The
Creation of Order
Introduction
The world is massively
chaotic. The complexity and complications of living within an
increasingly populated and technological world are steadily growing to
enormous proportions. The ability to create order in this chaos
is a prime criterion to individual and world sanity, and the mind is
that basic tool which can create such order.
Our educational systems today fill our minds with
massive knowledge and the preferred analysis. Then they reward
the student for "right" answers. The result is we no longer think
for ourselves.
Knowledge is important, but only when it is linked
with other skills. Einstein said, "Imagination is more important
than knowledge". Today with these educational institutions
supplying you with knowledge and their analysis at the price of
creativity and the ability to analyze, imagination is more important
than ever. It is not your job to simply answer questions, but
rather to do your own analysis, speculate, create order and solve
problems.
The very same tools
can be used as a structure for peacemaking conversation, a structure
which people can use to talk about important things without loss of
face by conceding "truth" to belief and belief to taste so that they
can merge ideas into better ones.
Peacemaking conversation is generally not
peaceful. It involves talking about disagreement. The tools
make it possible to disagree with understanding and without
anger. When we refuse to talk about politics and religion we give
up self-determination in favor of copying the beliefs of our political
and religious leaders.
Mathematics as a Science of Use
Intuition has been evaded as a mathematics for a
long time because generally people have chosen to deny that it can be
organized. This text will show how. Not only can
organization give intuitive people the power of mathematics, but it can
also give mathematical people the power of intuition.
The purpose of mathematics
is to aid people in dealing with real problems. It makes it
possible to predict a result without performing the corresponding
experiment. This ability to predict is accomplished through the
utilization of a method: first, to assume facts; second, to symbolize
these facts; third, to rearrange the symbols according to assumed
working operations; and fourth to re-associate the symbols with the
facts. Thus they can make a prediction of what would happen to
the facts if these facts, rather than the symbols, were
manipulated. The more accurately the facts can be symbolized, the
more useful is the symbolic outcome. The more the symbols can be
accurately manipulated, the more ways facts can be predicted.
Language is such a symbolism.
The usefulness of our
assumptions assumes that we have agreed to symbolize something in the
same way and have used the same set of rules. Since nouns are
arbitrarily associated with objects there must be a shared
understanding that we will call any given thing by the same name. For
instance we might agree to call an object “table “when speaking with
English people but call it” mesa “when speaking with Spanish
people. In the world one person would concede the word to the
other. The concession would only be in naming the object.
The same is true for all the other parts of
speech. The concession is the same in that we only agree to what
a word means.
The same is true for grammar. The concession
we make is in agreeing to how we structure the words.
Man, as he learns,
assimilates the symbols of his language and tends to force his various
observations to fit those already fixed symbols. Often highly
educated men have difficulty grasping a foreign idea. Difficulties in
translating books stem from the same limitations in language.
Literal translation of a phrase into another language often does not
convey the same meaning since; connotative elements in both languages
may be radically different.
Consider the difficulty in
translating an eastern philosophy for western readers. Every
idiom, phrase, or word which does not have a translation must be
altered to something which can be translated. The altered version
has lost some meaning and the westerner does not receive exactly what
the eastern writer wish to convey. If the westerner could free
himself from existing language limitations he would understand more.
The easterner must coin new words while the
westerner must forget his existing language. The easterner would coin a
new word, elaborately define this new word, and finally convey the idea
he wishes the westerner to have. He would be able to recall this idea
by writing the word when he needed it. By continued use, the
meaning of the word would become clear. The easterner has
invented a mathematics; he has coined a new symbol because it was
useful to do so.
If we have a simple declarative sentence written
within common (agreed to by both parties) grammar using words with
common definitions, it is usually easy to agree to what a sentence says
on the face of it. In problem solving where you begin with the
problem rather than a set of words, you create common definitions and
grammar. The declarative sentence is king.
In mathematics
anything can be symbolized; connotation, denotation, differences,
similarities, changes, abstractions and operations are only a
few. These symbols can be arranged in any way that helps solve a
problem. You are not limited by your initial
vocabulary or previously established rules of manipulation.
Sometimes language is a mathematics with the purpose
of understanding one another. Words and word sets are symbols.
When they are arranged according to rules they convey meaning. Symbols
have been manipulated according to a set of rules to aid people in
understanding one another.
In peacemaking conversation you frequently are
attempting to understand the same text. When confronted with
hyperbole, simile, metaphor, idioms, parables,
words with heavy connotation, poetic structure, stream of
consciousness, imagery, bad grammar or simply purposefully confusing,
then the parties must identify meaning. Sometimes it is enough to
say, "It is a parable" or "I am speaking figuratively". However,
it is usually necessary to bring in all their witness to find a common
understanding. It is not a common belief. The understanding
is only an agreement to the meaning of the phrase, paragraph, chapter,
book, et al.
