Quotes from Steven H. Strogatz

Although the role of serendipity is familiar, what's not so well appreciated is how different serendipity is from luck. Serendipity is not just an apparent aptitude for making fortunate discoveries accidentally, as my dictionary defines it. Serendipitous discoveries are always made by people in a particular frame of mind, people who are focused and alert because they're searching for something. They just happen to find something else.

If we're lucky and skillful enough — if we transform the equations in just the right way — we can get them to reveal their hidden implications. To a mathematician, the process feels almost palpable. It's as if we're manipulating the equations, massaging them, trying to relax them enough so that they'll spill their secrets. We want them to open up and talk to us.

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. Truth in, truth out.

With the star up above and the blackness of space, I can't avoid feeling awe. How could we, Homo sapiens, an insignificant species on an insignificant planet adrift in a middleweight galaxy, have managed to predict how space and time would tremble after two black holes collided in the vastness of the universe a billion light-years away? We knew what that wave should sound like before it got here. And, courtesy of calculus, computers, and Einstein, we were right.

and out comes another empirical truth, possibly a new one, a fact about the universe that nobody knew before (like the existence of electromagnetic waves).

In a nutshell, calculus wants to make hard problems simpler. It is utterly obsessed with simplicity.

Once that's done, it solves the original problem for all the tiny parts, which is usually a much easier task than solving the initial giant problem. The remaining challenge at that point is to put all the tiny answers back together again. That tends to be a much harder step, but at least it's not as difficult as the original problem was.

Pi is fundamentally a child of calculus. It is defined as the unattainable limit of a never-ending process.

Cubism Meets Calculus

divert the discussion from its main intent and fasten upon some statement of mine which lacks a hair's-breadth of the truth and, under this hair, hide the fault of another which is as big as a ship's cable.

Thus the hexagon argument demonstrates ? > 3.

So he used calculus not only to predict the existence of electromagnetic waves but also to solve an age-old mystery: What was the nature of light? Light, he realized, was an electromagnetic wave.

But we all know that, at least in principle, sync can be persistent without being periodic. Think of the musicians in an orchestra. All the violins come in at the same time, and stay in sync throughout. Yet they are not periodic: They do not play the same passage over and over again.

Calculus is the mathematics of change. It describes everything from the spread of epidemics to the zigs and zags of a well-thrown curveball. The subject is gargantuan—and so are its textbooks. Many exceed a thousand pages and work nicely as doorstops.

In colloquial usage, chaos means a state of total disorder. In its technical sense, however, chaos refers to a state that only appears random, but is actually generated by nonrandom laws. As such, it occupies an unfamiliar middle ground between order and disorder. It looks erratic superficially, yet it contains cryptic patterns and is governed by rigid rules. It's predictable in the short run but unpredictable in the long run. And it never repeats itself: Its behavior is nonperiodic.

What I'm trying to say is that his calculation of ? was heroic, both logically and arithmetically. By using a 96-gon inside the circle and a 96-gon outside the circle, he ultimately proved that ? is greater than 3 + 10/71 and less than 3 + 10/70.

Looking at numbers as groups of rocks may seem unusual, but actually it's as old as math itself. The word "calculate" reflects that legacy -- it comes from the Latin word calculus, meaning a pebble used for counting. To enjoy working with numbers you don't have to be Einstein (German for "one stone"), but it might help to have rocks in your head.

let's begin with the word "vector." It comes from the Latin root vehere, "to carry," which also gives us words like "vehicle" and "conveyor belt." To an epidemiologist, a vector is the carrier of a pathogen, like the mosquito that conveys malaria to your bloodstream. To a mathematician, a vector (at least in its simplest form) is a step that carries you from one place to another.

To grasp how different a million is from a billion, think about it like this: A million seconds is a little under two weeks; a billion seconds is about thirty-two years.

Actually, languages can be very tricky in this respect. The eminent linguistic philosopher J. L. Austin of Oxford once gave a lecture in which he asserted that there are many languages in which a double negative makes a positive but none in which a double positive makes a negative—to which the Columbia philosopher Sidney Morgenbesser, sitting in the audience, sarcastically replied, "Yeah, yeah.

Calculus succeeds by breaking complicated problems down into simpler parts. That strategy, of course, is not unique to calculus. All good problem-solvers know that hard problems become easier when they're split into chunks. The truly radical and distinctive move of calculus is that it takes this divide-and-conquer strategy to its utmost extreme — all the way out to infinity.

things that seem hopelessly random and unpredictable when viewed in isolation often turn out to be lawful and predictable when viewed in aggregate.

Those of us who teach math should try to turn this bug into a feature. We should be up front about the fact that word problems force us to make simplifying assumptions. That's a valuable skill—it's called mathematical modeling.

Infinity lies at the heart of so many of our dreams and fears and unanswerable questions: How big is the universe? How long is forever? How powerful is God? In every branch of human thought, from religion and philosophy to science and mathematics, infinity has befuddled the world's finest minds for thousands of years. It has been banished, outlawed, and shunned. It's always been a dangerous idea.