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Mathematics and the Physical World (Dover Books on Mathematics) Paperback – March 1, 1981
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Simple example. We've all seen dozens of ways to explain the first derivative (the rate of change) of say x^2 as 2x. These range from slope finding to falling objects to matrices. But nearly all I've seen assume a curve, tangents, etc. Kline blows the mind with a rarely seen example-- a circle!
He simply "animates" the circle and asks you to mentally extend the radius as it turns in the area-clock like the minute hand, only the size of the circumference. He then points out that the first derivative, the rate of change, is like extending the radius a bit, which "sweeps" out a larger circumference. Then he points out that y' in that case changes (pi)r^2 to 2(pi)r, instantly connecting radius, area and circumference in motion!
This may seem trivial until we remember that Feynman used the same clock face example to explain quantum mechanics verbally, which blew the math community away at the time. In fact, connecting Kline's example with Feynman's gives some really interesting "aha" moments of new understanding.
The whole book is like this, with many fun stories of how math reflects the natural world. You'll be amazed at the level of math he gets into with such simple assumptions as "very little math knowledge" before beginning. I mean, ALL applied math texts assume a LOT of math to begin. This is truly a rare gem, and a find for any High School STEM teacher wanting a real bridge to advanced concepts. Highly recommended, even in 2013 plus...
When earth was put apart
Granted that it was the superior mathematics of the new theory which inspired Copernicus and Kepler, and later Galileo, to repudiate religious convictions, scientific arguments, common sense, and well-entrenched habits of thought, how did the theory help to shape modern times?
First, Copernican theory has done more to determine the content of modern science than is generally recognized. The most powerful and most useful single law of science is Newton's law of gravitation. Without anticipating here the discussion reserved for a more appropriate place in this book we can say that the best experimental evidence for this law, the evidence which established it, depends entirely on the heliocentric theory.
Second, this theory is responsible for a new trend in science and human thought, barely perceptible, at the time but all-important today. Since our eyes do not see, nor our bodies feel, the rotation and revolution of the Earth, the new theory rejected the evidence of the senses- Things were not what they seemed to be. Sense data could be misleading and reason was the reliable guide. Copernicus and Kepler thereby set the precedent that guides modern science, namely, that reason and mathematics are more important in understanding and interpreting the universe than the evidence of the senses. Vast portions of electrical and atomic theory and the whole theory of relativity would never have been conceived if scientists had not come to accept the reliance upon reason first exemplified by Copernican theory. In this very significant sense Copernicus and Kepler began the Age of Reason, in addition to fulfilling the cardinal function of scientists and mathematicians, that is, to provide a rational comprehension of the universe.
By deflating the stock of Homo sapiens, Copernican theory reopened questions that the guardians of Western civilization had been answering dogmatically upon the basis of Christian theology. Once there had been only one answer; now there are ten or twenty to such basic questions as: Why does man desire to live and for what purpose? Why should he be moral and principled? Why seek to preserve the race? It is one thing for man to answer such questions in the belief that he is the child and ward of a generous, powerful and provident God. It is another to answer them knowing that he is a speck of dust in a cyclone.
Mathematics in Western culture, Morris Kline, 1953