Consistency
A mathematics is called internally consistent if,
when used accurately, it produces approximately the same results
regardless of the manipulator. For example, if two people read
the same paragraph and both apply the mathematics to summarize a
paragraph independent of each other, their summaries should be similar
or the mathematics is not internally consistent.
Internal consistency is destroyed most often by
statements that can be interpreted in more than one way relative to the
problem, or by leaving several paths open that do not lead to the same
conclusion. Imperfect internal consistency is frequent outside
classical mathematics. It should be kept in mind as a guideline.
External consistency is more elusive. For a
mathematics to be externally consistent it must be consistent with the
real world. Everyone has heard of the perfectly designed bridge
that collapsed or the perfect gambling system that bankrupts its
user. In these cases, no matter who used the system, they
achieved the same result. It appeared to solve the problem on
paper, but when applied to the real world, it failed. It was
internally consistent but externally inconsistent.
External consistency is a major guideline to
mathematics design. For mathematics to be externally consistent
all assumptions must be universally and completely
accepted. As there is nothing that is completely accepted
by everyone, then everything is not accepted by everyone. That
would be one complete universal truth, something I do not deal
with. Design your mathematics so that it is useful.
In peacemaking conversation consistency is
everything. Declarative sentences are rare, yet most
helpful.
Let us assume we wish to know (and agree upon) what
a sentence means. We would each bring our common (agreed to by
both parties) experience to the table. First, we would use our
common objective definition and grammar. The legal question for
this is, “what does it say on the face of it?”
If we have a simple declarative sentence written
within proper (common) grammar using words with common definitions, it
is usually easy to agree to what a sentence says on the face of
it. Most legal documents contain only simple declarative
sentences.
On the other hand one might say “da da da da
da”. Now to me it sounds like gibberish.
If communication is our goal my suggestion is to add
from our common witness our common grammar and definition. In
this case (“da da da da da”) we would probably agree to throw out
grammar and select instead to view it as verse. If we were to
agree that it expresses a rhythm of baby sounds, then it would require
that we bring in from our witness an understanding of rhythm and baby
sounds. We agree that it's a baby singing. That was easy.
Let's try something with some
meat.
Let’s look at the first line of the gospel according
to John, King James Version. "In the beginning was the Word and
the Word was with God and the Word was God."
You might accept it as true without attempting to
paraphrase and view it as part of the whole testament. It is
called “verse” and not subject to the laws of grammar.
On the face of it, its nonsense.
Correcting the grammar the sentence would become, "In the beginning was
the Word and the Word was with God and the Word was “God”. If the
“Word” is literal then the word is “God” and God possessed his
name. You would be hard pressed to find anyone that believed John
wished this to be the understanding.
If “the Word” is figurative such as a specific holy
knowledge, then God is this knowledge and God possesses this knowledge.
There would be a sizable body of people whose witness agreed with this
understanding.
If “the Word” is a metaphor for existential
existence, then in the beginning there was God and his existential
existence. Bringing some more witness you might paraphrase this as God
and the Holy Spirit.
If “the Word” is idiomatic as in “give me your
word”, then in the beginning there was a covenant with God.
If “in the beginning” refers to the time when Jesus
would come, then “the Word” might actually refer to the Old
Testament. Then at the time of Jesus there was the Old Testament
which was God’s Word.
The important point is that every understanding
requires that you bring what you have witnessed, judged and understood
in order to communicate your understanding of what John meant us to
understand rather than what we believe to be the truth.
The value in the pursuit is that we understand each
other’s understanding of "In the beginning was the Word and the Word
was with God and the Word was God." We may not have arrived at
consistency but we have discovered the details of our agreements and
disagreements and as a result have more comfort with and less fear of
each other. Like I have already said, consistency is a guideline
and mathematics is a science of use.
The
Axiom of Choice and the Theory of Unity
There is nothing and there is everything. Zero
and one. With algebra we fill in the middle by counting the
fractions. We numerate (count) the denominations (the sized
pieces) and relate them according to rules (there are only nine rules
in algebra). That is we choose units and manipulate them.
The axiom of choice says we get to choose how to
count and what we count. The theory of unity shows us how we size
the pieces. Choosing units and the rules of their manipulation
are the basis of all mathematics and possibly the basis of all thought.
Unitizing is the
process of representing something with a useful symbol. Nothing
and everything are not useful symbols.
