Steven Strogatz
Author of The Joy of x: A Guided Tour of Math, from One to Infinity
About the Author
Steven Strogatz is the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University. Also the author of Sync and The Joy of x, he lives in Ithaca, New York.
Image credit: Wikipedia
Works by Steven Strogatz
The Joy of x: A Guided Tour of Math, from One to Infinity (2013) — Author — 1,148 copies, 39 reviews
Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (1994) 492 copies, 3 reviews
The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math (2009) 133 copies, 6 reviews
O poder do infinito - Como o calculo revela os segredos do universo (Em Portugues do Brasil) (2019) 5 copies
The Great Courses: Chaos 1 copy
Associated Works
What Is Your Dangerous Idea? Today's Leading Thinkers on the Unthinkable (2007) — Contributor — 668 copies, 8 reviews
The Next Fifty Years: Science in the First Half of the Twenty-first Century (2002) — Contributor — 411 copies, 10 reviews
Tagged
Common Knowledge
- Legal name
- Strogatz, Steven Henry
- Birthdate
- 1959-08-13
- Gender
- male
- Education
- Princeton University (AB | Mathematics | 1980)
Harvard University (PhD | Mathematics | 1986) - Occupations
- professor of mathematics (Massachusetts Institute of Technology, 1989-1994)
professor of mathematics (Cornell University, 1994- ) - Organizations
- Cornell University
Massachusetts Institute of Technology
American Mathematical Society
American Physical Society - Awards and honors
- Euler Book Prize (2014)
Lewis Thomas Prize for Writing about Science (2015)
American Academy of Arts and Sciences (Fellow) - Nationality
- USA
- Birthplace
- Torrington, Connecticut, USA
- Associated Place (for map)
- Connecticut, USA
Members
Reviews
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 show more 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 [b:The Beginning of Infinity: Explanations That Transform the World|10483171|The Beginning of Infinity Explanations That Transform the World|David Deutsch|https://i.gr-assets.com/images/S/compressed.photo.goodreads.com/books/1311705051i/10483171._SY75_.jpg|15388653] 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. show less
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 [b:The Beginning of Infinity: Explanations That Transform the World|10483171|The Beginning of Infinity Explanations That Transform the World|David Deutsch|https://i.gr-assets.com/images/S/compressed.photo.goodreads.com/books/1311705051i/10483171._SY75_.jpg|15388653] 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. show less
Bello! Leggero, veloce e molto curato nei riferimenti. Citazioni di titoli di libri, pubblicazioni accademiche, video, articoli e siti interessanti dove approfondire. Rispetta tutte le aspettative che mi ero fatto nel leggere il retro di copertina. Una lettura quasi tutta d'un fiato (tre giorni) ma che mi ha rispolverato concetti affrontati nella corriera scolastica con quel che di tranquillità e serenità in più. Una bella lettura che ritengo possa piacere indipendentemente da età e show more background. Che si sia appassionati, simpatizzanti o semplicemente curiosi si può trovare qualcosa che fa per noi sui vasti meandri della matematica. Utile anche a chi volesse approcciare la materia nel modo giusto, non solamente tramite formule e definizioni applicate in modo meccanico e mnemonico. Strogatz lo conoscevo soprattutto per i suoi contributi alla teoria del caos in ambito accademico e devo ammettere che scoprire il suo lato paterno e divertente con cui presenta ogni argomento, mi ha colpito molto positivamente. Trasmette oltre che ottime informazioni, la sua passione.
Non posso che riportare l' essenza del libro, brillantemente descritta nel retro di copertina:
"Astratta, astrusa, sembra lontanissima dal nostro mondo e spesso mette paura, eppure la matematica è dappertutto, basta sapere dove guardare. Con giocosa semplicità Strogatz spalanca per noi una finestra sul mondo nascosto della matematica, svelando la meraviglia dei numeri insieme ad alcuni piccoli e grandi misteri della vita"
Oserei quasi dire che La Matematica è il Nostro Mondo show less
Many people assume that calculus began with Newton and Leibniz, but Strogatz, in this wonderful general audience maths book, makes clear that the history of calculus really starts nearly two thousand years earlier with the ancient Greeks, most notably Archimedes.
Similarly, there wasn't some vacuum between Archimedes and official inventors of calculus in the 17th century, but many bit players increasing our knowledge of the underlying tools needed for calculus. Strogatz describes them all in show more detail, and only turns to Newton past halfway through the book. Each section is described with a playful, skillful style, and made more engaging with ample, entertaining character descriptions and anecdotes.
