Physics Rediscovered #8: MAGA
Make Algebra Great Again
It is human nature to take things for granted. Different things for different people but the pattern is there and probably always will.
Here’s one:
It’s so obvious that you probably have no idea what I’m talking about. Well, you are in luck as I’m going to tell you in a long-winded way.
Time for some mathsplaining (with some terrible philosophy takes).
Everyone’s dreams are boring, except mine
It begins with a man with a dream.
Well, three dreams, more like. His dreams are of terrifying phantoms, earsplitting thunders and, well… books. A bit anti-climactic, I know.
However, these dreams is what jump-started our civilization into the age of reason.
For fucks sake, why is it that I cannot dream of anything revolutionary. All I have is the one where the fish I ate is my Uber driver, staring at me with its dead eyes and gulping away, while I try to get out of the car, only to find that I’m paralyzed. Not much to work with, is it?
Ok, enough about me - back to the man. He dreams his stupid dreams, wakes up and decides that everything sucks.
Yes, everything. Even the fact that everything sucks sucks.
With that inspiring thought to lead him, here’s what the man concluded:
It is already some time ago that I saw that, from my first years, I had accepted many false opinions for true ones, and that what I had since then founded on such ill-assured principles could only be strongly doubtful and uncertain; such that I would have to undertake seriously, once in my life, to undo all the opinions that I had taken into my belief since then, and to start anew from the foundations, if I wanted to establish something firm and constant in the sciences.
Now, this type of thinking might’ve led him to become an anti-establishment flat-earther. Fortunately, he became the most influential philosopher in history. We really dodged a bullet there. But where are this man’s manners. He still hasn’t introduced himself.
René, René Descartes. Enchanté, at your service and whatever people used to say back then.
If not for anything else, you know him for the clichéd
“Cogito, ergo sum”,
or
“I think, therefore I am”.
Wow, I’m getting nauseous just writing that. Ok, let me make this less regurgitative.
Why would Descartes say something like that and condemn us to suffering insufferable home-made philosophers trying to impress the rest of us with vacuous quotations?
Well, he didn’t feel very secure about his existence. No, he wasn’t worried that he’s not getting enough protein. He was worried that he might be a brain in a vat, in a simulation in the matrix, run by demons. Pretty reasonable if you ask me.
This is where professional doubting leads you. So then how do you get yourself out of this hole? Descartes figured that doubting existence itself, instead of taking it all at face value is proof that there exists something to be doubted.
Agree or not, this “even the sucking sucks“ is actually what makes it all better.
Death by geometry
Of all the things that suck, geometry sucks especially hard, Descartes thought.
You see, since the Greeks, geometry is a bit non-numerical, hand-wavy and pretty verbose. Here is a square and within that square is a circle with a radius equal to half of the side of the square, placed in the intersection of the diagonals and so and on bla bla bla. You get the point.
That’s not to say that this prevented brilliant work from happening. One example is the famous Gerolamo Cardano and his algorithm for the general solution of cubic equations.
While Cardano nailed it, his solutions where written out in words. In goddam words! Wanna have a taste of how mind-numbing that is? Try and have a go at Copernicus’ work here. Call an ambulance beforehand.
Descartes thought he can do better. He wanted the whole thing to be more systematic, universal and for the love of all that’s good, less tedious. He wanted to make it more XXI century, which is especially impressive since he lived in the XVII.
Silly modern reference? Not at all, as we use what he delivered to this day.
What he did is he came up and showed that: once you have two lines on a plane, you can represent everything else on that plane in relation to those two lines:
Thus, if we wish to solve some problem, we should first of all consider it solved, and give names to all the lines—the unknown ones as well as the others—which seem necessary in order to construct it. Then, without considering any difference between the known and the unknown lines, we should go through the problem in the order which most naturally shows the mutual dependency between these lines, until we have found a means of expressing a single quantity in two ways. This will be called an equation, for the terms of one of the two ways [of expressing the quantity] are equal to those of the other.
This is how analytical geometry was born.
He took naked geometry, dressed it in tight latex of algebra and never even thought about a safe word.
But that’s not all. He needed a place to do all that geometrical analysis in. Something with a reference point and precisely defined dimensions. What he came up with we now call the Cartesian coordinate system.
It’s a grid, numbers everywhere, variables are at the end of the alphabet, constants and parameters at the beginning.
Now, all of a sudden, where a straight line used to be a drawing with a bunch of words, it became:
Not only that. It was also easily relatable to other lines. To get a line parallel to that one, you no longer needed to apply moving rulers. You simply change the value of b. Brilliant.
More complex curves were also made simple. A parabola became:
The cosine of an angle is:
…and so on. I’m sure you’re asking what the big deal is. Isn’t this stuff obvious.
The only reason you are doing so is that our civilization has used this framework ever since. We are schooled in algebra since we are kids and it’s become second nature. Most of us have lost the geometrical mode of thinking that was prevalent in history. After all, who can now construct a hyperbola with sliding rulers. Not me.
However, Descartes was the one who lived in both worlds. He came up with various curves first and, through absolutely exhausting geometrical proofs, translated them into algebraic equations. So that we don’t have to.
Geometry to the people
In modern business language Descartes realized that geometry will not scale in its current form. Of all the people who were right in history of mankind, he was one of the most right.
What he did is what we would today label as the democratization of geometry through algebra. As Paul Valery put it:
“the most brilliant victory ever achieved by a man whose genius was applied to reducing the need for genius”
What Descartes didn’t know is that his work was the beginning of something even greater, something that would reshape our civilization. More on that in the next episode.
See you there!