Having read a few good books on Aristotelian and scholastic (meta)physics (and, thanks to Scholastic Metaphysics: A Contemporary Introduction by Edward Feser, maybe starting to get a grip on what a formal cause is supposed to be), I’d had a few idle thoughts about what a mathematical formalisation of Aristotle’s physics would look like. Aristotle, and his colleagues in the Middle Ages, distinguished between **pure mathematics**, which studies abstractions, and **natural philosophy**, which tries to understand the physical world. The problem child for this scheme was astronomy, which studied the natural world and yet did so in terms mathematical, quantitative abstractions. Astronomy was thus categorised as a **middle science**.

It was Descartes (as best I can tell) who first championed the audacious idea that all of physics could be as mathematical as astronomy. Newtonian mechanics gives us the first complete example of such a physical theory, mathematical from its very foundations.

It is no coincidence that these two names – Descartes and Newton – are pivotal in the development of mathematical physics. They were not just physicists. Descartes pioneered analytic geometry, and Newton developed calculus. In trying to understand the physics of motion, it is (for a modern physicist!) difficult to know where to start if you don’t know about using variables to represent space and time, and rates of change of position. (Although, just to put a spanner in this oversimplified account, Newton largely relied on clever arguments from geometry rather than calculus in developing his theory of mechanics.)

But what if we used the resources of modern mathematics to analyse Aristotelian physics? Having had this idle thought, I was very pleased to see someone else do all the hard work. Carlo Rovelli’s “Aristotle’s Physics: a Physicist’s Look” is great! Here’s the abstract:

I show that Aristotelian physics is a correct and non-intuitive approximation of Newtonian physics in the suitable domain (motion in fluids), in the same technical sense in which Newton theory is an approximation of Einstein’s theory. Aristotelian physics lasted long not because it became dogma, but because it is a very good empirically grounded theory.

Rovelli summarises the qualitative principles of Aristotelian physics as follows:

- There are two kinds of motion: violent (or unnatural) motion, and,
- Natural motion.
- Once the effect of the agent causing a violent motion is exhausted, the violent motion ceases.
- The natural motion of the Ether in the Heavens is circular around the centre.
- The natural motion of Earth, Water, Air and Fire is vertical, directed towards the natural place of the substance.
- Heavier objects fall faster: their natural motion downwards happens faster.
- The same object falls faster in a less dense medium.
- The speed v of fall is proportional to the weight W of the body and inversely proportional to the density ρ of the medium. (Technically, as a power law.)
- The shape of the body accounts for their moving faster or slower.
- In a vacuum with vanishing density a heavy body would fall with infinite velocity.
- From what has been said it is evident that void does not exist.

Rovelli’s claim is that “Aristotle’s physics is the correct approximation of Newtonian physics in a particular domain, which happens to be the domain where we, humanity, conduct our business. This domain is formed by *objects in a spherically symmetric gravitational field (that of the Earth) immersed in a fluid (air or water)* and the main celestial bodies visible from Earth.” The total force on such objects is given by:

Total Force = gravity + buoyancy + viscosity (fluid resistance) + external force

Within this Newtonian model, we understand Aristotle’s principles as follows.

- Violent motion is when there is an external force.
- Natural motion is when there is
**no**external force. - Because of viscosity, the effect of violent motion decays away in a finite time.
- The motion of planets under gravity (buoyancy and viscosity being negligible), viewed from a rotating Earth, can be described using circular orbits. Remember that, within the observational limits of the ancient Greeks, Ptolemy’s model works!
- A buoyant body, initially at rest and immersed in a fluid, “will immediately start moving up or down, according to whether its density is higher or lower than the density of the fluid in which it is immersed. Therefore Earth will move down in any case. Water will move down in Air. Air will move up in water.” And so on. “Furthermore, if a body is immersed in a substance of the same kind, as Water in Water, then it can stay at rest: it is at its natural place.”
- The terminal velocity of a buoyant body falling in a viscous fluid increases with its mass. Thus, ignoring the transient period of acceleration when gravity is dominant, heavy objects fall faster.
- The terminal velocity of an object (ignoring buoyancy) is inversely proportional to the (square root) of the density of the medium.
- Combining the two points above gives Aristotle’s law. In fact, the terminal velocity is proportional to the square root of the weight divided by the density. “What Aristotle does not have is only the square root … which would have been hard for him to capture given the primitive mathematical tools he was using. His factual statements are all correct.”
- The constant in the law in h) depends on the shape of the object.
- Somewhat surprisingly, this is a correct inference from the Newtonian model of Aristotle’s physics, in that as the density approaches zero, the terminal velocity approaches infinity. This doesn’t happen, of course, and identifies an unrealistic assumption: that the gravitational field is everywhere uniform. In reality, the falling object would eventually hit the mass originating the attraction.
- Following the conclusion of the model in j.), and supposing that infinite velocities are impossible, one must conclude that the vacuum is a physical impossibility. This is an interesting lesson in extrapolating physical theories beyond their domain of validity.

Aristotle’s physics is obviously not perfect – Rovelli lists all the important failings – but it does a very good job in the right regime, summarising the more complete physical model of Newton.

A few of Rovelli’s conclusions are worth quoting.

“Aristotelian physics is often presented as the dogma that slowed the development of science. I think that this is very incorrect. The scientists after Aristotle had no hesitation in modifying, violating, or ignoring Aristotle’s physics. … In the Middle Ages the physics of Aristotle was discussed and modified repeatedly, but it took Copernicus, Galileo, Kepler and Newton to find a more powerful theory. … The reason Aristotelian physics lasted so long is not because it became dogma: it is because it is a very good theory. … With all its limitations, it is great theoretical physics.”

“The bad reputation of Aristotle’s physics is undeserved, and leads to widespread ignorance: think for a moment, do you really believe that bodies of different weight fall at the same speed? Why don’t you just try: take a coin and piece of paper and let them fall. Do they fall at the same speed? Aristotle never claimed that bodies fall at different speed “if we take away the air”. He was interested in the speed of real bodies falling in our real world, where air or water is present. It is curious to read everywhere “Why didn’t Aristotle do the actual experiment?”. I would retort: “Those writing this, why don’t they do the actual experiment?”. They would find Aristotle right.”