The Economist has an article discussing the multiverse: Multiversal Truths. I’m mentioned in one of the figures. I’m not sure if Martin Rees would appreciate being called a string theorist, but otherwise the article is worth a read.
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.”
Another talk in Sydney tonight. Everything you could possibly want to know about cosmology!
The Macarthur Astronomy Forum proudly presents Dr Luke Barnes (USYD)
When: Monday July 20th.
Where: UWS Campbelltown, University of Western Sydney – Bldg 30, Main Theatre – School of Medicine: Goldsmith Avenue, Campbelltown.
Time: 7.30pm sharp
Topic: Act One, Scene Two: How the Universe blooms.
How do we know what the Universe is made of? And what shapes its parts into the stars, galaxies and clusters of galaxies that we see around us? Starting from the very early universe, I’ll discuss how the fundamental factors of our universe, its forces, particles, and the dynamical stage that they tread (space time), compete and cooperate to fashion the Universe we see around us.
A favourite XKCD:
Made out of Meat
Here’s a few idle, Friday afternoon thoughts. I study distant galaxies. I use mathematical models of the laws of nature (and a supercomputer) to try to predict the properties of light emitted by and scattered through swirling vortices of matter, each containing a thousand trillion trillion trillion tons of stars, gas and dark matter, almost a trillion trillion kilometers away. My discipline – cosmology – has taken as its object of study the universe as a whole. And we’re doing pretty well, thanks for asking. I’d like to think that I am an evidence collecting, theory discovering, model investigating, equation solving (with a little help from my computer) machine.
And then I hear a talk from a biologist. I am reminded that I’m a fighting, fleeing, feeding, and reproducing machine. The lump of stuff in my head was produced by causes that “see” survival and reproduction. My brain is the control centre of a biological organism, and there seems to be precious little overlap between survival, reproduction and astrophysical ability. (Unless my astrophysical brain has made me so attractive to the ladies that it significantly increases my chances of reproduction. I’ll ask my wife.) An accurate mental picture of the world, formed using mostly reliable senses and the ability to reason logically, creatively and flexibly, seems useful to survival. But to use a brain to do cosmology? Really? (If you haven’t read Terry Bisson’s wonderful short essay “They’re made out of meat“, then do it now: “Thinking meat! You’re asking me to believe in thinking meat!”.)
A parable. Suppose I call Toyota customer services.
Me: Hi there. I own a 1993 Toyota Camry. I have a question.
Toyota: Certainly, sir. Is the car running well, getting you from A to B in comfort?
Me: Sure. It’s doing all that nicely. I was thinking about using it to drive to the moon.
Toyota: … Right … Wouldn’t recommend that, Mr Barnes. No … uh … not really in the user manual, I’m afraid. Not what it’s made for.
Suppose I find a customer support label on the back of my brain.
Me: Hi there. I own and operate one of your brains. I have a question.
Support: Certainly, sir. Is it operating your body, correctly? Are you getting enough food? Have you found a mate?
Me: Sure. It’s doing all that nicely. I was thinking about using it to do theoretical physics, discover the fundamental laws of the universe and use them to understand the structure and evolution of the universe and all its contents.
Support: … Right … Wouldn’t recommend that, Mr Barnes. No … uh … not really in the user manual, I’m afraid. Not what it’s made for.
Let’s be clear about the point I’m making here. I don’t doubt that physicists in general and cosmologists in particular have discovered true facts about the universe. It’s just a tad amazing that we can do that sort of things with our brains. (We use computers and telescopes as well, of course, but they too are the products of human brains). To extend the analogy, it’s as if I find myself standing on the moon, wondering how I got there. And as I look around, all I can see is a 1993 Toyota Camry. It’s not that I doubt where I am; I’m wondering how I got here in that! I’m not asking: how do I know that our investigation of the universe is successful? I’m asking: why is our investigation of the universe successful? How does fighting/fleeing/feeding/reproducing machine manage to do theoretical physics?
Perhaps the boring answer is the right one: we do it bit by bit. If we view science as extended and refined common sense, then maybe we can understand how a brain “made for” understanding local terrestrial environments is able to understand the universe. We don’t directly grasp the universe, of course. We rely on mental pictures and analogies. Mathematical models of the universe are perhaps analogies with equations. Having a mental picture of the world is useful. Just add curiosity and get practicing.
It seems like the same problem arises for mathematics – how does a brain manage to investigate such abstract ideas as those of pure mathematics? The same answer suggests itself: abstract thinking is useful. Just add curiosity and get practicing.
The universe is easy
We seem to need another ingredient in this explanation. That a brain can do theoretical physics and cosmology suggests not only that it is a remarkably adaptable, programmable thing, but also that the universe is an easier problem than we might have expected. A great example of this is the so-called cosmological principle. (I discuss this in more detail in my Australian Physics article here.)
That the universe is rationally analysable at all, that there is order and reason waiting for us in the mathematical structure of the universe, is a remarkable fact. The intellectual problem we are presented with in nature is, in a very real and precise sense, solvable. It is one thing that the universe exemplifies such beautiful mathematics as Lagrangian dynamics; it is another, a fortiori, that the Lagrangians that describe our universe display numerous and deep symmetries. The universe is a complicated place, and the mathematics that describes it must be complicated at some level. The remarkable thing is that the complication is on top; there is simplicity underneath. To be more precise, the laws of nature are simple, their solutions can be complicated. Newton’s law of gravitation is simple, but for even three bodies, its solution cannot be written down analytically.
In physics’s search for the ultimate laws of nature, many physicists wouldn’t accept a proposed fundamental theory unless it were simple, elegant, and beautiful. Paul Dirac went so far as to say that “it is more important to have beauty in one’s equations than to have them fit experiment”. It follows that physics cannot explain why the laws of nature are simple, elegant, and beautiful. Now there’s a thought for the weekend.
Next month, I’ll be in Minnesota at the St. Thomas Summer Seminars in Philosophy of Religion and Philosophical Theology, presenting the science of fine-tuning to a bunch of philosophers. I presented similar lectures back in 2011 – they’re on YouTube. I’ve got a new section this time on Bayesian approaches to the multiverse and naturalism. I found a few guinea pigs to test it out on, so if you’re in Sydney on Saturday then come along. Here’s the details:
Title: Fine-tuning and Naturalism
Abstract: Christopher Hitchens stated that “At some point, we [the New Atheists] are all asked which is the best argument you come up against from the other side. I think every one of us picks the fine-tuning one as the most intriguing.” After introducing the fine-tuning of the universe for intelligent life, I will consider what it means for naturalism, the multiverse, and science’s quest for the ultimate laws of nature.
Time: Saturday 30th of May 2015, 4pm.
Address: St Philips Anglican Church, 402 Port Hacking Rd Sth Caringbah
If you’re in Sydney on Monday (18th May, 2015), then come along to The Royal pub in Darlington to see A Pint of Science! It’s an international science festival, with similar events in 9 countries. I’ll be speaking on:
Universes, one after the other)
Cosmologists are considering the idea that our universe is just one of a vast ensemble. I’ll give two reasons to take that incredulous look off your face, and two reasons to put that incredulous look right back again.
You’ll also hear “Quantum origin of galaxies, stars and life”, by Archil Kobakhidze (theoretical particle physicist), and Quantum Technologies of the Future by David Reilly (quantum physicist).
And, naturally, beer. All welcome!
Research question: Do telepathic powers exist? Such powers would be highly in demand, so highly in fact that telepaths might become paranoid and keep their abilities secret. Here, I propose a method to identify hidden telepaths. Continue Reading »