If we subdivided everything into Quantum’s of
energy, we would have an infinite number of easy to manipulate
units. However each unit would be useless because we could not
tell them apart.
If we combined all the Quantum’s of energy into one
thing we would have a unit with great meaning. However we can not
manipulate just one thing and it would be useless.
We must define a unit that has a bounded meaning and
can be usefully manipulated.
In peacemaking conversation and problem solving the
units are all the words in all the dictionaries plus all the words we
might invent. The rules are all the grammars of prose and poetry
plus those we invent. That's the everything.
For the sake of peace or problem solving we will
choose subsets to make the system useful. That is we will size
the pieces.
To make a useful unit, then, we must eliminate all
those parts of the idea that don’t concern us in the context of the
problem and include all those parts that do concern us in the context
of the problem.
For example, an apple in the eyes of an artist has a
different meaning than an apple in the hand of a cook. When the cook
and the artist are cutting down an apple tree, they don’t bother with
defining apple. They define cutting down the tree.
When two cooks talk
about an apple they differentiate between tart and sweet, crisp and
soft and more. Rather than talking about apples, they talk about
Macintosh, Granny, or Delicious.
When dealing with new
problems and new concepts we coin our own words by agreement. When two
people agree that they will agree, they are reasonable. They are only
agreeing on what to call something. They can change their mind
about the meaning and invent a new word later.
Now how do we invent the rules of
manipulation? Let's make some order.
The natural order of everything is called
chaos. I find it beautiful. However because I cannot tell
what part lies between what other parts, it is useless.
Back to the apple. We might view an apple as a set
of proteins, carbohydrates, vitamins and minerals. The proteins
we might view as a set of molecules. The molecules we might view
as a set of protons, electrons and neutrons. The protons we might
view as a set of Quantum’s. Putting these associations in an
order we have the symbols, quantum, proton, molecule, protein and
apple.
Starting with the apple again we might view it as a
subset of the tree, which is a subset of the forest, which is a subset
of the land, which is a subset of the earth, which is a subset of the
universe, which is a subset of everything. Putting these
associations in an order and combining them with the above set of
symbols we have the following order of symbols; quantum, proton,
molecule, protein, apple, tree, forest, land, earth, everything.
This is called a continuum. It is a set of symbols in an order.
The symbol "apple" lies at the intersection of many
continuums. It lies between tree and protein. On another
continuum it lies between fruit and food. On another continuum it
lies between freedom and sin.
By deferring to the problem with which we wish to
deal rather than some absolute knowledge of a symbols meaning, we can
establish a useful definition by selecting bounded subsets of
continuums which each contain the symbol "apple". We are freed from
preconceptions. We have a definition bounded by the needs of the
problem.
In all the previous illustrations I have used nouns
because they are simple. In contrast to nouns, “Assemble the
bicycle” has a well-defined meaning to every father who has ever
assembled a bike on Christmas Eve. The need to order the events is
clear to the father. If he follows the directions in the order
given he succeeds. It is a continuum with a beginning and an end
composed of instructions. We will next look at the kinds of
pieces that can be ordered.
Basic
definition, agreeing on the instinctive understanding of a word.
"There is nothing as difficult as the obvious"
(Einstein). A person without a language or other mathematics has
only one way to establish meaning; by using his senses.
Basic definition is the agreement to recall
non-verbal understanding by name. To establish basic definition
people must agree to a specific name for an agreed understanding.
It is an intuitive understanding independent of word.
Babies have only one way to establish meaning, by
using their senses. Therefore, to basically define something such
as mother, he must see it, touch it, smell it, and taste it. He
must experience mother before he can know what it is. Finally, because
mother repeats the word “mother” over and over he gets it. He
calls the experience “mother”. He says his first word.
The word has no functional value until another
person understands what “mother” means. My mother and your mother
are not the same mother. What we do then is show and tell.
I show you my mother and tell you “this is my mother”. You show
me your mother and tell me “this is my mother”. Finally when I
say “mother” you recall the experience of your own mother, not mine.
Through a process of show and tell we have a common understanding of
“mother“.
We all have our own mothers. Though the
experience of mother is different for each of us, we agree that
experiencing “mother” is common. This is not to say that our
understanding is the same for my mother may have disciplined me by
reward while your mother may have disciplined you by punishment.
Let us explore further the difference between common
and same.
“A rose by any other name would smell as
sweet”. When the word “rose” is exchanged there is usually some
basic understanding exchanged based on the smell of a rose. The
agreement between reasonable people is to call it a “rose”. Even
to a person who has smelled but never smelled a rose the word has use
because the person has a basic understanding of “smelling“.
If smelling were the only sensation that identifies
a rose, then a person who cannot smell would have no understanding of a
rose. But a rose has more. It has romantic imagery.
