Ancient Greek scientists studied the dynamics of falling bodies. Were “Galileo’s” discoveries anticipated in these treatises that have since been lost? This question leads to a bigger one regarding relativism versus universalism in the history of thought.
Opinionated History of Mathematics
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Quiz! Who said the following:
“The study of mechanics is eagerly pursued by all those interested in mathematics.”
What is the source of that quote? And here is another one:
“We attack mathematically everything in nature.”
Who said that? Surely it is after Galileo anyway, right? Since he is the one who invented the mathematisation of nature?
No. The quotes are from Pappus and Iamblichus. Greek authors, ancient authors. Well over a thousand years before Galileo.
Much modern scholarship would have you believe that such attitudes never existed. This is what you get if you read too much Plato and Aristotle and not not enough mathematicians. In Plato, mathematics is purer than snow. To apply it to the physical world is to defile it. So that’s quite an obstacle to science. If you glorify pure and abstract thought as the only worthwhile pursuit of rational beings, and deride empiricism as fit only for unphilosophical beasts, then you’re not going to get a whole lot of science done.
Aristotle too has many teachings antithetical to proper science. There’s his big book on “Physics”, so called, which is really metaphysical philosophy. Aristotle doesn’t care about the phenomena or laws of motion. Instead he cares about pseudo-profound philosophy puzzles like: Does any motion have a cause? Or: Does motion presuppose the existence of a cause of that motion? And by the way, there are four different kinds of causes, you see: the efficient cause, the material cause, and blah blah blah, whatever the other ones are, who cares. And you can sit around and split hairs between them all day long. Typical philosophy boilerplate stuff. Obviously a completely wrong turn on the road to science.
So that’s Plato and Aristotle in a nutshell. If that’s all you know about Greek scientific thought then Galileo is a breath of fresh air, sure enough. But Plato and Aristotle do not speak for “the Greeks”. You don’t get a very good idea of 20th-century science by reading Sartre and Heidegger. Likewise, if you want to understand Greek scientific thought, forget about Plato and Aristotle. On mathematical subjects they are derivative commentators at best.
There is zero evidence that those Platonic and Aristotelian ideologies that I outlined had any influence whatsoever on any mathematically competent person in the classical and Hellenistic eras of antiquity. People often refuse to accept this. Leading scholars. Their entire worldview is centered around the assumption that Plato and Aristotle are the alpha and the omega of Greek thought. Any suggestion to the contrary they will fight with tooth and nail as if their life depended on it. Which it actually kind of does, as I explained before.
But to take Aristotle’s ultra-philosophical physics as the state of the art of Greek science is ridiculous. It’s like saying that Wittgenstein was a leading 20th-century quantum physicist. Read the mathematicians, if you want to know what was really going on.
We know for a fact that even Aristotle’s own successors as heads of his own Lyceum departed right away from his teachings on a number of scientific questions. And that’s philosophers from his own school. Just imagine how little the mathematicians cared about Aristotle’s ideas. Only much later, in the Middle Ages, which was a time of vastly lower intellectual quality, did people come up with the imbecile notion of taking Aristotle to be an authority on physics.
But ok, enough ranting about that. Now let’s look at some actual Greek science instead. Let’s think about some of Galileo’s main results and investigate whether there was anything like that in Greek times.
Falling bodies, for example. The laws of motion of falling bodies, that’s a big Galilean triumph, right? Well, not so fast.
Maybe the Greeks already knew those things. We don’t know because a huge part of the Greek scientific corpus is lost. Burnt down in various library fires and what not. Or just disintegrated and discarded. It was a fragile thing. These works had to be hand-copied over and over to survive. Just think about books from the 19th century and what state they are in today. They’re already falling apart. Ancient Greek works had to beat long odds to make it.
Take Euclid’s Elements, for example. The oldest manuscript we have is closer to us in time than to Euclid. From ancient times we have only the tiniest scraps. A quarter of a page here and three lines there. Basically nothing.
It was a long time ago. How many of the books we print today are going to be around in more than 2000 years? Not many, I bet.
But we know something about what was lost through references in other works. And as far as the science of falling bodies is concerned we have some very intriguing indications.
We know for a fact that Strato wrote a treatise on falling bodies. It did not survive. But here’s what we know. Strato was an avid experimenter. To prove that falling objects speed up, he said: Pour water slowly from a vessel. At first it flows in a continuous stream, but then further down its fall it breaks up into drops and trickles. This is because the water is speeding up. So the water spreads out, like cars let loose on a highway after a congested area.
