Infinite Powers: A fun reminder of the power and elegance of calculus in understanding the physical universe.

Infinite Powers: How Calculus Reveals the Secrets of the Universe” by Steven Strogatz is a fun and interesting read about the history and basic foundations of calculus and its applications in the real world.

The fact that the universe’s behavior can be predicted through math equations is remarkable. Its a fact we accept readily today, but it is quite astounding and magical to revel in the history of how such understanding of physical phenomena were discovered and understood through math. Calculus, as stated by Richard Feynman, is “the language that God talks”.

While I wasn’t a math major in college, I was fortunate to take an elective course in partial differential equations with Professor Strogatz while he was still at MIT. Strogatz was known to be an excellent teacher and he didn’t dissapoint — it was one of my favorite courses. I remember the joy and wonder I felt as he shared the mathematical secrets of the universe.

This book is no different.

Infinite Powers provides a brief tour through the history of arithmetic, geometry, algebra, analytic geometry, and ultimately calculus explaining in plain language what problems mathematicians were trying to solve and how they came up with the solutions that they did.

The Basics and the Background

Here are some of the interesting notes (paraphrased) from the book:

  • Calculus began with [geometers] frustration with roundness
  • The idea of using infinity to solve difficult geometry problems has to be among the best idea anyone has ever had
  • Math evolved with fusion between different disciplines — the number line is an example of arithmetic consorting with geometry, where numbers are presented as points on a line
  • Newton was like a mash-up artist — combining geometry from the Ancient Greeks, infused with Indian decimals, Islamic algebra, and French analytic geometry
  • Calculus can be explained with three central problems: How can we figure out the changing slope of a curve? How can we reconstruct a curve from its slope? And how can we figure out the changing area beneath the curve?

Calculus, to me, is defined by its credo: to solve a hard problem about anything continuous, slice it into infinitely many parts and solve them. By putting the answers back together, you can make sense of the original whole. I’ve called this credo the Infinity Principle.

Applications of Calculus

Calculus is famous for its use in physics and science for good reason. It was the starting formality for multiple other branchs of math and formal language in physics:

Calculus was the Cambrian explosion for mathematics. Once it arrived, an amazing diversity of mathematical fields began to evolve. Their lineage is visible in their calculus-based names, in adjectives like differential and integral and analytic, as in differential geometry, integral equations, and analytic number theory. These advanced branches of mathematics are like the many branches and species of multicellular life.

There are numerous interesting applications of calculus explained in the book (microwave ovens, brain imaging, etc.) but a few stood odd:

General Relativity. Einstein of course had to use very sophisticated math in the development of his general theory of relativity. Its worth noting that while Newton’s laws of gravity predict most of our every day life very well — without Einstein’s correction for general relativity, errors in global positions would accumulate at about ten kilometers each day, and the whole system would become worthless for navigation in a matter of minutes.

Quantum. “Nothing that humanity has ever predicted is as accurate as the predictions of quantum electrodynamics [ the theory of how light and matter interact ]. 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 it 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.”

Positron. Dirac attempted to reconcile Einstein’s special theory of relativity with quantum mechanics as applied to an electron moving near the speed of light. The equation he came up with was derived substantially on an elegant/aesthetic basis — but an interesting outcome was the prediction of anti-matter. It was a few year later that the positron was experimentally determined.

My Take

Infinte Powers is a great and accessible read. Even if you are not a math lover like me, I think you’ll enjoy and be inspired by the super-interesting history and wonderfully woven web of stories, anecdotes, and insights that Strogatz shares.

References:

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