It functions well that a rose induces more feelings than just
sensations. This allows almost everyone to experience a
rose. As each person has their own experience it is by agreement
established through show and tell that gives common meaning to
words. Though I can never know your experience I can agree to
agree. We share common experience that we name by definition.
Basic verbs are also learned by show and tell.
When a baby suckles his mother he feels “she is”. When the baby chews
his hand he knows “it is“. When the baby sees a toy he knows “the toy
is”. When others say “she is, it is, the toy is“ the baby
understands what “is” means basically, that is without words.
Back to communicating.
To basically define something such as a rock, a
person must see it, touch it, smell it, and maybe even taste it.
He must experience a rock before he can know what it is.
At this point the rock
has meaning to the person. However every rock would be different.
Therefore, the next
step the individual must take is to experience many rocks and compare
them, attempting to collect a sense of commonness between them.
He must associate abstracted characteristics and compromise extremes of
dissimilarity until any rock may be called a rock. Now, for
himself, he has defined the concept, "rock". However, he cannot yet
communicate it.
In order to communicate “rock“ let us introduce a
second person and expose him to the same collection of rocks.
This second person by the use of his senses establishes a different
concept of “rock”. Let the two people meet and agree to say
“rock” for the collection of objects before them. They can now
exchange the word “rock“.
In “Gulliver's Travels” there exists a society in
which conversation takes place at a basic level.
“...Many of the most learned and wise adhere to the
new scheme of expressing themselves by things, which had only this
inconvenience attending it, that if a man’s business be very great, and
of various kinds, he must be obliged in proportion to carry a greater
bundle of things upon his back, unless he can afford one or two strong
servants to attend him. I have often beheld two of those sages
almost sinking under the weight of their pacts, like peddlers among us;
who, when they meet in the streets, would lay down their loads, open
their sacks, and hold conversation for an hour together; then put up
their implements, help each other to resume their burdens, and take
their leave.”
If I wished to convey "rock" to another person in
this society I would have to carry a lot of rocks in my sack. The word
"rock" is easier to carry than a bunch of rocks. After enough
experience I have a vocabulary of basically understood words.
Secondary
definition, the naming tool.
If a definition can be worded, than the definition
is a secondary definition. Secondary definitions have the form of
dictionary definitions. You are only agreeing on what to call some
understanding so that you can recall that understanding by saying the
word.
The foundation of
secondary definition is basic definition. To illustrate let us go
to the dictionary and select the word “obelisk”. Rather than
showing a series of obelisks as we would have to do to define the term
basically, we say an obelisk is “a four sided usually monolithic pillar
tapering as it rises, and terminating in a pyramid”.
If we understand the definition of “terminating”,
“pyramid”, “rise”, “taper”, “monolithic”, “four sided”, and “pillar”,
we know what an obelisk is.
Note that all these terms are understood as
secondary definitions based on other secondary definitions or basic
definitions. For example in my case I need the term “four sided” to
define obelisk. I use the secondary definition of “four sided” as
having four sides. Four I define as one plus one plus one plus
one. “One” and “plus” I understand basically. “Side” I define in
this case as a plane. “Plane” I define as a locus of points equidistant
from two fixed points. “Locus” and “point” I understand
basically. “Equidistant” I define as equally distant. “Equal” and
“distance” I understand basically.
You, on the other hand, may have seen a collection
of obelisks and understand what they are with no need to refer to a
dictionary. Another person may have a basic understanding of
"line" because he draws a lot of them. He uses a secondary
definition of "point" as the intersection between two lines.
Still another person may have no concept of "plus". But he knows
what four is because he has counted to four many times.
Because the dictionary uses words to define other
words exclusively (they never say; well, you just have to know what
this word means basically), the dictionary goes in circles. For
example the dictionary uses "point" in defining "line", and uses
"line" in defining "point". If you keep looking up all these
words you get circles of knowledge. By understanding at least one
point of the circle basically, you get to understand all the words in
the circle. This is not to say that my understanding is the same
as your understanding for I may basically understand a different point
of the circle because I have had different experiences. It does
imply that our understanding is common.
The dictionaries total dependence on secondary
definitions is well illustrated by its difficulty in conveying a basic
definition. For example "is" is basically understood by almost
everyone. We all know what it means yet the dictionary
(because it is obliged not to leave out a word) uses a lot of words to
attempt to convey the obvious.
This leads to a lesson to be learned; a middle
ground between basic and secondary understanding. How do you
convey basic understanding with words? You use the same technique
that dictionaries uses to define "is". You use hyperbole, simile,
metaphor, idioms, poetic license, et al. You use words as if they
were rocks taken from your "Gulliver’s Travels" bag that you carry.
