Book Review-Infinite Powers By Steven Strogatz

After having been trained in the dark arts of engineering, I considered myself, if not a full-blown expert, at least well versed in calculus: what it is, what the idea can be used for in my engineering specialty, and some of the history about calculus through my classes. I now happily admit to my unreasonable hubris. Steven Strogatz is the master of all that is calculus, what it was, how it came into being, where it is now, and possibly where it is going to be in the future.

My initial arrogance concerning my knowledge on the subject made the initial chapters of the book flow by quickly, I read the historical notes that Strogatz included as a sidelight to the main discussion — something to entertain the less informed of the general reader — I wondered why we would need to know the arcana of ancient mathematics, even I also fancied myself a nerd for history of mathematics? Little did I realize how important these historical notes will be: to drive later discussions as well as to form the foundations of the macro view of calculus.

Strogatz frames the story of calculus in ten chapters, creating the intricate scaffolds that allows the readers to follow the technical developments through history with added notes on the mathematicians that originated the ideas which drove calculus to where it is now. An eleventh chapter serves as his own peering-into-the-crystal-ball statement on what he believes will come in the future. He carefully builds up the structure of the development of calculus and seamlessly build the connections between subjects and shows the open questions that was left at the end of the previous chapters and how the topics covered in the new chapters serves to answers those open questions. It is this attention to the many loose ends and how they were resolved that held my interest.

As Strogatz observed, the teaching of calculus had been subdivided into many subtopics for the sake of convenience, but in so doing, the students had been sold a myth that these subtopics are standalone topic because it suited the purposes of teaching logistics rather than suiting the purposes of gaining a holistic view of what mathematicians throughout history had wrought, continuously.

Chapters 8, 9, and 10 were the chapters that had me holding my breath, for it is in these chapters that Strogatz pulled together all the work from the previous seven chapters, integrated them and brought the story to a denouement, for the moment. It brought together the differential and integral halves of calculus, showed the true powers of the calculus. True to the title of the book, he also forcefully made the point of just how the powerful idea of infinity allowed the method to flourish in the minds of mathematicians, scientists, engineers, and so many more specialties.

One cannot discuss calculus without discussions — many times heated ones — about the two men who are recognized as the progenitor of the largest leaps forward in the calculus: Newton and Leibnitz. Strogatz recognition of each, while nicely put their contributions into the historical context of calculus without delving into the bickering that happened between proponents of either men, which is as it should be, even though the human smallness in me wanted some juicy stories about the two.

Strogatz introduced us to the important women mathematicians which made contribution to the art and science of calculus, their contributions were most often ignored and if recognized, their works were slighted. He gave them credit where it was due, and the book is much better for the recognition.

As I was taking my time reading and enjoying the narrative, I thought about how this book should be made an integral part of the teaching of calculus, a required text taught in parallel with the technical aspects of calculus; a book that answers the “why” and “how did it get this way” questions in parallel with the technical training that answers the “how to do it” questions. I then realized that the reasons that I appreciated this book so much are not the same reasons that the young students in AP Calculus or in college level calculus would appreciate. It took me years of working with the calculus to ask those questions that Strogatz had sought to answer. It takes a certain level of maturity and appreciation for the context of the methods which built up the citadel that is the calculus. I still think that the material in this book has a critical role to play in motivating the understanding of the “how” while also building an appreciation for what our forebears had wrought. As Newton had said: We Stand on the Shoulders of Giants. This book would nicely illumnate that blind spot.

One note of interest. I was reading Mortimer Adler and Charles Van Doren’s book How To Read a Book (Adler 1972) when it was brought to my attention that until the end of the nineteenth century, scientific books were written for the layman, that the habit of having specialists writing only for specialists was necessitated by the increasing complexities that comes with the expansion of knowledge in each scientific topic, so that the necessary knowledge needed to understand scientific books became so broad as to be covered in a single tome. Which I thought was a shame, but I understand how daunting the task of writing science and mathematics books has become. Which makes this tome that much more impressive in that, whether Strogatz realizes it or not, he had accomplished a rare and difficult feat — to communicate this very specialized and complex topic to the general public — a general public that has varying levels of a priori knowledge to draw upon to aid in their comprehension. He has joined the pantheon of authors which serves the knowledge of everyone, if they chose to read the book. He has served the role of the public intellectual by writing this book.

