Gödel, Einstein, Mach: Casting Constraints on All-embracing Concepts

Can a theory turn back, as it were, upon itselfand vouch for its own features? That is, canthe derived elements of a theory be the veryprimitive terms that provide thepresuppositions of the theory? This form of anall-embracing feature assumes a totality inwhich there occurs quantification over thattotality, quantification that is defined bythis very totality. I argue that the Machprinciple exhibits such a feature ofall-embracing nature. To clarify the argument,I distinguish between on the one handcompleteness and on the other wholeness andtotality, as different all-embracing features:the former being epistemic while the latter –ontological. I propose an analogy between the Mach principleas a possible selection principle in generalrelativity, and the vicious-circle principle infoundations of mathematics. I finally concludewith a consequence of this analogyvis-à-vis completeness and totality,viz., both should be constrained if they wereto be valid concepts for a physical theory. The paper progresses chronologically. Itfocuses on the physical approach of Mach thatformed the background for Einstein's generaltheory of relativity. The solutions of thefield equations in the form of cosmologicalmodels set the scene for the view ofall-embracing concepts discussed in the paper.Specifically, the ideas encapsulated in whatEinstein called the Mach principle, constitutethe thread of this account. The principle isfound however to falter, in view of the factthat there are several different types ofsolution of the field equations that contradictit. One such important cosmological model withramifying consequences is the rotational masssolution of Gödel. The question arises asto whether there is an analogy betweenincompleteness in foundations of mathematicsand in physics? The analogy between the vicious-circleprinciple and the Mach principle demonstratesan affirmative answer which suggests in turnthat completeness and totality must becurtailed – that is, conditions and limitsshould be imposed on completeness and totalityto render them valid for physical theories.