What do you think?
Rate this book
Infinite Powers: How Calculus Reveals the Secrets of the Universe
Without calculus, we wouldn’t have cell phones, TV, GPS, or ultrasound. We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket.
Though many of us were scared away from this essential, engrossing subject in high school and college, Steven Strogatz’s brilliantly creative, down‑to‑earth history shows that calculus is not about complexity; it’s about simplicity. It harnesses an unreal number—infinity—to tackle real‑world problems, breaking them down into easier ones and then reassembling the answers into solutions that feel miraculous.
Infinite Powers recounts how calculus tantalized and thrilled its inventors, starting with its first glimmers in ancient Greec
e and bringing us right up to the discovery of gravitational waves (a phenomenon predicted by calculus). Strogatz reveals how this form of math rose to the challenges of each age: how to determine the area of a circle with only sand and a stick; how to explain why Mars goes “backwards” sometimes; how to make electricity with magnets; how to ensure your rocket doesn’t miss the moon; how to turn the tide in the fight against AIDS.
As Strogatz proves, calculus is truly the language of the universe. By unveiling the principles of that language, Infinite Powers makes us marvel at the world anew.
Though many of us were scared away from this essential, engrossing subject in high school and college, Steven Strogatz’s brilliantly creative, down‑to‑earth history shows that calculus is not about complexity; it’s about simplicity. It harnesses an unreal number—infinity—to tackle real‑world problems, breaking them down into easier ones and then reassembling the answers into solutions that feel miraculous.
Infinite Powers recounts how calculus tantalized and thrilled its inventors, starting with its first glimmers in ancient Greec
e and bringing us right up to the discovery of gravitational waves (a phenomenon predicted by calculus). Strogatz reveals how this form of math rose to the challenges of each age: how to determine the area of a circle with only sand and a stick; how to explain why Mars goes “backwards” sometimes; how to make electricity with magnets; how to ensure your rocket doesn’t miss the moon; how to turn the tide in the fight against AIDS.
As Strogatz proves, calculus is truly the language of the universe. By unveiling the principles of that language, Infinite Powers makes us marvel at the world anew.
360 pages, Hardcover
First published April 2, 2019
About the author
Steven H. Strogatz
12 books936 followersSteven Strogatz is the Schurman Professor of applied mathematics at Cornell University. A renowned teacher and one of the world’s most highly cited mathematicians, he has been a frequent guest on National Public Radio’s Radiolab. Among his honors are MIT's highest teaching prize, membership in the American Academy of Arts and Sciences, and a lifetime achievement award for communication of math to the general public, awarded by the four major American mathematical societies. He also wrote a popular New York Times online column, “The Elements of Math,” which formed the basis for his new book, The Joy of x. He lives in Ithaca, New York with his wife and two daughters.
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 30 of 727 reviews
December 17, 2019
"Everything becomes simpler at infinity."
I have a habit of seeing a book, realising it's on a subject I don't know much about, and finding myself overcome with a strong desire, a need even, to read the book. That was the case with Infinite Powers: How Calculus Reveals the Secrets of the Universe. I knew nothing about Calculus. The only math I took in school was Algebra and Geometry and though I loved both of them, I can't remember diddly-squat about them. As for Calculus? Never learned it, never knew a thing about it. Thus, when I saw the cover of this book, I suddenly thought, "I don't know anything about Calculus. I HAVE to read this book!"
Now, I would be doing a donald and lying out my ass if I said that upon reading this book I understand Calculus and can perform any number of its equations. Nope. Not even close. Unfortunately, my brain couldn't quite make sense of it though I often re-read parts in hopes of "getting" it. However, I did at least learn what calculus is, how it is based on infinity, how it takes complex problems and breaks them down into simpler pieces in order to solve them. How it is obsessed with simplicity.
Steven H. Strogatz is a professor of applied mathematics and it must be such a delight to take one of his classes. He writes exuberantly about Calculus. He makes it impossible to NOT see the beauty in it, so poetically he writes, so passionately. I couldn't wait to pick up this book each time, it was so exciting to read! Yes, even though I didn't understand it all. Even though it sometimes made my brain hurt. That didn't matter, or maybe that added to my enjoyment, the way it made my brain go into overtime trying to make sense of what I was reading.
I was surprised to learn how much of modern human life is made possible by Calculus. Everything from putting rockets into space to storing music on cell phones to treating AIDs patients. There are a myriad of ways in which Calculus makes our lives simpler. My microwave oven croaked last night and my partner and I had to rush to the store after 10 in order to buy a new one. How could we possibly live without it? Well, it's thanks to Calculus that we have this luxury. And did you know: The first microwave ovens stood over 6 feet tall and cost what today would be thousands of dollars? Our kitchen is small so I'm thankful we could buy a compact microwave costing much less than thousands of dollars!
But I digress.....
I found it fascinating to learn how GPS would not be able to work without the aid of Calculus. The author says, "The whole global positioning system requires nanosecond accuracy to work properly" and that errors in it "would accumulate at about ten kilometers each day". We wouldn't get anywhere close to where we want to go using GPS if it wasn't for Calculus.
The author traces the history of Calculus, discussing in length the contributions of such people as Archimedes, Newton, Leibniz, Bruno, Galileo, and more. He blends equations with examples and with stories of women and men who helped discover these equations and who used them to discover other great things. The book is never dry, and it made it become more alive to have this history incorporated into it.
I debated giving this book four stars because I wasn't able to learn how to actually do Calculus, but as that is not the fault of the author and because I nonetheless had such a wonderful reading experience, I feel it deserves a full five stars. If anyone can make Calculus beautiful and exciting, this author certainly can! I still might not know how to do Calculus but I can appreciate its beauty and simplicity thanks to what I learned in this book. As Mr. Strogatz writes, "It isn’t necessary to learn how to do calculus to appreciate it, just as it isn’t necessary to learn how to prepare fine cuisine to enjoy eating it." And appreciate it I do!
February 7, 2021
The Language of God
Steven Strogatz seems conflicted. He presents mathematical calculus as a description of reality and within a paragraph or two recants and calls it a useful fiction. He claims that it reveals the hidden structure of the universe; yet he admits that its fundamental presumptions contradict the findings of modern physics. He is proud of the strictly logical development of mathematical inference; but he recognises that the invention of calculus was not the result of mathematical rationality. He is a philosophical Platonist who is sympathetic to the idea of the existence of numbers quite apart from our use of them. But he is also pragmatically Aristotelian in his insistence that his devotion is not because numbers are the Truth but because they Work.
Strogatz is not embarrassed by these contradictions because his mission is to generate enthusiasm for his subject among non-mathematicians. He is willing to say whatever it takes to create interest and understanding about the usefulness, beauty, and profound insights of his branch of mathematics. Infinite Powers Reads like a sort of investment prospectus for Calculus Inc. It’s purpose is to sell, not to criticise, judge or evaluate. What we are meant to buy is not just the fact that calculus has been an immense human triumph historically, but also that it is an emblem of a culture that deserves to be more widely spread. The book is, in short, a gospel which seeks to spread the good news of real rationality.
I sympathise with Strogatz’s appreciation of mathematics in general and calculus specifically. They are obviously useful, beautiful for those who take the time to study them, and they enforce a discipline of reasoning which is unparalleled in any other intellectual endeavour. It is this last with which I have issue with Strogatz, however. Gospel-writing evangelists are notoriously unreliable when it comes to what they mean by rationality. I understand that he wants to present “the world according to calculus.” But he is constantly slipping in to the ‘world as calculus,’ which is a very different matter. In doing so he distorts both calculus and the world outside of calculus.
Infinite Powers lays out a simple and, on the face of it, plausible epistemological theory: “Calculus is an imaginary realm of symbols and logic; nature is an actual realm of forces and phenomena. Yet somehow, if the translation from reality into symbols is done artfully enough, the logic of calculus can use one real-world truth to generate another.” The implicit claim is that the universe possesses characteristics and properties that can be revealed through the relationships of the symbols and the logical connections among these symbols within the calculus itself. This is an apparently uncontroversial claim which he makes buoyantly and triumphantly.
The claim is also wrong and demonstrably so. The ‘realm’ of calculus is a language, the characteristics of which have no referent at all in the realms outside of calculus. The most obvious of these is the essential presumption of calculus that the world it deals with is continuous. That is, that every force, event, and movement is describable to an infinite level of precision. This presumption is indeed warranted when dealing with what mathematicians call the ‘number line’ which is infinitely dense - because it is defined as such. This means that there is always another number between any two numbers, no matter how small those numbers are (this is the reason Zeno can’t reach his wall when thinking mathematically). And many of these numbers are ‘real’ in calculus but have no existence elsewhere.
Strogatz recognises that this is a problem. The world outside of mathematics is not continuous, nor does it contain distances of indeterminate length (like those involving a multiple of π). Quantum science, for example, is based on the presumption that everything about the world - matter, energy, even space - is ‘lumpy’ and cannot be infinitely divided. The world is discontinuous. More accurately the world outside of calculus as described by physicists contradicts the presumptions of calculus. Strogatz dismisses this with a wave and calls continuity a useful fiction. And that it may be, but like Newton’s gravitational action at a distance and Democritus’s solid atoms, it is a fiction which has no certain existence outside the world of language, and is disputed by alternative scientific fictions.
This applies as well to all other mathematical ‘phenomena,’ from infinitely thin lines to infinitesimal points, to n-dimensional surfaces. These are aspects of the syntax of mathematical language. They exist because they are defined within and by the language itself. No one would claim that these things have a referent outside mathematics. Like continuity in calculus, they are all useful fictions. That is, mathematics, including the mathematics of calculus, have no reliable connections to anything non-mathematical. In other words mathematical symbols and relations cannot ‘represent’ the world much less reveal it.
But there is also a related issue of which Strogatz seems unaware. Probably the best way to expose this issue is by using another artificial language, the Law. As in any language, legal terms are defined in relation to each other. These terms define various legitimate ‘actors’ as well as things upon which they can act. In Roman Law, for example, there are two ‘persons’ specified - the emperor (a unique individual) and the paterfamilias (a class of individuals). The paterfamilias is further defined as a male head of household. The terms ‘male,’ ‘head,’ and ‘household’ in the way of legalese are then further defined. And so on, potentially ad infinitum, but practically until sufficient precision has been reached so that it can be determined if a party involved in a law suit is a paterfamilias or not by a normally intelligent judge.
In other words ‘paterfamilias’ is a sort of legal mask (the Greek root of the word person) or, perhaps more plainly, a categorical box in which a litigant is placed and which determines how he will be treated in law. It may not be obvious, but this categorisation is not in any way a ‘property’ of the litigant. Neither are the various other criteria - maleness, familial seniority, or indeed familial status - which underlie the definition of paterfamilias. All of these are the consequence of record-keeping not existential characteristics. They are ‘facts’ because they are recorded somewhere by someone (or testified to by someone) and are presented to the court. The documentation defines the litigant as a person or it doesn’t.
Legally we are our documents - birth certificate, driving licence, passport, etc. This is so even in a much more technologically advanced society than ancient Rome, in fact more so because of the technology. For example, it might be argued that DNA analysis now can determine sex, and therefore maleness, without the need for other records. But DNA analysis is a report about a chemical reaction that involves a number of stages and many procedural guarantees to ensure the ‘integrity’ of the result. These are all documents, that is to say, language-based evidence. In fact this modern evidence is even more embedded in language than a quick peek under the toga and a nod by a court official in a Roman tribunal (note that the nod is also language, just a bit more concise than required in the modern legal system).
We are our documents for legal purposes. Or, more accurately, the language that is associated with our proper name, determines which category we are assigned within the legal system. These documents are certainly not ‘us’ in an existential sense. We do not necessarily have the properties recorded about us on our driving license - blue eyes, brown hair, glasses. Neither do these descriptive characteristics exhaust the ways in which we could be described. They are also categories into which we are placed by the system. But they nevertheless constitute our identity.
This is why ‘identity-theft’ is such an increasing problem. Defining documents can be mistaken, forged, mis-laid, duplicated, or mixed up with others. Proving such a crime or mistake requires producing yet more documents and showing that these documents are somehow superior to those produced by the criminal or incompetent clerk. The more steps in the process of document-production, the more complex this proof becomes. This complexity is itself the result not of some existential mystery - ‘I’ may be standing physically in front of a judge and jury trying a case - but of the issues of language involved: error, fraud, intent (also a linguistic issue), etc.
The point of this legal digression is to make two things clear. First, the world of language, including the language of mathematics, is a very different world from that of ‘what happens’ or ‘events’ or what we casually call reality. Legal definition and classification is a kind of measurement. It is a less precise kind of measurement than that of science or engineering but in principle it is the same process. A recording of events is assigned to a place within the definitional framework at hand. In the law this framework is a status such as a ‘person.’ In science and engineering it is usually a position on a numerical scale, a metric. The measurement involved is never a property of the thing measured. Rather, the thing measured becomes a property of the category or the metric.
So, second, when Strogatz writes about “artfully translating” between reality and mathematics, he is playing to an audience for whom this seems plausible only because it hasn’t had to think seriously about this kind of claim before. He doesn’t want to confuse potential devotees. But as has been said before by philosophers “Things happen; everything else is literature.”
Scientific measurement is as literary as a novel by Jane Austen. The two may be written in different languages but unless ‘what happens’ to the scientist or to the novelist is recorded, that is finds its way into language, it does not exist. It is merely hearsay in both law and science; it has no evidential value. And once ‘what happens’ becomes language, it is part of its artificial world in which it will be compared, connected, and judged by other documents, recordings and measurements contained in that world.
This isolation of the world of language from the world of not-language is not news. Plato’s metaphor of the cave within which shadows are cast by a flame behind us we cannot see is an ancient expression of the condition we are in. Immanuel Kant’s permanently elusive thing-in-itself is the classic Enlightenment statement of the same issue. And 20th century philosophers like Heidegger and Wittgenstein have clearly shown how language itself, all language, is implicated in what is our unique human condition.
None of these thinkers would have thought of denying that something called reality exists (although a certain 18th century Irish philosopher, Bishop Berkeley, did), or that we experience it more or less accurately (mostly less as it turns out). Their assertion is merely that reality cannot be captured reliably in language, that the world of language is entirely separate from the world of not-language. Or as the American Pragmatist philosopher put it so concisely: Reality “is surface all the way down.” That surface is the language we use to talk about it.
Remarkably perhaps, it is the parallel existence of the worlds of language and not-language that is the source of the power of language in its various dialects from mathematics to books to music. Because language is purely symbolic and purely definitional, that is self-contained, it is relieved entirely of constraints in either physical resources or imagination. It is a human creation rather than a ‘naturally’ occurring thing. We know everything about it since we created it. So we can do with it what we like.
Strogatz is entirely correct about this. But language, including mathematical calculus ‘works’ not because it describes the way the universe is ‘in reality.’ It works because it permits our species to think, to argue, to make judgements, and to learn how to do things progressively over generations. Language frees us from reality. This is our strength and perhaps our ultimate weakness as we spread a dense cloud of language-based consciousness along with its social vices and its overwhelming technology into every part of the planet.