This is a remarkable book from my perspective, it filled in the gaps of my knowledge, technical, historical, and conceptual, without losing my interest nor overwhelmed me.

References

Adler, Mortimer. "How to Read Science and Mathematics." In How To Read A Book, by Charles Van Doren Mortimer Adler, 255-269. New York: Touchstone, Simon and Schuster, 1972.

 

 

 

 

Book Review-A Mathematician's Lament By Paul Lockhart

In the forward to this monograph. Kevin Devlin of Stanford University, a well renowned mathematician, tells the story of how Paul Lockhart, someone who had given up his career as a research mathematician to devote himself to the mission of improving K-12 mathematical education, turned an earnest but obscure essay into a resounding statement.

The first genesis of this book is as a 25-page document that was passed around in the mathematical education circles. It became a sensation because many felt that Paul Lockhart had hit the nail on the head with his observations; observations and beliefs that resonated with mathematics educators; indeed, he struck a very sensitive nerve. As this document was passed around, it became a clarion call to mathematicians, mathematics teachers, and anyone who has a passion for how mathematics is taught.

A Mathematicians Lament is short and compact. Paul Lockhart had a lot to say, and he says it with urgency and alarm. Part I of  the book is the lamentation, he goes into everything that he feels is wrong with mathematical education. He makes his argument progressively starting with a discussion on mathematics and culture,  then a discussion on mathematics in the school, a dive into the national mathematics curriculum — a chapter in which he was unsparing in his criticism. In the last chapter in Part I, Lockhart zeroes in on a well-known and well reviled target: high school geometry.  Lockhart gave it a subtitle: Instrument of the Devil. This is his coup de grâce, his pronouncement on the abysmal state of mathematics education in the United States.

He expounds on the insidious practice of limiting mathematics education to just computation, while emphasizing the mechanical and uninspiring practice of training skills without giving the students a vision of what true mathematics is. We don’t give the students enough credit for being perspicacious enough to sense the immutable and deep beauty of mathematics. We don’t give the allure of the mathematical abstraction enough credit for being able to inspire and elicit  passion from the students; we think that the average student could not fathom the depths of meaning of mathematics; and that the student can only appreciate mathematics in its most utilitarian and unimaginative incarnation. It is an insult to the students and to mathematics.

As an engineer by training, I managed to survive my formal mathematics training with my love of mathematics intact, even though I knew my talent for theoretical math is limited.  I recognize all the stated pitfalls and shortcomings of how mathematics is taught because I had experienced it firsthand.

Although I  appreciate the beauty of mathematics, as I had aspired to be an applied mathematician; unfortunately, I had made a mess of the higher math that I took as a grad student in engineering, I didn’t have the patience nor the curiosity to sustain my interest because I was studying to gain a degree rather than studying for the love of a discipline. I was resigned to take enough applied math to help me become an engineer even though I was always curious about doing pure mathematics. Even as I have  resigned myself to the fact that I won't ever be a pure mathematician nor  even be a good applied mathematician, I have come to appreciate and love the subject.

In the second part of A Mathematician’s Lament — titled Exultation — Paul Lockhart made his elevator speech  to  anyone and everyone reading about the beauty of mathematics. He assiduously avoided the equations, a smart decision in my estimate. He dealt with mathematics as a holistic entity. He is much more eloquent in stating his case than I will ever be, so I will let the reader  read the book rather than dilute his passion and his narrative.

He discusses the common sensical instinctive aspect of  mathematics. There are crude but effective sketches about the points that he wanted to make, adding to the intuitive charm of the narrative.  He refrains from delving into the dreaded and unwelcoming geometry that he wrote about in Part I;  he uses simple sketches to ease the reader into mathematical thinking.