But the book comes into its own, and really becomes exceptional with the explanation of the maths itself. For the first two thirds of the book, this is so clearly described, with multiple examples and step throughs, that makes otherwise complex concepts appear simple. This is the true gift of the book, to make maths appear inspiring, straight-forward and its history following a logical step-by-step progression. I'm not sure I've come across another popular maths book that has explained things so well as here.
The other area where this book shines is in capturing the deep concepts behind the topics, for instance that calculus really just boils down to splitting a hard problem into many tiny simpler problems and solving that way. This makes the book the opposite of dry - this is far from a series of equations to memorise. Strogatz makes the maths sing with resonance and connections between fields, in a similar way to the youtube channel 3blue1brown does.
That latter sections of the book, after Newton and Leibniz, discuss the more modern implications of calculus and its tools in the sciences, especially physics, but also in many other domains. Some books try to make a case for their topics to be one of the most important and central in the world. Strogatz succeeds on this aim with calculus. Having said this, it is always a good sign that the limits of a topic are mentioned, and here too the limits of calculus are described in fascinating ways that I wasn't previously aware of, both in terms of chaos theory, and in terms of the proved limits of calculus, for instance that it can't capture the motion of a spinning top.
My only very slight complaint is that whereas the former sections of the book are brilliant at making us understand the ins and outs of the mathematics, Strogatz doesn't even try when it comes to differential equations, particularly partial differential equations, towards the end of the book. This seemed a little incongruous with the way he'd tried so hard to describe the maths itself before, and I would have loved at least some attempt at this, even if in an epilogue or something.
Still, this little quibble aside, this is possibly the general audience maths book I've enjoyed the most, and learnt the most from. show less
Similarly, there wasn't some vacuum between Archimedes and official inventors of calculus in the 17th century, but many bit players increasing our knowledge of the underlying tools needed for calculus. Strogatz describes them all in show more detail, and only turns to Newton past halfway through the book. Each section is described with a playful, skillful style, and made more engaging with ample, entertaining character descriptions and anecdotes.
But the book comes into its own, and really becomes exceptional with the explanation of the maths itself. For the first two thirds of the book, this is so clearly described, with multiple examples and step throughs, that makes otherwise complex concepts appear simple. This is the true gift of the book, to make maths appear inspiring, straight-forward and its history following a logical step-by-step progression. I'm not sure I've come across another popular maths book that has explained things so well as here.
The other area where this book shines is in capturing the deep concepts behind the topics, for instance that calculus really just boils down to splitting a hard problem into many tiny simpler problems and solving that way. This makes the book the opposite of dry - this is far from a series of equations to memorise. Strogatz makes the maths sing with resonance and connections between fields, in a similar way to the youtube channel 3blue1brown does.
That latter sections of the book, after Newton and Leibniz, discuss the more modern implications of calculus and its tools in the sciences, especially physics, but also in many other domains. Some books try to make a case for their topics to be one of the most important and central in the world. Strogatz succeeds on this aim with calculus. Having said this, it is always a good sign that the limits of a topic are mentioned, and here too the limits of calculus are described in fascinating ways that I wasn't previously aware of, both in terms of chaos theory, and in terms of the proved limits of calculus, for instance that it can't capture the motion of a spinning top.
My only very slight complaint is that whereas the former sections of the book are brilliant at making us understand the ins and outs of the mathematics, Strogatz doesn't even try when it comes to differential equations, particularly partial differential equations, towards the end of the book. This seemed a little incongruous with the way he'd tried so hard to describe the maths itself before, and I would have loved at least some attempt at this, even if in an epilogue or something.
Still, this little quibble aside, this is possibly the general audience maths book I've enjoyed the most, and learnt the most from. show less
The story of a teacher and student maintaining correspondence about their shared interest in math on it's own is heartwarming, but The Calculus of Friendship goes above and beyond. The lovable character of Mr. Joffray (Joff) is the type of teacher that many would aspire to have because of his dedication to his students and uncanny ability to demonstrate the calculus of everyday life. Although I struggled with some of the math problems, I thought the book was well paced and organized, and the show more author's use of the original letters helped bring Joff's voice to life. Strogatz's writing reveals the immense personal investment he placed in this cathartic project, and I was moved by his personal reflection and the emotional development he underwent through the years of communicating with Joff. In the end, it's clear that although Strogatz passed on a great deal of complex mathematical knowledge to Joff, he actually learned more important lessons about approaching life's constant changes, good or bad, with an open mind and open heart. show less
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