It is this last point which Strogatz’s mathematical boosterism fails to note. This is what gets up my nose. Calculus is a powerful tool, a foundational technology, with a fascinating history. So is the hydrogen bomb. Strogatz wants us to believe that “there seems to be something like a code to the universe, an operating system that animates everything from moment to moment and place to place. Calculus taps into this order and expresses.” This is scientism not science. Scientism is an ideology which claims that there is a privileged way of understanding the world. It is a species of thought which operates under the more general heading of idolatry, the claim to know the essence of reality. What Strogatz is really selling is not calculus as a beautiful, useful, historically interesting tool, but as an inside track to the divine, the language of God (vying of course with Hebrew, Arabic, and Latin).
I’m not buying.
Steven Strogatz seems conflicted. He presents mathematical calculus as a description of reality and within a paragraph or two recants and calls it a useful fiction. He claims that it reveals the hidden structure of the universe; yet he admits that its fundamental presumptions contradict the findings of modern physics. He is proud of the strictly logical development of mathematical inference; but he recognises that the invention of calculus was not the result of mathematical rationality. He is a philosophical Platonist who is sympathetic to the idea of the existence of numbers quite apart from our use of them. But he is also pragmatically Aristotelian in his insistence that his devotion is not because numbers are the Truth but because they Work.
Strogatz is not embarrassed by these contradictions because his mission is to generate enthusiasm for his subject among non-mathematicians. He is willing to say whatever it takes to create interest and understanding about the usefulness, beauty, and profound insights of his branch of mathematics. Infinite Powers Reads like a sort of investment prospectus for Calculus Inc. It’s purpose is to sell, not to criticise, judge or evaluate. What we are meant to buy is not just the fact that calculus has been an immense human triumph historically, but also that it is an emblem of a culture that deserves to be more widely spread. The book is, in short, a gospel which seeks to spread the good news of real rationality.
I sympathise with Strogatz’s appreciation of mathematics in general and calculus specifically. They are obviously useful, beautiful for those who take the time to study them, and they enforce a discipline of reasoning which is unparalleled in any other intellectual endeavour. It is this last with which I have issue with Strogatz, however. Gospel-writing evangelists are notoriously unreliable when it comes to what they mean by rationality. I understand that he wants to present “the world according to calculus.” But he is constantly slipping in to the ‘world as calculus,’ which is a very different matter. In doing so he distorts both calculus and the world outside of calculus.
Infinite Powers lays out a simple and, on the face of it, plausible epistemological theory: “Calculus is an imaginary realm of symbols and logic; nature is an actual realm of forces and phenomena. Yet somehow, if the translation from reality into symbols is done artfully enough, the logic of calculus can use one real-world truth to generate another.” The implicit claim is that the universe possesses characteristics and properties that can be revealed through the relationships of the symbols and the logical connections among these symbols within the calculus itself. This is an apparently uncontroversial claim which he makes buoyantly and triumphantly.
The claim is also wrong and demonstrably so. The ‘realm’ of calculus is a language, the characteristics of which have no referent at all in the realms outside of calculus. The most obvious of these is the essential presumption of calculus that the world it deals with is continuous. That is, that every force, event, and movement is describable to an infinite level of precision. This presumption is indeed warranted when dealing with what mathematicians call the ‘number line’ which is infinitely dense - because it is defined as such. This means that there is always another number between any two numbers, no matter how small those numbers are (this is the reason Zeno can’t reach his wall when thinking mathematically). And many of these numbers are ‘real’ in calculus but have no existence elsewhere.
Strogatz recognises that this is a problem. The world outside of mathematics is not continuous, nor does it contain distances of indeterminate length (like those involving a multiple of π). Quantum science, for example, is based on the presumption that everything about the world - matter, energy, even space - is ‘lumpy’ and cannot be infinitely divided. The world is discontinuous. More accurately the world outside of calculus as described by physicists contradicts the presumptions of calculus. Strogatz dismisses this with a wave and calls continuity a useful fiction. And that it may be, but like Newton’s gravitational action at a distance and Democritus’s solid atoms, it is a fiction which has no certain existence outside the world of language, and is disputed by alternative scientific fictions.
This applies as well to all other mathematical ‘phenomena,’ from infinitely thin lines to infinitesimal points, to n-dimensional surfaces. These are aspects of the syntax of mathematical language. They exist because they are defined within and by the language itself. No one would claim that these things have a referent outside mathematics. Like continuity in calculus, they are all useful fictions. That is, mathematics, including the mathematics of calculus, have no reliable connections to anything non-mathematical. In other words mathematical symbols and relations cannot ‘represent’ the world much less reveal it.
But there is also a related issue of which Strogatz seems unaware. Probably the best way to expose this issue is by using another artificial language, the Law. As in any language, legal terms are defined in relation to each other. These terms define various legitimate ‘actors’ as well as things upon which they can act. In Roman Law, for example, there are two ‘persons’ specified - the emperor (a unique individual) and the paterfamilias (a class of individuals). The paterfamilias is further defined as a male head of household. The terms ‘male,’ ‘head,’ and ‘household’ in the way of legalese are then further defined. And so on, potentially ad infinitum, but practically until sufficient precision has been reached so that it can be determined if a party involved in a law suit is a paterfamilias or not by a normally intelligent judge.
In other words ‘paterfamilias’ is a sort of legal mask (the Greek root of the word person) or, perhaps more plainly, a categorical box in which a litigant is placed and which determines how he will be treated in law. It may not be obvious, but this categorisation is not in any way a ‘property’ of the litigant. Neither are the various other criteria - maleness, familial seniority, or indeed familial status - which underlie the definition of paterfamilias. All of these are the consequence of record-keeping not existential characteristics. They are ‘facts’ because they are recorded somewhere by someone (or testified to by someone) and are presented to the court. The documentation defines the litigant as a person or it doesn’t.
Legally we are our documents - birth certificate, driving licence, passport, etc. This is so even in a much more technologically advanced society than ancient Rome, in fact more so because of the technology. For example, it might be argued that DNA analysis now can determine sex, and therefore maleness, without the need for other records. But DNA analysis is a report about a chemical reaction that involves a number of stages and many procedural guarantees to ensure the ‘integrity’ of the result. These are all documents, that is to say, language-based evidence. In fact this modern evidence is even more embedded in language than a quick peek under the toga and a nod by a court official in a Roman tribunal (note that the nod is also language, just a bit more concise than required in the modern legal system).
We are our documents for legal purposes. Or, more accurately, the language that is associated with our proper name, determines which category we are assigned within the legal system. These documents are certainly not ‘us’ in an existential sense. We do not necessarily have the properties recorded about us on our driving license - blue eyes, brown hair, glasses. Neither do these descriptive characteristics exhaust the ways in which we could be described. They are also categories into which we are placed by the system. But they nevertheless constitute our identity.
This is why ‘identity-theft’ is such an increasing problem. Defining documents can be mistaken, forged, mis-laid, duplicated, or mixed up with others. Proving such a crime or mistake requires producing yet more documents and showing that these documents are somehow superior to those produced by the criminal or incompetent clerk. The more steps in the process of document-production, the more complex this proof becomes. This complexity is itself the result not of some existential mystery - ‘I’ may be standing physically in front of a judge and jury trying a case - but of the issues of language involved: error, fraud, intent (also a linguistic issue), etc.
The point of this legal digression is to make two things clear. First, the world of language, including the language of mathematics, is a very different world from that of ‘what happens’ or ‘events’ or what we casually call reality. Legal definition and classification is a kind of measurement. It is a less precise kind of measurement than that of science or engineering but in principle it is the same process. A recording of events is assigned to a place within the definitional framework at hand. In the law this framework is a status such as a ‘person.’ In science and engineering it is usually a position on a numerical scale, a metric. The measurement involved is never a property of the thing measured. Rather, the thing measured becomes a property of the category or the metric.
So, second, when Strogatz writes about “artfully translating” between reality and mathematics, he is playing to an audience for whom this seems plausible only because it hasn’t had to think seriously about this kind of claim before. He doesn’t want to confuse potential devotees. But as has been said before by philosophers “Things happen; everything else is literature.”
Scientific measurement is as literary as a novel by Jane Austen. The two may be written in different languages but unless ‘what happens’ to the scientist or to the novelist is recorded, that is finds its way into language, it does not exist. It is merely hearsay in both law and science; it has no evidential value. And once ‘what happens’ becomes language, it is part of its artificial world in which it will be compared, connected, and judged by other documents, recordings and measurements contained in that world.
This isolation of the world of language from the world of not-language is not news. Plato’s metaphor of the cave within which shadows are cast by a flame behind us we cannot see is an ancient expression of the condition we are in. Immanuel Kant’s permanently elusive thing-in-itself is the classic Enlightenment statement of the same issue. And 20th century philosophers like Heidegger and Wittgenstein have clearly shown how language itself, all language, is implicated in what is our unique human condition.
None of these thinkers would have thought of denying that something called reality exists (although a certain 18th century Irish philosopher, Bishop Berkeley, did), or that we experience it more or less accurately (mostly less as it turns out). Their assertion is merely that reality cannot be captured reliably in language, that the world of language is entirely separate from the world of not-language. Or as the American Pragmatist philosopher put it so concisely: Reality “is surface all the way down.” That surface is the language we use to talk about it.
Remarkably perhaps, it is the parallel existence of the worlds of language and not-language that is the source of the power of language in its various dialects from mathematics to books to music. Because language is purely symbolic and purely definitional, that is self-contained, it is relieved entirely of constraints in either physical resources or imagination. It is a human creation rather than a ‘naturally’ occurring thing. We know everything about it since we created it. So we can do with it what we like.
Strogatz is entirely correct about this. But language, including mathematical calculus ‘works’ not because it describes the way the universe is ‘in reality.’ It works because it permits our species to think, to argue, to make judgements, and to learn how to do things progressively over generations. Language frees us from reality. This is our strength and perhaps our ultimate weakness as we spread a dense cloud of language-based consciousness along with its social vices and its overwhelming technology into every part of the planet.
It is this last point which Strogatz’s mathematical boosterism fails to note. This is what gets up my nose. Calculus is a powerful tool, a foundational technology, with a fascinating history. So is the hydrogen bomb. Strogatz wants us to believe that “there seems to be something like a code to the universe, an operating system that animates everything from moment to moment and place to place. Calculus taps into this order and expresses.” This is scientism not science. Scientism is an ideology which claims that there is a privileged way of understanding the world. It is a species of thought which operates under the more general heading of idolatry, the claim to know the essence of reality. What Strogatz is really selling is not calculus as a beautiful, useful, historically interesting tool, but as an inside track to the divine, the language of God (vying of course with Hebrew, Arabic, and Latin).
I’m not buying.
March 2, 2022
Personally I really enjoyed this book. I think it did a good job of making something most people find inscrutable seem more accessible. With abstract concepts I think its always helpful to try to use analogies or demonstrate their practical application. I actually am currently about to take a higher level calculus course so I read this in hopes of hyping myself up and I think I feel sufficiently motivated now. I do understand some of the reviews complaining about his overzealousness assertions about calculus being the language of nature but it worked for me personally. I think it feels much more interesting to say that calculus will reveal the secrets of the universe than to tell me I could figure out the rate of change for something.
June 29, 2019
Calculus is one of those subjects that is so complicated that most people not only don’t understand it, they don’t even know what it is that they don’t understand. But that’s unfortunate, because calculus is one of humanity’s most impressive achievements, an accomplishment that unlocks the secrets of the universe and delivers our most profound and useful technology, from radio and television to GPS navigation and MRI imaging. Calculus is the main protagonist in the story of science, and is a subject every educated person should understand at least conceptually.
Fortunately, you don’t have to trudge through a thousand-page textbook to appreciate the story and power of calculus. Steven Strogatz, in his latest book Infinite Powers, has provided a clear, concise, and fascinating tour of the subject. In fact, if you don’t understand what calculus is all about after reading this book, then the prospects are not great that you ever will. There is simply no better, clearer presentation of the ideas available. Strogatz uses metaphors, illustrations, stories, and examples to guide the reader through the most difficult concepts. While this is not an easy read, it is as reader-friendly as possible; remember, you’re tackling the most sophisticated branch of mathematics, the underlying logic of all science, and a subject that the sharpest mathematical minds in history had to grapple with for thousands of years.
As Strogatz explains, calculus is difficult because it’s tackling the most difficult problems humans encounter, problems that necessitate complex equations, notation, and mathematical manipulation. But behind this computational complexity lies an obsession with simplicity, with breaking down hard problems into easier parts. The special innovation of calculus, as Strogatz explains, is that problems are broken down into infinitely small and manageable parts and then recombined back into the whole.
So what is calculus, exactly? It’s easier to describe calculus by the types of problems it solves than by standard mathematical definitions. When most people hear terms like “infinite series,” “limits,” “derivatives,” and “integrals,” they lose sight of the bigger picture of what calculus is trying to accomplish.
One type of problem calculus can solve is the area under a curved surface. Area is typically quite easy to solve for shapes with straight lines. For rectangles, for example, the area is simply length times width. But what about for shapes with curves where the slope is constantly changing? There is no simple formula to calculate the area in this situation.
You could approximate the area by overlaying rectangular objects over the curved shape (as shown below), but this would only be an approximation as the rectangles would not fit exactly in the curved shape. However, as you made the rectangles smaller (and increased their number) the fit inside the curved shape would keep improving and the approximation would keep getting more accurate. Since you can always keep dividing a number in half (you can always make a number larger or smaller), you can add an infinite number of smaller rectangles into the curved shape. You can never complete this process (which is why the concept of “completed infinity” is logically incoherent), but you could potentially keep adding rectangles forever, which is logically coherent and shows the difference between “completed infinity” and “potential infinity.” As you increase the number of rectangles, you get closer and closer to the area, which is the limit of the infinite series. The area of the curved shape becomes the sum of the infinite series of rectangles. Calculus is the set of equations and procedures to carry out this calculation precisely.
Calculus can also solve problems of motion. Straight-line motion at constant velocity is easy. If you know the speed of an object, then the distance traveled is simply speed times time. But how can you calculate the trajectory of, say, a planet, that not only continuously changes direction in orbit around the sun but that also speeds up or slows down depending on its distance from the sun? This is not so easy, but is solved in a similar way by breaking down the trajectory into infinitely smaller parts and then summing the series. Calculus provides the procedures and notation to carry this out in the most efficient way.
You’ll notice that both examples above solve for problems where some quantity is continuously changing. That means that calculus can solve any problem that involves a quantity that is continuously changing, like the spread of a virus, population growth, or continuously compounding interest in finance. Even without understanding the specific calculations, it’s amazing to contemplate the fact that we can harness the power of infinity to calculate with precision the area under any curved surface, the dynamics of any continuously changing variable, and the trajectory of any object anywhere in the universe!
Of course, this brief sketch is only a description of the subject in its simplest terms; there is much more to the subject and the mechanics of the calculations gets incredibly complex. If you’re interested in diving deeper into the subject, with examples and proofs, Strogatz delivers a nice mixture of pure mathematics, practical examples, and a history of the personalities behind the development of calculus. Of particular interest for me was Strogatz’s solution of Zeno’s Achilles and the Tortoise paradox, a solution that finally made sense to me (in brief, the solution is that an infinite amount of steps can be completed in a finite amount of time).
