A Theory of Knowledge
Science and technology develops from hypotheses (rules
and patterns) for testing directly or through the consistency of
implications (chains or reason) with observations, all in an empirical
repeatable and reproducible manner. The latter may imply the
limits of rules and patterns. Mixed or applied mathematics too is
an empirical subject built on assumed numerically and geometrical rules
and patterns - assumptions drawn from experience and consistent with
the most part with experience. While historical and
pedagogical path to the thought-based development of mathematics
skills and concepts goes through synthetic (coordinate-free)
drawings in geometry, the empirical limitations of the latter path
appear in diagrams whose faults are explained with the aid of analytic
geometry, and the empirical nature of pure mathematics appears in the
absence of an absolute basis for mathematical theories rich enough to
represent the infinite set of natural numbers. There are
stories to be told and repeated here about the development and
construction of skills and concepts in mathematics. The telling
and repetition of stories to understand and explain the development of
mathematical skills and concepts in a repeatable and reproducible
manner is most likely inconsistent with post-modern, rule and pattern
-rejecting developments in educational theories favoring subjective
learning and knowledge, and indirect instruction.
We have the ability to follow and present stories on paper and on stage.
Those stories may be fiction or not. Some stories may follow each
other, one at a time and one after another, or in parallel. Each person
has his or her story to tell. Mine is brief since I have forgotten
many of the details. Now the ability to follow and tell stories echoes in
the works of knowledge and fiction met in mathematics, science,
technology and society. Non-fiction is preferred.
In mathematics, each proof or deductive chain of reason in
represents a story or a sequence of stories to be told and
repeated. The telling and repetition of stories or proofs
links and develops skills and concepts in mathematics, one at a time and
one after another, all in a repeatable and reproducible manner.. In
each empirical theory, there are stories to be told and repeated in
the development, construction and testing of skills and
concepts, or skills and concepts, subject to the limitations of rule and
pattern based thought. There-in lies a gamble. So no all
certain. But many of the methods of mathematics appear to be
repeatable, reproducible and hence reliable tools in science and
commerce. So there is a chance, the methods are non-fiction.
Mathematics instruction may be given the task of providing students with
an operational command of the calculating and reasoning or proof methods
in mathematics, pure or applied or mixed, and an eventual awareness of
benefits, origins and limitations of the rules and patterns involved in
the subject and other disciplines. In education, the empirical hope
or hypotheses that a student has an operational command of one area of
proof or figuring can be tested by observing what a student writes or
produces. If a student fails, more instruction or study is required while
if a student passes the test, chances are he or she has master some
mathematics, enough to continue instruction without review. Mathematics
education is an empirical art in which instructor may observe the work of
each student, and provide feedback or correction while the student is
trying to follow the theories and methods of mathematics in a repeatable,
reproducible and objective manner, modulo the limitations of rule and
pattern based thought and processes.
Science, Mathematics and Education
Mathematics is called the Queen of Science. But mathematics is still an
empirical science. Historically, the thought-based development of
mathematics begins began with synthetic (coordinate-free) drawings in
geometry to arrive at conclusions with the aid of axioms (assumed
patterns). But the empirical limitations of the latter path, the
use of drawings, appear in diagrams whose faults are only explained with
the aid of analytic geometry, the use of coordinates. That use
turns the development historical development of mathematics upside
down. Synthetic geometry is now replaced by coordinate-based
geometry - models in drawings are codified or represented by points and
sets of points, models in which the properties of real numbers are now
employed to arrive at conclusions. None the less, the
empirical nature of pure mathematics stems in the origins of its axioms -
assumed patterns which are not given, they are chosen. Here they are
chosen to avoid inconsistencies met in previous attempts to provide a
consistent thought based development of mathematics from axioms for real
numbers - more precisely assumptions about sets that give a model of
mathematics in which real numbers are represented or
codified. Thus mathematics itself has an empirical origin,
albeit one sufficient to imply repeatable and reproducible, and hence
verifiable deductive chains of reason.
Hypothesis (Conjecture) Testing in mathematics: In a
mathematics theory or model based on axioms (assumed patterns),
we test of an statement or assertion by looking for a proof, that
is, a deductive chain of reason starting with and only involving
previously tested or proven deductive consequences of the axioms
(assumed patterns). If a valid proof is found, the statement is
considered to be tested and hence proven. That is subject to the
comments above about works of fiction and
non-fiction, consistent or otherwise.
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