Theory of Knowledge

Appetizers and Lessons for Mathematics and Reason (www.whyslopes.com) \|\|Définition d'une variable \|\| Algèbre \|\| Arithmetique \|\| Logique \|\|La raison basée sur les règles et modelés\|\|
Online Volumes 1, Elements of Reason. 1A. Pattern Based Reason 1B. Math Curriculum Notes 2. Three Skills for Algebra 3. Why Slopes & More Math (Optional Book Orders)	More Site Areas 1. Help Your Child or Teen Learn 2. Solving Linear Equations 3. Fractions Ratios Rates Proportions & Units 4. Euclidean Geometry 5. Analytic Geometry/Functions 6. Number Theory. 7. More Calculus	More Site Areas 8. Complex Numbers 9. Qc Maths Education 10. Secondary IV(?) maths 11. Real Analysis 12. LaTeX2HotEqn: 13. Electric Circuits Etc 14. Français 15. Algebra, Odds & Ends, Etc	More Site Areas 16. Math Education Essays 17. Telling & Working with Time 18. Maps, Plans & Drawings 19. Quantitative Skills for home, shopping and work 20. Statistics Useful, or Not. Test the Twiddla Whiteboard
[Site Entrance & Hub][ Back ] [ Area Entrance & Hub ] [ Next ][Site Exit] Theory of Knowledge

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.