We agree fully........ however, where I state that specific equations apply to specific cases, what I mean to say is that "current knowledge equations" apply to "current knowledge scenarios". Therefore I belive that there lurks somewhere an "all-encompassing" general theory.... we simply haven't discovered it yet.
To apply this to my reply to alex.... S = a x b, is a great equation for area in the specific case of negligable curvature. For greater curvature, however, we need an equation that accounts for such. So now we call on the "sphere surface area" formula..... and we say that a "flat" surface is simply a "special case" of this more general all-encompassing equation. But wait...... is this really all encompassing? Or is it itself a special case of a "two dimensional surface wrapped around a three dimensional space" (the sphere)? It seems evident that we need to include a variable number of dimensions in our so called "all-encompassing" equation. And we also need to include cases of varying degrees of curvature, correct? What about surface area on parabolicly curved surfaces, surely this is not a spherical degree of curvature....... WELCOME ladies & gentlemen, to the wonderful world of Riemannian Geometry! We have now found our all-encompassing, GENERAL model of the properties of spatial curvature in n-dimensions.
See where this is leading? The "Holy Grail" of physics, is not to find all equations that apply to all situations, but rather to find one equation that includes all situations as a specific case. This desire of physicists lies at the heart of the great search for the as of yet elusive TOE, or Theory of Everything. Many scientists are pointed in the direction of some Grand Unified Field Theory, that should theoreticaly describe all processes in the known universe with a single "all-encompasing" model. These equations are not yet "cureent knowledge" and they certainly encompass scenarios that failed as of yet to be witnessed or even conceived in some cases. i am a firm believer that mathematics will prevail, and that there is some all-encompassing mathematical model that will cover the general case. When the time comes that physicists unvail it...... we will no longer be required to "pattern match"; all "patterns" will be described by one, and one general theory only.