Secondary definitions are useful because they can be
tailored to our needs. Consider again the word “obelisk”. If we
need the definition in order to distinguish between a solid object and
a planar surface, it would be sufficient to define an obelisk as a
solid object.
Note that this definition cannot be freely used
outside the context of our particular need. It would be simpler
to use "solid object". The definitions are to be useful, that is
close to common meaning yet specialized enough to our need. A
thesaurus might help. Of course you can always invent a new word
if you cannot find an appropriate different word.
In peacemaking conversation the reverse is
true. We start with words and write their meaning. The
problem is frequently that each word carries much connotation and
denotation.
For example "abortion". First we must agree on
what it means objectively. We might start with "to interrupt
before completion". In the interest of exchanging our
understanding we may later agree to "the surgical removal of an
immature fetus". The first is the more accurate definition.
The second is what we want to talk about. It suits the need to
understand each other.
Now that we agree on what the word means we can go
on to the connotation of the word. Is it good or bad? This is not
a matter of definition. This is a matter of judgment.
Expressing judgment will be discussed later under "principles".
Principles
In the broadest sense, a principle is a statement of
equivalence. That is; a principle is a statement that different
things (or statements, or ideas, et al) have the same value. This
requires an understanding of why, how and what you are
evaluating. This requires judgment.
Principles in problem solving
In creating a mathematics to solve problems,
principles are facts. You create facts that are useful. The
test for usefulness is whether they are internally and externally
consist.
An illustration of a principle is the commutative
principle of multiplication in algebra (B times A equals A times
B). Certainly B times A is not identical to A times B. Yet
when you perform the identified operations you get the same
answer. That is 3 times 2 is 6 and 2 times 3 is 6.
You may have a mathematics involving screw
driving. A knife, a screwdriver and a dime all drive
screws. They are not identical. They all have the same
value in that they drive screws. In principle they are the
same. They are equivalent.
Another example, “A Ford is a Cadillac” is a
principle if they are being considered as ground transportation. “A
Ford is a Cadillac” is not a principle if their cost is the
consideration.
Sometimes we forget that our facts are
judgments. "It is true that for every action there is an opposite
and equal reaction". This is a principle. We are judging
truth. It is common to drop the judgment (it is true that) and
say “for every action there is an opposite and equal reaction”.
We call it a fact rather than judgment. As long as we agree,
there is no problem.
Remember the "fact" that the earth is flat.
Galileo disagreed. Remember the "fact" that matter cannot be
created or destroyed. Einstein disagreed. Now we know they
were judgments.
Sometimes our facts have strong evidence
contraindicating them, yet we use them because they are useful.
In physics, the Bohr model (the planetary model of masses for atoms)
has been superseded by quantum mechanics. Because the Bohr model
is so useful in understanding Newtonian physics it is presented as fact
in most high school physics courses.
With human morality it gets even trickier.
However, as long as your purpose is to solve a problem, you can
transcend unnecessary judgments.
Let us illustrate with a mathematics designed for
criminal recourse. You might start with a number of principles; a
thief is a bad person, a thief is a person who steals, bad people must
be punished, a person who steals bread to feed his family is not a bad
person, and more. All the morality questions can be put aside by
the following practical principle; the judge, whose authority is
established by due process, shall determine the course of action forced
upon a person that is determined guilty by a jury of his peers.
A more personal example; in biology we arrange all
the animals and plants in an order that is useful. The source of
the order ("intelligent design" or "Darwin's theory of evolution") can
be transcended.
Principles in peacemaking
conversation
Sometimes people will not allow their beliefs to be
transcended. In the example above; was the order a matter of
"intelligent design" or "Darwin's theory of evolution"?
In peacemaking conversation, principles are alleged
facts. A fact is a statement that somebody decided is true.
In peacemaking conversation the problem usually begins with conflicting
facts. The solution is not to agree on the facts but rather to so
understand them that the violation of disagreeing people is unnecessary
and unwanted.
A principle is a statement of judgment. They
are your proposed facts. Once they are accepted they become your
common facts. This rarely happens.
Once again we are working in reverse. We are
not creating a principle but rather we are starting with the principle
and the willingness to understand it. It is not that we wish to
arrive at the same principles but rather to arrive at a common
understanding of each other's principles.
There is a way of writing principles so that they
can be commonly understood. The way of writing is to include the
author of the fact.
For example we may not agree that "the sky is
falling". On the other hand we can agree that "Joe said, 'the sky
is falling'". By identifying the authority, in this case the
author of the judgment, and associating that authority with the
judgment we arrive at a common understanding.
