The most influential discovery in science didn’t happen in a lab.
James Maxwell was intrigued by other people’s work in magnetism and electricity, and decided to put his skill at calculus to work by summarizing what they had found.
The result was four equations that captured the close ties between both forces. It made clear that both were opposite sides of the same coin, better thought of as a single force that could be expressed in two ways. This opened up the idea of a “theory of everything,” that could describe the laws of the entire universe in a few lines of math. For this alone, Maxwell is noteworthy.
As he looked over his work, however, he spotted something. The equations predicted that a changing magnetic field would create an electric field that would create a magnetic field, and so on. The result was a blip of energy that expanded outward.
In other words, Maxwell discovered radio waves, using little more than a chalk board.
That changed civilization. Before, when we wanted to communicate electrically, we had to string up thin, delicate wires. After, all you needed was two antennas and an agreement on how to use them. As a result Rob Hall, alone and dying in a storm on Mount Everest, could have one last conversation with his wife back at home in New Zealand. This wireless bridge can span anywhere from two metres, via a cheap pair of wireless headphones, to 16,957,965,862,947 metres, the distance between the Voyager 1 space probe and Earth.
Science has exploited this to the fullest. If we send a probe into space we don’t care if we get it back, it will tell us what it’s seeing right until it smacks into a planet. In some cases we don’t even need to send probes; the rocky surface of Venus was mapped by bouncing radio waves off it from our home planet, a few million kilometres distant. The Sun, Jupiter, and lightning all spray out radio waves that tell us something about their underlying physics.
And yet those pale in comparison to Maxwell’s final discovery. He noticed that his equations set a limit on how fast these waves could travel. By plunking in a few constants and doing some simple math, he was able to calculate this speed.
It matched the speed of light.
The full impact of that match has been lost over the years. Maxwell’s discovery came in 1865, however; back then, most scientists thought electricity came in particles, light was a wave that rippled through some sort of aether, and magnetism was a field like gravity. No one thought light and electricity were linked, and yet a math geek armed with a blackboard had shown they must be. You know you’ve done good when Albert Einstein is moved to say:
The precise formulation of the time-space laws was the work of Maxwell. Imagine his feelings when the differential equations he had formulated proved to him that electromagnetic fields spread in the form of polarised waves, and at the speed of light! To few men in the world has such an experience been vouchsafed . . it took physicists some decades to grasp the full significance of Maxwell’s discovery, so bold was the leap that his genius forced upon the conceptions of his fellow-workers.
(Science, May 24, 1940)
Maxwell’s simple bit of math spawned a multitude of experiments that confirmed its hunch, which in turn led to the most successful scientific theories we’ve found yet: Quantum Mechanics, and General Relativity.
It’s a staggering legacy for four equations. And it’s not an isolated incident, either. Riemann manifolds were regarded as a weird, useless oddity of math when they were invented; 70 years later, General Relativity relied on them to describe how the universe was shaped. Complex numbers were treated with disdain, until they became essential for electromagnetism and Quantum Mechanics.
But why should these abstract bits of logic and math do such a good job of describing the universe? Doesn’t this point to an underlying order to the universe, a harmony that exists separately from the material world, which could only be provided by God?
The Connection to Reality
One problem with this proof is that it puts the cart before the horse.
I’m assuming you, the reader, are of the species Homo Sapiens Sapiens. Even if I’m wrong, it’s quite likely you take up a finite amount of space and time, and we share the same laws of the universe. You are only a small part of the greater whole, and don’t have complete knowledge of the remainder.
As a result, you interact with things outside of your immediate understanding on a regular basis. You cope with this through abstraction. The collection of wood, metal, and petrochemicals that I’m currently sitting on, for instance, is known as a “chair.”
This “chair” is a structure built to relieve the strain my lower half puts up with as it tries to keep my upper half from hitting the pavement. This abstraction is a big help; without it, every time I wanted to relieve said strain I would have to examine the surrounding area for a flat spot next to a vertical panel at a convenient height, and test its structural integrity and comfort level. With it, I scan for an object that looks like a “chair,” then sit on it. The time and energy savings are enormous!
Abstraction works because the laws of the universe allow it to. If physics somehow forced my lower half to forever carry all the weight of my upper half, no matter what position I put myself in, I’d never create the concept of “chair.”
So it is with numbers. Octopuses, whales, dolphins, parrots, elephants, dogs, and apes like myself can all do basic math. Why? Since all of these are social animals, it seems likely that numbers are handy in social situations. Perhaps we used them to keep track of food or gifts, so we can ensure our generosity is returned or that no-one is being a pig. Whatever the reason, the concept of counting is based on the physical reality that matter is a limited resource, and tends to stay in one place. If food was constantly available to all, or three apples turned into 20 apples before dissolving into mush, it’s doubtful any species would develop the abstraction called “numbers.”
A few things result from this abstraction. If numbers are distinct and unchanging, we could imagine ways to combine them, for instance “adding” and “multiplying.” Likewise, if matter tends to lump together and remain relatively constant, those extensions we developed for math will also work in the universe. Calculus is an extension that deals with the way numbers can change, based on a few assumptions about those numbers. If those assumptions are similar to the laws of the universe, then the discoveries and predictions of calculus will match reality very closely.
Confirming The Obvious
So there’s a good reason math seems to have an uncanny knack for describing our universe. It was built on the basic laws of said universe! Maxwell’s four equations were an excellent abstraction of the laws of electricity and magnetism, so good that they revealed some surprising connections no one had noticed before.
Sometimes, however, we make assumptions that don’t match the underlying laws. Ole Rømer spent a decade looking at Jupiter’s moon Io, and in 1678 noticed that the predictions of Newton’s “laws” of gravity wobbled from what he was seeing. Years of painstaking study showed that the moon seemed to slow down on one side of its orbit and speed up on the other, and yet it appeared to move the same speed when it moved in front of Jupiter as when it was moving behind. After a lot of head-scratching, he found an explanation. One of the assumptions behind his math was that light moved from one place to another instantly. This was reasonable, since it matched what scientists observed and what the math permitted.
If he instead assumed that light traveled at a fixed rate, his timing problems disappeared. He could estimate this speed from his numbers and the equations, and fortunately it was very, very, very fast. If it was not, that would conflict with our previous guess that it was infinitely fast, and a lot more assumptions would have to be tossed out.
This brings up another good point. We’re small beings in a big universe, so when we find out we’ve misunderstood some part of the greater whole, we’ve got no reason to be surprised. On the contrary, when we really nail down a part of it, we break out the champagne because we realize the odds of getting it right are small. Maxwell’s accomplishment was noteworthy because it goes against our expectations, while Rømer’s observation was just more proof that we don’t understand the universe, so we don’t celebrate it in the same way. A flip through any book on cosmology will show that while we’ve learned a staggering amount in the 330 years since Rømer, there’s still a lot more to know.
 His simple last words still move me: “I love you. Sleep well, my sweetheart. Please don’t worry too much.”
 As of May 2nd, 2010. Since Voyager is zipping away from us at 17km/s, it’s even further than that by now.
 While pigs are considered more intelligent than dogs, I can’t find any evidence that they understand numbers.