Appetizers and Lessons for Mathematics and Reason   
www.whyslopes.com          ( Français
 Logic mastery is key to easing or avoiding learning difficulties in work & studies. 

Online Volumes (Book 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

Site  Folders for Instructors & Adults
A. Public Policy Essays
B.  Mathematics  Education Essays  2006-7
C -Logic & Applied Math Program  
    for education,  June 22, 2008 
D. Quebec English Math Ed -  Standards to
 avoid  in course design & teacher education 
E. Help your child or teen
How TOs/ Ref.-08- 2008
1. Arithmetic Reference
2. Algebra 
3. More Algebra 
4. Geometry  
5. More Geometry
6. Calculus
7. Logics in Maths
16 End Notes      Back ] Up ]


Pattern
Based
Reason

understanding & explaining
Reason and Math
Volume 1A
Printed in Canada
ISBN 0-9697564-5-3

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. 


YOU are better than YOU think. Show yourself  how:

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 For better work & study skills, read logic chapters 1 to 5  in  Three Skills for Algebra. Sooner is better. Good luck.

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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.


Explore collaborative whiteboards from groupboardtwiddla  or scriblink.


End Notes and Review
Chapter 16

Previous: Statistical Inference and It Limitations

1. In a rule which suggests that whenever a first situation is made to happen, a second situation will follow, the first situation is called a possible cause of the second. The second situation is also said to be a possible effect or consequence of the first.

2. Human made rules or models for nature's behavior suggest or describe patterns without explaining why they occur. Science and technology are mixtures of facts, guidelines and recipes. Some parts are certain or almost certain. Other parts are less certain. The empirical approach to knowledge tries to identify those repeatable, reproducible processes: processes that work, accidentally or otherwise.

3. Scientific and technical knowledge can be viewed as a collection of theories or recognized patterns and recipes (implication rules). Details accompanying such rules should say when they do or don't apply – the range of applicability. Knowledge of this range can be unclear. Our knowledge of the physical sciences forms both a collection of recorded patterns or recipes for solving some problems and a collection of unsolved problems.

4. The unsolved problems (or mysteries) say or indicate that more work is required. We humans have discovered many skills and techniques, wonderful or not. In any area of application, only a few of these skills are pertinent, that is, applicable. In any area of application, further skills or techniques are often required. In technical areas, we find two kinds of knowledge: a knowledge of processes that work and a knowledge of processes that don't. So more work is required on them. Whether or not this work is feasible always remains to be seen.

5. Technical knowledge is based on repeatable and reproducible methods, along with some trial and error from deliberate experimentation (sometimes accidents) to find them. As human beings, we can spot or imagine patterns. From them, we try to predict what will happen.

6. Creativity and subjectivity (guesses, past knowledge and experience, guidelines/assumptions) are involved in deciding what chains of implications to form or investigate. Once a chain of implications with an interesting result or conclusion has been discovered, the result or conclusion and how it was obtained can be shown to others. The path to such a result or conclusion can then be repeated by others. Mathematics, engineering, science, chemistry, cooking, computer science, all these disciplines follow this pattern of discovery and repetition or reproducibility. Reporting how a conclusion or goal was obtained or missed is a result. It is a result which informs how something was done (or missed). We can learn from the experience and the errors of others and ourselves.


Chapter Sections: 16 Private Agreements ] 16 Public Laws ] 16 Physical Laws ] 16 Accidental Patterns ] 16 Reliable(?) Patterns ] 16 Scientific Method ] 16 Reaction to Failed Tests ] 16 Chaos ] 16 Statistical Inference ] [ 16 End Notes ]

Next: Chapter 17, Objective Ways (Trial and Error Discovery, Etc)

 


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 room for thought and refinement..

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