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.
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 both logic and 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.
Physics, which utilized the method “by first principles”, gave a mathematical form to itself at the same time. Nowadays the boundary between physical theories and applied mathematics seems not to be different from that between mathematical structures and their interpretations. Some other sciences tried to follow the model of physics more or less successfully. However other sciences, mainly in the scope of humanities, history, and philosophy implicitly or explicitly refuted the way of mathematization in principle.
On the contrary, philosophical phenomenology (e.g. Husserl’s doctrine) establishes an inherent link between: (a) logic and mathematics; (b) philosophy; (c) psychology: The link relates the three by means a kind of transcendental idealism in the German philosophical tradition. Thus a bridge for transfer and reinterpretation between notions of psychology, logic and mathematics is created under the necessary condition for those concepts to be considered as philosophical as referred to that kind of transcendental subject.
One can question about the mathemazability of one (or any) scientific theory formally of that historical and conceptual background.
Statement:
Any scientific theory admits isomorphism to some mathematical structure in a way constructive (that is not as a proof of “pure existence” in a mathematical sense).
Comments of the statement:
1 If any theory admits to be represented as the finite intension of a rather extended notion, the proof is trivial: Being finite, the intension can be always well-ordered to a single syllogism, the first element of which is interpretable as “first principles” (axioms): Those axioms generate a mathematical structure isomorphic to the theory at issue.
2 If one admits the axiom of choice, any intension can be well-ordered even being infinite. However, then the structure isomorphic to the theory would exist only “purely”, which is practically useless.
3 In fact any theory even as a description in humanities, philosophy or history is some finite text. This does not imply, though, that some finite extension corresponds to it for any text admits links to its context unlimitedly. Properly, this third case is what is worth to be proved mathematically.
A sketch of the proof:
The essence is the approach of Husserl’s phenomenology to be formalized and applied in both directions: to intension (“eidos”, “phenomenon”, intention) and to reality.
1 One can introduce “epoché” both to “phenomenological” and to “eidetic reduction”. As to the latter, it would mean the entire processes of removing one by one all free variables of the corresponding extension.
2 The induction in Peano arithmetic is not sufficient to be obtained the finite intension of any real thing having infinitely many dimensions in its extension in general: One needs transfinite induction (or bar induction in intuitionist mathematics) for that purpose in the case of “eidetic reduction” or the “cut-elimination rule” in the case of “phenomenological reduction”.
3 Peano arithmetic is able to generate only finite numbers: 1 is finite. Adding 1 to any finite number, one obtains a finite number. Consequently, according to the axiom of induction, all numbers in Peano arithmetic are finite.
4 One can complement Peano arithmetic to a complete model of reality adding to it a single bit, “R”, interpretable as “infinity”. That single bit is also interpretable as a second Peano arithmetic independent of the first one. The transition between the two Peano arithmetics in both directions needs transfinite induction or its equivalent. The completeness of Peano arithmetic is provable by transfinite induction.
5 One can interpret that model of reality naturally in terms of Husserl’s phenomenology if “epoche” is represented as removing “R”: adding “R” would be the reverse operation.
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