When he hit his stride talking about mathematics, it is a beauteous expression of passion he speaks of the raw beauty of mathematics that makes it so attractive, intoxicating,  and habit forming for so many. It is as if  mathematics is some kind of addiction. And to mathematicians that I know, and to a much lesser degree to me, that addiction is very real.

The second part of the monograph reminds me of the passion exuded by another book written by a mathematician. Francis Su wrote Mathematics for Human Flourishing, (Su 2020). I reviewed it in 2020. (https://polymathtobe.blogspot.com/2020/02/book-review-mathematics-for-human.html) Prof Su had the advantage of having a book to make his point about the allure of mathematics. It is perhaps a good companion book to buttress the second part of the argument.

The monograph is an extended essay identifying the problems with the way we have taught mathematics, how the math that is taught is contrary to what the mathematics lovers love about mathematics; what mathematics is in the eyes of those that are knowledgeable in the art; while  proposing in broad strokes what need to be done to change that paradigm. It is a timely and necessary clarion call to our society and our educators that we are irresponsibly squandering our opportunity to educate our society in the art of thinking, questioning, and creating. It is an attempt to reverse the trend, and more broadly, it is a valiant attempt to convince a math deficient public that they are missing the boat, and our society will suffer.

I hope that this is not just preaching to the choir, but the obstacles to universal understanding of the importance of the subject is quite high. I hope that Paul Lockhart is not too late.

Works Cited

Su, Francis. Mathematics for Human Flourishing. Yale: Yale University Press, 2020.

 

 

 

 

Book Review-Mind and Matter: A life in Math and Football. John Urschel and Louisa Thomas.

I read the bulk of this book, two hundred pages, in one sitting.

It was so engrossing partly because of how well written this book is, the co-author, Louisa Thomas is a well known writer; and partly because the book addresses two worlds that are dear to my heart: mathematics and sports. I didn’t engage either one of the worlds in the depth that the author does, I am an engineer and a youth coach, but the juxtapositions of the two worlds was held deep attraction for me.

For most of the general audience the two worlds are seemingly diametrically opposed, but the authors manage to portray the deep love that the two world engenders in John Urschel. Indeed, the authors did a magnificent job coupling the two seemingly disparate threads together into a cogent whole. At first, I feared that I was going to dislike the structure of the book: they chose to alternated math and football chapters, but the book was so well written that my perceived distraction evaporated as I dove into the book.

John Urschel’s story is widely reported in the popular press. He straddled the football and math worlds as an undergraduate, a graduate student, and a post grad while playing at Penn State and in the NFL. He was good enough to be drafted by the Baltimore Ravens and having a productive three years while also studying for his PhD in mathematics at MIT. This book roughly described his journey. The book tells a great story in an unselfconscious and natural way. John Urschel came through the account as a genuine and honest person, even as he addressed a few issues that could have been controversial: the fall out from the Sandusky affair at Penn State and the effect of repeated concussions on his potential as a mathematician, he honestly told his story focused on his own perceptions and thoughts, while assiduously avoided inflaming any nerves. He told the story through his eyes without extrapolating the facts to come to any indefensible conclusions, which is all we can ask for.

The other part of the book that could have been difficult is the mathematics. I have had the background and training to get through most of the mathematics, most of the concepts were on an advanced undergraduate to graduate level, John Urschel’s teaching ability was evident and shone through in his explanations of some of the more advanced mathematics topics. I moved away from any thoughts of majoring in mathematics after my initial experience with real analysis, so I was cognizant but not an expert in many of the areas; but I was able to understand his explanations of his work in Graph Theory, algorithm development, uncertainty, and spectral bisection. His explanations assume some background in math, but he was able communicate to the readers in an exceptionally clear fashion just in terms of concepts and intuition and without employing any mathematical language. In fact, intuition was his guiding light as he powered through his way through his mathematical explorations, and he was able to explain the role that intuition played in his mathematical thoughts.

