Chapter 13, Euclidean Model of Reason
Geometric Thinking
The road and door analogies in the previous chapter describe the division of human knowledge
into sections or islands. The knowledge in each section may be strongly or weakly linked
together by implications. 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 the 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
Deduction From First Principles
The aim of the axiomatic/deductive method is to gather and to organize an island or body of
knowledge so that all parts of it can be reached from a few basic, clear and self-evident
ideas or principles. This is the axiomatic goal. The simplicity of this goal, an ideal, is
appealing.
Where or with what should we begin? The starting points and the rules used are human
selections. If one point can be reached from another, and vice-versa, then each is as good as
the other as a starting point. Changing the starting place in this manner does not change the
destinations or results reachable. Finally, different starting points for the organization of
knowledge have different advantages. A central starting place may provide faster or easier
access to the various parts and results.
The axiomatic deductive method is used in mathematics, in the physical sciences and in human
laws. The first model of the axiomatic method comes from Euclidean geometry: the works of
Euclid and his school in mathematics some two thousand years ago. This Euclidean model of
reason is deductive. It is based on supposedly self-evident facts and implication
rules. Here the fewest possible rules are used to avoid conflicts and contradictions.
Euclidean models for reason in all disciplines has been an ideal and goal for some
philosophers and religious thinkers in Europe and possibly elsewhere. The framers of the Bill
of Rights in the United States Constitution were perhaps influenced by the Euclidean example
when they started by declaring certain rights self-evident.
The axiomatic, deductive, chain-of-reason approach to a subject requires a starting point. We
try to build our knowledge and our judgments and conclusions on a few laws, principles, rules
or facts that can be assumed, or viewed as self-evident. Self-evident rules and principles
represent starting points, sometimes held beyond debate.
The laws, principles or facts we start with and pretend or assume to be true are called
hypotheses, first principles, assumptions, postulates or axioms. Which word or phrase you
use is a matter of choice. For the sake of variety, we may use all these words and
phrases interchangeably.
A set of assumptions together with their consequences, that is, the conclusions which can be
obtained from them, form and define a theory. (We mortals will only see a finite number of
the consequences.) The set of assumptions on which a theory is built is called a foundation.
Again, the assumptions forming the foundation are supposed to be self-evident, clear and
credible. Identifying the self-evident ones has for mankind been a matter of trial and error,
and perhaps a matter of culture.
Clever Mortals
People (clever mortals) have tried to base mathematics, physical sciences and philosophy on a
few true (or reliable) first principles: assumptions, rules, laws, postulates, axioms or
whatever you would call them. One fear in the human selection of the rules is that they or
their consequences may conflict. That is, they may imply that one situation is and is not
possible. To avoid such contradictions, we are afraid to assume too much. [1]
[1] Contradictions and paradoxes may appear when
deductive reason leads from ideas assumed or thought to be self-evident, to
mutually exclusive claims or predictions. One way to avoid their appearance is to
assume fewer ideas, that is, be more selective in making assumptions. Yet how selective to
be is not always self-evident.
Proof is desired to show that all possible chains of reasons will avoid contradictions. A
contradiction is given by statements that say two exclusive situations must or will occur
simultaneously.
Despite these fears, we assume a few principles and implication rules to reach or make
conclusions. What to assume or take for granted is a question for philosophers and for
practical people as well. In practice, we need rules tested for the situation at hand. On
these we build, while being aware of their limitations and shortcomings. That is perhaps, the
most that can be done.
The search for a foundation of mathematical and physical knowledge has been a collective
effort. It has been pursued via trial and error. Like other human concepts, the resulting
laws/implication rules may be workable and acceptable. There is no guarantee of completeness
or consistency, no matter how much that is desired.
Caution
Before you rush out to look at Euclid's works, or a modern translation and formulation of it,
note the following. Euclid's works on geometry provided the model for reason by deduction
from clear, self-evident starting points. Our treatment of geometry and mathematics differs
today from that of Euclid. The presentation of mathematics in schools has changed, and is
continuing to change. So Euclid's works may be hard to read. None the less, Euclid's works
still represent the first example of the axiomatic method in practice. For that it is
remembered.
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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
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
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
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
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Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
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Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
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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