The Power of Human Cooperation
If you find calculus near impossible to learn, you won’t be happy to know that Isaac Newton invented the subject before he turned 25. But you might find some solace in the fact that Newton did little else; he had few friendships and no romantic relationships, so he had all the time in the world to devote to numbers and experiments.
Newton also couldn’t have done it alone. He was exactly right when he said that he was able to see further by “standing on the shoulders of giants.” As Strogatz explained:
“But again, he [Newton] couldn’t have done any of this without standing on the shoulders of giants. He unified, synthesized, and generalized ideas from his great predecessors: He inherited the Infinity Principle from Archimedes. He learned his tangent lines from Fermat. His decimals came from India. His variables came from Arabic algebra. His representation of curves as equations in the xy plane came from his reading of Descartes. His freewheeling shenanigans with infinity, his spirit of experimentation, and his openness to guesswork and induction came from Wallis. He mashed all of this together to create something fresh, something we’re still using today to solve calculus problems: the versatile method of power series.”
There are at least two lessons here; first, knowledge grows exponentially, not linearly, and there is no limit to what can be discovered. By standing on the shoulders of giants, each generation can build on the developments of the past, as Einstein was able to do by rejecting Newton’s ideas of space and time as absolute. Holding a person, idea, or generation in complete reverence inhibits progress, as when we followed Aristotle for 1,500+ years and maintained the belief that the earth was stationary. The best book I’ve read that elaborates on this point is The Beginning of Infinity: Explanations That Transform the World by the physicist David Deutsch.
Second, calculus demonstrates the power of human cooperation. No single mind could have developed calculus from scratch. People of diverse origin and circumstance collaborated to find solutions to common, tangible problems, because they didn’t waste their time thinking about arbitrary human divisions and other products of pure imagination, like religion. Newton borrowed from ancient Greek geometry, French analytic geometry, the Indian decimal system, and Arabic algebra. As a result, he discovered the mathematical logic and underlying laws of nature that applied equally to objects anywhere in the universe, thus uniting the entire cosmos. This universality, as Strogatz recognized, sparked the beginning of the Enlightenment.
A final point: in the concluding chapter, Strogatz describes Richard Feynman’s quantum electrodynamics (QED) theory, which, by using calculus, describes the quantum interaction of light and matter. Physicists use the theory to make predictions about the properties of electrons and other particles. As Strogatz said, “by comparing those predictions to extremely precise experimental measurements, they’ve shown that the theory agrees with reality to eight decimal places, better than one part in a hundred million.”
This means that QED is the most accurate theory anyone has ever devised about anything. A prediction with an accuracy of 8 decimal places is like, using Strogatz’s example, planning to snap your fingers exactly 3.17 years from now down to the second, without the help of a clock or alarm. As Strogatz further explains:
“I think this is worth mentioning because it puts the lie to the line you sometimes hear, that science is like faith and other belief systems, that is has no special claim on truth. Come on. Any theory that agrees to one part in a hundred million is not just a matter of faith or somebody’s opinion. It didn’t have to match to eight decimal places.”
You will also often hear that science can’t determine right and wrong actions, which in some sense is correct, but misses the point. The moral element of science does not lie in any particular factual claim; it lies in the orientation to forming beliefs. The scientific mindset is not about clinging on to and forming your identity around a set of unalterable beliefs. The scientific mindset is about curiosity, orientation to discovering truth, intellectual integrity, and revising beliefs in the face of new evidence. It’s also, as I believe calculus shows, about the recognition of the power of human cooperation and the pursuit of knowledge as a collective human endeavor.
Fortunately, you don’t have to trudge through a thousand-page textbook to appreciate the story and power of calculus. Steven Strogatz, in his latest book Infinite Powers, has provided a clear, concise, and fascinating tour of the subject. In fact, if you don’t understand what calculus is all about after reading this book, then the prospects are not great that you ever will. There is simply no better, clearer presentation of the ideas available. Strogatz uses metaphors, illustrations, stories, and examples to guide the reader through the most difficult concepts. While this is not an easy read, it is as reader-friendly as possible; remember, you’re tackling the most sophisticated branch of mathematics, the underlying logic of all science, and a subject that the sharpest mathematical minds in history had to grapple with for thousands of years.
As Strogatz explains, calculus is difficult because it’s tackling the most difficult problems humans encounter, problems that necessitate complex equations, notation, and mathematical manipulation. But behind this computational complexity lies an obsession with simplicity, with breaking down hard problems into easier parts. The special innovation of calculus, as Strogatz explains, is that problems are broken down into infinitely small and manageable parts and then recombined back into the whole.
So what is calculus, exactly? It’s easier to describe calculus by the types of problems it solves than by standard mathematical definitions. When most people hear terms like “infinite series,” “limits,” “derivatives,” and “integrals,” they lose sight of the bigger picture of what calculus is trying to accomplish.
One type of problem calculus can solve is the area under a curved surface. Area is typically quite easy to solve for shapes with straight lines. For rectangles, for example, the area is simply length times width. But what about for shapes with curves where the slope is constantly changing? There is no simple formula to calculate the area in this situation.
You could approximate the area by overlaying rectangular objects over the curved shape (as shown below), but this would only be an approximation as the rectangles would not fit exactly in the curved shape. However, as you made the rectangles smaller (and increased their number) the fit inside the curved shape would keep improving and the approximation would keep getting more accurate. Since you can always keep dividing a number in half (you can always make a number larger or smaller), you can add an infinite number of smaller rectangles into the curved shape. You can never complete this process (which is why the concept of “completed infinity” is logically incoherent), but you could potentially keep adding rectangles forever, which is logically coherent and shows the difference between “completed infinity” and “potential infinity.” As you increase the number of rectangles, you get closer and closer to the area, which is the limit of the infinite series. The area of the curved shape becomes the sum of the infinite series of rectangles. Calculus is the set of equations and procedures to carry out this calculation precisely.
Calculus can also solve problems of motion. Straight-line motion at constant velocity is easy. If you know the speed of an object, then the distance traveled is simply speed times time. But how can you calculate the trajectory of, say, a planet, that not only continuously changes direction in orbit around the sun but that also speeds up or slows down depending on its distance from the sun? This is not so easy, but is solved in a similar way by breaking down the trajectory into infinitely smaller parts and then summing the series. Calculus provides the procedures and notation to carry this out in the most efficient way.
You’ll notice that both examples above solve for problems where some quantity is continuously changing. That means that calculus can solve any problem that involves a quantity that is continuously changing, like the spread of a virus, population growth, or continuously compounding interest in finance. Even without understanding the specific calculations, it’s amazing to contemplate the fact that we can harness the power of infinity to calculate with precision the area under any curved surface, the dynamics of any continuously changing variable, and the trajectory of any object anywhere in the universe!
Of course, this brief sketch is only a description of the subject in its simplest terms; there is much more to the subject and the mechanics of the calculations gets incredibly complex. If you’re interested in diving deeper into the subject, with examples and proofs, Strogatz delivers a nice mixture of pure mathematics, practical examples, and a history of the personalities behind the development of calculus. Of particular interest for me was Strogatz’s solution of Zeno’s Achilles and the Tortoise paradox, a solution that finally made sense to me (in brief, the solution is that an infinite amount of steps can be completed in a finite amount of time).
The Power of Human Cooperation
If you find calculus near impossible to learn, you won’t be happy to know that Isaac Newton invented the subject before he turned 25. But you might find some solace in the fact that Newton did little else; he had few friendships and no romantic relationships, so he had all the time in the world to devote to numbers and experiments.
Newton also couldn’t have done it alone. He was exactly right when he said that he was able to see further by “standing on the shoulders of giants.” As Strogatz explained:
“But again, he [Newton] couldn’t have done any of this without standing on the shoulders of giants. He unified, synthesized, and generalized ideas from his great predecessors: He inherited the Infinity Principle from Archimedes. He learned his tangent lines from Fermat. His decimals came from India. His variables came from Arabic algebra. His representation of curves as equations in the xy plane came from his reading of Descartes. His freewheeling shenanigans with infinity, his spirit of experimentation, and his openness to guesswork and induction came from Wallis. He mashed all of this together to create something fresh, something we’re still using today to solve calculus problems: the versatile method of power series.”
There are at least two lessons here; first, knowledge grows exponentially, not linearly, and there is no limit to what can be discovered. By standing on the shoulders of giants, each generation can build on the developments of the past, as Einstein was able to do by rejecting Newton’s ideas of space and time as absolute. Holding a person, idea, or generation in complete reverence inhibits progress, as when we followed Aristotle for 1,500+ years and maintained the belief that the earth was stationary. The best book I’ve read that elaborates on this point is The Beginning of Infinity: Explanations That Transform the World by the physicist David Deutsch.
Second, calculus demonstrates the power of human cooperation. No single mind could have developed calculus from scratch. People of diverse origin and circumstance collaborated to find solutions to common, tangible problems, because they didn’t waste their time thinking about arbitrary human divisions and other products of pure imagination, like religion. Newton borrowed from ancient Greek geometry, French analytic geometry, the Indian decimal system, and Arabic algebra. As a result, he discovered the mathematical logic and underlying laws of nature that applied equally to objects anywhere in the universe, thus uniting the entire cosmos. This universality, as Strogatz recognized, sparked the beginning of the Enlightenment.
A final point: in the concluding chapter, Strogatz describes Richard Feynman’s quantum electrodynamics (QED) theory, which, by using calculus, describes the quantum interaction of light and matter. Physicists use the theory to make predictions about the properties of electrons and other particles. As Strogatz said, “by comparing those predictions to extremely precise experimental measurements, they’ve shown that the theory agrees with reality to eight decimal places, better than one part in a hundred million.”
This means that QED is the most accurate theory anyone has ever devised about anything. A prediction with an accuracy of 8 decimal places is like, using Strogatz’s example, planning to snap your fingers exactly 3.17 years from now down to the second, without the help of a clock or alarm. As Strogatz further explains:
“I think this is worth mentioning because it puts the lie to the line you sometimes hear, that science is like faith and other belief systems, that is has no special claim on truth. Come on. Any theory that agrees to one part in a hundred million is not just a matter of faith or somebody’s opinion. It didn’t have to match to eight decimal places.”
You will also often hear that science can’t determine right and wrong actions, which in some sense is correct, but misses the point. The moral element of science does not lie in any particular factual claim; it lies in the orientation to forming beliefs. The scientific mindset is not about clinging on to and forming your identity around a set of unalterable beliefs. The scientific mindset is about curiosity, orientation to discovering truth, intellectual integrity, and revising beliefs in the face of new evidence. It’s also, as I believe calculus shows, about the recognition of the power of human cooperation and the pursuit of knowledge as a collective human endeavor.
May 24, 2020
A Must Read!
This is one of the most beautiful books I have ever had the utmost pleasure to read. How I wish Steven Strogatz had been my calc teacher. There are authors who let you know that they are smart and there are authors who write with a definite intention to make the readers smart. Strogatz falls into the latter category. He will infuse you with such a love for math, no matter what level of math you have reached in your studies.
Strogatz main message is that calculus extends far beyond what is labeled as Calc I, II, III, or IV. We call all of the other calculus courses by other names because we don't want to just call them calc 1-32. At its essence, Strogatz wants you to know, that calc is utterly obsessed with simplicity. It uses what Strogatz calls the “infinity principle,“ where problems are broken down into tiny parts, which can become infinite parts, and then are fit back together to generate an answer that is easier than trying to answer the original equation. We humans are only as good as our current scientific tools. Once Issac Newton discovered -- or more accurately, allowed the public a reluctant glimpse into his work on-- calculus, life changed in a revolutionary way. In fact, calculus was so revolutionary, Richard Feynman claimed it is indeed the language God speaks. Using calculus as a tool, we humans were able to model an effective treatment for AIDS that stopped the deadly virus from killing a significant portion of our population. We used calculus to listen to unseen objects called black holes. It was the calculus that allowed us to truly hear and see those invisible objects.
Strogatz's introduced his reader to the world of calculus by illustrating some examples that show how math surrounds us humans in nature and is eventually uncovered. For example, music itself is mathematical. Since he chose this subject to draw the reader into the world of maths, it was hard not to fall in love with the book. While I wish he had gone a bit more into the neuroscience of music, math, and the brain, he did a lovely job of helping the reader understand how music can be boiled down to ratios. If you pluck a guitar string at the bottom of the string and then move your hand halfway up the string and pluck again, the new note is exactly half as long as it used to be, a ratio of 1:2, and sounds precisely one octave higher than the original note . If instead the vibrating string is 2/3 of its original length, the note goes up by fifth.
I once had a neuroscience professor, Mike Kaplan, who taught us the neuroscience of music and the brain by delivering the entire lecture by playing a keyboard and singing. It was fucking amazing. I particularly enjoyed the portions of the lecture on how the brain can process and enjoy a little bit of dissonance or can instead be agitated by dissonance. And why? Because the inner ear is basically a calculator and it likes mostly straight math, going up octaves with only a little dissonance. If you use more than a little dissonance, as some artists do, you better do it well or it will be like nails on a chalkboard. The long and short of the neuroscience is getting reward and emotional centers to activate and how it's actually the math sequences that can do that. So, if he updates to a new version of this book someday, I really hope he extends his music and the brain discussion.
Strogatz marveled at the scholars who were obsessed with understanding numbers and shapes. He provided a quick tour of how using numbers came about -- mostly to keep track of livestock. Numbers were mere scratches on an animal bone and let the king know how many cows you owed him. Eventually numbers evolved into the concept of a number line, something we think nothing of as elementary school children today, became a concept. He marveled even more at how Archimedes came to understand the seemingly simple shape of the circle. He derived pi and wondered how many tiny shapes could fit inside a circle. From those ancient studies came todays understanding of pixilation and the achievement of making Shrek's round belly out of 1,000s of tiny little polygons.
I remember reading Clockwork Universe by Edward Dolnick. It is still one of my favorite books. No one, and I mean no one, does a better job of detailing Kepler's maddening obsession with understanding the shape of planetary motion (what would eventually be Kepler's Law of planetary motion). Infinite Powers was every bit as beautiful as Clockwork Universe and did a better job making the math digestible by providing a sweeping and general understanding and then solidifying that understanding with tons of logical and intuitive concrete and local examples of the various forms of calculus (what other authors would call maths related to calculus).
Strogatz painted vivid portraits of Tyco Brahe, Galileo, Kepler, (not as much about Copernicus) Newton, and Leibniz. But what I am most grateful for is Strogatz very clear invitation to women to join the maths by highlighting some very important women who made significant contributions to math. I was surprised not to see Emmy Noether among his examples (maybe because she is known for abstract algebra). Strogatz brought to life stories that literally made my heart flutter with absolute joy. Strogatz, you are the fucking bomb!
As a young girl, Sofya Kovalevskaya -- who eventually became the first woman in the world to obtain a PhD in mathematics in 1874 --had an entire wall of her bedroom wallpapered with the calculus notes from when her father was a student at university. She said when she was about 11, she spent hours just trying to make sense of one line of the equations. She had a predilection for math but it was against the law for her to go to college in Russia. When she entered into an unhappy arranged marriage, the one positive aspect of this union was that it allowed her to travel to Germany where she could finally attend math classes. Germany did not not officially allow her to attend university, but she impressed several professors and was able to unofficially attended classes.
