|
YOU are better than YOU think. Show
yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
|
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. |
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
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;
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
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.
| |
Euclidean Model for Physics
Full Axiomatic Foundation or Codification Unlikely
Chapter 14, subsection
Previous: Before and After Set
Theory in Mathematics
Hilbert also posed the problem of the axiomatic foundation of physics. The
solution is not near. Advances in physics – a description of nature — have
come from experiment and suggestive calculations. These calculations represent
patterns (islands of knowledge) that work well, but comprehension of why is
inadequate. Physics and much of technology form collections of patterns that
work. These subjects remain far from a complete axiomatic organization in which
all their results can be deduced rigorously from a few clear principles. Islands
of knowledge remain.
Operational Views
Today with the aid of electronic computers, engineers, physicists and now
numerical analysts explore, refine and invent mathematical calculations. They
may try to find reproducible calculations in accord with experience. The
calculations are found or discovered through a mixture of deductive reason,
knowledge of the efforts of others, and trial and error (guesses). This yields
an empirical knowledge or view of computation. This empirical approach to
computation may use deductive reason where it can, and trial and error, after
that. Here engineers and physicists, sometimes using and sometimes in ignorance
or defiance of the rigorous, rule-based parts of mathematics, often find
calculations that work. See [4]
[4] Anyone applying mathematics in their specialty is
looking for the simplest mathematics that can be employed to solve their
problems, but the search for the simplest solution may require a vast
knowledge of mathematics. The simplest solution method that works may be
explained to people with a minimal mathematical background, but such a
background may not be enough to find it in the first instance – exceptions
here are always possible.
Within the empirical approach just described, never-disobeyed rules (or rules
we have confidence in) may be combined with less sure rules to propose methods
that might work. The reliability of a proposed method can then be examined and
the circumstances in which it works or it fails can be recorded. Of course,
reason should be based on the most reliable implication rules.
In the application of mathematics to other fields, physical observation and
physical theories may suggest or imply the initial equations. Physical arguments
and observation are needed to obtain the equations of science and technology,
but after the formulation of these equations, their solution and manipulation
(the accounting) should rely only on the more secure assumptions about
arithmetic.
To find or determine the motion of objects observed in nature may be the
motivation for constructing an equation. But the construction process may be
inexact. Due to this inexactness, whether or not an equation has solutions is a
question whose answer may be suggested by physical consideration, but not
guaranteed. In contrast to mathematicians, engineers and scientists, through
exposure to calculations that work, may believe that modeling a system or
physical process by equations is sufficient to guarantee the existence and
computability of a solution.
None the less, engineers, scientists and applied mathematicians may proceed
by trial and error; and in this use any equation they have constructed as a
guide. This trial and error may fail or may succeed and go ahead of the
rule-based reason, the formal justification that might provide a place
for its successes in the set theoretic, rule-based framework for mathematics. In
previous centuries and in the absence of the set-based framework for arithmetic
and computation, calculation had to proceed in a manner secured by geometric
arguments, by physical considerations or by more speculative means.
The discovery or formulation of schemes for the solution of the equations or
problems may be guided by physical considerations or expectations. Yet the use
of physical arguments and mathematical approximations to draw conclusions of a
mathematical nature is suspect — represents weak links in the chains of reason
followed. The best that can be done is to recognize the weak links and look for
replacements. Again chains of reason, even those with weak links, may provide a
guide or a clue to a more rigorous and secure approach to computations that
might work. To minimize uncertainty, a solution method and the justification of
its steps should depend as much as possible on rigorous mathematical arguments
or accounting. But an engineer or scientist whose calculations have given
results in accord with observation and measurement may feel that the links, weak
or not, in the chain of reasoning are strong enough. They worked. From this
limited (and sometimes very practical) perspective, there is no immediate need
to look for and replace the weakest links.
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: Applied Mathematics apart from
Modern Mathematics (Without or Apart from Sets)
| |
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.
.
|