Mathematics Defined as 'Ideometry' -- The Deductive, Formal-Logical Moment of Its Meaning.
In the definition, given below, we do not define mathematics as ‘ideometry...’, only in some non-standard sense, that excludes the usual account of mathematics as “axioms” and “postulates” finding, as “rules of inference” and “primitives” deciding, and as “theorems proving”, from the former as the foundations of that proving activity.
The ‘ideometry’, the ‘measurement of the ideas’, that are the “primitives”, the “rules of inference”, and the “axioms”/“postulates” of a [e.g., of a candidate, or in-development] axioms-system of mathematics, e.g., of a newly-developing branch, or application, of mathematics, is accomplished -- using the word “measured” in the expanded, F.E.D. sense -- by drawing out their [conjoint] deductive, formal-logical consequences, and by the usefulness of those consequences -- consequences in the form of theorems, lemmas, and corollaries; usefulness even if only in terms of the cognitive-esthetic pleasure of the mathematician/author in the system that (s)he has created, but, hopefully, also, usefulness in the less narcissistic sense of the scientific and technological, human-societal self-reproductive self-force contributions/benefits/utility consequences of that axioms-system, e.g., due to the capability of its equations to model/predict-for salient aspects of nature/human experience.
In the real human praxis of mathematics, as opposed to its mystified myths and fables,
axioms and postulates do not descend, or “fall from the heavens”, as “immaculate conceptions”, from above, from some “transcendental realm”. Candidate axioms are tried-out, to see whether or not, and to what extent, their conjoint implications achieve the goals and motives to fulfill which the axioms-system is to be created in the first place.
Axioms, postulates, primitives, and rules of inference, are not themselves accessible by deductive logic, as Plato pointed out, so long ago, but are the logical «arché» -- derived by dialectical means, according to Plato -- from which all else is to be deduced.
In the process of the development of a new axiomatic system, the implications of the candidate «arché», in effect, feed back upon, and act back upon -- reflect/reflex upon, and reflux upon -- those candidates, via the active, living mediation of the human agent(s) of this development.
This process results in changes to those candidate «arché» when their implications fall short of their desiderata, and, thus, also change the deductive consequences, in an iterative process, that continues as long as, e.g., the human agents of mathematics see need for, and hope for, the improvement of the axioms-system, relative to its desiderata.
Thus, the very development of a mathematical, axiomatic system is a self-reflexive function, and a self-refluxive function, conducted by human mathematicians, and is, in that sense, also a ‘‘‘nonlinear process’’’ -- a process of ‘ideometry’, mediated by formal deduction.