In most cases the individual is not the author of
the fact. Rather he agrees with the group to which he belongs.
For example some proposed principals might be “life
is ongoing and should not be prevented by contraception”, “life begins
at conception”, "life begins in the second trimester of pregnancy" or
“life begins at birth “. A mixed group would reject these
principles.
Let us include the judgers (the authorities) with
the judgments. “The Pope insists that life is ongoing and should
not be prevented by contraception, pro-lifers believe that life begins
at conception, the Supreme Court of the United States established that
legally life begins in the second trimester of pregnancy,
and a plurality of people believe that life begins at
birth" would be a principle accepted by the group. Note that each
judgment now contains its authority; that is, “the pope, pro-lifers,
the Supreme Court, and a plurality of people”. We have given each
authorities "truth" equal value. That is; since the value of each
authority is the same each belief carries equal weight. Not only
have we established common facts (a listing of who believes what) but
we have also removed our own individual superiority.
It is possible that you may need to do even more
refining. You might propose that a minority of people believe
that life begins at birth. A change to "many people believe that
life begins at birth" would probably be acceptable.
Understanding does not require that we all believe
the same thing. It does require that we find a common
truth. In this case it might be "there is no common agreement
about when sacred human life begins".
Postulates
Postulates establish order. It can be a rule
that determines order or an actual listing of items (events, ideas,
things, even postulates, et al).
Some postulates establish first and last. We
might agree that “the chicken came before the egg“(but probably not).
Another might be “the election of Lincoln proceeded the civil
war”. These postulates help us agree on an organized past (when,
where and what happened). When there is agreement postulates become our
accepted history. When there is disagreement the postulate can
become a matter of principle. An example; “according to the book
'Outline of US History' the election of Lincoln preceded the civil
war“.
Some postulates establish counting order.
Items are in counting order when comparing items you always know which
one is first, which one is second, which one is third, which one is
fourth, continuing until you know which one is the last one. An
example;
A postulate to wash your hands in a sink.
1. Turn on the water.
2. Pick up the soap.
3. Lather your hands.
4. Put down the soap.
5. Scrub the hands together.
6. Rinse hands.
7. Turn off water.
8. Dry hands.
Some postulates use rules. An example; the
"right" size is between 2 and 6 inches. It is bounded by a rule
but has an uncountable number of right answers.
There are many other kinds of order. We will
be using first and last, counting order and (bounded) betweeness.
Postulates in problem solving
In problem solving the order is created. We
use personal experience to guess what might work and alter it until it
does work. Like it is with problem solving principles, the
standard is internal and external consistency.
Every mathematics contains at least one performance
postulate for every mathematics has a purpose and to accomplish this
purpose you must instruct the user on what to do. "A postulate to
wash your hands in a sink" is an example.
Another example of such a postulate is the law of
order in algebra which says that in order to compute the value of an
expression, first raise to powers and take roots, second multiply and
divide, and third add and subtract. If you follow this
instruction, you’ll always get the same answer. It is completely
internally consistent.
Postulates can occur in
postulates. An example;
A postulate to determine if a person is good.
1. Determine the moral standards of the time. (This would
require a postulate to determine the moral standards of the time)
2. Determine the moral standards of the person. (This would
also require a postulate to determine the moral standards of the
person.)
3. Compare the moral standards of the person and the society.
(This would require a postulate to perform the comparison.)
4. Draw a conclusion.
The nesting of postulates turns into an
outline. Looking only at the first item;
A postulate to determine if a person is good.
1. Determine the moral standards of the time.
a. Contact religious leaders and ask them to
write a statement on what is a moral person.
b. Survey existing laws that are being
enforced regularly that protect individual rights.
c. Compile a compromise between these views
and call it the moral standards of the times.
2. Determine the moral standards of the person. (as above)
3. Compare the moral standards of the person and the
society. (as above)
4. Draw a conclusion.
Sometimes the mathematics fails. When
different people applying the postulate come to conflicting conclusions
and the need to determine if a person is good is essential to the
purpose of the mathematics, then rewrite the postulate.
As in principles if you focus on the problem
solution you can frequently transcend differences in beliefs. For
example, "Vote for presidents who will appoint pro-life judges."
This avoids which comes first; a mother’s right to choose or preserving
a fetus?
Postulates in peacemaking
conversation
Disagreements over order of importance are generally
the problem. In peacemaking conversation you don't get to rewrite
the postulate. Consider the abortion question above.
Who makes the judgment enters the picture.
