Tools which may be used in such a comprehensive research, including molecular biology applied to field studies, require a methodological shift from the reductionism that has been used until recently as a basic axiom in interpretation of data.
One of the more general philosophical questions that have emerged from this research is the following: which conditions have to be satisfied in order for a principle to be a putative basic axiom of mathematics?
Although a basic axiom in orthodontics is that "bone traces tooth movement," the ratio of bone modeling to tooth movement varies according to the direction of orthodontic tooth movement.
Kant formulated the basic axiom of the public law: "All actions relating to the right of other men are unjust if their maxim is not consistent with publicity".
This is actually a very puzzling situation which was first experimentally verified by Michelson and Edward Williams Morley, another American scientist, in 1887 and which is the basic axiom of Albert Einstein's theory of relativity.
In fact, he went on to show that an impressive attempt by logician Gottlob Frege to derive all of arithmetic from a few basic axioms of logic, was susceptible to a logical, fatal flaw.
One was the fact that B Fuchssteiner [15] had recently proved that my 1965 theorem on fixed points of nonexpansive mappings, which made explicit use of the Axiom of Choice via Zorn's lemma, did in fact have a proof wholly within the basic axioms of Zermelo-Fraenkel.
In fact, N Brunner [18] has shown that any proof of (E) requires at least the basic axioms of Zermelo-Fraenkel (ZF) plus a form of the Axiom of Choice called the Axiom of Dependent Choices (DC), whereas R Mańka has shown in [19] that (C) holds wholly within (ZF).
Regardless of one's position on this undoubtedly political issue, two facts remain undeniable: 1) Supporting, or in this case, not actively protesting against, a complex international treaty does not violate any of the basic axioms of faith that could theoretically disqualify one from legitimately claiming an Orthodox identity.
In the second half of the nineteenth century Dedekind proved that the basic axioms of arithmetic have, up to isomorphism, exactly one model, and that the same holds for the basic axioms of Real Analysis.