And here’s another experiment Strato used. Stones dropped into a sand bed from various heights. The stone makes craters of different depths depending on height fallen. Again showing that the thing is picking up speed as it goes. Did Strato make actual numerical measurements of this? Of the sand craters and other ways of quantifying fall? Maybe. Who knows?
And here’s another guy who wrote on falling bodies: Hipparchus. That’s an author who commands respect. Probably the greatest mathematical astronomer of antiquity, though his works are lost. Certainly a way better mathematician than Galileo, that’s for sure.
So we know that he wrote a treatise on falling bodies, which is lost. Of course he didn’t follow Aristotle’s ridiculous theory of falling bodies. In fact, a commentator on Aristotle explicitly says: “Hipparchus contradicts Aristotle regarding weight.”
But what exactly did he say? The indications are that he argued that weight depend on distance from center of the earth. He seems to have been engaged in questions like: if there was a tunnel through the center of the earth and an object fell down it, what would happen as it approached the middle? Superficial commentators have picked up on these striking aspects of his work that can be presented as a kind of gimmick in isolation. In the original they would surely have been incorporated in a mathematical treatment.
These things are very much in line with 17th-century physics. Dropping stones into sand, thinking about how gravity varies on a super-terrestrial scale and inside the earth. That is literally exactly what scientists spent a lot of serious effort on in the 17th century.
So here’s what we know. Greek scientists, who were excellent mathematicians and keen experimenters, wrote several lost treatises on falling bodies. What are the chances that this included good chunks of so-called Galilean science? Maybe including for example “Galileo’s” law of fall? They are appreciable. It is squarely within the realm of possibility and then some.
Here’s what a modern scholar says about Galileo’s law of fall: “The ease of stumbling upon this discovery renders it highly improbable that natural philosophers had ever searched for the law of fall [before the 17th century].” Yes, either that, or: they did search and they did find it. If it’s easy to stumble upon, what are the chances that these very sophisticated first-rate minds, who wrote entire treatises on this exact subject, somehow missed it? It’s food for thought.
The law I’m talking about here is the one that say that the acceleration of a freely falling object is constant. Equivalently, its velocity is proportional to time, and the distance fallen is proportional to time squared. Motion on an inclined plane is closely related to this. A ball rolling down a slope is basically a slow-motion version of falling. The ball will acquire the same speed as it would have in free fall through the same vertical distance. As we would say today, in anachronistic terms, since all balls covering the same vertical distance trade in the same amount of potential energy, they get the same amount of kinetic energy out of it.
This slow-motion of falling is easier to deal with experimentally. Here’s Stillman Drake the Galileo scholar again; he says: “In a way it is surprising that the law for the spontaneous descent of heavy bodies had not been recognized long before the 17th century. Measurements sufficient to put the law within someone’s grasp are quite simple. Equipment for making them had not been lacking---a gently sloping plane, a heavy ball, and the sense of rhythm with which everyone is born.”
The sense of rhythm is like a natural clock. You can tell when a musician is off beat. We can use this innate skill to verify the law of fall. No stopwatch required.
Just get your inclined plane, and put markers at interval that the ball ought to cover in equal times according to your hypothesised law, and then put little bells there at those points, and roll the ball down the slope. Do the bells ring at equal intervals or not? That’s quite easy to tell.
I did this experiment for your benefit. I’m going to play it for you. I used a very simple and primitive setup, using just some stuff I had around the house. For my inclined plane or ramp I found a suitable piece of wood in my basement; it used to be part some furniture or other. So I used a kitchen knife and cut some markings into it at places a rolling ball should cover in equal times. That just means putting a measuring tape down and marking at all the squares. I made marks at 4, 9, 16, and 25 units from the starting point. You can use whatever unit you like. There is no need to know anything about a gravitational constant or anything like that. Just put your ruler down and make these marks at the square numbers and that’s it.
So then I put the ramp in position on my living room coffee table. Then I got four wine glasses that I placed along the ramp at the markings I had made. I just propped them up with some books as needed. Then I had a glass marble lying around that happed to be suitable to the purpose. I positioned everything so that the marble would just about touch the wine glasses as it rolled down the ramp.
So, here we go, I’m going to play it for you. The idea is that you will hear four equally spaced clinking sounds.