Kovalevskaya arranged to privately study with Karl Weierstrass And he awarded her a PhD for her outstanding workIn analysis, dynamics, and partial differential equation. She eventually got a position teaching at the University of Stockholm and taught there for eight years before dying of Influenza at the age of 41.
French mathematician Sophie Germain never earned a degree. It certainly was not because she didn't earn it. Indeed she did. Germain had to pretend to be a man just so she could sneak into math classes. Her work was significantly better than that of the man she replaced. Because of this, she was called into a meeting with Lagrange. When he found out she was a woman, he took her under his wing and championed her. I love the progressive men of the past who tried their best to help women. She also caught the attention of the famous Gauss, who was so impressed with her work, he engaged in a regular correspondence with her, thinking all the while she was a man. He eventually found out she was a woman (an interesting tale told by Strogatz in this book). For her work on number theory, and Fermat's Last Theorem, Gauss recommended Germain to receive an honorary degree. But, it was not enough pull to earn her the well deserved degree.
Strogatz's awe infused ending to his beautiful book marveled at how, because of the fact that we had calculus as a tool and because we humans and our detectors were listening at the right time, we heard the sound of two black holes colliding. He spoke of how it made him feel when he heard the news. I could not help but tear up because I felt the same way. This book is about the best of what humans can achieve when they uncover the secrets of the world and the universe. I am bursting with love for this book.
This is one of the most beautiful books I have ever had the utmost pleasure to read. How I wish Steven Strogatz had been my calc teacher. There are authors who let you know that they are smart and there are authors who write with a definite intention to make the readers smart. Strogatz falls into the latter category. He will infuse you with such a love for math, no matter what level of math you have reached in your studies.
Strogatz main message is that calculus extends far beyond what is labeled as Calc I, II, III, or IV. We call all of the other calculus courses by other names because we don't want to just call them calc 1-32. At its essence, Strogatz wants you to know, that calc is utterly obsessed with simplicity. It uses what Strogatz calls the “infinity principle,“ where problems are broken down into tiny parts, which can become infinite parts, and then are fit back together to generate an answer that is easier than trying to answer the original equation. We humans are only as good as our current scientific tools. Once Issac Newton discovered -- or more accurately, allowed the public a reluctant glimpse into his work on-- calculus, life changed in a revolutionary way. In fact, calculus was so revolutionary, Richard Feynman claimed it is indeed the language God speaks. Using calculus as a tool, we humans were able to model an effective treatment for AIDS that stopped the deadly virus from killing a significant portion of our population. We used calculus to listen to unseen objects called black holes. It was the calculus that allowed us to truly hear and see those invisible objects.
Strogatz's introduced his reader to the world of calculus by illustrating some examples that show how math surrounds us humans in nature and is eventually uncovered. For example, music itself is mathematical. Since he chose this subject to draw the reader into the world of maths, it was hard not to fall in love with the book. While I wish he had gone a bit more into the neuroscience of music, math, and the brain, he did a lovely job of helping the reader understand how music can be boiled down to ratios. If you pluck a guitar string at the bottom of the string and then move your hand halfway up the string and pluck again, the new note is exactly half as long as it used to be, a ratio of 1:2, and sounds precisely one octave higher than the original note . If instead the vibrating string is 2/3 of its original length, the note goes up by fifth.
I once had a neuroscience professor, Mike Kaplan, who taught us the neuroscience of music and the brain by delivering the entire lecture by playing a keyboard and singing. It was fucking amazing. I particularly enjoyed the portions of the lecture on how the brain can process and enjoy a little bit of dissonance or can instead be agitated by dissonance. And why? Because the inner ear is basically a calculator and it likes mostly straight math, going up octaves with only a little dissonance. If you use more than a little dissonance, as some artists do, you better do it well or it will be like nails on a chalkboard. The long and short of the neuroscience is getting reward and emotional centers to activate and how it's actually the math sequences that can do that. So, if he updates to a new version of this book someday, I really hope he extends his music and the brain discussion.
Strogatz marveled at the scholars who were obsessed with understanding numbers and shapes. He provided a quick tour of how using numbers came about -- mostly to keep track of livestock. Numbers were mere scratches on an animal bone and let the king know how many cows you owed him. Eventually numbers evolved into the concept of a number line, something we think nothing of as elementary school children today, became a concept. He marveled even more at how Archimedes came to understand the seemingly simple shape of the circle. He derived pi and wondered how many tiny shapes could fit inside a circle. From those ancient studies came todays understanding of pixilation and the achievement of making Shrek's round belly out of 1,000s of tiny little polygons.
I remember reading Clockwork Universe by Edward Dolnick. It is still one of my favorite books. No one, and I mean no one, does a better job of detailing Kepler's maddening obsession with understanding the shape of planetary motion (what would eventually be Kepler's Law of planetary motion). Infinite Powers was every bit as beautiful as Clockwork Universe and did a better job making the math digestible by providing a sweeping and general understanding and then solidifying that understanding with tons of logical and intuitive concrete and local examples of the various forms of calculus (what other authors would call maths related to calculus).
Strogatz painted vivid portraits of Tyco Brahe, Galileo, Kepler, (not as much about Copernicus) Newton, and Leibniz. But what I am most grateful for is Strogatz very clear invitation to women to join the maths by highlighting some very important women who made significant contributions to math. I was surprised not to see Emmy Noether among his examples (maybe because she is known for abstract algebra). Strogatz brought to life stories that literally made my heart flutter with absolute joy. Strogatz, you are the fucking bomb!
As a young girl, Sofya Kovalevskaya -- who eventually became the first woman in the world to obtain a PhD in mathematics in 1874 --had an entire wall of her bedroom wallpapered with the calculus notes from when her father was a student at university. She said when she was about 11, she spent hours just trying to make sense of one line of the equations. She had a predilection for math but it was against the law for her to go to college in Russia. When she entered into an unhappy arranged marriage, the one positive aspect of this union was that it allowed her to travel to Germany where she could finally attend math classes. Germany did not not officially allow her to attend university, but she impressed several professors and was able to unofficially attended classes.
Kovalevskaya arranged to privately study with Karl Weierstrass And he awarded her a PhD for her outstanding workIn analysis, dynamics, and partial differential equation. She eventually got a position teaching at the University of Stockholm and taught there for eight years before dying of Influenza at the age of 41.
French mathematician Sophie Germain never earned a degree. It certainly was not because she didn't earn it. Indeed she did. Germain had to pretend to be a man just so she could sneak into math classes. Her work was significantly better than that of the man she replaced. Because of this, she was called into a meeting with Lagrange. When he found out she was a woman, he took her under his wing and championed her. I love the progressive men of the past who tried their best to help women. She also caught the attention of the famous Gauss, who was so impressed with her work, he engaged in a regular correspondence with her, thinking all the while she was a man. He eventually found out she was a woman (an interesting tale told by Strogatz in this book). For her work on number theory, and Fermat's Last Theorem, Gauss recommended Germain to receive an honorary degree. But, it was not enough pull to earn her the well deserved degree.
Strogatz's awe infused ending to his beautiful book marveled at how, because of the fact that we had calculus as a tool and because we humans and our detectors were listening at the right time, we heard the sound of two black holes colliding. He spoke of how it made him feel when he heard the news. I could not help but tear up because I felt the same way. This book is about the best of what humans can achieve when they uncover the secrets of the world and the universe. I am bursting with love for this book.
February 2, 2024
Simplicity is a quality largely undermined in our society today. At times, humans consider the cacophony of chaos as a quotidian normality. What are the products of chaos? Do these promote intellectual, physical, spiritual growth or the opposite? Algorithms may educate us by using the mathematics of nature to bring a cure for a calculus based "katzenjammer." Strogatz identifies the solutions calculus brings and the language God speaks (via differential equations).
"A few decades before Fermat and Descartes, Galileo understood the power of this deliberate simplification of reality. He meticulously changed just one thing at a time in his experiments while holding everything else constant. He let a ball roll down a ramp and measured how far it went in a certain amount of time. Nice and simple—distance as a function of time."
---Steven H. Strogatz
It may be infantile to think that human civilization has no relation to calculus: it makes possible things like cellphones, radios, pharmacology, human genome technology, and CT scans. Sans calculus, modern civilization would not be a part of our reality according to Strogatz. Behavior is a pattern, just like the calculations relating to the the movements of heavenly spheres and the relationship of linear ( y = mx + c) and non-linear things in our universe (ax2 + by2 = c).Read.
"A few decades before Fermat and Descartes, Galileo understood the power of this deliberate simplification of reality. He meticulously changed just one thing at a time in his experiments while holding everything else constant. He let a ball roll down a ramp and measured how far it went in a certain amount of time. Nice and simple—distance as a function of time."
---Steven H. Strogatz
It may be infantile to think that human civilization has no relation to calculus: it makes possible things like cellphones, radios, pharmacology, human genome technology, and CT scans. Sans calculus, modern civilization would not be a part of our reality according to Strogatz. Behavior is a pattern, just like the calculations relating to the the movements of heavenly spheres and the relationship of linear ( y = mx + c) and non-linear things in our universe (ax2 + by2 = c).Read.
March 26, 2019
TL;DR
In Infinite Powers,Dr. Steven H. Strogatz teaches us how to use our microwaves to calculate the speed of light. I’m not kidding. That’s all the recommendation this book needs. Highly Recommended.
Review cross-posted at Primmlife.com
Review
When I tell people that I’m an engineer, my wife likes to follow up that comment with, “He does math all day long.” A common response is, “Oh, you must really like math. I didn’t enjoy it in [insert level of schooling here].” To keep the conversation moving I agree, and while I do like math, I didn’t always. Until I started studying calculus, math bored me. Algebra existed as a set of rules; geometry, though my introduction to proofs, seemed too abstract. But when I first solved a derivative, my indifference turned to frustration and intrigue. My plan to take only enough math to get an engineering degree changed to a serious contemplation of switching career paths to applied mathematics (with an eye towards physics grad degrees). Ultimately, I stuck with the engineering curriculum but ended up studying higher level mathematics, and to this day, I still read about and love math. Part of my studies now involves going back and filling in what I missed during previous years. One of the voices to which I turn is Dr. Steven Strogatz, and his latest book, Infinite Powers, fills in details about calculus that I lacked. His explanations don’t rely on the familiar equations but, instead, root themselves in history, in logic, and in excellent prose. Infinite Powers transforms calculus from equations into meaning.
The Story
In Infinite Powers, Dr. Strogatz starts with Archimedes from ancient Greece and carries on to some of today’s most unique challenges. It is the story of calculus told as a continuum of human learning. Often, the public thinks that scientific breakthroughs happen when lone geniuses discover something new, but in reality discoveries occur when people improve upon the work of others. In Infinite Powers Dr. Strogatz traces the methods Archimedes used to Newton and Leibniz, who are the inventors or discoverers of calculus. Along the way, we learn about contributions from Fermat, Galileo, Descartes, Arabic, and Chinese mathematicians. But we don’t stop at the discovery era. Infinite Powers continues on to Fourier and Sophie Germain. We even get to see how calculus is being used today to treat HIV patients, to create microwaves, and, near to my own heart and pocketbook, help the 787 fly.
The Writing
Math possesses a strong language of its own rooted in symbols and logic. While I view this as a strength, I also know others view the equations, Greek letters, and symbols to be inscrutable. Others have said that math texts tend to be dry reads. For anyone who thinks this, Infinite Powers is the book for you. While equations do exists, they are few. Dr. Strogatz takes the time to explain, in detail, what each of the symbols means. But the majority of the book reads more like a history text than a mathematical treatise. While it doesn’t spoon feed the reader, it doesn’t bog down in jargon. Clarity and simplicity are the descriptors I have already used talking about this book with friends. Dr. Strogatz does an excellent job describing what the math is actually doing. The reader will NOT be able to do any calculus after reading it, but he/she will understand how powerful a tool it is.
There are graphs and pictures throughout the book. In my advanced reader’s copy (ARC), the graphs didn’t show up. So, I cannot speak to their quality; however, with my background and the detail of Dr. Strogatz’s descriptions, I could picture what his intent was with the graphs. That should be an indicator of success for the prose of this book.
Ugh, Math, Really?
Bear with me here as I get on my soapbox for a minute. One of the other responses that I get when I’m introduced as an engineer is, “You must be really good at math.” And compared to most people, yes, I am good at math. But I’m good at math for one reason only, I’ve been practicing it in one form or another for the last 23 years. In the martial arts, there’s a saying that a black belt is simply a white belt who didn’t quit. To me, that’s all that math is. I’m good at math and calculus because I didn’t quit doing math. The general public often thinks that math requires a certain mindset or, even, a certain person. No, it requires practices and tenacity. The reason that I stuck with math is because of teachers in high school that showed the same enthusiasm that Dr. Strogatz shows in this book. Teachers and professors who care that students understand a subject make this world a better place. After reading Infinite Powers, I have no doubt Dr. Strogatz is a teacher than inspires students. I can’t help but wonder what could happen if a book like this gets into the hands of someone who thinks they have to be good at math to understand it.
Math as Art
Though we use math in the sciences, I’ve come to view it more as an art. The mathematician, engineer, chemist, or whoever must know and understand the tools math gives us in order to solve problems, and like a painter picks and chooses the right brush to add to the painting, the problem solver picks and chooses the correct mathematical tool. It’s a creative process that, instead of being hung in a gallery or museum, zips down the road, flows through our veins, or launches a satellite into space. Dr. Strogatz demonstrates the versatility and creativity that we are capable of when using calculus. Whether putting satellites in space or determining how viruses spread, calculus is a tool for delving into nature’s mysteries. Infinite Powers stirred that creative sense, that feeling of awe at being able to see into the universes internal mechanisms. At the same time, it reminded me of the ingenuity of the human animal to seek out and explore the world around us. Dr. Strogatz conveys the beauty that one can find in math, and I felt that thrill of discovery again as I read this book.
Infinity
Originally, I requested this book because I thought it was about infinity. That mathematical concept that looks like an 8 fell asleep, ∞. Instead, it was about calculus; so, I went into the first few chapters with the wrong expectations. Dr. Strogatz discusses infinity but not enough to satisfy me. And throughout the book, he does reference back to the topic of infinity, but it feels more like a forced attempt to tie the later chapters to the theme. I’m still hoping that Dr. Strogatz gives us a book about infinity in the same detail and manner that he gave us a book about calculus.
Conclusion
Dr. Steven Strogatz’s Infinite Powers details the history and development of calculus. Dr. Strogatz’s ability to relate complex mathematical concepts in clear and precise language is at peak form in this book. For anyone curious about calculus, this book provides answers in delightful, easy to understand prose that will awaken your curiousity.
In Infinite Powers,Dr. Steven H. Strogatz teaches us how to use our microwaves to calculate the speed of light. I’m not kidding. That’s all the recommendation this book needs. Highly Recommended.
