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Quick Links:
Foreword: This work Pattern
Based Reason surveys rule and pattern based thought in daily life, society,
science and technology.
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1
Introduction
2 Communication
3 Elements of Reason
4 Implication Rules
5 Deception
6 Chains of Reason
7 Longer Chains - mathematical induction with
a Romeo and Juliet perspective.
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8 A Language Change
9 The Next Chapters
10 Responsibility
11 Accidental Patterns
12 Knowledge Islands
13 Euclidean Logic
14 Views of Math
15 Objectivity
16 Origin of Patterns
17 Objective Ways
18 Sense+Knowledge |
20
Pronouns in Logic
21 Occurrence Tables,
22 The Contrapositive
23 Truth Tables
24
Direct and Indirect Reason
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| 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 imagination] [Links for
reason, logic and critical thinking] [ Three
Remarks] |
| Summary:
Volume 1A, Pattern Based Reason, describes
logic, critical thinking and problem solving skills for many arts and
disciplines. Read it to learn about the benefits, origins,
limits and risks of rule- and pattern-based activities and
explanations; to develop a critical command and understanding of science
and technology before defending or attacking any part; to learn how patterns
are suspected or recognized, and learn what patterns can be tested before
jumping to conclusions or alternatives. . This work provides base for work and
studies, decision-making, in many arts and disciplines at work and
school. |
Chapter Descriptions:
Chapter 1 Introduction:
To reason often means to persuade someone of the need for an idea or action.
That someone could be yourself. In the latter case, reasoning may mean following
a line or pattern of thought to arrive at a conclusion, action or decision.
Chapter 2 Communication
: No area of knowledge is properly mastered until it can be
readily explained to others. Each subject needs paths (or curricula) passing
through easily described and easily repeated ideas and skills. Each such path
permits those who have traveled along it to tell others what to expect and
hopefully why. The existence of such paths may show that an area is
well-understood.
Concludes with Inductive Principles for Instruction
Chapter 3 Elements of Reason
: Chapters four to eight describe the basic elements of rule- and
pattern-based thought and hint at their benefits and limitations. In particular,
the next three chapters, Implication
Rules, Deception and Chains
of Reason describe basic ideas in reason and logic which everyone should
master.
Chapter 4 Implication Rules:
Are you a careful thinker? Can you understand exactly the meaning
of a rule or pattern? Instructions for building or creating provide rules and
patterns which say and suggest that when this is done, that should happen.
Every cook and dressmaker knows the importance of following instructions
carefully. Instructions and suggestions which are not repeatable and results
which are not reproducible are not of interest to a cook or dressmaker.
[ Chapter
Entrance ] [First Puzzle] [Second
Puzzle ] [ One-
Versus Two-Way ] [ Talking About Logic
] [ Implications versus or as Suggestions
] [ Implications Versus Suggestions ]
[ Repeatable & Reproducible ] [
Limits and Benefits ] [ Accidental
Rules ] [ Steps for Better Reason ]
Law of the Excluded
Middle
essay Dec 1, 2008
Logic I of Partial Inclusion : Let B
be a region in a one occupant house. We say B is true when the
occupant is partially in B. Likewise we say Not B is true when the
occupant is partially in the rest of the house. Now for that
occupant, the statement
B or Not B
holds, but the two events B and Not B
may occur simultaneously. That is the assertion
B and Not B
may be true. They are not mutually
exclusive. Whence to say B holds is not to say Not B does not, and
vice-versa.
Logic II of Full Exclusion:
Again, let B be a region in a one occupant house. We say B is true
when the occupant is fully in B. Likewise we say Not B is true when
the occupant is fully in the rest of the house. Now for that
occupant, the statements B and Not B are mutually exclusive.
Thus
B and Not B never both hold.
However the
statement
B or Not B
fails when the occupant is partially in
both.
A More Careful Logic III: Yet
again, again, let B be a region in a one occupant house. But this
time, let A be the statement that
the occupant is partially in B.
Then Not A would be the statement
the occupants is not partially in B - the
occupant is fully out of it.
Then statement A and the Not A are mutually
exclusive: The statement
A and NOT A
can never hold. Moreover, the
statement
A or Not A
will be hold as well. So Not (Not A)
implies A and A implies Not (Not A). That be said, even though the latter
holds, we still may be a state of ignorance which one occurs and
when.
