At first blush, people think scientists experiment and measure and observe and analyze and then experiment some more....working with eyes, ears, and hands. Then they summarize it all with one of those compact scientific laws like f=ma or E=mc².
It doesn't work that way. Instead, from a very few measurements or observations, scientists work with mathematics, deriving equation after equation, confident that such equations will find application in physical reality. And it turns out to be that way. Thus Galileo obtained his laws of motion, Kepler his laws of planetary movement, and Newton his laws of gravitation. The close correlation of math with the physical world greatly impressedthese scientists, as we should be impressed by it today, but largely aren't. Does it not show the design in nature? Galileo summed it up with "God wrote the universe in the language of mathematics."
It's convenient for mathematics to work that way since there is only so much experimenting one can do with objects millions of miles away. Not to mention things very tiny....things we can't even see, let along poke our fat fingers into. One hundred years after Galileo, Kepler & Newton figured out the big things, others focused on the small. Magnetism and the flow of electrons [electricity], for example. At first the two were thought to be unrelated phenomena, buy later they were linked. In 1785, Charles Coulomb published the law of force between two electrically charged bodies, q1 and q2:
F =- k(q1·q2/r²) where k is a constant and r is the distance between the two bodies.
What even the dumbest person in class must note is the law's identical form to that of gravity, a wholly different phenomena. The gravitational attraction between two masses (m1 and m2) is
F = k(m1·m2/r²)
The only difference is that electrical force can attract or repulse, depending on whether the two bodies have equal or opposite charges; gravity always attracts.
Along came Michael Faraday, who discovered electromagnetic induction. He found that if you rotate a rectangular frame of wire through a magnetic field, you generate electricity in that wire, it's intensity rising and falling as the frame rotates, mathematically described by the sine function.
Furthermore, when Faraday passed current through a coil of wire, a magnetic field was produced which would induce current in a separate coil of wire. But how far apart could the coils be? What if he tried a longer coil or a tighter coil? How would that change things?
Since you can't see any of this, it was left to a mathematical physicist, Robert Maxwell, to figure the results. He worked out through math that current flowing through a coil of wire produced an electrical field in the surrounding space. And that electrical field gave rise to a magnetic field. Which gave rise to an electrical field, which gave rise to a magnetic field, and so on. When "pushed" by current flowing though the originating wire, these electromagnetic waves traveled great distances, and did so, he calculated, at 186,000 mi/sec.
But wait! Light had already been measured as traveling at just that speed. You don't suppose....Yes! Light was part of this electromagnetic spectrum, only at a much different frequency. And how is it generated? By passing electricity through tightly wound tungsten filaments, same as Faraday generated lower frequency waves from coils of different materials and dimensions.
In time, other "slots" of the spectrum were filled in: radio waves, infrared rays, x-rays, gamma rays. We now make endless and routine use of the spectrum, yet nobody knows just "what it is" that travels through space. It is described mathematically. As Alfred North Whitehead put it: "The originality of mathematics consists in the fact that in the mathematical sciences connections between things are exhibited which, apart from the agency of human reason, are extremely unobvious."
Physicist Heinrich Hertz observed: "One cannot escape the feeling that these equations have an existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them." [italics mine]
Yes, we do get a lot out of them. So much so that we've become completely oblivious to what was "put into them" in the first place and who did the "putting." "One cannot escape the feeling," Hertz stated. Yet today's materialistic society has managed to do just that.
For since the creation of the world God's invisible qualities—his eternal power and divine nature—have been clearly seen, being understood from what has been made, so that men are without excuse [for ignoring him]. Rom 1:20 NIV
Many of the particulars here are found in the book Mathematics and the Search for Knowledge, by Morris Kline.