It worked pretty well, I’d say. The thump at the end is the marble crashing into a bundled-up t-shirt that I put at the end of the ramp. I also made some recordings with the glasses in the wrong positions. I put them about equally spaced instead of spaced like square numbers. So I will play that recording as well and see if you can tell the difference.
I think the difference is pretty clear. Considering that I threw this experiment together very sloppily in about ten minutes I’d say it is viable and easy to build a quantitative theory of falling bodies this way. Obviously it would be easy to improve the accuracy of the experiment a lot from my extremely primitive setup. The way I did it the marble whacked into the wine glasses pretty good. That probably slowed it down by perhaps a non-trivial amount. If I was doing this for real I would of course use a heavier ball, a longer ramp, and, for the markers, tiny bells or something that isn’t heavy enough to impact the ball.
But I think even my setup kind of worked. I made a number of takes with each configuration in fact. I’ll play a couple of them and then as a blind test we will see if you can hear which are the correct configuration with the equal intervals and which are the bad ones with unequal intervals. Here we go.
We don’t know if the Greeks did this something like this. There is no evidence that they did. They certainly could have. But let’s put that question aside.
There’s a more general point to be made here. This stuff about the inclined plane, it’s very simple. The law is simple; the mathematics is simple; the experiment is simple; the idea is simple. You can explain everything to a child. If Galileo was the first to discover stuff like this, it must be considered a revolution or breakthrough of a conceptual nature. Certainly not a mathematical one, because mathematically it’s trivial. This goes for all or almost all of Galileo’s so-called achievements.
The underlying question is: Are there conceptual revolutions like that? Is the history of science a story of conceptual revolutions that made previously unimaginable things suddenly obvious? Some people are very willing to accept this. Many scholars are. It’s quite fashionable even, I would say, to insist on this.
Historians make a big mistake here in my opinion. They start from a reasonable premiss. They observe that those who have little or no expertise in history often interpret historical episodes in a naive way. These historical beginners take current ways of thinking for granted, and small-mindedly interpret past thought in terms of those narrow categories. That’s a massive mistake of course. And it is very stupid because it defeats the whole purpose of studying history in the first place, namely that history shows us other perspectives, other ways of thinking.
So that’s all fine and well. Modern historians are very much correct in condemning this kind of thing. However, they then go too far. They think to themselves: since it is bad to involve too much current ways of thinking when looking at the past, the best historian must be the one who does the exact opposite of that. That is to say, the best historian must be the historian who always emphasises conceptual differences whenever possible, and who always blows the tiniest indications of differences way out of proportion as if it was a very big deal. Since this is the opposite of the naive approach, it must be the pinnacle of sophistication, right? That’s what many historians seem to think.
This is of course a bad idea, since it is just as naive and dogmatic as the erroneous mentality it is supposed to counter. But it’s not my purpose to criticise it now, only to consider what it means in the case of Galileo.
I said I could explain the rolling ball and the law of fall to a child. Any child; modern or ancient. This is what the modern historians are eager to deny. They do not believe in such universalism. They say: No, the Greeks lived in a different conceptual universe; their way of thinking is just fundamentally, qualitatively different from ours. Even the most obvious or evident thing to us may very plausibly have been completely outside of their conceptual sphere. They couldn’t even think it, because the way they approached the world was just inherently and profoundly different from ours.
Galileo’s status stands and falls with our willingness to accept this radical relativism. His discoveries are so basic and obvious that the only way to consider them profound is to maintain that they were once *not* basic and obvious. In other words, that they are fundamentally different in character from anything the Greeks were doing, for example.
If we think there is only one common sense, and that mathematical truth and thought is the same for everyone, then we are strongly inclined to see Galileo’s achievements as quite basic, and we are strongly inclined to think that the works of for example Hipparchus on falling bodies was probably very similar to what 17th-century authors said on the same subject.
On the other hand, if we reject the very notion of a universal scientific common sense, then we are primed to think that Galileo opened up an entirely new world with his style of science.
So studying Galileo is a mirror to much larger questions. Either you are a cultural relativist and you think Galileo was a revolutionary, or you think mathematical thought is the same for you, me and everybody who ever lived, and then you think Galileo was just doing common-sense stuff. Those are the two possibilities. You have to pick sides. You can’t mix and match. You can’t have both mathematical universalism *and* Galileo being a revolutionary. These two basically contradict one another.
So once again Galileo is at the heart of fundamental questions about history. All the more reason to study him in detail. Which is exactly what we are going to do. Next episode it’s into the weeds on Galileo’s work on falling bodies.