(Introduction) [A dichotomic option of the kind of mathematics? From this problem concerning the whole of theoretical physics to the same problem in electromagnetism] Recognition of either continuous or discontinuous descriptions of physical reality. In particular, [this is the problem] is Maxwell’s continuous electromagnetic theory of universal validity? If not, the electromagnetic theory is also discrete and thus one may hypothesize energy quanta although [at present time] without the support of any evidence.

(Section 1) [A first unit of reasoning (including an indirect proof) on the relationship between Maxwell’s theory and statistical mechanics] In order to solve the above problem, the application of Maxwell’s continuous electromagnetism plus the equipartition theorem of statistical mechanics to the case of molecules and electrons interacting through radiation; the calculations lead to a distribution [Rayleigh-Jeans’], whose formula gives a divergence; this result does not correspond to experience [i.e., it is an absurdity. Implicit conclusion: The hypothesis of a discrete electromagnetism is not impossible].

(Section 2): [A second unit of reasoning for obtaining an introductory result] The formula of Planck’s law [already obtained by means of the mathematical notion of quanta] produces an exact evaluation of the Avogadro constant. [Implicit conclusion: It is not impossible to theorize a discrete electromagnetism].

(Section 3): [A third unit of reasoning obtains an analogy between the two formulas concerning the thermodynamic behaviours of respectively radiation and an ideal gas] From the well-established Wien’s law some thermodynamic arguments lead to define the entropy S for the radiation of a particular frequency v.

(Section 4): [The analogy] From this definition is obtained for the radiation the function S = S(V), which is the same as the functions S = S(V) for an ideal gas [Implicit logical conclusion: It is not impossible that some electromagnetic phenomena are discrete].

(Section 5): [A fourth unit of reasoning obtains an analogy between the microscopic behaviours of the two physical systems at issue] Probabilistic definition of Boltzmann’s formula for the entropy of a gas. Deduction of the function S = S(V) for the molecules of a gas.

(Section 6): [Analogical conclusion of a universal nature] The probability of occurrence of the elements in space, obtained by the previous function is the same as that obtained by the formula S = S(V) of Section 4 for radiation. Conclusion: “Monochromatic radiation of low density… behaves thermodynamically as thought it consisted of a number of independent energy quanta…” (emphasis added), i.e. like the particles of a gas. [The implicit logical conclusion: It is not impossible that quanta exist as independent entities].

(Sections 7, 8 and 9): [Three experimental verifications] The previous conclusion is changed into the corresponding affirmative proposition, i.e. independent energy quanta do exist. The mathematical deductions from it are successfully tested on three electromagnetic phenomena [The argument of the entire paper implicitly constitutes an ad absurdum proof for the existence of quanta].