The Standard Model is hands down the most precisely and thoroughly tested physical theory of all time [See the introduction to Dr. Steven Weinberg’s three volume “The Quantum Theory of Fields”]. The mathematical structure of the Standard Model is based on a branch of mathematics called “group theory” [NOT “set theory”] and is formally described as a Lie group SU(3)xSU(2)xU(1). Without going into the formal mathematics, this means that calculations involving strong nuclear interactions involves 3×3 matrices, those involving weak nuclear interactions involve 2×2 matrices and electrodynamics is described with a scalar factor. Of course this is a simplified representation since those types of physical interactions cannot be so neatly separated but the combination works amazingly well. When I say, the Standard Model works, what I mean is that when we physicists try and do a calculate with it- trying to predict what we will see in any given physical situation- to date the calculations have always agreed with what is actually measured to the limit of the precision with which we are physically able to measure!
This is actually frustrating to physicists like myself because we are LOOKING for limitations of the Standard Model. Only from instances where the Standard Model does not quite work right could we learn more about the physical world, such as how gravity fits into the picture of the quantum world. So far, we have been completely unable to find any such discrepancy. That alone establishes that so far at least the Standard Model is the best possible description of the physical world.
Will that change in future? We physicists hope so. Part of our job is to find or figure out phenomena in the physical world that we do not yet understand and then TRY and understand those phenomena. This has been rightly described as the “privilege of being a physicist” by Dr. Victor F. Weisskopf who so entitled a book I was given upon finishing one stage of my formal studies.
No one says the Standard Model is the best possible physical theory ever but it IS certainly the best possible physical theory to date and any future theory will have to incorporate it as a limiting case, just as for example general relativity incorporates Newtonian gravity as a limiting case.