The Law of Excluded Middle. This law
holds when the statement when an assertion C and Not C are (a) mutually
exclusive and (b) at least one of the two statement C or Not C
occurs. The law of excluded middle fails for logic I and II, but
holds for logic III.
is always false. The falsity of the latter
is equivalent (check this) to saying the statement
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| Consistency in Developing a story or
Theory
Our dreams and nightmares need not to be
consistent. Contradictory events may appear in them.
Contradictory events may also appear in a work of fiction. In composing a
story or a developing theory, the teller or composer may require
consistency. As the plot or theory expands, the teller or composer may
exclude events or statements that are directly inconsistent with prior
elements or which imply via chains of reason an inconsistency.
As the plot thickens or theory expands, an event or pattern A may be
possible. In the case where A and Not A are mutually exclusive, and at
least one is required to occur, the inconsistency of event or pattern A
with the prior elements of the story or theory implies not A, and the
inconsistency of Not A with the prior elements of the story or
theory implies A. The foregoing represents an indirect argument for
A or Not A. That direct chains of reason may also imply one or
another.
In the telling or development of a story, the writer may
have many chooses. For each possibility A where both A and not A are
consistent with prior elements of the story, neither being implied by
prior elements, the story developer may choose to include A,
to include not A in the story. Not A might be represented by alternative A
or nothing. A work of fiction may result. That work may come alive
on stage in play or a further kind of show. And the fiction may be told in
a repeatable and reproducible fashion. And in fiction, if there
happens to be an inconsistency, despite the best efforts of the writer,
the world survives it.
In developing a theory which is attempting to be
non-fictional, or which attempts to correspond to reality, that is what we
see and observe with a our senses, the developer of an art,
discipline or theory for such may try to follow the Euclidean model for reason
present in geometry. That requires the identification of key elements in
the environment and the assumption or inference of patterns and practices
to assume and employ in developing theory, the art or the discipline
in a deductive or constructive (practice-based) manner.
Recorded constructions and chains of reason in the theory, art or
discipline may then be followed in a repeatable and reproducible manner to
repeatable and reproducible results. Those results may be reached in
more than one in sufficiently rich theories, arts and practices.
Different ways may come and go, be recorded and/or forgotten, or lose
popularity.
In the case of applied mathematics, rules, patterns,
conventions and practices may come ahead of or besides pure theory.
One role of theory is to provide a fuller or richer understanding of the
practices and the links between them. Even if the theory is wrong or
inconsistent, the practices may continue. A role of mathematics education
from first steps to the end of calculus may be to develop a knowledge of
the geometric, numerical and algebraic practices and habits sufficient by
themselves for applications and further thought, and sufficient if
possible and if wanted for the further study of mathematics theory.
Just a computer starts (or use to start) with an machine level code, very
basic and not high level, a student needs to master geometry, numerical
methods, and the algebraic-deductive way of writing and reasoning,
before modern mathematics may be seen and understood.
That being said, the Euclidean model or idea for an
axiomatic (assumed pattern) mono-theory development of an art or
discipline appears to be very, very successful in pure mathematics,
but not in applied mathematics. In the latter, practices are explore
numerically and formally beyond axiomatic rhyme and
reason. And in physical science or chemistry, there are
competing theories. Litmus paper provides the principal way to
classify an ionic solution as as a base or acid. Molecular formulas
or notation for ionic and (?) polar substances, and further methods, ,
provide more acid-salt-base classification schemes or theory not fully
consistent in practice with the Litmus test nor each other. Whence
rules and conventions in physical science are based on overlapping partial
theories, and not just a single theory, with each partial theory being
practical in its own domain. And each theory provides a partial guide to
what can be done or tried, and in need of experimental verification. Thus
practice may advance in a technical, combinatorial manner, with each step
or advance be subject to a check, to ensure it is repeatable and
reproducible, and hence verifiable.
We will stop here. The site author has run out of
steam. To learn more, see the chapters and postscripts in this
volume on direct and indirect reason - proof by absurdity and
contradiction included. Visit too Mathematics
Education Essays site folder and find the essays comparing and
contrasting mathematics and science education. Is a theory but a story
told in a effort to represent reality and to provide model of it, reliable
enough for decision making?
telling, repeating & following stories and theories, fiction or
not, consistent or not, checked
or not, may be the basis for
learning, teaching and sharing
a common knowledge of ideas & methods,
In writing a story or telling a theory, we may reject an
extension or a possibility for the sake of consistency if it implies
an inconsistency. Stories may be based on independent elements.