Once again the author of the alleged truth must be included to arrive
at a common truth. The principle containing two postulates
becomes “pro-choice people believe that a woman’s right to choose is
more important than preserving a fetus and pro-life people believe that
preserving a fetus is more important than a woman’s right to choose”.
Sometimes by specifying precisely where we disagree
leaves us open to sharing why we disagree. Let me introduce the
Chinese government's view. They have a recent history of
overcoming starvation through forced population control. They
postulate that controlling the population is more important than a
woman's right to choose. They limit the number of children a
family can have. They would have us understand starvation.
Then the world principle contains three postulates
and becomes "population control people believe it is most important to
abort birth, pro-choice people believe that a woman’s right to choose
is more important than preserving a fetus, and pro-life people believe
that preserving a fetus is more important than a woman’s right to
choose”.
Axioms
An axiom is a statement of cause and effect.
Axioms in problem solving,
creating branching order
An axiom is frequently in the if/then form.
One of its purposes is to establish alternative paths based on
alternate conditions.
For example, assume you want to sort apples.
Each apple can be hard, soft, or rotten. You know that hard
apples are best for supermarkets, that soft apples are used for
canning, and that rotten apples can be sold for pig slop.
Therefore you may employ the following postulate of three axioms;
A postulate to sort Apples
1. If you select a rotten apple, then put it in the slop
bin.
2. If you select a soft apple, then put it in the sauce
bin.
3. If you select a hard apple, then put it in the market
shipping bin.
Another purpose of an axiom is to describe possible
alternatives. For example, assume you wish to make applesauce and
canned sliced apples from your soft apples. You have an apple
grinder and an apple slicer. Two axioms that would serve as
guidelines might be;
If you place soft apples in the grinder, then you
will make applesauce.
If you place soft apples in the slicer, then you
will make sliced apples.
By inserting axioms in performance postulates you
can branch to other performance postulates or you can put them in
outline form.
A postulate to process Apples
1. Pick the apples.
2. Sort the apples.
3. Process the apples.
This becomes;
A postulate to process Apples
1. Pick the apples.
2. Sort the apples.
A. If you select a rotten apple, then put it in the slop
bin.
B. If you select a soft apple, then put it in the sauce
bin.
a. If the demand for applesauce is
higher than the demand for sliced apples, put it in the grinder.
b. Put the apple in the slicer.
(if not a.)
C. If you select a hard apple, then put it in the market
shipping bin.
3. Process the apples.
Axioms in peacemaking
conversation, the principled axiom
Sometimes the purpose is to agree (or understand the
disagreement) on what caused what. An example might be “the
election of Lincoln caused the civil war”. This might become
“according to 'Outline of US History' competition between a slave
economy and a free economy led to a balance of power in Congress that
was offset by Lincoln’s election which caused the South to secede which
caused Fort Sumter... “
Sometimes you can transcend the middle and agree on
the judgment. In the above case “according to 'Outline of US
History' the election of Lincoln caused the civil war” may be
enough. This is a principled axiom.
Sometimes you can transcend the middle and agree on
the axiom. In the above case “the election of Lincoln caused the
civil war”.
A major function of axioms in peacemaking
conversation is to speculate on cause and effect. It is an
expression of one's belief of what will happen given a specific cause.
A religious example; "if you do not accept Jesus as
your savior, then you will go to hell" is an axiom only if everyone
agrees. "Joe believes that if you do not accept Jesus as your
savior, then you will go to hell" is a principled axiom. Like
every principle, it contains its author.
As with all principles, a principled axiom
need not identify the author as a single person. I could have
said "the catholic church believes that if you do not accept Jesus as
your savior, then you will go to hell".
Two political examples. Joe believes that
the removal of all our troops from Iraq will cause a
bloodbath. Bob believes that the removal of all our troops
from Iraq will cause an end to the present bloodshed. By
stating their positions Joe and Bob might notice that their positions
are not mutually exclusive. They may wish to consider bloodshed
now with bloodshed in the future, whose bloodshed and how much
bloodshed.
Two historical examples. Joe says Hitler's
rise to power was the result of the wealth generated by stealing Jewish
wealth. Bob says the stealing of Jewish wealth was the result of
Hitler's rise to power. Note that their positions are mutually
exclusive.
Sometimes it is the cause that needs
understanding. As an example let us look at the passionate
principle "pro life people believe sanctified life begins at
conception". How do you know?
One might say they had an epiphany. Another
might say it is in the teachings of Buddha.
Let us look at the one who says the bible
establishes the fact. How does he know the bible establishes the
fact? We are exposing branch after branch of understanding and
the source of that understanding.
By searching for common ground, we arrive back at an
understood definition of a group who call themselves pro life and have
a common understanding that sanctified life begins at conception.
