Mathematics Defined
as 'Ideometry' -- The Deductive, Formal-Logical Moment of Its
Meaning.
Dear Reader,
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.
Regards,
Miguel
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