13. Euclidean Model

Euclidean Model of Reason
Chapter 13

The road and door analogies in the previous chapter describe the division of human knowledge into sections or islands. The knowledge in each section may be strongly or weakly linked together by implications. Knowledge in one section may touch or not touch that of another. All depends on what implication rules are known. Our minds can explore each section of knowledge as we meet it.

In this chapter, the Euclidean model for organizing reason and knowledge is discussed. In this Euclidean model for reason and knowledge, each area or segment of knowledge is derived via chains of reason from a few secure first principles or assumptions about data and implication rules. This Euclidean model is an ideal which we would like to attain. Can we?

Deduction From First Principles

The aim of the axiomatic/deductive method is to gather and to organize an island or body of knowledge so that all parts of it can be reached from a few basic, clear and self-evident ideas or principles. This is the axiomatic goal. The simplicity of this goal, an ideal, is appealing.

Where or with what should we begin? The starting points and the rules used are human selections. If one point can be reached from another, and vice-versa, then each is as good as the other as a starting point. Changing the starting place in this manner does not change the destinations or results reachable. Finally, different starting points for the organization of knowledge have different advantages. A central starting place may provide faster or easier access to the various parts and results.

The axiomatic deductive method is used in mathematics, in the physical sciences and in human laws. The first model of the axiomatic method comes from Euclidean geometry: the works of Euclid and his school in mathematics some two thousand years ago. This Euclidean model of reason is deductive. It is based on supposedly self-evident facts and implication rules. Here the fewest possible rules are used to avoid conflicts and contradictions. Euclidean models for reason in all disciplines has been an ideal and goal for some philosophers and religious thinkers in Europe and possibly elsewhere. The framers of the Bill of Rights in the United States Constitution were perhaps influenced by the Euclidean example when they started by declaring certain rights self-evident.

The axiomatic, deductive, chain-of-reason approach to a subject requires a starting point. We try to build our knowledge and our judgments and conclusions on a few laws, principles, rules or facts that can be assumed, or viewed as self-evident. Self-evident rules and principles represent starting points, sometimes held beyond debate.

The laws, principles or facts we start with and pretend or assume to be true are called hypotheses, first principles, assumptions, postulates or axioms. Which word or phrase you use is a matter of choice. For the sake of variety, we may use all these words and phrases interchangeably.

A set of assumptions together with their consequences, that is, the conclusions which can be obtained from them, form and define a theory. (We mortals will only see a finite number of the consequences.) The set of assumptions on which a theory is built is called a foundation. Again, the assumptions forming the foundation are supposed to be self-evident, clear and credible. Identifying the self-evident ones has for mankind been a matter of trial and error, and perhaps a matter of culture.

Next: 13. Clever Mortals - the challenges of forming explanations.

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster.
The Rest © 1995 onward by site author, Alan Selby, All Rights Reserved.

Euclidean Model of Reason Chapter 13

Deduction From First Principles

Euclidean Model of Reason
Chapter 13