In my last post, Incantations and Language Games in Harry Potter, I compared our use of language to the casting of magic spells, and used the example of learning geometric shapes to illustrate what I meant. I want to expand that idea to see what it means for the mathematical equation. This is part two in a three part series on metaphysics, mathematics, and magic. Check back in next Monday for the final installment.
An equation is little more than a language game, and yet it has the power to show truth, solve puzzles, build bridges, and so much more. Equations are mere symbols and signs that have no meaning unless we understand them, and yet we consider them to be real in their own right. Mathematical truths can be invented, in some ways, and yet still maintain an almost untouchable ontological status. They are squiggles on a page, and yet have the power to describe and predict happenings in our universe with an accuracy otherwise unimaginable. Math allows us to find information we cannot observe, and then act on that knowledge – to make engines go and pendulums swing.
In this way, math is more than a little like magic, and equations are a bit like magic spells. With an equation, you can take something seemingly unknowable, and with a bit of effort, reveal the truth. With the right equation, you can get the answer with very little effort at all – the right equation can make you feel like a wizard. I explained in my post last week that words (and magic spells) serve to anchor metaphysical abstraction from experience and give it conceptual substance, free of any ties to particular phenomena, making the lessons learned reusable in a variety of circumstances.
Math is the language of that abstraction, and as such it can help us understand that move to the metaphysical. Beyond that, once we’ve made that move, we can still abstract further – and this gives us more than description: it gives us shortcuts that let us skip all the hard steps of that initial abstraction, and lets those who are adept in the art of it transcend mortal dependency on practical observation.
As we learn from Arthur C. Clarke, any sufficiently advanced technology will look like magic to those who don’t understand it. I’m still relatively convinced that speakers are magic, because I just can’t get myself divested of my dependence on the material particularities of sound.
It’s easiest for me to see this through examples.
In math class, when we first learn to find a derivative, it’s a long drawn out process with a huge equation. Later on, we learn a shortcut for it, and it feels like magic, and you might have wondered (I know I did) why your teacher bothered with that harrowing Newtonian mess at all in the first place. At this point in our mathematical education we have already established a level of abstraction that allows us to understand problems in terms of graphs, lines, slopes, etc. What we’re doing now is abstracting even further to find something even harder to see in image form – we’re trying to find the slope of a curve.
If we skip right to the shortcut, we’ll give ourself the language to solve problems at a higher level, but we won’t understand how it connects to what we knew before. It would be like we hyperspace-jumped to the answer rather than plotted a course for ourselves. We get the answer to the present problem, but only in isolation.
Newton’s difference quotient allows us to turn the derivative problem into a line we can observe on a graph at our current level of abstraction. But this line is only an approximation – we need to use limit theory to take the line one abstraction further, giving us the concept of a slope of a curve. If we skip this step and go right to the shortcut, we’d only give ourselves the solution to a meaningless puzzle. The difference quotient allows us to derive not only this shortcut, but also other shortcuts we’ll need for other problems that we have not yet encountered. Understanding the underlying principles shows us how to abstract to new principles, freeing the derivative from the graph and enabling us to power lightbulbs, build bridges, and launch rockets.
And there are many other examples of this. Lagrange’s and Hamilton’s reformulations of Newtonian mechanics (and more) made them simpler and thus more powerful, leading to the general concept of energy and quantum mechanics, respectively. Imaginary numbers allowed electrical engineers to turn complex differential equations into mere arithmetic. The Fourier transform allowed people to move from the time domain to the frequency domain in spite of our empirically linear experience of the world.
The point is that equations are powerful. They’re not just puzzles with set answers, but magic spells that can elevate the mind to higher levels of understanding, making the metaphysical realm something workable and moldable such that we can do the impossible – we can see the unseeable and use it for our own purposes.
The discovery of Neptune was not the result of empirical observation, but of exploring logical necessity and mathematical equations. Following Alexis Bouvard’s prediction that there had to be another planet beyond Uranus in order to account for its orbit according to Newton’s theory of gravity, John Couch Adams and Urbain Jean Joseph Leverrier calculated its location and mass. For the math to work, Neptune had to be roughly where they said it was. And as Johann Gottfried Galle observed in 1846, it was. Adams and Leverrier couldn’t see Neptune, but were able to discover it through the manipulation of signs and symbols. The equations they used allowed them to “see” the planet in the abstract even though it had not yet been observed. The math told them where to look, and they were right – as if they had magically conjured the planet where they needed it to be.
People often think that metaphysical reflection focuses solely on unprovable spiritual revelation based in feeling and vague philosophical questions with no answers, but such a rendering is reductive of both deductive reason and speculative religion. As Galileo put it, “Mathematics is the language with which God has written the universe.” The deeper we delve into this language, the higher we can ascend metaphysically, untethering ourselves from the limitations of our physical finitude, knowing more, seeing more, and doing more than we ever thought possible.