## Friday, July 11, 2008

Epicycles have become one of the labels that the laymen often use to identify what they consider to be a bad science.

However, it turns out that virtually every single layperson - and, in fact, not only a layman - misunderstands what the epicycles were, what was bad about them, what was good about them, and what was similar and dissimilar about them to the theories in the modern era. This essay will try to clarify some issues about the history of science but the essential points will be about philosophy of science - something that is relevant even today.

Achievements of ancient astronomy

Many ancient civilizations became extremely good in astronomy. I am absolutely sure that the astronomers of their epoch - and even many ordinary people who used to live with them - were much better than the contemporary public (including your humble correspondent) in the understanding of trajectories of the celestial bodies. They actually knew where the Sun, the Moon, the stars, and the planets (especially Venus) were going to move tonight.

I find it incredibly ignorant and arrogant when a person in 2008 who has no idea about these matters attacks the ancient astronomers who were so much better in this business - or if he even claims that their theories were disconnected from the observations. Nothing could be further from the truth.

Millenia ago, people realized that the celestial bodies were moving along quasi-periodic orbits. Because the circle is the simplest non-trivial "periodic curve", they were sensibly trying to describe the observations in terms of circles: every good physicist would start in this way. The simplest "circular models" could be easily falsified so they were obviously forced to invent non-minimal models of the motion.

This "falsification" comment of mine sociologically oversimplifies the actual historical situation. It wasn't that easy to falsify these things because the ideal circular motion was a part of many ancient guys' belief system. Observations were often insufficient to kill their wrong belief that the orbits had to be exact circles. Believing otherwise was equivalent to a heresy - a belief that God and Nature were sloppy prostitutes. At any rate, they eventually accepted that the orbits were not exact circles - and whether or not it implies that God was sloppy was no longer discussed. ;-) Maybe, He is.

Ptolemy: a good phenomenological theory

Such early scientific reasoning took place in many societies: astronomy was the simplest "exact science" whose observations and predictions were easily accessible to the humans which is why many nations spent a lot of time with it and became very good in it. Many of the best mathematicians used to be astronomers, too. Because of my European heritage, I will focus on ancient Greece. In this context, these important breakthroughs are most often associated with Ptolemy (83-161 AD) even though the hard work goes back to Apollonius of Perga who lived 300 years earlier. Ptolemy was great in the global context, anyway.

Just to be sure, his life belongs to a much more recent era than the lives of people like Socrates, Plato, and Aristotle.

Now, you should realize that during Ptolemy's life, wise people were already fully aware that the Earth was round and it was spinning around an axis. So this subtlety that makes the apparent motion of all celestial bodies more complex could have been "subtracted".

When they did so, both the Moon and the Sun were apparently moving along circles surrounding the Earth. The planets had more complicated orbits. But the orbits, encoded in functions x(t),y(t), could have been parametrized in terms of many circles. In terms of a complex variable z=x+iy (of course, they didn't use complex numbers at that time), you have
z(t) = a exp(ift) + b exp(igt) + ...
where "a,b" are complex pre-factors and "f,g" are real frequencies. This kind of a parametric curve is called an epitrochoid, especially if the frequencies "f,g" are related in a particular way. The signs of f,g usually coincide.

Actually, I am oversimplifying another aspect of the description: the shape of the parametric curve above in the 3D space was what the theory was actually predicting but the variable "t" wasn't quite the time. The actual speed of planets was declared to be such that it visually looked constant from the viewpoint of the equant, an extra point that will be mentioned further in the text again.

The most important circle with the larger radius - let's imagine that it's the first term proportional to "a" because |a| > |b| - was called a deferent. The smaller circles - all the corrections - are the (not so) well-known epicycles. The center of the circle (the origin of my coordinates) was actually not the Earth. It was a different point C and just in the middle between this center C and the Earth, you find another important point that was called the equant (that we announced in the previous paragraph). The importance of this point was in determining the speed of the celestial bodies: the apparent speed of the bodies without the epicycle corrections - i.e. the centers of epicycles - as seen from the equant viewpoint was postulated to be constant.

Depending on the celestial bodies and the required accuracy, they needed different numbers of epicycles. If you look at the formula above and allow arbitrary frequencies, you may encode an arbitrary motion as long as you use the integral: this fact is called the Fourier transformation. If all the frequencies are multiples of a basic one, any periodic motion can be expressed in this way because you get the Fourier series (sums). The closer you are to a quasi-circular, smooth motion, the more convergent and more useful the Fourier expansion becomes.

There is nothing wrong whatsoever with this description. Before you understand Kepler's laws and Newton's laws, the epicycles are the most obvious strategy for a high-precision description of the observed orbits. And Ptolemy figured out many details that most readers and kibitzers don't even realize. For example, it is damn important that the center of the deferent is in the middle between the equant and the Earth. This assumption properly approximates the equal-area Kepler's law (the conservation of the angular momentum), up to the first subleading order.

