Newton's Apple

A New Standard for the Rest of Science?

Two Laws of Inertia

Deduction of Kepler's Laws in Newton's Principia

Theorem 1, Proposition 1 of Newton's Principia says: The areas which revolving bodies describe by radii drawn to an immovable centre of force do lie in the same immovable plane, and are proportional to the times in which they are described.

Proof: Multimedia demonstration.

This argument, and arguments like it, are what made Newton famous.

Derivation of Kepler's 2nd law: Newton uses this result to show that (a) uniform linear motion obeys the area law, and (b) any uniform motion with an added motion (change in velocity, or acceleration) towards the center also obeys the area law. Part (b) covers the case of elliptical orbits, but is actually more general than that.

Derivation of Kepler's 1st law: In Proposition XI, Problem VI, Book I, Newton proves that if the radius from a focus of an ellipse to a body on the ellipse sweeps out equal areas in equal times then the magnitude of the acceleration towards the focus is inversely proportional to the square of that distance. His also shows a converse to this theorem: that if the acceleration is inversely proportional to the square of that distance, then the path is either elliptical, parabolic or hyperbolic.

Derivation of Kepler's 3rd law: He derives Kepler's third law by showing that if the planets are much lighter than the sun, then D3/T2 is a measure of the mass of the sun, and is therefore constant.

Newton's Scientific Research Program

The Apple Fable

According to the myth, Newton thought of the idea of gravity when an apple fell on his head as he was gazing at the moon in the day-time sky. While the story is untrue, there is an important grain of truth behind it. For it suggests that the key element behind Newton’s theory is the unification of celestial and terrestrial motion. Note that unification is not mentioned in the hypothetico-deductivist story.

The Deduction of Galileo's Law

How did Newton account for Galileo’s law of projectile motion? Newton’s analysis comes in two stages.

The Unification of Terrestrial and Celestial Motion

K = Keplerian model of the moon's motion around the Earth.

G = Galilean model of terrestrial motion on the surface of the Earth.

N = Newtonian idealization of motion near the Earth.

What is the relationship amongst N, K, and G?

We could ask why N is better than K or G, but this has an easy answer—N has broader scope than G and N has broader scope than K. The correct question is ask is why N is better than a combination of K and G, which we denote by K & G.

There are at least four ways of describing the virtue of N over K & G.

  1. N is more falsifiable than K & G.
  2. K & G can accommodate any observational facts that N can accommodate, but there are some predictions that N could make that K & G could not. For example, Given data about terrestrial motion, we can infer the mass of the Earth and use this information to predict the acceleration of the moon (given its distance from the Earth).
  3. N explains the agreement in the independent measurements by saying that they arise from a common cause (namely, the Earth's gravity). K & G dismisses this as a mere coincidence.
  4. N has fewer adjustable parameters than K & G.

Therefore, N does not do "just as well" as K & G, it does better. And the model N was generated by scientific research program that has a strong positive heuristic (Lakatos's term)—namely a recipe for constructing new models that involve minimal complications of the previous ones.

Four Problems with Hypothetico-Deductivism

  1. Hypothetico-deductivism assumes that observational statements are derived from theories. However, Kepler's laws, or Galileo's laws are not observation statements.
  2. Even when observation statements are derived, fit is not measured by the truth of predictions, but by their degree of fit. It is a statistical degree of fit and not one of logic (e.g., whether two independent measurements agree).
  3. The problem of idealization: Popper's falsificationism (being a kind of hypothetico-deductivism) implies that idealizations should be abandoned. However, idealizations are not abandoned if they provide sufficiently good predictions. This suggests that it is good prediction, rather than true models, that is the goal that scientists pursues.
  4. It does not mention, or place any significance in, things like simplicity or unification.