Review cross-posted at Primmlife.com
Review
When I tell people that I’m an engineer, my wife likes to follow up that comment with, “He does math all day long.” A common response is, “Oh, you must really like math. I didn’t enjoy it in [insert level of schooling here].” To keep the conversation moving I agree, and while I do like math, I didn’t always. Until I started studying calculus, math bored me. Algebra existed as a set of rules; geometry, though my introduction to proofs, seemed too abstract. But when I first solved a derivative, my indifference turned to frustration and intrigue. My plan to take only enough math to get an engineering degree changed to a serious contemplation of switching career paths to applied mathematics (with an eye towards physics grad degrees). Ultimately, I stuck with the engineering curriculum but ended up studying higher level mathematics, and to this day, I still read about and love math. Part of my studies now involves going back and filling in what I missed during previous years. One of the voices to which I turn is Dr. Steven Strogatz, and his latest book, Infinite Powers, fills in details about calculus that I lacked. His explanations don’t rely on the familiar equations but, instead, root themselves in history, in logic, and in excellent prose. Infinite Powers transforms calculus from equations into meaning.
The Story
In Infinite Powers, Dr. Strogatz starts with Archimedes from ancient Greece and carries on to some of today’s most unique challenges. It is the story of calculus told as a continuum of human learning. Often, the public thinks that scientific breakthroughs happen when lone geniuses discover something new, but in reality discoveries occur when people improve upon the work of others. In Infinite Powers Dr. Strogatz traces the methods Archimedes used to Newton and Leibniz, who are the inventors or discoverers of calculus. Along the way, we learn about contributions from Fermat, Galileo, Descartes, Arabic, and Chinese mathematicians. But we don’t stop at the discovery era. Infinite Powers continues on to Fourier and Sophie Germain. We even get to see how calculus is being used today to treat HIV patients, to create microwaves, and, near to my own heart and pocketbook, help the 787 fly.
The Writing
Math possesses a strong language of its own rooted in symbols and logic. While I view this as a strength, I also know others view the equations, Greek letters, and symbols to be inscrutable. Others have said that math texts tend to be dry reads. For anyone who thinks this, Infinite Powers is the book for you. While equations do exists, they are few. Dr. Strogatz takes the time to explain, in detail, what each of the symbols means. But the majority of the book reads more like a history text than a mathematical treatise. While it doesn’t spoon feed the reader, it doesn’t bog down in jargon. Clarity and simplicity are the descriptors I have already used talking about this book with friends. Dr. Strogatz does an excellent job describing what the math is actually doing. The reader will NOT be able to do any calculus after reading it, but he/she will understand how powerful a tool it is.
There are graphs and pictures throughout the book. In my advanced reader’s copy (ARC), the graphs didn’t show up. So, I cannot speak to their quality; however, with my background and the detail of Dr. Strogatz’s descriptions, I could picture what his intent was with the graphs. That should be an indicator of success for the prose of this book.
Ugh, Math, Really?
Bear with me here as I get on my soapbox for a minute. One of the other responses that I get when I’m introduced as an engineer is, “You must be really good at math.” And compared to most people, yes, I am good at math. But I’m good at math for one reason only, I’ve been practicing it in one form or another for the last 23 years. In the martial arts, there’s a saying that a black belt is simply a white belt who didn’t quit. To me, that’s all that math is. I’m good at math and calculus because I didn’t quit doing math. The general public often thinks that math requires a certain mindset or, even, a certain person. No, it requires practices and tenacity. The reason that I stuck with math is because of teachers in high school that showed the same enthusiasm that Dr. Strogatz shows in this book. Teachers and professors who care that students understand a subject make this world a better place. After reading Infinite Powers, I have no doubt Dr. Strogatz is a teacher than inspires students. I can’t help but wonder what could happen if a book like this gets into the hands of someone who thinks they have to be good at math to understand it.
Math as Art
Though we use math in the sciences, I’ve come to view it more as an art. The mathematician, engineer, chemist, or whoever must know and understand the tools math gives us in order to solve problems, and like a painter picks and chooses the right brush to add to the painting, the problem solver picks and chooses the correct mathematical tool. It’s a creative process that, instead of being hung in a gallery or museum, zips down the road, flows through our veins, or launches a satellite into space. Dr. Strogatz demonstrates the versatility and creativity that we are capable of when using calculus. Whether putting satellites in space or determining how viruses spread, calculus is a tool for delving into nature’s mysteries. Infinite Powers stirred that creative sense, that feeling of awe at being able to see into the universes internal mechanisms. At the same time, it reminded me of the ingenuity of the human animal to seek out and explore the world around us. Dr. Strogatz conveys the beauty that one can find in math, and I felt that thrill of discovery again as I read this book.
Infinity
Originally, I requested this book because I thought it was about infinity. That mathematical concept that looks like an 8 fell asleep, ∞. Instead, it was about calculus; so, I went into the first few chapters with the wrong expectations. Dr. Strogatz discusses infinity but not enough to satisfy me. And throughout the book, he does reference back to the topic of infinity, but it feels more like a forced attempt to tie the later chapters to the theme. I’m still hoping that Dr. Strogatz gives us a book about infinity in the same detail and manner that he gave us a book about calculus.
Conclusion
Dr. Steven Strogatz’s Infinite Powers details the history and development of calculus. Dr. Strogatz’s ability to relate complex mathematical concepts in clear and precise language is at peak form in this book. For anyone curious about calculus, this book provides answers in delightful, easy to understand prose that will awaken your curiousity.
June 10, 2019
I certainly wish I had read this book while in high school or college. We grilled all the basic technical parts of calculus and yet unsure what was the point. Well certainly you don't need to know it if you do not work in science and research. Life can go on just as well. But being able to appreciate the beauty of it is an added bonus. And then who knows, seeing that beauty could change the path you take in life.
Strogatz takes the same approach with his earlier pop science book, The Joy of x, i.e. trying his best to explain calculus via intuition, allegories, and graphs. It works up to some extent. Some concepts require a lot more, but as the author tries to avoid discussing the technicals, the discussion becomes a bit shallow. That is still fine, this is not a textbook.
What I'm a bit uneasy about is Strogatz's reverence for calculus to the point of being religious. Maybe I have taken it wrong, taken his comparison/allegory too much at face value when he cites Feymann over and over again that it is the language of God. Just as the title says, Strogatz argues that calculus is so special, nature acts according to the rules of calculus, everything can be described using these types of equations. Perhaps I've taken calculus for granted. Of course it is behind everything. Modern science is built upon math and ultimately one of the most powerful tools ever invented. Yes, calculus is a tool, a language. It is a lens of looking at the world and of course whatever picture one sees would be described in that medium. Is it so surprising? Could there be another language that provides a different lens to see the world? I'm an illiterate here, I have no idea. My take is that calculus has been so successful to deal with so many problems of interest, and there are so many things to extend and develop, that one isn't incentivized to go through all the pain to develop another language to compete with it. But I would prefer to think that there are possible options out there, so that calculus is not necessary the one (and only) language of "God."
Strogatz also emphasizes that calculus is behind everything, behind all sorts of important equations and inventions in human history, driving progress and success. It is true that calculus is used, but I feel that the writing is tilted towards overstressing the importance of calculus and downplaying all other types of ingenious ideas and inventions that work together. Perhaps one needs to exaggerate to attract attention. It is the way of writing not the idea.
And of course any language has its limitation. Strogatz gives a lot of praises and does not so much tell readers what are the stuffs that calculus cannot deal with. Sure, towards the end he does say there is a limit to what mathematicians can solve, such as nonlinearity. He hints at the need for another type of method to approach such problems, such as that from Poincare. The discussion is not as comprehensive as I would like.
Don't get me wrong, I love calculus. I'm more of an abstract thinker and do not need much to be convinced that calculus a wonderful and elegant tool. Yet I have problems with the writing. I prefer writings with more nuance. I ended up looking forward to the bits talking about mathematicians and their private lives. But Professor Strogatz is not a historian, so I was naturally also not satisfied either.
Strogatz takes the same approach with his earlier pop science book, The Joy of x, i.e. trying his best to explain calculus via intuition, allegories, and graphs. It works up to some extent. Some concepts require a lot more, but as the author tries to avoid discussing the technicals, the discussion becomes a bit shallow. That is still fine, this is not a textbook.
What I'm a bit uneasy about is Strogatz's reverence for calculus to the point of being religious. Maybe I have taken it wrong, taken his comparison/allegory too much at face value when he cites Feymann over and over again that it is the language of God. Just as the title says, Strogatz argues that calculus is so special, nature acts according to the rules of calculus, everything can be described using these types of equations. Perhaps I've taken calculus for granted. Of course it is behind everything. Modern science is built upon math and ultimately one of the most powerful tools ever invented. Yes, calculus is a tool, a language. It is a lens of looking at the world and of course whatever picture one sees would be described in that medium. Is it so surprising? Could there be another language that provides a different lens to see the world? I'm an illiterate here, I have no idea. My take is that calculus has been so successful to deal with so many problems of interest, and there are so many things to extend and develop, that one isn't incentivized to go through all the pain to develop another language to compete with it. But I would prefer to think that there are possible options out there, so that calculus is not necessary the one (and only) language of "God."
Strogatz also emphasizes that calculus is behind everything, behind all sorts of important equations and inventions in human history, driving progress and success. It is true that calculus is used, but I feel that the writing is tilted towards overstressing the importance of calculus and downplaying all other types of ingenious ideas and inventions that work together. Perhaps one needs to exaggerate to attract attention. It is the way of writing not the idea.
And of course any language has its limitation. Strogatz gives a lot of praises and does not so much tell readers what are the stuffs that calculus cannot deal with. Sure, towards the end he does say there is a limit to what mathematicians can solve, such as nonlinearity. He hints at the need for another type of method to approach such problems, such as that from Poincare. The discussion is not as comprehensive as I would like.
Don't get me wrong, I love calculus. I'm more of an abstract thinker and do not need much to be convinced that calculus a wonderful and elegant tool. Yet I have problems with the writing. I prefer writings with more nuance. I ended up looking forward to the bits talking about mathematicians and their private lives. But Professor Strogatz is not a historian, so I was naturally also not satisfied either.
January 30, 2020
Time spent in the math world is one of the best ones. I am of a strong opinion that the Universe not only talks in the language of math, the Universe is math. Whether math is the langue of the universe, the universe itself, or just a construct that exist only in a human brain, it's the mysterious thing we've been given helping us to reveal many secrets of the universe and making our lifes improved in so many ways and distinquished on this planet.
Steven Strogatz can obviously write a book about math that becomes a page turner, an exelent job when it comes to math related texts. A mathematiacian who can write is worth his weight in gold. I wish I had come across such a book at high school. Several passages were a bit murky, but overall, it was an inpiring read. I've filled some gaps in my math brain and got some new and important insights. Strongly recommend the book to anyone.
Steven Strogatz can obviously write a book about math that becomes a page turner, an exelent job when it comes to math related texts. A mathematiacian who can write is worth his weight in gold. I wish I had come across such a book at high school. Several passages were a bit murky, but overall, it was an inpiring read. I've filled some gaps in my math brain and got some new and important insights. Strongly recommend the book to anyone.
March 1, 2021
This is in part intro into calculus for the general public, part – a history of this branch of math, part – unusual current application of calculus, from GPS to drug regiments for HIV-infected to using wavelets to achieve fingerprints for FBI. I read is as a part of monthly reading for February 2021 at Non Fiction Book Club group.
The book starts with famous (in narrow circles of math fans) quip by Richard Feynman that “Calculus is the language God talks.” That was said to the writer Herman Wouk about the Manhattan Project. The latter made several attempts to learn it but failed each time, so the goal of the book is to allow ‘not-math’ people to understand main concepts.
Historically it starts with Archimedes, Zeno and Pythagoreans, shifting to Galileo and Kepler, then to Fermat and Descartes, and finally to Newton and Leibniz; several later mathematicians are mentioned, like Fourier, Sophie Germain, Gauss, Kovalevskaya and Poincaré. Non-European contributions are mentioned but quite briefly. This reads as a very interesting research about people and problems they face and even if you are not interested in calculus, it is still a great non-fic history of science and discoveries.
The book starts with famous (in narrow circles of math fans) quip by Richard Feynman that “Calculus is the language God talks.” That was said to the writer Herman Wouk about the Manhattan Project. The latter made several attempts to learn it but failed each time, so the goal of the book is to allow ‘not-math’ people to understand main concepts.
Historically it starts with Archimedes, Zeno and Pythagoreans, shifting to Galileo and Kepler, then to Fermat and Descartes, and finally to Newton and Leibniz; several later mathematicians are mentioned, like Fourier, Sophie Germain, Gauss, Kovalevskaya and Poincaré. Non-European contributions are mentioned but quite briefly. This reads as a very interesting research about people and problems they face and even if you are not interested in calculus, it is still a great non-fic history of science and discoveries.
April 12, 2019
This book does not make calculus interesting
Calculus is widely perceived as important part of science in understanding basic laws of physics. But it also has important applications in advanced physics; relativity and quantum mechanics, cosmology, astronomy, biology, chemistry, medicine, geology, ecology and in everyday life. In this book, the author discusses calculus as catch-as-catch-can story in an historical context without giving some ideas of how calculus helped physics to evolve. This is not a recipe book and at the same time it is not overwhelming. But in the absence of clear mathematical methods or its applications, this is a slapdash story that does not make calculus interesting.
Calculus is widely perceived as important part of science in understanding basic laws of physics. But it also has important applications in advanced physics; relativity and quantum mechanics, cosmology, astronomy, biology, chemistry, medicine, geology, ecology and in everyday life. In this book, the author discusses calculus as catch-as-catch-can story in an historical context without giving some ideas of how calculus helped physics to evolve. This is not a recipe book and at the same time it is not overwhelming. But in the absence of clear mathematical methods or its applications, this is a slapdash story that does not make calculus interesting.
July 5, 2022
I missed this one when it came out, possibly because the cover looks somewhat amateurish. Stephen Strogatz starts by exploring the prehistory of calculus - arguably the most widely applied mathematical tool in physics and engineering. We tend to think of calculus starting with Newton and Leibniz, but there was a long prehistory stretching back to the Ancient Greeks. This involved using methods that might, for instance, mentally cut something up into smaller and smaller pieces, then rearranged those pieces in order to work out, for instance, the relationship between the area of a circle and its circumference. This background is delightfully introduced.
Strogatz takes us through some, though not all, of the intervening history before the real thing bursts on the scene, but oddly then gives up on the historical context, so we don't hear about Newton and Leibniz until we have absorbed a whole host of detail, including where necessary some equations, ranging from functions to the natural logarithm and its exponential function before we get on to the basics that lie behind differentiation.
Uncovering the fundamentals of the mathematics is the kind of thing Strogatz does brilliantly. He can really dive into what makes calculus tick. Things are less effective on the history front. We do eventually get both Newton and Leibniz's side of the story, but I found the way it was mixed up with mathematical detail made it difficult to absorb the message. Again we then lose the historical structure - no Bishop Berkeley and not much on the way that limits were introduced to fix the problem of infinitesimals (though this is touched on early on in the book). Partial differential equations get an introduction but with less detail, as does Fourier analysis. Along the way, Strogatz introduces a wide range of real world applications, and finally looks at future possibilities.
