Quotes from Steven H. Strogatz

For reasons nobody understands, the universe is deeply mathematical. Maybe God made it that way. Or maybe it's the only way a universe with us in it could be, because nonmathematical universes can't harbor life intelligent enough to ask the question. In any case, it's a mysterious and marvelous fact that our universe obeys laws of nature that always turn out to be expressible in the language of calculus as sentences called differential equations.

As the writer Arthur Koestler astutely observed, "Johannes Kepler became enamored with the Pythagorean dream, and on this foundation of fantasy, by methods of reasoning equally unsound, built the solid edifice of modern astronomy. It is one of the most astonishing episodes in the history of thought, and an antidote to the pious belief that the Progress of Science is governed by logic.

It relied on measures of geometrical objects: lengths of lines, areas of squares, volumes of cubes. All of these they called magnitudes. They thought of them as distinct from numbers and superior to them. This, I believe, is why Archimedes held pi at arm's length. He didn't know what to make of it. It was a strange, transcendent creature, more exotic than any number.

The insights provided by the model overturned the prevailing view that the virus was lying dormant in the body; in fact, it was in a raging battle with the immune system every minute of every day. With the new understanding that calculus helped provide, HIV infection has been transformed from a near-certain death sentence to a manageable chronic disease—at least for those with access to combination-drug therapy.

Yet twenty-two trillion is nothing compared to the infinitude of digits that define the actual pi. Think of how philosophically disturbing this is.

Naturally, the place to start is at infinity.

They exist in some Platonic realm, along with abstract concepts like truth and justice.

Each year surveyors had to redraw the boundaries of farmers' fields after the summer flooding of the Nile washed the borderlines away. That activity later gave its name to the study of shape in general: geometry, from the Greek g?, "earth," and metr?s, "measurer.

Mathematically, circles embody change without change.

Whenever some aspect of nature doesn't care about direction, circles are bound to appear.

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.

the idea of using infinity to solve difficult geometry problems has to rank as one of the best ideas anyone ever had.

Winfree wrote equations for his system of oscillators, describing how fast each one moves through its cycle. At any instant, an oscillator's speed is determined by three contributions: its preferred pace, which is proportional to its natural frequency; its current sensitivity to any incoming influences (which depends on where it is in its cycle); and the total influence exerted by all the other oscillators (which depends on where they all are in their cycles).

Creation is intuitive; reason comes later. In the story of calculus, more than in other parts of mathematics, logic has always lagged behind intuition. This makes the subject feel especially human and approachable, and its geniuses more like the rest of us.

What I've just described is called a system of differential equations. Such equations arise whenever we have rules for speeds depending on current positions.

More generally, when a nonchaotic system is disturbed slightly, the disturbance either doesn't grow at all or else grows very mildly, increasing in proportion to how much time has passed. One says that the errors grow no faster than linearly in time.

Though we couldn't see how to explain these results mathematically, an intuitive explanation suggested itself: The shortcuts were providing high-speed communication channels, enabling mutual influence to spread swiftly throughout the population. Of course, the same effect could have been achieved by connecting every oscillator directly to every other, but at a much greater cost in wiring. The small-world architecture apparently fostered global coordination more efficiently.

pack and a disorganized band of fringe oscillators. When the system was self-synchronizing, Winfree found that no oscillator was indispensable. There was no boss. Any oscillator could be removed and the process would still work.

From a purely mathematical perspective, a power law signifies nothing in particular—it's just one of many possible kinds of algebraic relationship. But when a physicist sees a power law, his eyes light up. For power laws hint that a system may be organizing itself. They arise at phase transitions, when a system is poised at the brink, teetering between order and chaos. They arise in fractals, when an arbitrarily small piece of a complex shape is a microcosm of the whole.

The amount of time we can successfully predict the state of a chaotic system depends on three things: how much error we're willing to tolerate in the forecast; how precisely we can measure the initial state of the system; and a time scale that's beyond our control, called the Lyapunov time, which depends on the inherent dynamics of the system itself.

In a chaotic system, the required precision in the initial measurement grows exponentially, not linearly.

Lyapunov time sets a horizon beyond which acceptable prediction becomes impossible. For a chaotic electrical circuit, the horizon is something like a thousandth of a second; for the weather, it's unknown but seems to be a few days; and for the solar system itself, five million years.

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.

In particular, the creeping advance of an improbable cascade near the second tipping point is reminiscent of a low-budget hit that starts out slowly and builds by word of mouth.