tl;dr: For 2000 years humanity’s best mathematicians stymied by their inability to conceive of the concept of *multiplying four things together*, some guy you’ve never heard of named Descartes solves the problem in half a page.
A bunch of odd conversations about mathematics after Secular Solstice– mostly picking apart the strange and autodidactic opinions of @winged-light-blog, which I still don’t understand, (this happens when you do the right thing and don’t trust your maths teachers)– reminded me of the little I know about the Ancient Greeks’ approach to mathematics. Their worldview was interesting enough that I’d thought I’d share them with the world, at least so we can appreciate how much about maths we moderns take for granted and don’t realise involve nontrivial insights.
To Euclid and Archimedes and the rest of them, a number was a length, and, usually, any operations you were allowed to perform on numbers you had to be done with the old fashioned tools of a compass and straightedge, which led to problems when they tried to do things like trisect angles or square circles.
But where this way of thinking really got them stuck was how they thought about multiplication.
You could multiply a length by a length, but you’d have to satisfy yourself with the answer being the area of a rectangle – an entirely different object. You’d prove that two areas are the same by some kind of dissection argument or something.
You could multiply a length by an area, and get a volume. But if you tried to multiply a volume by something the response would be TYPE_ERROR, because geometry is about the real world and there are only 3 dimensions in the real world.
And if to you tried to add a length to an area, or any funny business like that, the response would also be TYPE_ERROR, which is, I admit, a reasonable thing to do.
(You could also, I think, multiply things by positive integers simply by adding them to themselves enough times, so that Euclid’s proof of the infinitude of primes still worked.)
And for about 2000 years or so this is the way that western mathematicians thought, and this really screwed them over. They couldn’t talk about trying to solve equations with powers higher than the third – (anything fancy with exponentials or power series was, of course, right out). It was much easier to think of curves as loci of fancy constructions or conic sections than anything you could remotely compute anything with. And don’t forget that this was before modern mathematical notation, so you’d have to write all your operations out laboriously in Ancient Greek. It’s a wonder they got anything done at all.
This whole complicated and annoying mess was finally resolved by Descartes. Yes, he of the fancy coordinate axes and “I think therefore I am”. On the way to basically inventing algebraic geometry and rendering almost all of ancient Greek geometry obsolete he deals with this problem in a few lines and a diagram.
First he declares by fiat that some random line segment is one (pretty much the same way that modern physicists declare that c, G, ħ and any other random constants they don’t like are 1 and that they can just rescale the units)
Then he does some geometry with a pretty pair of similar triangles.
[Attempted translation: For example, letting AB be unity and wanting to multiply BD by BC, we need only join the points A & C and then draw DE parallel to AC. BE is then the product of the multiplication.]
And somehow you’ve multiplied a line by a line and gotten a line.
And from then on he just declares that he is allowed to do this as many times as he likes, and if he wants to take the 691st power of some line segment, so be it. And from there he was comfortable saying “This curve is the set of points satisfying x^23+ 59x^31y^3+y+47=0″ which would have made the ancient Greeks’ heads explode.
Descartes wasn’t terribly modest about his new invention of coordinate geometry, saying that “it compares to the Ancient Greek geometers like the rhetoric of Cicero compares to the ABCs of children”. And he probably should have considered the consequences of unleashing a new branch of maths onto the brains of unsuspecting university students.
(What algebraic geometry looks like today. This is scary and I don’t understand it and I don’t want to think about it)
But however much of a douchebag he might have been about it, I do appreciate the ability to take 4th powers.