An equation doesn't always have to be a bad thing. In fact, an equation can never be a bad thing!
For a while now, I’ve been intent on explaining stuff from particle physics.
A lot of it is intuitive if you go beyond the mathematics and are ready to look at packets of energy as extremely small marbles. And then, you’ll find out some marbles have some charge, some the opposite charge, and some have no charge at all, and so forth. And then, it’s just a matter of time before you figure out how these properties work with each other (“Like charges repel, unlike charges attract”, etc).
These things are easy to explain. In fact, they’re relatively easy to demonstrate, too, and that’s why not a lot of people are out there who want to read and understand this kind of stuff. They already get it.
Where particle physics gets really messed up is in the math. Why the math, you might ask, and I wouldn't say that’s a good question. Given how particle physics is studied experimentally – by smashing together those little marbles at almost the speed of light and then furtively looking for exotic fallout from the resulting debris – math is necessary to explain a lot of what happens the way it does.
This is because the marbles, a.k.a. the particles, also differ in ways that cannot be physically perceived in many circumstances but whose consequences are physical enough. These unobservable differences are pretty neatly encapsulated by mathematics.
It’s like a magician’s sleight of hand. He’ll stick a coin into a pocket in his pants and then pull the same coin out from his mouth. If you’re sitting right there, you’re going to wonder “How did he do that?!” Until you figure it out, it’s magic to you.
Theoretical particle physics, which deals with a lot of particulate math, is like that. Weird particles are going to show up in the experiments. The experimental physicists are going to be at a loss to explain why. The theoretician, in the meantime, is going to work out how the “observable” coin that went into the pocket came out of the mouth.
The math just makes this process easy because it helps put down on paper information about something that may or may not exist. And if really doesn’t exist, then the math’s going to come up awry.
Math is good… if you get it. There’s definitely going to be a problem learning math the way it’s generally taught in schools: as a subject. We’re brought up to study math, not really to use it to solve problems. There’s not much to study once you go beyond the basic laws, some set theory, geometry, and the fundamentals of calculus. After that, math becomes a tool and a very powerful one at that.
Math becomes a globally recognised way to put down the most abstract of your thoughts, fiddle around with them, see if they make sense logically, and then “learn” them back into your mind whence they came. When you can use math like this, you’ll be ready to tackle complex equations, too, because you’ll know they’re not complex at all. They’re just somebody else’s thoughts in this alpha-numerical language that’s being reinvented continuously.
Consider, for instance, the quantum chromodynamic (QCD) factorisation theorem from theoretical particle physics:
This hulking beast of an equation implies that *deep breath*, at a given scale (µ) and a value of the Bjorken scaling variable (x), the nucleonic structure function is derived by the area of overlap between the function describing the probability of finding a parton inside a nucleon (f(x, µ)) and the summa (Σ) of all functions describing the probabilities of all partons within the nucleon *phew*.
In other words, it only describes how a fast incoming particle collides with a target particle based on how probable certain outcomes are!
The way I see it, math is the literature of metaphysics.
For instance, when we’re tackling particle physics and the many unobservables that come with it, there’s going to be a lot of creativity and imagination, and thinking, involved. There’s no way we’d have had as much as order as we do in the “zoo of particles” today without some ingenious ideas from some great physicists – or, the way I see it, great philosophers.
For instance, the American philosopher Murray Gell-Mann and the Israeli philosopher Yuval Ne’eman independently observed in the 1960s that their peers were overlooking an inherent symmetry among particles. Gell-Mann’s solution, called the Eightfold Way, demonstrated how different kinds of mesons, a type of particles, were related to each other in simple ways if you laid them around in an octagon.
A complex mechanism of interaction was done away with by Gell-Mann and Ne’eman, and substituted with one that brought to light simpler ones, all through a little bit of creativity and some geometry. The meson octet is well-known today because it brought to light a natural symmetry in the universe. Looking at the octagon, we can see it’s symmetrical across three diagonals that connect directly opposite vertices.
The study of these symmetries, and what the physics could be behind it, gave birth to the quark model as well as won Gell-Mann the 1969 Nobel Prize in physics.
What we perceive as philosophy, mathematics and science today were simply all subsumed under natural philosophy earlier. Before the advent of instruments to interact with the world with, it was easier, and much more logical, for humans to observe what was happening around them, and find patterns. This involved the uses of our senses, and this school of philosophy is called empiricism.
At the time, as it is today, the best way to tell if one process was related to another was by finding common patterns. As more natural phenomena were observed and more patterns came to light, classifications became more organised. As they grew in size and variations, too, something had to be done for philosophers to communicate their observations easily.
And so, numbers and shapes were used first – they’re the simplest level of abstraction; let’s call it “0”. Then, where they knew numbers were involved but not what their values were, variables were brought in: “1”. When many variables were involved, and some relationships between variables came to light, equations were used: “2”. When a group of equations was observed to be able to explain many different phenomena, they became classifiable into fields: “3”. When a larger field could be broken down into smaller, simpler ones, derivatives were born: “4”. When a lot of smaller fields could be grouped in such a way that they could work together, we got systems: “5”. And so on…
Today, we know that there are multitudes of systems – an ecosystem of systems! The construction of a building is a system, the working of a telescope is a system, the breaking of a chair is a system, and the constipation of bowels is a system. All of them are governed by a unifying natural philosophy, what we facilely know today as the laws of nature.
Because of the immense diversification born as a result of centuries of study along the same principles, different philosophers like to focus on different systems so that, in one lifetime, they can learn it, then work with it, and then use it to craft contributions. This trend of specialising gave birth to mathematicians, physicists, chemists, engineers, etc.*
But the logical framework we use to think about our chosen field, the set of tools we use to communicate our thoughts to others within and without the field, is one: mathematics. And as the body of all that thought-literature expands, we get different mathematic tools to work with.
Seen this way, which I do, I’m not reluctant to using equations in what I write. There is no surer way than using math to explain what really someone was thinking when they came up with something. Looking at an equation, you can tell which fields it addresses, and by extension “where the author is coming from”.
Unfortunately, the more popular perception of equations is way uglier, leading many a reader to simply shut the browser-tab if it’s thrown up an equation as part of an answer. Didn’t Hawking, after all, famously conclude that each equation in a book halved the book’s sales?
That belief has to change, and I’m going to do my bit one equation at a time… It could take a while.
*Here, an instigatory statement by philosopher Paul Feyerabend comes to mind:
"The withdrawal of philosophy into a "professional" shell of its own has had disastrous consequences. The younger generation of physicists, the Feynmans, the Schwingers, etc., may be very bright; they may be more intelligent than their predecessors, than Bohr, Einstein, Schrodinger, Boltzmann, Mach and so on. But they are uncivilized savages, they lack in philosophical depth -- and this is the fault of the very same idea of professionalism which you are now defending."