In other words, it is responsible for the fact that when a celestial body is closer to another one, the apparent (angular) velocity not only increases but it increases more than you would expect from a uniform circular motion, because of the reduced distance. In fact, the increase of the angular velocity is twice as large. Recall that 1/(1+x)=1-x... etc., up to the o(x) terms.

The people who have calculated the actual motion of the planets at some moment in their life should be aware of these approximations and appreciate the clever and successful approach of the ancient astronomers.

Copernicus and epicycles

There is a common misconception that Copernicus, the father of the heliocentric paradigm, did away with the need for epicycles. The epicycles are therefore incorrectly identified with geocentrism.

However, it shouldn't be shocking that even Copernicus badly needed them because they were the cutting-edge tools to describe the motion of planets accurately. The only improvement in this respect was that the circle corresponding to the relative motion of the Earth and the Sun could have been subtracted from the motion of all other planets because the Sun was more naturally placed at the center. But he kept the mathematical tools and parametrization of the Ptolemaic system.

There used to be three (wrong) dogmas of old astronomy:
1. the Earth is at the center
2. the trajectories are based on perfect uniform circular motion
Copernicus abandoned (1) which also weakened (3) - because the Earth became just another planet and it was natural to think that the stuff is similar for all planets. But he didn't get rid of (2) simply because it was the well-established, quantitative, non-trivial, predictive core of astronomy of his era and no functioning replacement was known.

As far as the shape of the orbits goes, the "upgrade" came with Johannes Kepler who used very accurate data extracted by Tycho Brahe. Superpositions of circles - epicycles - were replaced by ellipses. Epicycles expanded up to a given order are enough to describe the data as accurately as Kepler's ellipses. But Kepler simply noticed that the parameters of the epicycles were not arbitrary and a more specific trajectory - an ellipse that satisfies three Kepler's laws - was equally good. If a theory depends on a smaller number of parameters and arbitrary assumptions, and Kepler's theory did, it should be preferred as long as it agrees with the observations as accurately as the previous one.

So even before Kepler's laws were derived from a more complete theory and even before its predictions that go beyond simple epicycles could have been validated by observations, it was reasonable to replace the Ptolemaic system by the heliocentric (Copernicus...) elliptic system governed by Kepler's laws. But the geocentric/heliocentric debate is independent from the epicycle/ellipse debate.

Newton and epicycles

From the viewpoint of physics, Isaac Newton's breakthrough was the most important one, of course. He realized that the motion of apples and the motion of the Moon is governed by the same law. That led him to sketch the most natural equations that could control such laws: he had to invent the derivatives and integrals along the way. There were various uncertain parameters in his equations - for the example the exponent that dictates how the gravitational force decreases with the distance - but he could have easily determined these unknown parameters by the requirement that his theory agreed with Kepler's laws.

Newton's laws actually imply the Kepler's laws (elliptical orbits) exactly and if you have never checked it, you should.

Recall that Kepler had done something very analogous. For example, the law that the areas per unit time are constant is directly related to Ptolemy's law that the center of the large circle (deferent) is in the middle between the Earth and the equant. Up to the first subleading order in the eccentricity expansion, these two laws are completely equivalent. Kepler has, of course, recycled and reinterpreted the previous knowledge, building on Ptolemy's insights. Newton did pretty much the same thing.

Revolutions in physics don't mean that all the old experimenters and theorists are proven to be complete idiots. Their theories and observations are usually justified - they have to be - but they are interpreted from a newer and more complete perspective that the old guys were missing. The newer theory finds previously unknown relationships between phenomena and it allows one to make more far-reaching predictions than those accessible to the older, approximate theories.

Effective field theory and epicycles vs string theory

Some of the irrational, ill-informed critics of modern physics often say that the
1. epicycles were bad science disconnected from observations
2. epicycles were analogous to string theory, as far as their predictivity goes.
In both respects, the reality is just the opposite:
1. epicycles were high-precision science firmly rooted in observations, trying to match them as faithfully as possible
2. despite this virtue, epicycles were not analogous to string theory: string theory is completely analogous to Newton's laws in this story; instead, the effective field theory is the right counterpart of the epicycle model.
Let me describe these two qualitatively isomorphic situations in detail.

First, the epicycles were designed to properly describe the observed motion of celestial bodies. All their subtle features were simply needed because it was known that the simplest theory - one based on the perfect circular orbits - was apparently not quite correct.

This situation is analogous to effective quantum field theories. We have to include various small interaction terms simply because we know that the simpler theory without these terms doesn't quite agree with observations. For example, we have to include Fermi's four-fermion interaction (later refined by Feynman and Gell-Mann). Why? Because we observe the beta decay.

