a fresh perspective and beyond the confines of (1) adolescence and (2) the structured classroom. I am grateful, because his act is something that I often daydream about doing myself, and now I know, after reading "A Divine Language", that I don't have to. So many kids struggle with math in school, and for many generations, this struggle was considered a character flaw, and has filled millions with shame. Alec dispels those silly notions and replaces them with a heartfelt, honest self-assessment of his own process against the difficulty of learning this subject at any age. I also enjoyed his literary side-trips, first about Zitang Zhang, who teaches and studies math simply for the love of it, and second, about the particular type of collaboration that exists within the profession.
February 10, 2023
An older man doggedly attempts to understand Algebra, High School Geometry, and Calculus. Algebra is the point in mathematics where the things you study get abstract. You solve for unknowns and other unexpressed quantities. Geometry is proofs and demonstrations. I haven’t had it for over twenty years. Calculus is considered the big one, but it is merely a stepping stone to further studies.
None of that information helps Alec Wilkinson, though. The book is more memoir than math at the moment. Wilkinson discusses his shortcomings and looks to other people with the same problem. Wilkinson quotes many people throughout the book.
I don't have any issues with the book. Wilkinson is a master wordsmith and explains his shortcomings and reasoning well. Furthermore, Wilkinson asks deeper questions about why people do mathematics. As a pure science, mathematics didn’t have defense applications, say, as physics did. It underlies internet security, but that wasn’t always the case. Perhaps it boils down to being there and mocking people who fall victim to its enticements. I mean, why do people climb Mount Everest?
Thanks for reading my review, and see you next time.
None of that information helps Alec Wilkinson, though. The book is more memoir than math at the moment. Wilkinson discusses his shortcomings and looks to other people with the same problem. Wilkinson quotes many people throughout the book.
I don't have any issues with the book. Wilkinson is a master wordsmith and explains his shortcomings and reasoning well. Furthermore, Wilkinson asks deeper questions about why people do mathematics. As a pure science, mathematics didn’t have defense applications, say, as physics did. It underlies internet security, but that wasn’t always the case. Perhaps it boils down to being there and mocking people who fall victim to its enticements. I mean, why do people climb Mount Everest?
Thanks for reading my review, and see you next time.
October 27, 2022
I very much enjoyed the philosophy and history of math parts (i.e., most of the book), perhaps because my father was a mathematical logician and I miss him! But I agree with the reviewer who said that the author did seem unusually dense on the subject of math, and I also agree with the New York Times reviewer who questioned why the author didn't get a serious tutor. As it was, I often felt frustrated with the author and the book and its somewhat negative tone. I was like, "C'mon -- get some help! Go to a class! Don't just sit there struggling on alone and whining about it!" He had various rationales for not doing those things, but if it was OK to ask help from his math prof niece, why not get tutoring from a high school teacher or someone better equipped to help a beginner?
Sigh. I guess when I set out to learn something, I actually want to succeed at it, whereas it seems like the author's true goal was to write a book about how hard math is. Or something.
Sigh. I guess when I set out to learn something, I actually want to succeed at it, whereas it seems like the author's true goal was to write a book about how hard math is. Or something.
January 29, 2023
I really commend Alec for tackling a subject as challenging as math as an adult. The book appealed to me because I am also actively learning calculus for the first time as an adult.
My one criticism is that he went to high level mathematicians for guidance instead of other sources that may have been better able to communicate on his level. For example, seeking out others learning math topics as adults. From reading about their (our) struggles, he may have been less defeated by his own.
I think that seeking guidance from professional mathematicians may have set him up for unrealistic expectations based on the relative ease with which they handled the topics he struggled with.
I was a little disappointed that he was content to end his math journey once essentially completing calc I. It's his own choice for sure but I just thing there are so many interesting topics to learn when we get beyond calculus.
My one criticism is that he went to high level mathematicians for guidance instead of other sources that may have been better able to communicate on his level. For example, seeking out others learning math topics as adults. From reading about their (our) struggles, he may have been less defeated by his own.
I think that seeking guidance from professional mathematicians may have set him up for unrealistic expectations based on the relative ease with which they handled the topics he struggled with.
I was a little disappointed that he was content to end his math journey once essentially completing calc I. It's his own choice for sure but I just thing there are so many interesting topics to learn when we get beyond calculus.
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