Dear Demond,
If when you say we as theorists, you are referring to you and me, you may be right in your belief, but I don't believe that applies to all theorists. There are many in this world (both those who base their theories primarily on observations and those who base them mostly on mathematical models) who do not believe in a balanced approach especially if such an approach would disprove their pet theory or if their careers are at stake because they are dependent on maintenance of current scientific beliefs, etc.
I believe that the best approach is to first observe as much as we can of the world around us and all of the results of the interactions of its parts and then logically look at the possible structures and the rules of operation of those structures that are required to make a world that works in the way that this world does. The structures and rules of operation that are needed may include some that we can't presently observe where necessary for everything to work the way that it does (These could be considered to be predictions to be looked for in future observations). This will likely lead to the production of many possible hypotheses that attempt to explain this world. Math will be needed at all of the above steps to quantify measurements of things observed and the relationships that are observed to occur between them, the rules of their operation that we observe, and the interactions between them, etc., but at this stage I would discourage the building of complete math models as they would be very incomplete and could misguide efforts to understand the basic structures involved. All of the proposed hypotheses should then be compared to identify their differences and also the elements that are the same in all of them. At this point a math model can be built to model all of the parts that are common to all proposed hypotheses. Then the goal should be to identify and remove any parts that:
1. are unnecessary duplications of structures and rules or where they can be combined under one structure and/or rule that turns out to describe more elements of observed reality than was originally envisioned,
2. don't actually perform in agreement with observed reality,
3. conflict with other proposed structure(s) or rule(s), etc.
Those structures or rules that conflict with each other should then be analyzed and modified and/or replaced with structures or rules that are internally consistent and also agree with observed reality. The end result of this stage of development should be a math model that is internally consistent and agrees with observed reality, but will likely not explain all of it. The next step would be to look at and make math models of the parts of each hypothesis that do not agree with all other hypotheses. Those that are proved to be both consistent with observed reality and the main math model and add new understanding of or explain areas of observed reality that are not covered by the main math model can then be added to that model. In some cases multiple sets of structures and rules may be found that are both internally consistent with the math model and what can be presently observed of reality, but all describe the same parts of reality only in somewhat different ways. In such cases all such sets should be included in the math model and continually be compared to new observations and the required changes in the math model to accommodate such new observations with specific attempts being made to make observations that will discern which one actually most closely fits observed reality. The others can then be removed from the math model when they are proven incompatible with it and/or observed reality. Observations should also be made to confirm that the structures and rules that were needed to make the hypotheses and the math model, but hadn't yet been confirmed to be valid by previous observations are actually valid predictions. Any changes needed to bring them into conformance should be made and any that prove to be false should be removed. Because of the complexities involved in constructing and maintaining a complete and accurate logical hypothesis and math model, there should be 3 divisions made in the scientific community to be sure that a proper balance is maintained between work on the logical hypothesis structure and the math model. One section would be made up of those who are expert in making and recording the data results of observations. The next section would be composed of those who are expert at pattern recognition, logical reasoning, and interpretation of the data produced by observation. These would work on the production and maintenance of the logical hypothesis. The third group would be those who are expert in mathematics who would produce and maintain the math model. There would also be those who would be cross trained in 2 of the 3 areas who would act to transfer and coordinate the information between the 2 groups for which they are trained. Finally there would be those who are trained in all 3 areas that would handle overall coordination between the three groups and also transfer information to the general public, etc. This, of course, is only a general guideline and would need to be fleshed out much to be implemented into practice. As an example, there might be a division that would look at all new observations, new logical hypothesis concepts, and new math developments to determine how they might be further developed and then integrated into the total information structure as appropriate to ensure maximum overall development. Such new inputs of information could come from anyone, not just those who are currently working in the area that the new input is associated with.
You are right about the dimensions. This is because even though we cannot directly observe the extra dimensions, we can observe their outputs, such as the frequency, wavelength, and dynamic mass effect of energy photons that are generated by their fourth dimensional motions.
You are right. Einstein used both logical reasoning about observed reality and math structures to model it.
You are right again. All methods are required to get a full true understanding of reality to man's best ability.
Sincerely,
P. Butler