Physics, Mathematics, Calculation, Experiment

As a mathematician and outsider to the world of physics, I feel that gives me a perspective which to some extent is above the fray which is taking place in theoretical physics. Experience shows that successful physical theories follow a fairly well defined sequence of major steps. The first step is the elucidation of the essential physical ideas in purely physical terms which one needs to describe the theory. The second step is to describe mathematically a system which corresponds with the physical ideas resulting from the first step. The third step is to use the mathematical system resulting in step two to make calculations which make predictions of physically interesting quantities. The fourth step is to experimentally test the predictions resulting from the third step. One might introduce a fifth step which is to modify physical ideas in step one and go through the sequence of steps again in order to make improvements in the theory. In short, the steps which can be considered as a cycle, are  

(1) Physics

(2) Mathematics

(3) Calculation

(4) Experiment 

Some comments on these steps are now in order. It is important that the physical ideas of (1) not be overly influenced by mathematics. In step one, physics must be the main consideration. In (2) on the other hand, the level of rigour should not be too high as it can prevent progress and get in the way of progress to step three. Likewise, in (3), for calculations, similarly sometimes physical ideas can be used to aid in calculation where it would not be permitted in pure mathematics. For instance, if the Weierstrass level of rigour had been required of Newton, it could have prevented the development of Newtonian mechanics. The reason that the requirements of rigour can be relaxed here is due to the final step (4) which will be the final arbiter of success. Notice that this means that if (1), (3), and (4) are omitted, all that remains is sloppy mathematics... 

THE STRUCTURE OF LOCAL THEORIES OF GRAVITY 

Maurice J. Dupré, Department of Mathematics New Orleans, LA 70118 

18 September 2013