top down vs. bottom up

At any moment in time, there is always more than one scientific theory that can explain the available data. How one formalism is preferred over another one depends on historical factors (e.g. physicists avoiding models that employ mathematics that they are not familiar with), on sociological factors (e.g. one theorist being a lot more famous than the other), on metaphysical criteria (e.g. Occam's Razor), or on aesthetic criteria (e.g. one model being more beautiful than the other). Because of this state of affairs, it is said that scientific theories are, by their very nature, under-determined.

Here the term "under-determined" can be confusing. Scientific theories are under-determined from the point of view of nature. (A theory is never specific enough to be the only possible explanation that does not contradict the available data.) However they are over-determined from the point of view of available evidence. Not only do they make predictions about the future dynamics of the examined phenomenon, they also predict the existence of unrelated phenomena. Predictions of the second type are not supported per se by any current data. (This is the sort of stuff that lends credulity and prestige to a new model: Prediction and subsequent confirmation of entirely new and uninvestigated phenomena.)

There are two types of models:

1) Top-down ones that contain certain universal principles putting constraints on what can possibly take place. (Examples: Thermodynamics, General Relativity).

2) Bottom-up ones that start out with a theory about what the constituents of the reality are and define how they interact in ensembles to generate the phenomenon that our experiments can detect. (Examples: Newtonian Mechanics, Standard Model)

The top-down models are more under-determined in the sense that they do not uniquely explain everything. They just constrain the number of theories that can do the job. Depending on the nature of the principles involved, they can be more or less over-determined in the other sense of the term. (The principle postulating the finiteness of speed of light makes just a single prediction, namely the principle itself. On the other hand, the second principle of thermodynamics makes many simultaneous predictions.) In any case, compared with bottom-up models, they are less likely to be rendered obsolete due to unnecessary metaphysical assumptions with regards to the constituents of reality.

Two speculative remarks:

1) Just as science emerges from the interplay of bottom-up and top-down models, mathematics emerges from the interplay of set theory and category theory. Think of science and mathematics as sandwiches. The top and bottom buns are respectively the top-down and bottom-up approaches. In the case of science, in between lies the physical universe. In the case of mathematics, in between lies the mathematical universe.

2) The list of alternative foundations for mathematics has been exhausted. Structures can be examined and characterized inside-out or outside-in. There exists no other possible approach.