Teaching Calculus Using History (Part 1: Aristotle)
I have long harbored a fantasy of teaching intro calculus using the history of science reaching all the way back to Aristotle. Here is an outline:
In the time before Aristotle, people believed silly things like the earth rested on the back of a turtle. That is to say, they tried to explain their world, but they didn’t have a tradition of using rigorous arguments from data. In fact … I’m not even really sure what the turtle was supposed to explain. I guess they thought that the earth must rest on something, otherwise it would fall?
Anyway, a more interesting example is that they thought that bees came from lions. Yep, buzz-buzz bees. From roar-bite lions.
Remember that part of the Torah or Old Testament, wherein Samson gave a riddle to the Philistines, and the answer to the riddle was that he killed a lion and scooped out all the bees and honey? It seems like a totally unfair riddle until you learn that, back in the days when the Torah was written, it was common belief that bees came from lions.
Ok … but why did they think this? It’s hard to answer any question like that, but I think the following gives some helpful context. Flies seem to magically appear from rotting flesh, a fact that was probably pretty familiar to ancient people. If you stumbled upon a dead animal and notice the flies, you might think they came from somewhere else. But if a person died in a sealed room, as must have happened to I dunno like Pharaohs and stuff, and STILL somehow flies started emerging from the body? Well the flies definitely didn’t come from anywhere, they weren’t already in the room. (Ok, we now know that flies just have an amazing ability to find their way through just about any barrier; and also barriers are generally not as impenetrable as ancient people probably thought they were.) So the flies must have already been in the body and when the person died the flies hatched out of the body.
By a similar sort of logic, since humans had never been able to identify how eels reproduce, ancient people believed that eels grew right out of mud. They just couldn’t identify any other origin. In general there was a fairly wide-spread belief in spontaneous generation. And there was probably some amount of logic to it. I imagine that people might have reasoned, we don’t know exactly where bees come from, so probably they emerge from another creature the way that flies do. Which creature? Well it would have to be an aggressive and noble animal, because bees are courageous fighters. I guess it must be a lion!I bring all of that up, in order to introduce you to Aristotle. He is perhaps the most famous of the ancient Greek philosophers. He is especially known for bringing rigorous reason and an emphasis on a commitment to evidence when understanding the world.
So what, did he overturn the bee-lion theory?
Nope! He bought it. Hook-line-and-sinker. He believed a bunch of other spontaneous generation type theories too. He had lots of silly beliefs besides. He thought that if a woman looked into a mirror while she was menstruating, the mirror would fill with a red stain that cannot be easily removed. He had a whole theory about how it’s easier to clean this red stain from a dirty mirror than it is from a clean mirror. … So like I said, he had some bonkers ideas.
And yet! And yet I still persist in telling you that he did in fact increase the world’s respect and attention for scientific observation. He was imperfect, but like all other humans, we live in an imperfect world where the people around us can convince even very smart people of very dumb ideas. (Sorry, I did not mean to suddenly get political.)
For one thing, he got lots of stuff right. He created a very important first taxonomy of living beings. This was among the most important inventions in the history of biology, until the theory of evolution. But another very important thing that he did, which finally starts getting us on the path to talking about calculus, is: He attempted to formalize physics.Aristotle had several ideas about physics, many of them interesting, but not many of them would ultimately turn out correct. Yes, you may be noticing that Aristotle did not have a great batting average. He got a lot of things very wrong. Let me again remind you, though, that his batting average should be compared to the people around him at the time. In that measurement, he represents an enormous spike in the graph of the history of science.
Anyway, back to the point. Aristotle was the first person to attempt to precisely describe the motions of objects. In doing so he did not write down any mathematical formulae. But there were a few basic principles which could be understood to be mathematical.All objects tend to rest, unless motivated by something like a life force.
Objects in free-fall move at a constant rate. The force with which they impact the ground is determined by the height at which they fall.
The motion of celestial bodies (what we now call planets and stars) obey physical laws that are not the laws which govern the earth.
Every one of these would prove to be false. And yet this was still a monumental breakthrough in the history of science. Because for the very first time in history, someone had put down in writing a formal, testable theory of physics. Amazing as it may be that nobody had thought to actually study physics carefully, the act of identifying it as an object of study was a crucial first step that nobody in recorded history had made up to that moment. And without it we would never be on a path to getting a correct theory of physics.
To be continued.
P.S. Yes I know this is long and rambly. A whole post and no calculus even on the horizon. Still stuck 2000+ years in the past.
But it has colorful stories, which is always nice. It has narrative arc. It has characters. It has connections to other things that we know about.
More importantly it makes it intelligible how we got from being dumb old ancient Greeks to big smart modern people. You don’t just state the definition of the derivative like you just thought of it for no reason. You get to see the challenges that people encountered, the thoughts and the observations which informed their thinking, and how these were used to solve the challenge.
In effect, you get to answer students when they say “what does this mean” and “what is this used for?” and “Why did people invent this? Just to torture students?” Well, maybe they make you learn it now to torture students, but that wasn’t how it got started.