However, despite its being adequate to describe all known beta-decay phenomena of the 1960s, the four-fermion interaction suffers from a lot of theoretical problems. It doesn't explain why the coefficients for different groups of 4 fermions are related and seem to be similar. And equally importantly, this theory behaves badly at higher energies.

These two bugs are related and both of them are solved by the electroweak theory where the beta decay is induced by intermediate W bosons. The theory with the W bosons is renormalizable and well-behaved at much higher energy scales and it does explain why the muon and neutron decay amplitudes are related.

Newer, more complete theories often explain patterns that were unexplained by the previous theories. There are hundreds of examples. For instance, general relativity explains (by the equivalence principle) why the gravitational mass is proportional (or equal, with a natural choice of units) to the inertial mass, a fact that was accepted but unexplained by Newton's theory of gravity.

While the electroweak theory improves the situation of one particular formerly non-renormalizable interaction (just like Kepler's laws improve epicycles), it belongs to a broader theory - the Standard Model - that suffers from similar problems. In other words, the electroweak theory has a deeper understanding of the motion than the four-fermion theory but it doesn't seem to be maximally deep. The electroweak theory doesn't explain the values of its own parameters. And when you incorporate gravity into the Standard Model, which you should because gravity is also observed, it also breaks down at sufficiently high energy scales, namely near the Planck scale.

Again, this breakdown is a sign that there exists a deeper layer of knowledge beneath the effective field theories such as the Standard Model. And indeed, we know what it almost certainly is - it is string theory. In string theory, the parameters of various interaction terms in the Lagrangian are no longer independent. Much like Newton's theory relates the values of parameters needed for accurate epicycle models to match the observations, string theory relates many parameters and other features of the effective field theories.

If we were able to observe very massive particles whose masses are comparable to the string scale, string theory's explanation of their masses and couplings would be empirically spectacular.

We don't have the full picture (and we can't directly observe these high-brow physical phenomena) but we know that in all respects we can check with our current limited experimental gadgets and the current limited knowledge (especially in the vacuum selection business), it works.

For example, if both grand unification and supersymmetry are relevant for the reality, the values of the coupling constants may be related by the condition that the unification occurs. This prediction seems to be confirmed by observations. String theory also predicts a lot of things that are independent of the not-quite-understood vacuum selection subtleties, such as the behavior of scattering amplitudes at very high energies (dominated by black hole physics).

Again, string theory is a structure that implies the approximate validity of the previous approximate theories in various limits. Even though it uses a smaller set of elementary building blocks and tighter, more constraining assumptions, it seems to agree with all the predictions we are able to check at the current level of knowledge. Most importantly, it generates all qualitative features of the previous description of reality, including relativity, quantum mechanics (i.e. quantum field theory if you combine them), gravity, gauge forces, and both chiral and non-chiral fermions. If history is a good guide, is seems obvious that string theory is correct and its gnoseological situation will be interpreted analogously to Newton's laws in the future.

And the people in the future who will love mudslinging but who will be ignorant about physics will use the term "effective field theory" as an insult even though effective field theories are indisputably a priceless framework to scientifically study and describe the world from the very bottom, at least at the level of physics between 1960 and 2010.

If you allow me to summarize: both bottom-up theories, such as epicycles or effective field theories, and (relatively or fully) top-down theories, such as Newton's laws or string theory, are essential in the development of a scientific understanding of the Universe.

The advantage of the bottom-up theories is that they are closer to many detailed observations. The advantage of the top-down theories is that they don't fail to see the forest for the trees. They know how phenomena are related and why they are what they are. They can typically predict more general and "extreme" things that cannot be predicted by the bottom-up theories - which often includes predictions that can't be tested with the existing experimental technologies. Because our knowledge is gradually moving towards the "top", the correct top-down theories are ultimately accepted as the superior ones and the loads of data they imply are often yet unfortunately viewed as "trivial".

But they were not trivial in the past and bottom-up theories such as epicycles and effective field theories were usually necessary to explain these data and to make progress.

And that's the memo.

1. Thank you for this excellent posting, and sharing your knowledge of science history.

2. Dear Lubos:

An excellent post. I think you would find the following article to be informative and stimulative:

"An Essay on Thomas Kuhn’s First Scientific Revolution, The Copernican Revolution" by N. M. Swerdlow of the University of Chicago in Issue 1 of Volume 148 of the the Proceedings of the American Philosophical Society published March 2004 [PDF].

The article, although nominally about Thomas Kuhn, spends most of its length explaining Copernicus, Kepler, Galileo, and Newton.

Prof. Swerdlow was a member of the History and Astronomy departments at U of C, and is probably the worlds leading authority on Copernicus. He recently retired from Chicago and took a research position at CalTech.

3. I'm curious as to why, given how thoroughly Popper's philosophy of science has been "falsified" in the philosophical literature, you unscientifically keep employing it?