In writing a story or a theory, we are creating a
virtual reality, one that need not follow any laws. In literature,
one might say a story or a play has an existence all of its own apart from
any reality it may try to emulate or not. Yet a good story, even a
work of fiction, may hold lessons for reality - include models.
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Chapter 5 Deception:
People try to persuade us in many ways. We need to recognize the fair and unfair
ways, or the sensible and nonsensical ways. In persuading ourselves and others,
we need to recognize and appreciate or reward careful logic. Efforts to persuade
and lead us are met in advertising, public relations, political campaigns,
religion, law, business, mathematics courses (yes), and even your family
Chapter 6 Chains of Reason:
This chapter shows how reliable rules and patterns can be directly
employed repeatedly, one at a time, or one after another, to get conclusions or
further reliable rules and patterns. The question of what rules are reliable is
considered in the following chapters.
[Chapter
Entrance] [From a Single Rule] [Linking
and Chaining - Two Rules] [Putting
Several Rules Together] [Deductive,
Inductive and Empirical Reason]
Chapter 7 Longer Chains:
This chapter explains one version of
inductive reason: the recursive or repetitive approach to putting one-way
implication rules together, one after another. This chapter ends with a
description of the principle of mathematical induction – another method for
obtaining conclusions used only in mathematical arguments or computations. There
is more to mathematics than just doing arithmetic.
Chapter 8. A Language Change
(or two). The foregoing development of logic coins the terms one and two-way
implication. The latter can be identified with conditional and
biconditional statements. That being said, if we write B IF A for
the implication A implies B or IF A THEN B, the phrase B IF AND ONLY IF A means
there is no other situation or condition C with B IF C (unless C implies A
as well).
Speculation: Mathematics and logic education, and its
choice of words in North America, or outside of the UK, was influence by
Europeans, expert in subject matter, without a poetic command of English, that
needed to make technical concepts more accessible to students and teachers.
Chapter 9 The Next Chapters:
The problem of identifying reliable implication rules and reliable
information is described but not solved, except for the description of empirical
methods of coping in science and technology. This identification problem touches
many subjects. Students of critical thinking, persuasion, philosophy,
mathematics, science and technology should find its discussion in these chapters
helpful.
Chapter 10 Responsibility:
In this chapter, we give a short story: a conflict between the owners of
a cat and a dog about who or what is responsible for an accident. The murky
situation leads into a discussion of cause and effect, and then responsibility
versus freedom (the limits of freedom) and the absence of liability. Finally,
first principles or patterns for the assignment of responsibility and liability
are stated or suggested last.
[Chapter
Entrance - Felix versus Suzy] [Limits to Freedom ]
[Where does Responsibility
begin or end? who is to blame? Principle to Consider? ]
Chapter 11 Accidental Patterns:
What do we mean, when we say you have caused something to occur? In life
we may see a pattern that whenever a first situation occurs, so does a second.
The pattern could hold true accidentally. There may be no relationship between
the two situations or events. Alternatively, there might be some relationship.
We need in a sense to measure this relationship. We need to measure how much one
event forces, pushes or contributes to the occurrence of another event. This
measurement signals to what extent the first event is a cause or is the
cause of the second. Observation by itself is suggestive but not conclusive.
Examples to support this view follow.
Chapter 12 Knowledge Islands:
Whenever the building we are exploring has sections closed off or
unreachable, we can ignore all maps of those sections. Making a map of the
unreachable sections is not possible, except by guessing. Guessing is
suggestive, yet not reliable.
[ Chapter Entrance
] [12. Two Analogies or Metaphor for the division and
organization of know-how and even know-why ]
Chapter 13 Euclidean Logic:
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?
Chapter 14 Views of Math:
This chapter provides several perspectives on mathematics.
[ Chapter
Entrance - Set Theory ] [ Before
& After Set Theory in Pure Mathematics ] [ Euclidean
Model for Physics ] [ Applied
Maths and Electricity Apart from Sets ] [ Decimals
Absent From Pure Mathematics ]
[Modern
Mathematics Education ]
Some are slightly at odds. Some are slightly technical. The next chapter Objective
Processes returns to some simpler material.
