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YOU are better than YOU think. Show
yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
Learn to read notes and textbooks like
a lawyer, so that no nuance, no subtlety and no clause escapes your
attention. |
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
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
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
| |
Apart from Set Theory
Chapter 14, Subsection
Previous: Euclidean Model for
Physics, Axiomatic Foundation Unlikely
A body of mathematical knowledge, a growing acquaintance with calculations
and mathematical reasoning that worked and some which did not, has been found or
built before and besides a single axiomatic foundation for mathematics.
Mathematical knowledge in the 19th century, pure and applied, was a patchwork of
assumptions and consequences. This patchwork has been extended in the 20th
century despite and besides the presence of the set theoretic framework for
mathematics. Three examples will be briefly given. A more detailed description
would require a knowledge of calculus.
In the 1930s for the study of electricity, Heaviside developed a useful tool,
the concept of a generalized function. His description of it was apart from the
set theoretic concepts and the analytic reasoning then favored by mathematics.
But his use of it worked well. In the 1950s, a cleaner mathematical theory of
generalized functions was developed. Heaviside's calculations or calculus was
essentially incorporated into and justified within the set theoretic framework
– albeit users of Heaviside calculus, that is, faculty members in
electrical engineering departments, saw no immediate need to master a
complicated mathematical justification for calculations that worked and were
familiar to them. This attitude has slowly faded away with the passage of time,
the retirement of faculty, and the mathematical training of younger engineers by
mathematics departments.
Since the 1920s, the computations of quantum physics have provided another
example of applied mathematics. Physical consideration and heuristic reasoning
led, by trial and error, and the elimination of approaches that did not work, to
equations of motion for subatomic particles. The physicists developed
their own rules of calculation. They obtained calculations that worked well.
Predicted values of constants agreed amazingly well to those measured to several
decimal places. Quantum physics or mechanics has gone through few conceptual
transformations. With each transformations, theoreticians have decried that the
reasoning that led to some of their previous computations was fortuitous. So
apparently there is a collection of computations that work, but no strict
thought-based framework for it. Suffice to say that quantum mechanics is still a
mysterious subject with a patchwork of rules for computation that have not (to
the best of my knowledge) been organized in a fully coherent, Euclidean
axiomatic fashion.
Since the 1950s and the advent of the electronic computers, engineers have
developed so called Finite Element computer models for ships, planes,
buildings, etc., via numerical experimentation along with corroboration via
observation. Here, computations were conceived and done without a rigorous
mathematical justification. Mathematical theories have been develop since the
1960s to offer some refinement or justification.
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: Decimals Absent From Pure
Mathematics
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www.whyslopes.com
Volume 1A, Pattern Based Reason
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
To reason often means to persuade someone of
the need for an idea or action. That someone could be yourself. So be
careful.
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
There is a difference between
knowing how to spend money,
and having money to spend.
There is likewise a difference
between mastering a skill
and having meeting a situation in which it applies.
.
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