A principle, according to
which any scientific theory can be mathematized, is investigated. Social
science, liberal arts, history, and philosophy are meant first of all. That kind
of theory is presupposed to be a consistent text, which can be exhaustedly
represented by a certain mathematical structure constructively. In thus used,
the term “theory” includes all hypotheses as yet unconfirmed as already
rejected. The investigation of the sketch of a possible proof of the principle
demonstrates that it should be accepted rather a metamathematical axiom about
the relation of mathematics and reality.
The main statement is formulated as follows: 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).
Its
investigation needs philosophical means. Husserl’s phenomenology is what is
used, and then the conception of “bracketing reality” is modelled to generalize
Peano arithmetic in its relation to set theory in the foundation of
mathematics. The obtained model is equivalent to the generalization of Peano
arithmetic by means of replacing the axiom of induction with that of
transfinite induction.
The
sketch of the proof is organized in five steps: (1) a generalization of epoché; (2) involving transfinite induction in the transition between Peano arithmetic
and set theory; (3) discussing the
finiteness of Peano arithmetic; (4) applying
transfinite induction to Peano arithmetic; (5) discussing an arithmetical model of reality.
Accepting
or rejecting the principle, two kinds of mathematics appear differing from each
other by its relation to reality. Accepting the principle, mathematics has to
include reality within itself in a kind of Pythagoreanism. These two kinds are
called in paper correspondingly Hilbert mathematics and Gödel
mathematics. The sketch of the proof of
the principle demonstrates that the generalization of Peano arithmetic as above
can be interpreted as a model of Hilbert mathematics into Gödel mathematics
therefore showing that the former is not less consistent than the latter, and
the principle is an independent axiom.
The
present paper follows a pathway grounded on Husserl’s phenomenology and
“bracketing reality” to achieve the generalized arithmetic necessary for the
principle to be founded in alternative ontology, in which there is no reality
external to mathematics: reality is included within mathematics. That latter
mathematics is able to self-found itself and can be called Hilbert mathematics
in honour of Hilbert’s program for self-founding mathematics on the base of
arithmetic.
The
principle of universal mathematizability is consistent to Hilbert mathematics,
but not to Gödel mathematics. Consequently, its validity or rejection would
resolve the problem which mathematics refers to our being; and vice versa: the
choice between them for different reasons would confirm or refuse the principle
as to the being.
An
information interpretation of Hilbert mathematics
is involved. It is a kind of ontology of information. The Schrödinger equation in quantum mechanics is involved to illustrate that ontology. Thus the problem which of the two mathematics is more relevant to our being (rather than reality for reality is external only to Gödel mathematics) is discussed again in a new way
is involved. It is a kind of ontology of information. The Schrödinger equation in quantum mechanics is involved to illustrate that ontology. Thus the problem which of the two mathematics is more relevant to our being (rather than reality for reality is external only to Gödel mathematics) is discussed again in a new way
A few directions for future work can be: a rigorous formal proof of the principle as an
independent axiom; the further development of information ontology consistent
to both kinds of mathematics, but much more natural for Hilbert mathematics; the
development of the information interpretation of quantum mechanics as a
mathematical one for information ontology and thus Hilbert mathematics; the
description of consciousness in terms of information ontology.
Key
words: axiom of choice; axiom of induction; axiom of
transfinite induction; eidetic, phenomenological and transcendental reduction;
epoché; Gödel mathematics; Hilbert mathematics; information; quantum mechanics,
quantum information; phenomenology; principle of universal mathematizability
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A related post at Blogger or WordPress
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