Volume 2, Chapter 19, Functions and Sets, and Volume 1B,
Mathematics Curriculum Notes, and the rest of this site, material written
later, give further views on mathematics education, what was, what is and what
could be]
Chapter 15: Objectivity:
Recipes and rule-based processes, when carefully done, give results independent
of who obtains them. In this situation, the results cease to be subjective —
that is dependent on the person getting them – and they depend only on the
context. In this situation, the results are said to be objective. ... The main
advantage of objective (rule-based) reason and processes is as follows. Once we
have agreed upon the rules and recipes and on the evidence or ingredients to
use, the results obtained are independent of who or what obtains them.
[Chapter
Entrance] [ The search for Repeatable and Reproducible
Results
Chapter 16 Origin of Patterns:
A rule, law or agreement may say that when one event happens, another event
should also happen or may also happen. Most physical and legal theories, if not
all, use rules which are approximately correct. The rules are like all human
discoveries and creations; some are more reliable than others. The formulation
of laws and rules and agreements by people leads to the chance of error and
incompleteness. Even with uncertainty, once rules or laws or agreements have
been stated, we can use them tentatively, to reach conclusions or judgments.
Locating the weakest links in our reasoning gives us a chance to strengthen or
replace them.
[ Chapter
Entrance - Origin of Patterns ] [ Private
Agreements ] [ Public Laws
] [ Physical Laws ]
[ Accidental Patterns ]
[ Reliable(?) Patterns ]
[ Scientific Method ] [
Reaction to Failed Tests
] [ Chaos ] [ Statistical
Inference ] [ End Notes ]
Chapter 17 Discovery
of Objective Ways: Knowledge of what others have done or tried
to do may help and guide our actions. Without previous know-how and knowledge,
we need to improvise and look for patterns, rules and recipes that work. This is
where the search for objective reason, or simple rules to follow, becomes
subjective. Each may have a different idea of where to look. This is because
each person has a different background and varied preferences. The road to
objectivity is in part subjective and creative.
[Discovery of
Objective Ways - Yours Objectively in Creative and Subjective Manners] [17.
Discovery Process - Trial and Error Discovery)]
Chapter 18
Sense+Knowledge: Consciousness and thought appears in infancy or
childhood. There they may be initially taken for granted or not explicitly
noticed. Only later are they questioned, if they are questioned at all.
Vagueness of memory may hide the days when consciousness and thought began. A
few speculative remarks follow.
More About Logic:
The last five chapters 20 to 24 give a technical view of logic and also enter
the discussion of direct and indirect methods for reason. The latter discussion
is continued in online postscripts - material not in the printed or printable
version of Volume 1A.
Chapter 20, Shorthand or Pronouns in
Logic, introduces the use of letters A and B, and possibly others
first to represent situations that can occur or not, and second to represent
phrases or statements that may be true or false (or neither). Talking about
pronouns, the pronoun metaphor, and talking about shorthand, represent one or
two ways to introduce the the shorthand role of letters in logic and more
generally in mathematics.
The online Volume 2, Three Skills For Algebra, in Chapters 8
and 9, and in the online postscript, What is a Variable, go further in
Euclidating or clarifying the shorthand role of letters and symbols in logic
and algebra, or symbol based, shorthand paths, for arriving at conclusions
with implication rules and formulas (or numbers)
Chapter 21 coins or introduces Occurrence
Tables. for three phrases A AND
B; A OR B; and NOT A; for one
way implications B IF A, and for two-way
implications B IF and ONLY IF A. The last section of Chapter 21 defines Converses
to One Way Implications and so digresses from the earlier content of the
chapter.
The occurrence (or obedience) tables invented and introduced
in Chapter 21, Occurrence Tables,
identify those situations in which implication rules are obeyed, disobeyed or
not disobeyed. The latter notions are intended to simplify or justify the
explanation of truth tables for the implication B IF A, or if you prefer, the
implication, IF A THEN B.
Chapter 22, The Contrapositive
shows the equivalence of an implication rule with its contrapositive
formulation - meaning B IF A holds when and only when NOT A IF NOT B
holds. The analysis is based on the three notions of a rule being (i)
obeyed, (ii) disobeyed or v(iii) not disobeyed. An implication rule B IF A
or IF A THEN B is Vacuously True
when and only when it never applies - that is when situation A never occurs.
