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August 26, 2021 - January 4, 2022
Calculus is one of humankind’s most inspiring collective achievements. 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.
That rotational symmetry may be why circles are so ubiquitous. Whenever some aspect of nature doesn’t care about direction, circles are bound to appear.
Where there is circular motion, there are sine waves.
Patterns are what make compression possible in the first place. Only patterned data can be compressed. Random data cannot. Happily, many of the things people care about, like songs and faces and fingerprints, are highly structured and patterned.
This reasoning explains why 100 is defined as 1 (and has to be defined that way), a definition that many people find puzzling.
We somehow sense frequency logarithmically.
Whenever we study a relationship of any kind, we want to know: If one variable changes, how much does a related variable change? And in what direction, up or down? These are questions about derivatives.
A derivative is defined only if a curve approaches a limiting straight line as we zoom in on it.
In mathematical modeling, as in all of science, we always have to make choices about what to stress and what to ignore. The art of abstraction lies in knowing what is essential and what is minutia, what is signal and what is noise, what is trend and what is wiggle. It’s an art because such choices always involve an element of danger; they come close to wishful thinking and intellectual dishonesty. The greatest scientists, like Galileo and Kepler, somehow manage to walk along that precipice.
“Art,” said Picasso, “is a lie that makes us realize truth.” The same could be said for calculus as a model of nature.
An infinitesimal is a hazy thing. It is supposed to be the tiniest number you can possibly imagine that isn’t actually zero. More succinctly, an infinitesimal is smaller than everything but greater than nothing.
Elegant solutions are valued in math in part because they’re pretty but also because they’re powerful.
More generally, an ordinary differential equation describes how something (the position of a planet, the concentration of a virus) changes infinitesimally as the result of an infinitesimal change in something else (such as an infinitesimal increment of time). What makes such an equation “ordinary” is that there is exactly one something else, one independent variable.
it doesn’t matter how many dependent variables there are. As long as there is only one independent variable, the differential equation is considered ordinary.
Being differentiated is a traumatic experience for most curves. But not for a sine wave. After its derivative is taken, it dusts itself off and appears unfazed, as sinusoidal as ever. The only injury it suffers—and it isn’t even an injury, really—is that the sine wave shifts in time. It peaks a quarter of a cycle earlier than it used to.
These two properties—the proportionality between cause and effect, and the equality of the whole to the sum of the parts—are the essence of what it means to be linear.
The great advantage of linearity is that it allows for reductionist thinking. To solve a linear problem, we can break it down to its simplest parts, solve each part separately, and put the parts back together to get the answer.
determinism does not imply predictability.
