Pattern
Based
Reason
understanding & explaining
Reason and Math
Volume 1A
Printed in Canada
ISBN 0-9697564-5-3
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To reason often means to persuade someone of
the need for an idea or action. That someone could be yourself. So be
careful.
Learn More: If this work is too
your liking, you may also like the foreword of Volume 1, Elements of
Reason. with its description of all site volumes.
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YOU are better than YOU think. Show
yourself how:
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Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful,
Edifying, Strengthening Elixir.
It eases work & learning difficulties. Makes the hard easier. Opens
eyes. Leads to greater precision.
in reading and writing
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and
writing, and give a firmer base for work and studies at many levels.
Good luck.
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Explore collaborative whiteboards
from groupboard,
twiddla or
scriblink.
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Set Theory & Euclidean Model
for the codification of mathematics
Chapter 14, subsection
Previous: Chapter 14 Intro, Deductive
And Empirical Views of Mathematics,
Philosophers and pure mathematicians are aware of the human origin and growth
of mathematics. In past centuries, the rules and patterns followed in
mathematics were invented and verified in imaginative ad hoc ways — the
reliability of the rules and patterns discovered was sometimes unclear. In
everyday speech, more can be suggested, said or imagined than proven. Similarly,
in mathematics the algebraic way of writing allows more to be written than
shown. This leads to statements for which proofs of truth or falseness may be of
interest and a challenge.
In the middle and late 19th century, members of the then small mathematical
community began to look for a more certain and more rigorous foundation for the
description and manipulation of calculations. In accordance with the Euclidean
Model for Reason, the ideal foundation consists of a few simple, clear
principles on which the rest of knowledge could be built via rigorous, that is,
firm and reliable, thought free of contradictions. But what assumptions should
be made and what operations should be allowed in mathematical reasoning was not
clear. Despite this, a firm, or nearly firm foundation Zermelo-Fraenkel set
theory for mathematical computations, that is, arithmetic (analysis,
advanced calculus) was formulated around the period 1905-1920.
The theory itself is almost an accidental outgrowth of investigation by Georg
Cantor (1845-1918) and others of the set concept and what set-formation rules
should be permitted. Too much freedom (too freely adopted methods for set
formation) led to paradoxes.
Prior to the development of set theory, Giuseppe Peano (1858-1932) had
given axioms for the whole numbers. From the whole numbers with the aid of
coordinates (ordered pairs of numbers) and the idea of equivalent ordered
pairs, can be successively developed the integers, rational numbers (signed
fractions), the real numbers and the complex numbers. Beyond this, the
representation of functions by their graphs, here sets of ordered pairs,
implied mathematical operations could be represented within set theory.
With a selective assumption of rules for set formation, the set theoretic
representation (codification) of ordered pairs and Peano's axioms for whole
numbers become feasible, and this in turn implies a set theory basis for
arithmetic with whole numbers, integers, rational numbers, real numbers and
complex numbers.
The set theoretic foundation relied on the thought-based methods of logic,
that is, on algebraically written rules and patterns, and not on arguments
employing physical concepts or diagrams. The movement from a previous reliance
on diagrams and physical concepts and geometry concepts or intuition represents
the assumption that the most secure chains of reason are based on the rules and
properties of arithmetic or sets.
The Zermelo-Fraenkel set theory provides an axiomatic (assumption and
logic-based) treatment of numbers and arithmetic computations.
These axioms and their arithmetic consequences have no reliance on geometry or
physical arguments or motion. This theory is built on written implication
rules, direct and indirect chains of reason and the algebraic way of writing
and reasoning.
The foundation of knowledge and mathematics was also a concern of Gottlob
Frege (1848-1925), Bertrand Russell (1872-1970) and Alfred North Whitehead
(1861-1947). Ernst Zermelo lived from 1871 to 1953. Abraham Fraenkel lived
from 1891 to 1965.
The development of the set theoretic approach was motivated by the need to
provide an objective, thought-based organization for mathematical knowledge and
arguments with the fewest possible assumptions to avoid contradictions. The set
theoretic foundation gives a framework, a starting point which is more strict,
sure and rigorous than in previous centuries for mathematical computations. [2]
[2] The set theoretic foundation of
mathematics after arithmetic is a human creation or discovery. There are
alternative axiomatic foundations (frameworks) for mathematics, but I am not
familiar with their details and so cannot discuss them further.
On this set theoretic foundation, mathematical conclusions about sets and
computations can be derived through long chains of deductive reason.
Chapter Subsections: [ 14 Set Theory ] [ 14 Before & After Set Theory in Pure Mathematics ] [ 14 Euclidean Model for Physics ] [ 14 Applied Maths and Electricity Apart from Sets ] [ 14 Decimals Absent From Pure Mathematics ] [ 14 Modern Mathematics Education ]
Next: Before and After Set
Theory in Pure Mathematics
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Foreword +
Chapters 1 to 24
FOREWORD
Three Remarks
1 Introduction
2 Communication
3. Elements of Reason
4 Implication Rules
5. Deception
6 Chains of Reason
7 Longer Chains
For & From Consistency
8. Language Change
9 Next Chapters
10 Responsibility
11 Accidental Patterns
12 Knowledge Islands
13 Euclidean Logic
14 Deductive
& Empirical
Views of Mathematics
15 Objectivity
16 Origin of Rules
and Patterns
17 Objective Ways
18. Waking up
19. Symbols & Logic
20. Pronouns or Symbols
21. Truth Tables I.
22. Truth Tables II
22. Biconditional
22. Contrapositive
23. IF-THEN table
24. Indirect Reason Again
1A Logic Postscripts
- online only
+Proof by
Absurdity alias proof by contradiction
+How the demand
for consistency supports the law of the excluded middle
+Reality versus or with the aid of Imagination
+Links for reason, logic and crtical thinking
+Three Remarks
+History
Lost or Missing
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Caution: Site advice
is approximately correct, for some circumstances, not all.
Site How-TOs are
logically developed, but not tried and tested. That leaves
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