In the latter case B or NOT B implies NOT A is a tautology.
Chapter 24, Direct
and Indirect Reason describes and explains direct and indirect methods for
reaching or proving conclusions. Among the indirect methods, this chapter
describes in particular, how an implication rule can be shown to always hold by
(a) showing its contrapositive form always hold, or by (b) looking for
absurdities that would occur if the implication rule did not hold. The second
method (b) is more indirect than the first method (a).
Online Postscripts: While we may not know that a
theory is consistent, we use the requirement for consistency as part of
the reasoning process without loss of generality or harm we
hope. See Proof by
Absurdity alias proof by contradiction and see How
the demand for consistency supports the law of the excluded middle
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Professor Whyslopes
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<| (o) (o) |>
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says
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Site value lies in the difference
between its ideas and yours.
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If one site explanation is too full
for your liking, try another.
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Visit the site entrance whyslopes.com
before you go.
Bon
Appetit |
Two common gaps to fill or avoid
- The Old Algebra Gap: Algebra
appears with too few words of explanation in high school and college
mathematics. Chapters
8 to 12 in online Volume 2 put more words into the
explanation and comprehension of algebra. Chapter
14 in Volume 2 with its explicit discussion of the direct and
indirect use a formulas identifies a unifying theme for mathematics
and logic - all rules and patterns will be used forward and backwards.
Chapters
2 to 6 and 12 to 18 in Volume 3 may further ease or avoid the very
challenging use of algebra in the high level mathematics: calculus.
Calculus requires earlier high school mathematics at full strength: (i)
This logically complete but long lesson on complex
numbers shows how to simplify the senior high school
exposition of circular trig functions upto to formulas in the plane
for vectors dot and cross-products. The lesson provides the
route that would have been taken in course design if the key element
of the lesson, a December 2009 invention, had been available in
the 1950s. For further algebra skill development. See the site
coverage of fraction
with units, proportionality,
ratios and rates,
polynomials, quadratics
functions
and straight
line slopes and equations.
- The Arithmetic Gap: An exact and efficient
mastery of arithmetic with decimals and fractions is best (required)
for the high level study of mathematics alone and in science,
technology and business. Pages here on arithmetic
with decimals and integers, on fractions
and solving
linear equations with fractional
operations on stick diagrams may help fill the gap. That
exact and efficient command should be obtained in the last years of
primary school and the first years of secondary school.
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Would you believe: Skill
mastery in mathematics has to be seen to believed. To that
end, learn or teach how-to write and draw the steps in
mathematical figuring or reasoning clearly. Do not try
to save space by doing a sequence of step in one place. Instead, do
or record the steps in sequence on a separate lines to make each
step obvious and verifiable.
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Hire the site
author, as an online instructor, a technical support for
teachers, or advisor for curriculum review.
Site
Reviews may serve as references. See how online
whiteboards with voice and real-time writing make
online help possible - board content printable. Text
or written work scanned or saved to a pdf may be
uploaded for discussion in the whiteboard. |
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www.whyslopes.com
Parents: Help
your Child/Teen Learn
Online Volumes (orders)
1, Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3 .Why.Slopes.&.More.Math.1995
Math
How-TOs etc 2008
1. Arithmetic
2. Algebra
3. More
Algebra
4. Geometry
5. More
Geometry
6. Calculus
Site
Description/Reviews by 3rd parties
Site
Math Lessons
1. Arithmetic
Flash Videos 11-2008
2. Algebra Videos (to appear)
3. Fractions
and More
4.. Solving
Linear Equations 04-2005
5. Euclidean-Geometry
To Complex No.s
6. Analytic
Geometry/Functions 2006
7. Number
Theory. 2006-7
8. Exponents,
Radicals & logs. 2008
9 Calculus
2005
10..Real
Analysis 1995
11 Electric
Circuits Etc 2007
12. .Algebra,
Odds & Ends, HS level-2001
For Math
Instructors/Tutors/
Curriculum Committees
1. K0-11Applied Math Program Outline
2. Mathematics
education essays
3. LAMP
- an earlier applied math program.
4. Maps,
Plans, Similarity &Trig, with
Complex Numbers, 12-2009. (150 pages)
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