I had a couple of problems with the book. Strogatz sometimes gets carried away with floridity. For example, when talking about dividing a circle into quarters and arranging them in a line: ‘It’s certainly not a rectangle, so its area is not easy to guess. We seem to be going backward. But as in any drama, the hero needs to get into trouble before triumphing. The dramatic tension is building.’ He also commits the science writer's heresy of telling us 'During the Inquisition, the renegade monk Giordano Bruno was burned alive at the stake for suggesting that God, in His infinite power, created innumerable worlds.’ Not only was Bruno a friar, he was burned for conventional religious heresy, not his (often pseudo-) scientific views.
This was a book that couldn't decide what it was supposed to be. It started off as history of maths, but that petered out to be replaced by random historical snippets mixed in with an excellent exploration of what calculus is all about. I think it would be better to have either taken the historical approach throughout, fitting in the explanation of the maths, or to have based it purely around the maths with just passing references to the historical context. Yet despite that strange hybrid approach, there is so much to like in Strogatz's ability to bring the maths alive.
Strogatz takes us through some, though not all, of the intervening history before the real thing bursts on the scene, but oddly then gives up on the historical context, so we don't hear about Newton and Leibniz until we have absorbed a whole host of detail, including where necessary some equations, ranging from functions to the natural logarithm and its exponential function before we get on to the basics that lie behind differentiation.
Uncovering the fundamentals of the mathematics is the kind of thing Strogatz does brilliantly. He can really dive into what makes calculus tick. Things are less effective on the history front. We do eventually get both Newton and Leibniz's side of the story, but I found the way it was mixed up with mathematical detail made it difficult to absorb the message. Again we then lose the historical structure - no Bishop Berkeley and not much on the way that limits were introduced to fix the problem of infinitesimals (though this is touched on early on in the book). Partial differential equations get an introduction but with less detail, as does Fourier analysis. Along the way, Strogatz introduces a wide range of real world applications, and finally looks at future possibilities.
I had a couple of problems with the book. Strogatz sometimes gets carried away with floridity. For example, when talking about dividing a circle into quarters and arranging them in a line: ‘It’s certainly not a rectangle, so its area is not easy to guess. We seem to be going backward. But as in any drama, the hero needs to get into trouble before triumphing. The dramatic tension is building.’ He also commits the science writer's heresy of telling us 'During the Inquisition, the renegade monk Giordano Bruno was burned alive at the stake for suggesting that God, in His infinite power, created innumerable worlds.’ Not only was Bruno a friar, he was burned for conventional religious heresy, not his (often pseudo-) scientific views.
This was a book that couldn't decide what it was supposed to be. It started off as history of maths, but that petered out to be replaced by random historical snippets mixed in with an excellent exploration of what calculus is all about. I think it would be better to have either taken the historical approach throughout, fitting in the explanation of the maths, or to have based it purely around the maths with just passing references to the historical context. Yet despite that strange hybrid approach, there is so much to like in Strogatz's ability to bring the maths alive.
May 15, 2019
I need to psyche myself up to do some math for work. And I have a math sherpa and I arranged to meet him so he can take me through the paper I must tackle. But I’m old and only really remember my high school math well, so there is a genuine task at hand here.
So I duck and dive between the paper and my notes from my MSc thesis from at least fifteen years ago and I work out the answer to lesser problems and I write out my questions for my sherpa and I also need to be thinking math the whole time; I need to be in a mood, basically.
That’s the task.
So I did the sensible thing and went on a bit of a binge and bought a whole bunch of popular math books in one go to read in the tube. “Infinite Powers” I read first, because it looked like it would not challenge me at all and it gets good writeups.
It’s bloody awesome!
It’s more than an anthology of results and it’s more than a series of mini-portraits of mathematicians, it’s almost got a plot. Surprisingly often, even the obligatory corny applications of the math are (somewhat) related to what the author’s talking about.
Huge caveat: I knew both the math and even many of the stories upfront, so perhaps it’s not very well explained. I have no way of knowing. But I bet you it is. Perhaps not well enough that you could hope to learn calculus from here, Jordan Ellenberg’s praise on the back cover notwithstanding. (For that I can refer you to “Quick Calculus” by twin gods Kleppner and Ramsey.) But probably well enough to be a companion to anybody taking calculus for the first time.
Steven Strogatz had me from “hello,” of course, because he starts with the Greeks, on whom he lavishes immense praise. He could have left it there and I’d still be basking in the warm glow of my ancestors’ work. Needless to say, it does not stop there, he takes you from them to Fermat and Descartes, before introducing you to Newton and Leibniz, a couple words on Fourier and from him straight to Einstein, taking special care to erase all traces of evil men like the unspeakable inventor of delta-epsilon proofs. You won’t find the C-word here.
So there’s a massive hole in the nineteenth century, somewhere, but I’m sure you can buy another book to find out about that. Here you’ll discover a decent definition of e, an intuitive explanation of general relativity, the common cause of death of Leibniz and Newton, a fun game to play with your microwave oven, the first and second derivative of the sine wave, the dimension of the three-body problem, a strong defense of infinitesimals, WHAT’S NOT TO LIKE?
Enough from me, I’ll now go buy some extra copies for a few boys and girls I know. If one of them likes it, my job is done.
Oh, sorry, one more thing. About the plot: it’s a history of how mathematicians throughout time have sliced hard problems into infinite infinitely-thin slices where the problem has a clear answer and then dealt with infinity to sum up the solutions to the easy problem in order to come up with an answer to the hard problem.
Whenever you do that, you’re doing calculus, you’re putting together the answer granule by granule.
So I duck and dive between the paper and my notes from my MSc thesis from at least fifteen years ago and I work out the answer to lesser problems and I write out my questions for my sherpa and I also need to be thinking math the whole time; I need to be in a mood, basically.
That’s the task.
So I did the sensible thing and went on a bit of a binge and bought a whole bunch of popular math books in one go to read in the tube. “Infinite Powers” I read first, because it looked like it would not challenge me at all and it gets good writeups.
It’s bloody awesome!
It’s more than an anthology of results and it’s more than a series of mini-portraits of mathematicians, it’s almost got a plot. Surprisingly often, even the obligatory corny applications of the math are (somewhat) related to what the author’s talking about.
Huge caveat: I knew both the math and even many of the stories upfront, so perhaps it’s not very well explained. I have no way of knowing. But I bet you it is. Perhaps not well enough that you could hope to learn calculus from here, Jordan Ellenberg’s praise on the back cover notwithstanding. (For that I can refer you to “Quick Calculus” by twin gods Kleppner and Ramsey.) But probably well enough to be a companion to anybody taking calculus for the first time.
Steven Strogatz had me from “hello,” of course, because he starts with the Greeks, on whom he lavishes immense praise. He could have left it there and I’d still be basking in the warm glow of my ancestors’ work. Needless to say, it does not stop there, he takes you from them to Fermat and Descartes, before introducing you to Newton and Leibniz, a couple words on Fourier and from him straight to Einstein, taking special care to erase all traces of evil men like the unspeakable inventor of delta-epsilon proofs. You won’t find the C-word here.
So there’s a massive hole in the nineteenth century, somewhere, but I’m sure you can buy another book to find out about that. Here you’ll discover a decent definition of e, an intuitive explanation of general relativity, the common cause of death of Leibniz and Newton, a fun game to play with your microwave oven, the first and second derivative of the sine wave, the dimension of the three-body problem, a strong defense of infinitesimals, WHAT’S NOT TO LIKE?
Enough from me, I’ll now go buy some extra copies for a few boys and girls I know. If one of them likes it, my job is done.
Oh, sorry, one more thing. About the plot: it’s a history of how mathematicians throughout time have sliced hard problems into infinite infinitely-thin slices where the problem has a clear answer and then dealt with infinity to sum up the solutions to the easy problem in order to come up with an answer to the hard problem.
Whenever you do that, you’re doing calculus, you’re putting together the answer granule by granule.
November 18, 2019
For an ordinary layperson, this is perhaps the most accessible history of the development of Calculus one could hope for. In easily readable language Strogatz has provided a fascinating narrative covering the ideas behind Calculus, its history from the earliest Greek mathematicians, its “dismissal” from the formal geometric/mathematical canon for some two thousand years, until its resurgence in the 17th-c with the work of Newton and Leibnitz, and on to its amazingly extensive application to just about every sphere of activity in modern civilisation.
Some knowledge of basic mathematics is required, but not much more than that: Strogatz is more concerned with explaining what Calculus is all about, and pointing out that certain precise questions about specific physical problems can only be best answered by its application. Calculus is the most accurate mathematical tool ever developed for answering such questions.
Some things I particularly liked about this lie in the conceptual (philosophical?) matters that are implied or suggested: the idea, for example, that the modern understanding of the Space-Time continuum is maintained (for me, the “problem” between the continuum idea and the idea of infinitesimals remains, and probably should remain so); and the fact that, no matter how “infinitesimal” the divisions of differential calculus are made, there will always remain “a little bit” extra that is unaccounted for. The latter is difficult enough when dealing with the interrelationship of two “entities” or “bodies”, let alone the far more complicated problem relating to three (or more?) such entities, so it remains a potentially intriguing matter.
While the cynics among us might consider this enough to disparage its absolute applicability, Calculus remains our most successful, accurate and practical measurement tool so far. I like the idea that there is still the possibility for new approaches to be made: but that being said, the current “solution” to such errors will still most probably be found through the re-application and refinements of its processes.
An ironic example of this relates to Strogatz’s use of the Boeing 787 Dreamliner aeroplane as a proud example of the achievements made under the auspices of this form of mathematics. This may very well be the case, but at the same time, some bad reports coming in 2019 about Boeing’s 737 MAX line of planes suggested that, among other matters, computer design flaws were partly responsible for two of these planes crashing with a total loss of 364 lives: a sober reminder that the more intricate the maths, the more care needs to be taken in its applications and consequences in the physical world.
Even so, it is Calculus which is at the heart of most of the technological developments we already enjoy, and it will undoubtedly have much more to contribute, particularly in the areas of medicine, society and politics in the years to come. This book will at least provide a basic understanding of what it is that has become indispensable in our societies, whether we like it or not.
Some knowledge of basic mathematics is required, but not much more than that: Strogatz is more concerned with explaining what Calculus is all about, and pointing out that certain precise questions about specific physical problems can only be best answered by its application. Calculus is the most accurate mathematical tool ever developed for answering such questions.
Some things I particularly liked about this lie in the conceptual (philosophical?) matters that are implied or suggested: the idea, for example, that the modern understanding of the Space-Time continuum is maintained (for me, the “problem” between the continuum idea and the idea of infinitesimals remains, and probably should remain so); and the fact that, no matter how “infinitesimal” the divisions of differential calculus are made, there will always remain “a little bit” extra that is unaccounted for. The latter is difficult enough when dealing with the interrelationship of two “entities” or “bodies”, let alone the far more complicated problem relating to three (or more?) such entities, so it remains a potentially intriguing matter.
While the cynics among us might consider this enough to disparage its absolute applicability, Calculus remains our most successful, accurate and practical measurement tool so far. I like the idea that there is still the possibility for new approaches to be made: but that being said, the current “solution” to such errors will still most probably be found through the re-application and refinements of its processes.
An ironic example of this relates to Strogatz’s use of the Boeing 787 Dreamliner aeroplane as a proud example of the achievements made under the auspices of this form of mathematics. This may very well be the case, but at the same time, some bad reports coming in 2019 about Boeing’s 737 MAX line of planes suggested that, among other matters, computer design flaws were partly responsible for two of these planes crashing with a total loss of 364 lives: a sober reminder that the more intricate the maths, the more care needs to be taken in its applications and consequences in the physical world.
Even so, it is Calculus which is at the heart of most of the technological developments we already enjoy, and it will undoubtedly have much more to contribute, particularly in the areas of medicine, society and politics in the years to come. This book will at least provide a basic understanding of what it is that has become indispensable in our societies, whether we like it or not.
April 20, 2019
A few centuries ago some clever people noticed that nature is in an ever-changing state, notably Galileo (1564-1642) studying objects in free fall and Kepler (1571-1630) studying the motion of planets around our sun. Then Newton (1643-1727) and Leibniz (1646-1716) invented a mathematical tool to get closer and closer to the changing system at hand. Steven did a great job explaining how Calculus uses divide-and-conquer to the extreme taming infinity to describe the universe. It changed civilization; this book travels from Archimedes (-212) computing pi to today’s design of airplanes. And Calculus is still evolving like a living organism after an explosion of diversity to explain CHANGE everywhere. For example, Einstein (1879-1955) used Calculus to play with space (say x, y, z) and time, at least four things changing at the same time.
February 14, 2021
4.5 stars
It is such a pleasure to listen to a mathematician who loves what he's doing. Strogatz brings across the enthusiasm about his topic in a lively, interesting and above all very digestible way. A clear prose and many intuitively accessible examples make this book about the history of calculus approachable to laypersons.
I listened to it on audio - which is admittedly not the best way to consume formulas, even though the narrator did an excellent job - and with 2 exceptions of longer lines of arguments where I would have needed to actually see the written form, I could follow without problem.
Highly recommended.
It is such a pleasure to listen to a mathematician who loves what he's doing. Strogatz brings across the enthusiasm about his topic in a lively, interesting and above all very digestible way. A clear prose and many intuitively accessible examples make this book about the history of calculus approachable to laypersons.
I listened to it on audio - which is admittedly not the best way to consume formulas, even though the narrator did an excellent job - and with 2 exceptions of longer lines of arguments where I would have needed to actually see the written form, I could follow without problem.
Highly recommended.
August 11, 2019
4++
Wow! I wish I had had this book back in college 50+ years ago. Who knew calculus could be this interesting to read about. I only endured my class with a boring professor. No joy then, but I thoroughly enjoyed this book. All the applications of calculus through the years were fascinating. Amazing stories about the scientists and mathematicians. Very easy to read and understand. I only needed to back up to reread a few paragraphs.
Wow! I wish I had had this book back in college 50+ years ago. Who knew calculus could be this interesting to read about. I only endured my class with a boring professor. No joy then, but I thoroughly enjoyed this book. All the applications of calculus through the years were fascinating. Amazing stories about the scientists and mathematicians. Very easy to read and understand. I only needed to back up to reread a few paragraphs.
January 31, 2024
I found this book very valuable. I have studied Calculus in a couple of scenarios. I am not great at mathematics, but I enjoy studying the field, the principles and the mechanics. This book provided me with what I had been missing in the past, the history around the development of Calculus and an excellent overview of the problems and concepts that Calculus was designed to tackle. Previously I had focused on the details and missed the big picture, frequently not understanding or losing sight of what the purpose was of what I was doing. In reading this there were times when I was stretched to understand what was being explained, but I would have been disappointed if that were not the case.
My hope is to read some additional works by this author. I think this book would be an excellent companion in conjunction with any study of Calculus.
My hope is to read some additional works by this author. I think this book would be an excellent companion in conjunction with any study of Calculus.
January 4, 2023
Some who know calculus, like myself, would probably like more history and epistemology, others might want more mathematical explanations.
You’re not going to satisfy everyone here and the book can get a bit esoteric but the author has a lot of verve and his love of the quantifiable and unquantifiable is evident throughout.
