Fleeting thoughts: First principles formalized: First principles in science as the initial element of well-ordering

Prehistory and background: The research of first principles of the being conditioned the beginning of philosophy in Ancient Greece many millennia before the experimental science of the modern age. Euclid’s geometry was built successfully for decades of centuries starting from a few axioms and postulates and deducing all rest statements in it as theorems logically. Thus philosophy and geometry created a paradigm for constructing science from first principles, conserved until now. Set theory introduced the principle of abstraction allowing of the generation of all elements of a set (interpretable as the extension of a notion) from a finite property featuring that set (as the definition or intension of the corresponding notion). Yet the initial development of set theory generated the axiom of choice equivalent to the so-called well-ordering theorem or “well-ordering principle” in contemporary terms. According to it, any set can be mapped one-to-one into some subset of the set of natural numbers and thus well-ordered. Semantics and semiotics of scientific theory elucidated it as a set of semantic units, which is well-ordered (“vocabulary of words”) and thus designating the investigated area of objects (“things”) in a way as ordered as possible, also well-ordered as an ideal by a one-to-one mapping between “words and things”. 

Thesis: Scientific theory admits two kinds of exceptionally important formalizations: (i) as the intension of a rather extended notion, in which (i+ii) can be added well-ordering. The latter case corresponds to the deduction from first principles, which can be interpreted as the initial element of that well-ordering. 

A few main arguments:
1 The introduction of first principles independently of their relevance, from which the rest statements can be logically or otherwise deduced, completes the logical structure of the investigated area giving it the mathematical structure of lattice and thus, of ontology: Indeed, the first principles are the least element of the lattice, and the being as a whole or at least the investigated area are its biggest element.
2 This construction furthermore having the formal structure of logic is extended from the “beginning” of our knowledge in the first principles to the “end” itself of the being “by itself” (i.e. the biggest element of the lattice). Consequently, the first principles would complete our cognition as both logic and ontology.
3 The organization of knowledge in notions is much more economical, efficient and convenient for retaining, reproducing, transferring, and utilization. However, if the notion is as extended as a scientific theory, it is too huge even as a notion and needs a secondary compression to first principles and rules of deduction. 4 The organization by first principles being a secondary ordering refers to some description or definition of a notion, which can be a fiction or description of non-existing objects. Both notion about real things (or facts) and fiction can be equally well deduced from or compressed to first principles. 5 Thus that organization turns out to be especially relevant to mathematics