The pedagogy of physics begins with a presentation of energy as a dichotomy between kinetic energy and potential energy. This dichotomy presupposes a "configuration space" of "points" presumably correlated with a three dimensional real space. There is no such thing as an "absolute potential." Potential energy is understood as a system of differences correlated with the configuration space by virtue of a coherent vector field of force. Kinetic energy is understood as a system of arcs correlated with the configuration space by virtue of an unwitnessable algebraic dimension parameterizing the assumption of arc connectedness.
In the fourth bulletted example of the link,
http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/gauge%20space#examples
you will find that every topology defines a quasigauge space (not the same "gauge"). Importantly, the formulas used to make this definition contrast "discernibility" with a denial of discernibility. I emphasize this because the use of differences to represent potential energy must be coherent with the role of the law of identity implicit to the use of a real space as the configuration space.
Note that the denial of the discernibility relation yields the reflexive order of a conditional relation without expressing the singular character of an "individuated" "point." To my knowledge, which could certainly be in error, gauge equivalence in the usual sense is not "singular." Manifolds are understood with respect to charts and charts are coherently overlapped by stipulating smoothness conditions. And the fact that this denial is associated with an order is compatible with the fact that the witnessed continuum is dynamic.
With this mention of a quasigauge and charts, one sees a subtle change from the original pedagogy. Nevertheless, the basic idea of a real space is retained for the configuration space.
At present, forces different from gravity are understood with respect to a specific multiplication between group representations. The factors of such multiplications are presumably irreducible. So, the first simplification toward the original pedagogy is to consider the factors separately.
The relationship between a group representation and a group realization is that the representation is simply a form of the group which can act on the configuration space because its elements are specifically chosen to be linear tranformations over that space.
Again, one sees a subtle change from the original pedagogy. Some constructions used to explain force along these lines invoke complex spaces. But, I am fairly certain that stipulations involving Hessian forms secure applicability to real spaces.
What is important about acknowledging the use of group realizations, however, is to understand the relationship between an abstract group, the realization of an abstract group, and what is meant by a "quantity."
Cayley's theorem asserts that every abstract group is realizable by a transformation group. Transformation groups are understood as actions applied to "sets" or "domains of discourse (using 'discourse' rather than 'definition' to emphasize ontology)." A "quantity" is any object from this underlying set. And, the proof of Cayley's theorem is based upon taking the parameters of the abstract group, itself, as "quantities." In combinatorial group theory, parameters are not "denoting symbols."
While I am not a physicist, I see nothing in this analysis of the pedagogy corresponding to "material objects." The only "objectual ontology" involved refers to mathematical representations whose definiteness is made questionable by the independence of the continuum hypothesis.
It is true that relativistic physics and quantized energy alters things somewhat. In both cases, the mathematics attributing momentum to light has had its consequences. The evolution from Newtonian laws of motion through Lagrangian laws of motion to Hamiltonian laws relating laws of motion directly to energy occurred before the modern theories. But, they do not alter the basic pedagogy, and, the current theories are not yet reconciled.
And, if Susskind is to be trusted, the evolution of paradigms introduces mathematical artifacts without physical meaning in some cases.
Conifold asks why energy cannot be construed like velocity, momentum. or angular momentum. All three of these are understood as vectors. The relationship between position, momentum, and the uncertainty principle in quantum physics has an analogue with time and energy. However, there is no time observable in the equations of quantum physics to the best of my knowledge. This difference suggests that energy is incomparable with these vector fields because of its involvement with the pedagogical presuppositions. The transfer of force, mediated by bosons, is signaled by changed momentum (kinetic energy). This measurement can only be performed in a manner cohering with the fact that potential energy is only understood through differences at different positions. It is not that energy "jumps." It is that energy cannot be understood in the sense of an "objectual ontology" simply because it can be assigned numerical valuation.
Frog suggests that material objects are properties of energy. This analysis of the pedagogy, if reasonably correct, portrays energy as an essential undefinable required as a prerequisite to any description of the witnessable trajectories we attribute to the phenomena we call material objects. If energy has properties, it cannot be such an essential undefinable (if my understanding of such notions in philosophy is correct).
Causative has given a very good answer --- much better than mine. But, my understanding of physics is severely limited. I do not believe my answer is in conflict with that one.
I believe this answer probably supports your disagreement with what you perceive Bunge's article (which I have not read) to have said.
With regard to the role of real numbers, you might find "sets of uniqueness" from descriptive set theory of interest. This is what Cantor had been studying before going down the path of completed infinities.