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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Before and After Set Theory
in pure Mathematics
Chapter 14, subsection
Previous: Set Theory & an
Euclidean Model for mathematics
Before the advent of the Zermelo-Fraenkel basis for arithmetic, mathematics
could be viewed as two subjects: geometry and algebra. Algebraic thought
included arithmetic and computations of all forms. Since the advent, mathematics
may be regarded as three subjects: geometry, algebra and analysis. The modern
subject of algebra has been restricted to a study of the form of computation and
of what kind of objects, if any, will satisfy a given set of rules for
computation. The modern subject of analysis examines numerical computation
within the set theoretic framework for arithmetic.
The Spread of Set-Theory
In universities, the set theoretic foundation for mathematics went (see [3])
from being a curiosity in the 1920s to an essential part in the 1950s – more
and more areas of mathematics were viewed from a set theoretic view. And in the
1950s and 1960s, this approach spread from the universities into high schools.
[3] According to the article Development of Modern
Mathematics, an overview, by R. L. Wilder, in the book Historical
Topics for the Mathematics Classroom, edited and written by J. K. Baumgart
et al. The book was published by the National Council of Teachers of
Mathematics, 1969, second edition 1989, 1906 Association Drive, Reston,
Virginia, USA 22091.
Godel's Disappointment
Zermelo-Fraenkel set theory with its rules (assumptions, restrictions) on what
sets could be formed from others prevented the appearance of paradoxes, those
known at the time of its construction. From the 1920s to the present (1995) no
further paradoxes or contradictions have found. Thus the theory has stood an
empirical test of time for over seventy years. But this does not guarantee in a
thought-based manner that paradoxes or contradictions within this set theory
framework will not be found.
The reliability and logical self-consistency (absence of contradictory,
mutually exclusive results) of a framework for mathematics was one of the
guiding problems for the mathematical sciences posed by David Hilbert
(1862-1943) at the turn of the 20th century. But in the 1930s, Kurt Godel showed
that a proof of the consistency was not possible in the sense wanted. In a
rule-based system for mathematics large enough to include counting with the
whole numbers, the consistency of the system could not be deduced, that is,
concluded from a chain of implications. Consistency, the absence of
contradictory, mutually exclusive assertions, could not be proven within such a
system.
The foregoing represents an approximate phrasing of Godel's results.
Godel's conclusion represents a loss for the Euclidean ideal for reasoning
with certainty from first principles: axioms or assumptions. Our assertion that
certain axioms or assumptions could be taken as self-evident was not enough to
guarantee consistency in any framework for mathematic rich enough to include the
whole numbers. This implies perhaps that the Zermelo-Fraenkel set theoretic
framework for mathematics is presently just a very refined branch of empirical
art or science. Mathematics, the queen of science, is thus but an aloof, yet
very thoughtful, commoner.
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: Euclidean Model for
Physics, Axiomatic Foundation Unlikely
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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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