5.9. What Kind of Explanation is that?¶
If you ask someone why planetary orbits are elliptical, they might start by modeling a planet and a star as point masses; they would look up the law of universal gravitation and use it to write a differential equation for the motion of the planet. Then they would either derive the orbit equation or, more likely, look it up. With a little algebra, they could derive the conditions that yield an elliptical orbit. Then they could argue that the objects we consider planets satisfy these conditions.
People, or at least scientists, are generally satisfied with this kind of explanation. One of the reasons for its appeal is that the assumptions and approximations in the model seem reasonable. Planets and stars are not really point masses, but the distances between them are so big that their actual sizes are negligible. Planets in the same solar system can affect each other’s orbits, but the effect is usually small. And we ignore relativistic effects, again on the assumption that they are small.
This explanation is also appealing, because it is equation-based. We can express the orbit equation in a closed form, which means that we can compute orbits efficiently. It also means that we can derive general expressions for the orbital velocity, orbital period, and other quantities.
Finally, this kind of explanation might be appealing because it has the form of a mathematical proof. It is important to remember that the proof pertains to the model and not the real world. That is, we can prove that an idealized model yields elliptical orbits, but we can’t prove that real orbits are ellipses (in fact, they are not). Nevertheless, the resemblance to a proof is appealing.
By comparison, Watts and Strogatz’s explanation of the small world phenomenon may seem less satisfying. First, the model is more abstract, which is to say less realistic. Second, the results are generated by simulation, not by mathematical analysis. Finally, the results seem less like a proof and more like an example.
Many of the models in this book are like the Watts and Strogatz model: abstract, simulation-based and (at least superficially) less formal than conventional mathematical models. One of the goals of this book is to consider the questions these models raise:
What kind of work can these models do: are they predictive, or explanatory, or both?
Are the explanations these models offer less satisfying than explanations based on more traditional models? Why?
How should we characterize the differences between these and more conventional models? Are they different in kind or only in degree?
This book will attempt to offer answers to these questions, but they are tentative and sometimes speculative. You are encouraged to consider them skeptically and reach your own conclusions.
Q-1: Considering the passage above, what is the difference between a Watts and Strogatz model and the like, and a conventional mathematical model?