The football portions of the story were told somewhat matter-of-factly. I would imagine that this was intentional, as the authors may have assumed that the general public who would read this book are thoroughly engaged in the intensity and passions of football in America. Two parts of the football story engaged me: his freshman year workouts with his strength and condition coach at Penn State and the Raven’s win over the Pittsburgh Steelers and loss to the New England Patriots in the NFL playoffs. Those stories captured and conveyed the passion that John Urschel had of the game of football as well as the mindset he employed to become successful in football.

In possibly one of the great acts in self-awareness and honesty comes in the last chapter when he describes why he walked away from football and is devoting his considerable intellect to mathematics. Unlike most great athletes, he recognized his shortcomings and he was able to explain his logic and reasoning for walking away with aplomb and honesty.

I was a nice easy read but the book talks about the mathematics that he is doing as well as taking the reader though his life so far. I think that our culture’s preoccupation with specialization drives our internal narrative. We are expected to focus and be great at one thing, that one thing should give us all a good life while contributing to the orderly conduct of our life in society, but we all know that human beings are complex, and our intellect can be multi-faceted. What John Urschel’s story illustrates is that by exceeding societal expectations in terms of what his role is in life, he is staking his claim as a polymath.

This was, a very enjoyable and entertaining read.

Book Review-The Lady Tasting Tea By David Salzburg

This was not a book I had envisioned as being something that I would read, let alone grow to love. My experience with statistics had been limited to some courses I took in graduate school and then exposed to when I was on my first job, we were all exposed to statistical process control (SPC) and six sigma. My background in statistics only went so far as knowing some of the SPC tools. As I grew more mature I began to appreciate the usefulness of statistics but I had a hard time connecting the SPC tools I was exposed to with the mathematic heavy statistics that are taught in the textbooks. As I tried to parse through the dense formal statistical curriculum I grew frustrated with my own inability to get through to the kernel of the topic. As I struggled I kept seeing this particular book being recommended by a number of people, so I bought it and prepared for the worst, yet another dense explanation of rudimentary statistics that had very little to do with what I wanted.

To my surprise and amazement, this book was so different, different from any other book that I had ever read. It was a love paean to the study of statistics, it was a gossipy and information laden history of the evolution of the art of probability and statistics, it was a summary of the important developments in statistics, it was an invaluable primer in the methods used in the practical application of statistic, and finally, it was a hefty philosophical discussion of the problems and issues that are still plaguing the researchers in statistics. I think you get the idea that I kind of liked reading this book.

David Salzburg is a practitioner of the art of statistics, he has the ability to explain the very dense concepts in statistics, both the applied tools and the mathematical conundrums with adept ease. Most importantly he did this without employing any mathematics. Which in some ways is very impressive and in other times it was frustrating because it would have been more enlightening to resort to the bare bones mathematics, but no matter.

Prof. Salzburg clearly has a great love for the story as well as for the subject, he has a great sense of history as well as a deft touch for the internecine nastiness that occurred with the giants of statistics. His descriptions of the relationship, or lack thereof between Pearson and Fisher kept me riveted to the narrative. His description of some of the great mathematicians who were caught in the destructive totalitarian regimes during and after World War II added the human dimension to these stories. I don’t know which aspect of the book I appreciated more, the historical perspective or the unraveling the mystery of the functional relationship between statistical tools and ideas.

There is a clear devotion in his writing that reflects his devotion to giving credit where credit is due, even though he apologized for his inability to give credit to all that had contributed, the breadth and depth of the book was astounding and gratifying to someone who appreciates a truly “Big Picture” look at the statistical landscape from the 10,000 feet view. I particularly enjoyed the discussions regarding the contributions of Deming and Shewart to the SPC branch of the vast tree of statistical evolution. I was able to make the connections from those chapters to untie the knot that was in my mind.

The piece de resistance was the final chapter where he discusses his own views on the unexplained philosophical contradictions still existing in statistics. It felt like I was in the midst of the discussion even though I am a dilettante in the art of statistics.

This is a book that comprised of some very dense concepts and it was difficult to focus at times but it was well worth the effort in my mind.