You’re not going to satisfy everyone here and the book can get a bit esoteric but the author has a lot of verve and his love of the quantifiable and unquantifiable is evident throughout.
August 25, 2019
Infinite Powers won't teach you calculus but it'll gently familiarise you with the subject without oversimplifying it. You'll finish the book wanting to dive into the actual maths.
February 13, 2022
میدانستید که حساب دیفرانسیل سبب شده است که دانشمندان بتوانند بیماری ایدز را از یک بیماری کشنده تبدیل به یک بیماری مزمن کنند؟
این کتاب سرگذشت حسابان را از ابتدا تا حال بررسی میکند. قسمت جالب و خواندنی آن مربوط به دانشمندانی نظیر ارشمیدس، کپلر، نیوتون، فرما، دکارت و ... میشود. بخشهایی که خود مطالب حسابان را آموزش میدهد چندان برایم جذابیت نداشت. در کل برای شهود دادن، آشنا شدن با کاربرد حساب دیفرانسیل و انتگرال در زندگی واقعی، و سفر به عصر هر یک از دانشمندان معروف البته که ارزش خواندن دارد.
این کتاب سرگذشت حسابان را از ابتدا تا حال بررسی میکند. قسمت جالب و خواندنی آن مربوط به دانشمندانی نظیر ارشمیدس، کپلر، نیوتون، فرما، دکارت و ... میشود. بخشهایی که خود مطالب حسابان را آموزش میدهد چندان برایم جذابیت نداشت. در کل برای شهود دادن، آشنا شدن با کاربرد حساب دیفرانسیل و انتگرال در زندگی واقعی، و سفر به عصر هر یک از دانشمندان معروف البته که ارزش خواندن دارد.
June 8, 2019
"Dividing by zero summons infinity in the same way that a Ouia board supposedly summons spirits from another realm. It's risky. Don't go there." This sentence gives a good idea of the fun and rigor that Steven Strogatz brings to this book that explains what the big deal is to people who, let's face it, are unlikely to learn calculus. "The desire to harness infinity and exploit its power is a narrative thread that runs through the whole twenty-five hundred year story of calculus."
By weaving examples of what's so damn useful about calculus with stories of great minds and the problems they overcame personally and mathematically, Strogatz wrote a readable, yes, fun book about math. His is not a big ego, he gives pioneers their due and busts some myths along the way, "the Pythagorean theorem did not originate with Pythagoras, it was known to the Babylonians for at least a thousand years before him." We learn about the Chinese genius Liu Hui who improved on Archimedes's method for calculating pi as well as Zu Chongzhi who pushed the study of polygons further than anyone before him. Strogatz has a special affinity for the Sicilian Greek Archimedes. The Hindu contribution is enormous and seems worth an entire book on its own. The Arabic scholar Al-Hasan Ibn al-Haytham gets recognized as one of the many giants upon whose shoulders Newton and Leibniz stood.
We don't stop there but in several great chapters we learn about the absolute latest breakthroughs and applications of calculus, to Quantum Electrodynamics and Chaos theory, medicine and many other non-linear problems. Stretching my inelastic brain to the the snapping point, he discusses Eintstein's partial derivative theories.
"matter tells space-time how to curve, while curvature tells matter how to move. The dance between them makes the theory nonlinear."
I love that sentence! Strogatz is a true writer, an artist with words who, happily for me, applies that skill to explaining his profession, and love, calculus.
By weaving examples of what's so damn useful about calculus with stories of great minds and the problems they overcame personally and mathematically, Strogatz wrote a readable, yes, fun book about math. His is not a big ego, he gives pioneers their due and busts some myths along the way, "the Pythagorean theorem did not originate with Pythagoras, it was known to the Babylonians for at least a thousand years before him." We learn about the Chinese genius Liu Hui who improved on Archimedes's method for calculating pi as well as Zu Chongzhi who pushed the study of polygons further than anyone before him. Strogatz has a special affinity for the Sicilian Greek Archimedes. The Hindu contribution is enormous and seems worth an entire book on its own. The Arabic scholar Al-Hasan Ibn al-Haytham gets recognized as one of the many giants upon whose shoulders Newton and Leibniz stood.
We don't stop there but in several great chapters we learn about the absolute latest breakthroughs and applications of calculus, to Quantum Electrodynamics and Chaos theory, medicine and many other non-linear problems. Stretching my inelastic brain to the the snapping point, he discusses Eintstein's partial derivative theories.
"matter tells space-time how to curve, while curvature tells matter how to move. The dance between them makes the theory nonlinear."
I love that sentence! Strogatz is a true writer, an artist with words who, happily for me, applies that skill to explaining his profession, and love, calculus.
May 27, 2019
A fantastic book about calculus. A blend of the history of the development of calculus, its applications, and intuitive explanations of its power filled with nicely intuitive explanations that will either provide a refresher or a different way of understanding what you have already learned.
Steven Strogatz proceeds in (sort of) chronological order, defining calculus not as what you learn in school but any technique that breaks things apart into infinitesimal pieces and puts them back together again in order to solve problems. Rather than describing an immaculate conception of calculus by Leibniz and Newton, Strogatz starts with Archimedes, shows several geometric applications, and even spends a lot of time on Descartes and Fermat before even getting to what we consider calculus today. In all of these he shows how a combination of abstract ideas but also in many cases practical problems led to the development of calculus.
The chronological order is interrupted (in a good way) by Strogatz’s many descriptions of the applications of calculus to different practical problems, most of which are in the analytically relevant chapter. These include GPS, AIDS drugs, rocketry, and more. In all of these cases Strogatz shows his pedigree as an applied mathematician, going into significant but highly readable detail about the models and discoveries underlying these areas.
Overall, the book is very nicely written and highly recommended.
Steven Strogatz proceeds in (sort of) chronological order, defining calculus not as what you learn in school but any technique that breaks things apart into infinitesimal pieces and puts them back together again in order to solve problems. Rather than describing an immaculate conception of calculus by Leibniz and Newton, Strogatz starts with Archimedes, shows several geometric applications, and even spends a lot of time on Descartes and Fermat before even getting to what we consider calculus today. In all of these he shows how a combination of abstract ideas but also in many cases practical problems led to the development of calculus.
The chronological order is interrupted (in a good way) by Strogatz’s many descriptions of the applications of calculus to different practical problems, most of which are in the analytically relevant chapter. These include GPS, AIDS drugs, rocketry, and more. In all of these cases Strogatz shows his pedigree as an applied mathematician, going into significant but highly readable detail about the models and discoveries underlying these areas.
Overall, the book is very nicely written and highly recommended.
February 13, 2021
What a delightful excuse to revisit calculus, a subject I loved in school but haven't seen in 20 years. Strogatz did an extraordinary job of making the subject accessible; he has a knack for finding analogies that turn complex mathematical concepts into tangible ideas that are easy to visualize. The book covers the history, function (pun intended), and practical application of calculus, and even dares to imagine the future. Above all, Strogatz succeeds in communicating the extraordinary beauty of calculus. The final paragraph sent shivers up and down my spine and left me wondering what the universe will whisper to us next.
April 9, 2020
The essence of mathematics lies in its beauty and its intellectual challenge. It is a triumph of the imagination. In high schools and universities, we rarely grasp the core ideas in many branches of Mathematics. I thought I understood Euclidean geometry well in school. Much later, I stumbled upon non-Euclidean geometry for the first time. I learnt that Euclid’s fifth axiom - the axiom of parallels - is not a self-evident truth. It is not empirically verifiable. This makes other geometries possible. I had a similar experience with calculus. I grasped the core ideas behind infinitesimal calculus much after I left school. Often, I get fundamentals clarified later in life by popular science books. This book by Dr. Steven Strogatz showed me what was missing in teaching Maths in high schools. The soul of creative mathematics is intuition and imagination. Intuition comes first. Rigor comes after. The author says we leave this out in teaching Maths.
Most of us believe that Isaac Newton and Gottfried Leibniz invented Calculus. In this book, Dr. Strogatz traces the ideas of infinitesimal calculus from the days of Archimedes. We then meet Galileo, Kepler, Descartes and Fermat through to Newton and Leibniz. Along the way, we find the contributions of Arabic and Indian mathematicians. This book explains the beauty of calculus to any interested person, even if one is not well-versed in high-school mathematics. We follow the history of calculus. We find its key ideas with engrossing anecdotes of the scientists and mathematicians who developed the field.
What is calculus? Calculus is a branch of Mathematics which solves problems in three steps. It splits a complicated but continuous problem into an infinite number of simpler pieces. Then it solves each of the simple piece separately. Finally it puts them together as the solution. For instance, we can visualize motion at changing speed as composed of infinitely many, infinitesimally brief motions at constant speed. Similarly, one could pretend that curves comprise an infinite number of infinitesimally short straight lines. We call the splitting ‘differentiation’ and the putting together ‘integration’. Together, it becomes the Infinitesimal calculus. The author explains these ideas beautifully with many examples.
The first chapters in the book lay the foundation by showing us how ancient Greece approached the computation of Pi. Archimedes calculated the perimeter of a many-sided polygon inscribed in a circle. The polygon approached the circle as the number of sides approached infinity and the length of those sides approached zero. He also used the same approach in representing a smooth segment of a parabola. He did it through a mosaic of an infinite number of triangular shards. From here, we move on to see how this principle plays a big role in our life experience. Modern-day animators at DreamWorks used tens of thousands of polygons in creating Shrek’s round belly and trumpet-like ears. In the film ‘Avatar’, animators used millions of polygons to render each plant in the imaginary world of Pandora. Scientists have used the same principle in doing facial surgery for patients with misaligned jaws and other congenital malformations by creating Virtual surgery simulators. In modern-day CT Scanners, x-rays pass through tissue, bone and organs producing different levels of absorption. To compute the reduction of intensity in this process, the CT software calculates step by infinitesimal step as the x-rays travel through the tissue, and then combines all the results as the integral.
Calculus provides the language for describing waves and the tools for analyzing them. The vibration of strings produces music. It led to the discovery of the wave equation. James Clerk Maxwell used it to predict the existence of electromagnetic waves. It is calculus which enabled the invention of vacuum tubes, transistors, computers, radar, microwave ovens, drugs for HIV and the design of Boeing 787 wings.
The book contains many philosophical musings of the author which are important to understand the illusions which calculus creates to resolve reality. Calculus models Reality through an infinite number of divisions. If we look at any phenomenon in excruciatingly fine detail in time or space, we will see the breakdown of smoothness on which calculus depends on. The book illustrates this in explaining Usain Bolt’s record-breaking 100-meter dash in the 2009 World Championships in Berlin. Conversely calculus also takes its creative license by treating discrete objects as if they are continuous. The modeling is approximate but useful. For example, the DNA is a discrete collection of atoms and not continuous. Molecular Genetics treats it as a continuous curve, like a perfect elastic band. This enables Genetics to apply two spin offs of calculus - elasticity theory and differential geometry - to calculate how DNA deforms when subjected to the forces from proteins, from the environment and from interactions within itself.
The author is at his best in the final two chapters, namely ‘The Future of Calculus and Conclusion’. He begins with a defence of what he believes the essence of calculus is. The 19th century saw infinity, and the infinitesimal expunged from calculus. Scientists clarified what limits, derivatives, integrals and real numbers meant. They believed that they arrested the rampage of infinitesimal calculus. Dr. Strogatz differs. He is firm in his belief that the credo of calculus is the Infinity Principle. It is this principle that allows us to study any pattern, curve, motion, natural process, system or phenomenon that changes smoothly and continuously. He explains the role of calculus in linear, non-linear and chaotic systems using everyday experiences with amazing simplicity. Biology and sociology are nonlinear. When a system is nonlinear, its behavior may be impossible to forecast with formulas, even though that behavior is deterministic. Determinism does not imply predictability.
In this chapter, he further discusses Complex systems and higher Dimensions. He says that natural selection has tuned our nervous systems to perceive directions as up, down, front, back, left and right of ordinary space. We cannot visualize the fourth dimension, try as we might. We can only comprehend it through mathematical abstractions. This is a huge hurdle in Complex systems where we meet higher dimensions. Hence, the author says that it will be hard, if not impossible, to make progress on the most difficult problems of our time. They are in the behavior of economies, societies, cells, workings of the immune system, genes, brains and consciousness. Reading this, I felt that the Earth’s climate is a highly complicated, nonlinear system. Hence, humility and caution is necessary in making predictions for a hundred years.
Dr. Strogatz concludes the book by contemplating on what he calls the greatest mystery of all. Why is the Universe comprehensible and why is calculus in sync with it? He provides three eerie examples of the effectiveness of calculus. The differential equations and integrals of Quantum Electrodynamics (QED) is one. They help predict the properties of electrons and other particles to a precision of eight decimal places. Such precision is comparable to planning to snap one’s finger 3.17 years from now and getting it right to the nearest second without a clock or an alarm. The second example takes us back to 1928. Then, Paul Dirac wrote a differential equation for the electron which implied something new, true and beautiful about nature. Its logic and beauty demanded the existence of a new particle, the Positron. In 1932, the experimental physicist, Carl Anderson, found it, albeit unaware of Dirac’s prediction. The last example takes us even further back to 1915, to the differential equations of General Relativity. These equations predicted the expansion of the Universe, existence of black holes, gravitational waves, gravity’s effect on time, how matter curves space-time and how the curvature tells matter to move. It is a celestial dance of calculus and nature, in the author’s words.
This marvellous book leaves us in no doubt that Calculus is the language that God speaks. Everyone interested in the beauty of Mathematics, and Nature must read this book.
Most of us believe that Isaac Newton and Gottfried Leibniz invented Calculus. In this book, Dr. Strogatz traces the ideas of infinitesimal calculus from the days of Archimedes. We then meet Galileo, Kepler, Descartes and Fermat through to Newton and Leibniz. Along the way, we find the contributions of Arabic and Indian mathematicians. This book explains the beauty of calculus to any interested person, even if one is not well-versed in high-school mathematics. We follow the history of calculus. We find its key ideas with engrossing anecdotes of the scientists and mathematicians who developed the field.
What is calculus? Calculus is a branch of Mathematics which solves problems in three steps. It splits a complicated but continuous problem into an infinite number of simpler pieces. Then it solves each of the simple piece separately. Finally it puts them together as the solution. For instance, we can visualize motion at changing speed as composed of infinitely many, infinitesimally brief motions at constant speed. Similarly, one could pretend that curves comprise an infinite number of infinitesimally short straight lines. We call the splitting ‘differentiation’ and the putting together ‘integration’. Together, it becomes the Infinitesimal calculus. The author explains these ideas beautifully with many examples.
The first chapters in the book lay the foundation by showing us how ancient Greece approached the computation of Pi. Archimedes calculated the perimeter of a many-sided polygon inscribed in a circle. The polygon approached the circle as the number of sides approached infinity and the length of those sides approached zero. He also used the same approach in representing a smooth segment of a parabola. He did it through a mosaic of an infinite number of triangular shards. From here, we move on to see how this principle plays a big role in our life experience. Modern-day animators at DreamWorks used tens of thousands of polygons in creating Shrek’s round belly and trumpet-like ears. In the film ‘Avatar’, animators used millions of polygons to render each plant in the imaginary world of Pandora. Scientists have used the same principle in doing facial surgery for patients with misaligned jaws and other congenital malformations by creating Virtual surgery simulators. In modern-day CT Scanners, x-rays pass through tissue, bone and organs producing different levels of absorption. To compute the reduction of intensity in this process, the CT software calculates step by infinitesimal step as the x-rays travel through the tissue, and then combines all the results as the integral.
