In the mid 1800s, two of the planets were discovered to move slightly differently from what was expected from Newtonian mechanics: Uranus and Mercury. Although the discrepancy in the data looked similar, the underlying cause was very different. In one case, it meant that there is an additional planet. In the other, it meant that Newtonian mechanics was wrong.

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Prerequisites: The idea I’m describing came from Two Dogmas of Empiricism, § VI. “Empiricism without the Dogmas” by Willard Van Orman Quine (1951). Ideas from The Structure of Scientific Revolutions by Thomas Kuhn (1962) sometimes appear too. You don’t have to read either to understand this post, but they are certainly worth your time.

Originally Written: January 2019.

Confidence Level: My example of a well established idea in the philosophy of science.

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Any paradigm in science is a complicated tangle of ideas, equations, rules to manipulate ideas and equations, experiments, interpretations of experiments, predictions and postdictions, measuring equipment, technology, observations filtered through ontology, and open problems. A simplified view of some these connections can be see in Figure 1. The empirical content of the paradigm can be thought of as the boundary conditions along the bottom of this diagram. The rest of the paradigm is built on top.

We continually test the links between the components of a paradigm. The paradigm suggests open problems to be solved, methods that might be used to solve them, and what would count as a solution. If the solution is successful enough, it becomes incorporated into some technology and mass produced.

While working on the increasingly specialized and esoteric problems suggested to us by the paradigm, we occasionally find problems. This doesn’t falsify the paradigm. We are willing to overlook a few problems in the system, especially if there are no better alternatives. Eventually, we hope, these problems will be resolved.

It is extremely non-obvious how the problems will be resolved. I will demonstrate this by comparing two examples of problems with planetary motion.

Kepler’s first law states that planets move in ellipses with the sun at one focus of the ellipse. Newton’s theory that explains why this is true for a single planet orbiting a sun much larger than itself. Any modifications to this theory would result in orbits that are not quite ellipses. If the modification is small, the orbit is almost an ellipse, but the ellipse gradually rotates around the sun, as shown in Figure 2. This slow rotation of the ellipse is called the perihelion drift. The perihelion is the point on the ellipse that is closest to the sun.

In our solar system, the primary cause of perihelion drift is the presence of other planets, especially Jupiter. Newton’s derivation of Kepler’s first law assumed that the only gravitational force on the planet was due to the star. If there are gravitational forces from other objects, then the calculation must be modified. As long as the planets stay far apart from each other, their actual orbits can be calculated using perturbation theory.^[1]If two planets ever get close, their orbits will be dramatically altered and will not be even close to an ellipse. This is called a flyby and can involve a significant transfer of energy between the … Continue reading The result of the calculation is the rate at which each planet’s perihelion drifts.

By the mid nineteenth century, these calculations had been done for all of the known planets: Mercury, Venus, Earth, Mars, Jupiter, Saturn, and Uranus. The perihelion drift can also be measured by carefully by tracking the locations of the planets relative to the fixed stars. The predicted and measured perihelion drifts for each planet were extremely close for all of the planets except two: Mercury and Uranus.

Why couldn’t we predict the perihelion drift for Mercury and Uranus?

One possibility is that there is an extra planet that is also exerting a gravitational force. This is what happened for Uranus. We were able to calculate where the additional planet would have to be to have that effect on Uranus’s motion. We pointing our telescope in that direction and … discovered Neptune.

Neptune doesn’t influence any of the other planets’ motions significantly because it is much farther away from them than it is from Uranus.

This false prediction of Newtonian gravity was resolved at a low level. The problem was that our observations were not complete. Once Neptune is included in the calculations, everything else worked. No changes were need to the theory of Newtonian gravity.

Since that worked for Uranus, why wouldn’t it work for Mercury as well? Neptune is too far away to explain the problems with Mercury’s motion. But maybe there is another planet that we haven’t seen that accounts for the problem with Mercury’s motion. Mercury is hard to see because it is so close to the sun that it is only visible during sunrise and sunset. If this new planet were even closer to the sun, we might not be able to see it from earth at all. The new planet was even given a name: Vulcan.^[2]This is where StarTrek got this name from.

Vulcan was never discovered. Mercury’s perihelion drift was one of the little mysteries of Newtonian gravity.

In 1915, Einstein wrote a new theory of gravity. The Newtonian theory of gravity wasn’t in a crisis. It had fewer unsolved problems than most scientific paradigms. Einstein’s main motivation was that Newtonian gravity was inconsistent with his theory of special relativity. Both special and general relativity are examples of paradigms being replaced, even though the paradigms had few problems and were not in a crisis. Relativity did solve the open problems of the Michaelson-Morley experiment and the perihelion drift of Mercury. It also made the dramatic prediction that the path of light should be bent by a gravitational field. This prediction was confirmed in 1919 when Eddington led an expedition to the South Atlantic to watch a solar eclipse. Einstein’s genius was to be able to find improvements on paradigms that did not look like they needed improvement.

General relativity doesn’t influence any of the other planets’ motions because the other planets are farther from the sun. The sun’s gravitational field there is weaker and so looks more Newtonian.

This false prediction of Newtonian gravity was resolved at an extremely high level. The theory of gravity had to be completely rewritten to explain Mercury’s motion. The very laws of geometry were rewritten and gravity became a fictitious force in this new geometry.^[3]I have written a description of how Gravity Is Geometry.

Although the false predictions about the perihelion drifts of Mercury and Uranus looked very similar, they had very different resolutions. Uranus’s perihelion drift was explained by the discovery of a new planet. Mercury’s perihelion drift was explained by a completely new theory of gravity.