One might be led by my posts so far on Ernst Mach, notorious demander of experimental evidence and doubter of multi-layered abstractions, to think that he was anti-abstraction or anti-logical-deduction. This is not at all true. Mach very, very much valued the place of mathematics in physics.
For just one of countless examples of how math helps science, consider this: much of what a modern scientist does, Mach tells us, involves studying how one thing varies with another thing… in other words, how one thing is a function of another thing. Every current or former calculus student will hear right away in those words a clue that math may, indeed, be quite relevant to the study of science.
“The language of everyday life,” Mach tells us in The Science Of Mechanics, “has not yet grown enough to be sufficiently accurate for the purposes of so exact a science as Mechanics.” That is why we need the language of mathematics. Or, as Galileo pointed out, mathematics IS the language of Nature.
Because of this intimate connection between math and Nature, Mach explains that we can sometimes take our experimental observations and formulate them in the symbols and formulas of mathematics and thereby be led to discover new facets of the phenomena. In this way, mathematics is good “for completing what has been given.”
The application of mathematics to observables allows us to abstract from concretes, and then to go even farther and initiate “a new period” in the area being studied which Mach calls the “formal development.” During the formal development of branch of science, we can combine experiment with mathematical logic in order to deduce new facts which are not directly observable, or facts which we did not even think to look for.
Sometimes, Mach admits, the connection between mathematical formulas and Nature is mysterious, a happy coincidence wherein someone notices that “the relations between the quantities investigated were similar to certain relations obtaining between familiar mathematical functions.” In fact, he notes, “natural phenomena whose relations are NOT similar to those of familiar functions are at present very difficult to reconstruct.”
It was Euler, Mach informs us, who emphasized the expansive applicability of derivatives to physics. Euler believed, according to Mach, that “the purposes of the phenomena of Nature could afford as good of a basis of explanation as their causes.”
This beautiful backward thinking of Euler’s is a sort of reverse engineering. If we discover which outcomes a process is producing at a “maximum,” Euler contends that this can tell us something about the purpose of the process being studied.
Another application of derivatives comes form the “minimum” side: in a Universe which appears to want to be as lazy as possible, finding values for which Work or Distance or Energy-Expense are minimal will give us a good heads-up as to which path Nature is likely to travel and why.
And there’s yet another way maximums and minimums can be helpful in physics… Often, when testing a new hypothesis, the most illuminating information can be gleamed from studying the phenomena at extremes… at or near highest speeds and lowest speeds, highest temps and lowest temps, et cetera. Therefore, knowing relative maxima and minima is a tremendous aid when studying what makes Nature tick (or unravel, or explode…