Calculus provides the language for describing waves and the tools for analyzing them. The vibration of strings produces music. It led to the discovery of the wave equation. James Clerk Maxwell used it to predict the existence of electromagnetic waves. It is calculus which enabled the invention of vacuum tubes, transistors, computers, radar, microwave ovens, drugs for HIV and the design of Boeing 787 wings.
The book contains many philosophical musings of the author which are important to understand the illusions which calculus creates to resolve reality. Calculus models Reality through an infinite number of divisions. If we look at any phenomenon in excruciatingly fine detail in time or space, we will see the breakdown of smoothness on which calculus depends on. The book illustrates this in explaining Usain Bolt’s record-breaking 100-meter dash in the 2009 World Championships in Berlin. Conversely calculus also takes its creative license by treating discrete objects as if they are continuous. The modeling is approximate but useful. For example, the DNA is a discrete collection of atoms and not continuous. Molecular Genetics treats it as a continuous curve, like a perfect elastic band. This enables Genetics to apply two spin offs of calculus - elasticity theory and differential geometry - to calculate how DNA deforms when subjected to the forces from proteins, from the environment and from interactions within itself.
The author is at his best in the final two chapters, namely ‘The Future of Calculus and Conclusion’. He begins with a defence of what he believes the essence of calculus is. The 19th century saw infinity, and the infinitesimal expunged from calculus. Scientists clarified what limits, derivatives, integrals and real numbers meant. They believed that they arrested the rampage of infinitesimal calculus. Dr. Strogatz differs. He is firm in his belief that the credo of calculus is the Infinity Principle. It is this principle that allows us to study any pattern, curve, motion, natural process, system or phenomenon that changes smoothly and continuously. He explains the role of calculus in linear, non-linear and chaotic systems using everyday experiences with amazing simplicity. Biology and sociology are nonlinear. When a system is nonlinear, its behavior may be impossible to forecast with formulas, even though that behavior is deterministic. Determinism does not imply predictability.
In this chapter, he further discusses Complex systems and higher Dimensions. He says that natural selection has tuned our nervous systems to perceive directions as up, down, front, back, left and right of ordinary space. We cannot visualize the fourth dimension, try as we might. We can only comprehend it through mathematical abstractions. This is a huge hurdle in Complex systems where we meet higher dimensions. Hence, the author says that it will be hard, if not impossible, to make progress on the most difficult problems of our time. They are in the behavior of economies, societies, cells, workings of the immune system, genes, brains and consciousness. Reading this, I felt that the Earth’s climate is a highly complicated, nonlinear system. Hence, humility and caution is necessary in making predictions for a hundred years.
Dr. Strogatz concludes the book by contemplating on what he calls the greatest mystery of all. Why is the Universe comprehensible and why is calculus in sync with it? He provides three eerie examples of the effectiveness of calculus. The differential equations and integrals of Quantum Electrodynamics (QED) is one. They help predict the properties of electrons and other particles to a precision of eight decimal places. Such precision is comparable to planning to snap one’s finger 3.17 years from now and getting it right to the nearest second without a clock or an alarm. The second example takes us back to 1928. Then, Paul Dirac wrote a differential equation for the electron which implied something new, true and beautiful about nature. Its logic and beauty demanded the existence of a new particle, the Positron. In 1932, the experimental physicist, Carl Anderson, found it, albeit unaware of Dirac’s prediction. The last example takes us even further back to 1915, to the differential equations of General Relativity. These equations predicted the expansion of the Universe, existence of black holes, gravitational waves, gravity’s effect on time, how matter curves space-time and how the curvature tells matter to move. It is a celestial dance of calculus and nature, in the author’s words.
This marvellous book leaves us in no doubt that Calculus is the language that God speaks. Everyone interested in the beauty of Mathematics, and Nature must read this book.
May 14, 2022
Плекає мою безмежну закоханість в математику
Читала першу The Joy Of X ще років 5 назад, коли працювала вчителем математики. Вона охоплює як раз шкільну математику, тому ще й пішла по всіх моїх учнях)
Ця частина покриває матаналіз, який я вчила з 9 класу по 6 курс, тобто весь зміст був знайомий, але дуже елегантно викладений, десь на дві третини був присвячений відповідям на найчастіше питання - де це треба в житті
Читала першу The Joy Of X ще років 5 назад, коли працювала вчителем математики. Вона охоплює як раз шкільну математику, тому ще й пішла по всіх моїх учнях)
Ця частина покриває матаналіз, який я вчила з 9 класу по 6 курс, тобто весь зміст був знайомий, але дуже елегантно викладений, десь на дві третини був присвячений відповідям на найчастіше питання - де це треба в житті
January 22, 2020
Understanding how things move together and how to evaluate seeming disparate data and events are at the core of many of modern life's problems. Steven Strogatz presents a wonderful history of the methodology of those techniques that are embodied in Calculus in a very readable fashion.
He does two things which the non-math person should appreciate. First, he presents a good history of the development of thinking about advanced math. From Xeno's paradox (which stipulates it is impossible to get anywhere if you assume that all movement is conditioned on moving half the distance to the goal incrementally) to Arrow's Law (which tried to argue that conditions for decisions in a social setting are hard to establish) I have always been bothered by these mind games. In part, because, we do figure out how to get from here to there and at the same time to vote, even if the results are not perfect. But Strogatz's history is well done.
At the same time he presents some real life examples of the uses of calculus to understand some things that are commonly misunderstood. I'll present only one. He presents data that the length of sunlight in a day is not a linear function (each day the number of minutes of sunlight expands or contracts by a common amount). He shows, using calculus, that the days extend from the Winter Equinox to the Spring in an advancing pattern (each day's sunlight is a bit longer) and then through the Summer Equinox the period of extension of sunlight is shortening) - at the Fall Equinox the pattern begins to slow down again until the cycle repeats.
One of the problems of many analytical models we assume to be correct in all sorts of fields is their linearity. A good example was the Club of Rome Report which postulated that in the last couple of decades of the 20th Century life on earth would be severely compromised. But many of the things we deal with require (because in part human activity is multi-faceted) configural logic.
I will admit that while I am pretty good at math I never took Calculus. I've always wondered whether it would be worth the effort to teach myself the skills. This book helped me decide that there are two opportunities in Calculus - the first is to understand a set of techniques for understand all sorts of things - from how the aids virus interacts to why we get phantom traffic jams. With lots more work we might even be able to understand things like the complex interactions of climate. The second opportunity is to be thankful for all the people willing to go through the process of learning the functions and to appreciate the power of this tool.
He does two things which the non-math person should appreciate. First, he presents a good history of the development of thinking about advanced math. From Xeno's paradox (which stipulates it is impossible to get anywhere if you assume that all movement is conditioned on moving half the distance to the goal incrementally) to Arrow's Law (which tried to argue that conditions for decisions in a social setting are hard to establish) I have always been bothered by these mind games. In part, because, we do figure out how to get from here to there and at the same time to vote, even if the results are not perfect. But Strogatz's history is well done.
At the same time he presents some real life examples of the uses of calculus to understand some things that are commonly misunderstood. I'll present only one. He presents data that the length of sunlight in a day is not a linear function (each day the number of minutes of sunlight expands or contracts by a common amount). He shows, using calculus, that the days extend from the Winter Equinox to the Spring in an advancing pattern (each day's sunlight is a bit longer) and then through the Summer Equinox the period of extension of sunlight is shortening) - at the Fall Equinox the pattern begins to slow down again until the cycle repeats.
One of the problems of many analytical models we assume to be correct in all sorts of fields is their linearity. A good example was the Club of Rome Report which postulated that in the last couple of decades of the 20th Century life on earth would be severely compromised. But many of the things we deal with require (because in part human activity is multi-faceted) configural logic.
I will admit that while I am pretty good at math I never took Calculus. I've always wondered whether it would be worth the effort to teach myself the skills. This book helped me decide that there are two opportunities in Calculus - the first is to understand a set of techniques for understand all sorts of things - from how the aids virus interacts to why we get phantom traffic jams. With lots more work we might even be able to understand things like the complex interactions of climate. The second opportunity is to be thankful for all the people willing to go through the process of learning the functions and to appreciate the power of this tool.
November 21, 2021
[20 Nov 2021]
When I was in high school -- many, many years ago -- I was really good at math, so when I got to college I thought I would major in math. However, after two semesters of calculus I changed my mind. I didn't flunk. Got Bs both semesters. Because I was able to regurgitate the calculations correctly. But I didn't really understand. I didn't get it in my soul the way I understood algebra. So I changed to French, an easy major.
My professor was really smart. He had worked at NASA during the early years of the space program, probably during the time of Katherine Johnson. But I don't think he was a particularly good math instructor. He was rather boring. It's not that I blame him for my failure, but I can't help but wonder if I'd had some different help if I might have done better. I want to understand. I've tried other authors on calculus with no particular success. But this book got such good reviews that I decided to try again.
I can't say that Strogatz was able to make me truly get calculus. His explanations are friendly and it's obvious that he understands it very well. He might be a very good instructor in a classroom but I still just get fogged up in my head when trying to understand differentials, integrals, logarithms.
However, this book is not really about teaching the reader to "do" calculus. It's about the history of calculus, and understanding the essential importance of calculus to so many things that impact our lives, and exploring the future of calculus. And the author achieves those goals quite well. If you don't let yourself get bogged down in formulas and calculations, the writing is quite accessible and interesting. I enjoyed the history, and especially appreciated the predictions of the last chapter.
I definitely recommend this book to anyone even slightly interested in math.
When I was in high school -- many, many years ago -- I was really good at math, so when I got to college I thought I would major in math. However, after two semesters of calculus I changed my mind. I didn't flunk. Got Bs both semesters. Because I was able to regurgitate the calculations correctly. But I didn't really understand. I didn't get it in my soul the way I understood algebra. So I changed to French, an easy major.
My professor was really smart. He had worked at NASA during the early years of the space program, probably during the time of Katherine Johnson. But I don't think he was a particularly good math instructor. He was rather boring. It's not that I blame him for my failure, but I can't help but wonder if I'd had some different help if I might have done better. I want to understand. I've tried other authors on calculus with no particular success. But this book got such good reviews that I decided to try again.
I can't say that Strogatz was able to make me truly get calculus. His explanations are friendly and it's obvious that he understands it very well. He might be a very good instructor in a classroom but I still just get fogged up in my head when trying to understand differentials, integrals, logarithms.
However, this book is not really about teaching the reader to "do" calculus. It's about the history of calculus, and understanding the essential importance of calculus to so many things that impact our lives, and exploring the future of calculus. And the author achieves those goals quite well. If you don't let yourself get bogged down in formulas and calculations, the writing is quite accessible and interesting. I enjoyed the history, and especially appreciated the predictions of the last chapter.
I definitely recommend this book to anyone even slightly interested in math.
Want to read
December 29, 2020Got it because Hannah Fry jumped with joy while reading it. I liked it overall, but at times it felt a bit lengthy. More aimed at people who haven't read much history of maths I think.
The book tells the story of how the central ideas from calculus developed; beginning with Archimedes who approximated the surface area of a circle by covering it with an infinite series of smaller and smaller triangles; and ending with GPS and CT scanners.
Strogatz retelling revolves around the "Infinity principle":
To shed light on any continuous shape, object, motion, process, or phenomenon — no matter how wild and complicated it may appear — reimagine it as an infinite series of simpler parts, analyze those, and then add the results back together to make sense of the original whole.
Looking at the Wikipedia article of calculus, it seems surprising that the books hammering of this idea was useful for me, because my math courses at uni failed to even leave a trace of a memory of this in my brain (probably by focussing way too much on proofs and not so much on ideas and intuitions).
Along the way there are colorful characterizations of the mathematicians, that sometimes seem a bit too colorful to be believable:
The book tells the story of how the central ideas from calculus developed; beginning with Archimedes who approximated the surface area of a circle by covering it with an infinite series of smaller and smaller triangles; and ending with GPS and CT scanners.
Strogatz retelling revolves around the "Infinity principle":
To shed light on any continuous shape, object, motion, process, or phenomenon — no matter how wild and complicated it may appear — reimagine it as an infinite series of simpler parts, analyze those, and then add the results back together to make sense of the original whole.
Looking at the Wikipedia article of calculus, it seems surprising that the books hammering of this idea was useful for me, because my math courses at uni failed to even leave a trace of a memory of this in my brain (probably by focussing way too much on proofs and not so much on ideas and intuitions).
Along the way there are colorful characterizations of the mathematicians, that sometimes seem a bit too colorful to be believable:
The secrets of heat were unraveled by a man who often felt cold. Orphaned at the age of ten, Jean Baptiste Joseph Fourier was a sickly, dyspeptic asthmatic as a teenager. As an adult, he believed heat was essential to health. He kept his room overheated and swathed himself in a heavy overcoat, even in the summer. In all aspects of his scientific life, Fourier was obsessed with heat. He originated the concept of global warming and was the first to explain how the greenhouse effect regulates the Earth’s average temperature.
December 27, 2022
Is quite irritating to read “language of the gods” over and over again. I don’t see why writing for the popular audience should entail using really bad comparisons and unnecessarily misleading statements.
Here’s a hilarious passage that “compares” geometry vs algebra.
“ Geometry appealed to the right side of the brain. .. it called for a certain ingenuity. .. Beginning an argument required strokes of genius.
Algebra, however, was systematic. Equations could be massaged almost mindlessly, peacefully…. But algebra suffered from emptiness…
“
As a history, the book is incomplete and very eurocentric( just a disclaimer that you’re not Eurocentric and dropping a couple names is not enough). It presents a very simplistic, “certain geniuses did everything” style of history. For example,
“When Archimedes died, the mathematical study of nature nearly died along with him. Eighteen hundred years passed before a new Archimedes appeared.”
Really? Nothing happened in between?
To summarize,
The selection of applications i n the book is good and the Infinity principle is well explained. The rest of it is an ordeal.
Here’s a hilarious passage that “compares” geometry vs algebra.
“ Geometry appealed to the right side of the brain. .. it called for a certain ingenuity. .. Beginning an argument required strokes of genius.
Algebra, however, was systematic. Equations could be massaged almost mindlessly, peacefully…. But algebra suffered from emptiness…
“
As a history, the book is incomplete and very eurocentric( just a disclaimer that you’re not Eurocentric and dropping a couple names is not enough). It presents a very simplistic, “certain geniuses did everything” style of history. For example,
“When Archimedes died, the mathematical study of nature nearly died along with him. Eighteen hundred years passed before a new Archimedes appeared.”
Really? Nothing happened in between?
To summarize,
The selection of applications i n the book is good and the Infinity principle is well explained. The rest of it is an ordeal.
Displaying 1 - 30 of 727 reviews