Contents
According to Steven Strogatz in his book Infinite Powers about the history and theory of calculus, “Calculus is the language of the universe as well as the logical engine for extracting its secrets.” Strogatz sums up the fundamental insight behind calculus as “the infinity principle”:
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.
We all know a handy way to solve hard problems is to break them up into many smaller problems, but the idea that you could break really hard problems up into infinitely many smaller, simpler problems—and then put the solutions back together!—requires a leap of imagination. Infinite Powers traces the history of this imaginative leap from its earliest stirrings in Archimedes in the 3rd century BCE through its mathematical formalization by Newton and Leibniz in the 17th century and beyond.
“[a derivative] defines a rate of change as a function” the slope of a line is a number, while the slope of a curve is a function. (This is higher-order functions in functional programming!)
“the area [under a curve] problem is about predicting the relationship between anything that changes at a changing rate and how much that thing builds up over time” (stocks and flows!)
“the reason why integration is so much harder than differentiation has to do with the distinction between local and global. Local problems are easy. Global problems are hard.” (in programming, local is easier to deal with than global, too, hence the value of functional programming)
chaos, complexity, three-body problem and multi-dimensional state-space
Reality is Curved
In Einstein’s theory, matter tells space-time how to curve, while curvature tells matter how to move. The dance between them makes the theory nonlinear.
Sources
- Infinite Powers: How Calculus Reveals the Secrets of the Universe by Steven Strogatz