Logic in Mathematics

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Multiply Kinds of Reason in Mathematics - Essay II

First, rules and patterns may be accepted because they work in a repeatable, reproducible and thus verifiable manner. What is right or wrong is thus clear, or can be checked. The careful mastery of rules and patterns, one at a time and one after another, with repeatable and reproducible results, is a sign of intelligence and gives an operational viewpoint of mathematics. Explanation in mathematics may be based on giving examples to suggest or illustrate and confirm such rules and pattern. There-in lies one motivation for rote learning. And some students will say that there is no need for any thought-based development as indicated below, since they assume courses are presenting reliable rules and patterns to follow.

Second, rules and patterns in mathematics may follow and be accepted since they stem from combining earlier rules and patterns to arrive at new ones. The thought-based development of mathematics begins with the appearance or development of the ability to combine rules and patterns to arrive at results or further rules and patterns, even before the direct and indirect use of implication rules IF A then B.

Students may appreciate the use of logic or a thought-based development that gives new results or patterns, but the thought based development or proof of previously accepted mathematics will be seen as redundant, at best a confirmation of what has worked before, and not strictly necessary.

Students, all or most, will see no need for a thought-based development or explanation of a rule pattern which has worked or been accepted before.

The combination of earlier patterns to obtain new ones - deductive reason with or without explicit mention of implication rules - can be introduced as tool for further skill and concept development. That leads to a hierarchy of skills and concepts in which later ones depend on early ones.

Proofs: When a statement is made in mathematics, the question of whether or not there is a chain of deductive reason leading to it, built on prior knowledge gives a test. If such a chain of reason exists, the statement is accepted, and the chain of reason is recognized as a proof. After an operational command of that deduction, the possibility or option of an Euclidean style derivation of theorems from a minimal set of axioms (assumed rules/patterns) - prior knowledge can be mentioned as technical solution to the chicken and egg question of what comes first.

In the Euclidean-style logical development, derivation and codification of a mathematics or a body of knowledge, a few key patterns are assumed. A further pattern is accepted as (judged to be) part of that body of knowledge if its pass a test, namely, there is at least one chain of reason employing the key patterns which implies the further The latter chains of reason provides a proof and give a further reason for logic mastery mastery - besides its development of precision writing and reading, two must for work and study.

Mentioning the possibility of an axiomatic development may be sufficient for many students - as a logical or axiom (assumption-based) codification of mathematics take time and effort, and interest too. Whence some streaming according to interest may be required. However, seeing how rules and patterns, steps and methods, combine to give further ones connects course material (mathematics) in a way that helps students who find that learning with comprehension in preferable or easier than learning by rote. There-in lies a connections or connections which may favour meeting and mastering proofs and proof techniques.

The Question of Context and Motivation for Proofs

Opposition: Students may say its it the job of the teacher to give them facts and reliable methods. Therefore they view facts and methods provided by a teacher in preparation say for final examinations need to be mastered without need for justification or proof.

Counterpoint 1. Seeing and even mastering the justification may be a course objective, one that may be required to answer some questions on the final examination.

Counterpoint 2. Seeing how rules and patterns, steps and methods, combine to give further ones connects course material (mathematics) in a way that helps students who find that learning with comprehension in preferable or easier than learning by rote.

Counterpoint 3. Justification (proofs) may be required on final examinations.

The second counterpoint above, namely

Seeing how rules and patterns, steps and methods, combine to give further ones connects course material (mathematics) in a way that helps students who find that learning with comprehension in preferable or easier than learning by rote.

provides a justification for offering proof or deductive connective arguments and support in the development of Euclidean Geometry, trigonometry and calculus. But the full, logical codification of mathematics can be left for post-calculus studies in undergraduate programs covering or including some pure mathematics.

Further Reading: Logic chapters 1 to 5 (Français) in Volume 2, Three Skills for Algebra introduce the Euclidean logic methods and questions in mathematics free manner. . The use of logic in the form of direct or indirect use of implication rules B if A or equivalently, If A then B, informally or within axiomatic (assumed rules and patterns) frameworks leads to further rules and patterns to accept and use. See to the last chapters and postscripts of Volume 1A, Pattern Based Reason, for a further discussion of consistency questions and indirect chains of reason in general, and not just in mathematics.