Appendix 3, Consistency
as a Tool for Reason
While we may not know that a theory or set of assumptions is consistent,
or free of contradiction, we may use the requirement for consistency as
part of the reasoning process without loss of generality or harm we hope.
That is a gamble. Logician may have more say.
Example One: A detective in solving a crime may have a suspect.
Then he may found the suspect has an alibi which directly or indirectly
implies she did not committed the crime. So the alibi and suspicion are
inconsistent - that is incompatible. The detective may drop the
suspicion or challenge the alibi. Lawyers for the prosecution and
defense may erect competing chains of reason, and leave it to a jury or
judge to decide which one, if any, appears to be true. A conclusion may
follow or not.
Example Two: In developing a story or theory, we may require its
elements to be consistent. We writing or developing a story, we note
that a situation A is inconsistent with what has so far been written or
determined, we will not add situation A to the plot or theory. We will
instead accept the situation did not occur and may employ that in the
further composition of the plot, story or theory. Likewise, if that the
non-occurrence of a situation A is inconsistent with what has so far
been written or determined, we will be forced to add the occurrence of
situation A to the plot or theory. Finally, if the occurrence of A or
its non-occurrence has is independent of the plot, story or theory, we
extend the story or theory to include A or not as we like, if we are
writing fiction or describing what may be possible. On the other hand,
if we are trying to write non-fiction then the addition of statements
of patterns A to plot depends on their correspondence with reality.
In telling a story or developing a theory, we may look at the
consequences of our assumptions - the situations we tend to assume as
holding or being true. If a chain of reason implies that a situation C
occurs and does not occur, then the story or theory is inconsistent -
becomes absurd. For the sake of consistency, the story or theory needs to
be revised or abandoned.
That being said in developing a theory of how matters work, we hope that
the theory will be logically consistent. That we hope that there will be
no contradictions as a consequence of our assumptions. The consistency of
a theory is hard to prove or test, and impossible in a mathematical
theory large enough to include counting with whole numbers 1, 2, 3, 4,
...
In telling a story or developing a theory, there is an inconsistency if a
situation A and its negation Not A both occur. While we are developing a
theory from assumptions, we cannot be certain that an inconsistency A
and Not A will not occur. However, the assuming the law of excluded
middle in the development of a theory or is equivalent to the statement
that the situation A and Not A does not happen. It equivalent to
the assumption that the theory under development is consistent. That be
said, when we are developing a theory from logic and underlying
assumptions, we may not be able to prove that the theory is consistent.
If the theory is consistent, the law of excluded middle holds and so we
may used in our logical development of theory without loss of
consistency. But if the theory under development is inconsistent,
assuming or using the law of excluded middle in its development may lead
to the discovery of the inconsistency, sooner rather than later, if at
all. Assuming the law of excluded middle, simply adds another
inconsistency. .
Example Three: Assume any infinite decimal expansion locates a
point or distance on a real number line. Assume further that each ratio
of two whole numbers can be expressed a ratio of two whole numbers with
no common divisors? The Pythagorean theorem then suggests in an isosceles
right triangle, the ratio of the hypotenuse to each of the others sides,
the legs, by length given by the square root of 2. Is that square root
equal to a rational number? The suspicion or assumption that YES, the
square root of 2 equals a rational number implies an inconsistency.
Namely, that in any ratio or fraction that represents the square root of
2, the denominator and numerator will both be multiples of 2. So the
square root of two cannot be rational.
The Pythagoreans in finding the inconsistency in example 2 had a problem.
They assume lined segments in the plane represented numbers and they
assumed all such lengths were rational multiples of each other. When
these assumptions or their consequences clashed, reconciliation was not
obvious. Their view of numbers collapsed and a replacement was not
available. That was a serious problem for the Pythagorean school in their
theory of knowledge was based on and assumed rational numbers and only
rational numbers.
Today, however, we have an advantage or two. One advantage is a our
assumption that infinite decimals expansions represent rational and
irrational numbers. Physically, if you imagine ruler with a unit length
and its division into tenths, hundredths, thousandths and so on, then you
can count the maximum number of units, tenths, hundredths, thousandths
and that may fit in a line segment. The result is a sequence of numbers
which provide a better and better approximation to the line segment
length. You can further imagine that sequence of approximations given by
two, three, four, five and more decimal places in a decimal expansion
locate the end of a line segment. The decimal representation of numbers
does not depend on nor require the "numbers" to be rational.
Aside: The number 1.0000 represent a single unit
of length. It also represents the limit of the sequence 0.9, 0.99,
0.999, 0.9999 and so on. The sequence of decimal approximations 0.99999
(9 recurring) is shorthand for a sequence of lengths L(j) =1 -
10-j < 1 where the lengths are increasing, and where the
differences d(j) = 1 - L(j) = 10-j is getting smaller and
smaller as j increases. So the sequence approaches 1. Because the
difference tends to zero, the number 1 has several decimal
expansions
-
1, 1.0, 1.00 (0 recurring finitely many times)
-
1.000 (0 repeating indefinitely)
-
0.99999 (9 recurring)
where the last one represents 1 as the limit of a
sequence of approximations 0.9, 0.99, 0.999, 0.9999, 0.99999, ...
.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
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protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
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McCainian: drill, drill, drill then Toronto
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Skills with take
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time-date-calendar Matters; money matters; map, plan and
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Is your child able to add, subtract and multiply amounts
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Arithmetic
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Algebra
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Geometry
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More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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Text.
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Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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