The fact that pro lifers come from many places has been transcended.
Some will find it necessary in order to understand
to deal with the cause of all belief. An axiomatic investigation
helps but care must be taken to end at understanding. In
peacemaking conversation you can say "I believe". You cannot say
it is just plain true without the agreement of all.
Criteria,
placing boundaries on words
A criteria is a boundary that uses words. The
words can be a declaration, an instruction, a bounding postulate, an
axiom or a secondary definition.
A criteria of the specific is generally a list of
criteria. To distinguish it from a postulate which orders things,
a criteria is only a list. The order of the criterion in the
criteria is not a consideration.
A criteria needs naming in order to identify what
you are trying to specify. For example; if you wish to specify
what a pencil is write "a criteria for a pencil".
The purpose of a criteria is more specificity.
For example; if you wish to define a pencil as a practical writing
implement, you may do so by making several observations. You may
note that a pencil shorter than two inches or longer than nine inches
loses much of its effectiveness. You have limited the length of
your “pencil” by defining the endpoints two and nine inches and stating
that anything falling between these points is a pencil. This
first observation must be supplemented with additional observations
before it becomes specific enough to be a useful definition since a
tumbler, a small book and even a Chihuahua dog fall between two and
nine inches in length and are obviously not pencils. Our
observations are standards against which we measure the object we are
trying to define.
The criteria for a pencil might be as follows;
1. It contains something that marks surfaces that have a
coefficient of friction greater than .1.
2. It is lighter than 2 ounces and heavier than .1 ounce.
3. Its diameter is less than 3/4 inch but greater than 3/16
inch.
4. Its mark remains visible under normal wear and tear.
5. It is longer than two inches and shorter than nine
inches.
Notice that pencils, pens, crayons, and chalks are
all pencils under this criteria. This would create problems only
if it were necessary to differentiate between various writing
implements. If this differentiation is not important, then this
criteria is useful.
We can add meaning to words by changing tense or by
adding a suffix. For example if we know what the word “see”
means, we can with criteria establish what the word “saw” means.
However we may choose to write a criteria for past tense instead.
Criteria for the past tense of the verb:
1. The meaning of the present tense of the verb applies.
2. The meaning is applied in the past.
Every time you use an adjective or adverb you refine
a definition. You can write a criterion (a definition) for
"girl". You can write a criterion (a boundary such as over 6
feet) for "tall". You can write a criteria for "a tall
girl". Now "a tall girl" is no longer a judgment. It is a
girl who is over 6 feet tall.
Most of the time the judgments expressed by
adjectives and adverbs can be made more specific by writing a
criteria. Sometimes it is more practical not to do so. "She
is a pretty girl" is a principle. It is most often more practical
to accept "pretty" as a judgment then to write a criteria for it.
In peacemaking conversation a criteria may help you
specify your differences. A criteria has excellent use when we
wish to differentiate shades of meaning. We can handle words with
different connotations by establishing their criteria.
Sometimes criteria can help us focus on the
connotation. For instance teach and brainwash may mean the same
if our only criterion is conveying information, but they have different
meanings. Using two criteria will clarify the difference.
To teach is to convey the truth. To brainwash is to convey
falsehood. Now it is possible to compare the processes (how we
convey information) unencumbered by content (truth or falsehood).
Sometimes criteria can help us transcended a
connotation. Consider "prejudice". A criterion for both
faith and prejudice is prejudgment. Before any judgment is ever made
there is prejudgment. We all have our prejudgments.
A criteria for faith;
1. It is a prejudgment.
2. It is believed to be true.
A criteria for prejudice;
1. It is a prejudgment.
2. It is believed to be false.
This levels the playing field. Now we can see
another's prejudice (prejudgment) as we see our own faith
(prejudgment). Faith is my prejudgment while prejudice is
yours. We have transcended the connotation of faith and
prejudice. Now we can compare our beliefs without being insulted
by the connotation attached to the word "prejudice". We can talk
about our prejudgments.
Sometimes in peacemaking conversation transcending
the problem is counterproductive, the purpose being clearer
understanding. In this case writing a criteria is usually not
practical. For example instead of writing a criteria for a good
person, you could write a postulate to determine what a good person is
(though it might contain some terms determined by criteria).
Criteria forces upon you a timeless and universal standard for what is
a good person, while a postulate instructs an individual how to
determine a good person at any one time and place.
Whether engaged in problem solving or peacemaking
conversation, a criteria can be altered like a secondary
definition. You are only agreeing on a specific meaning of a
word. You are not agreeing to any belief.
Summary
of the creation of order
A mathematics to solve a problem looks like an
outline with footnotes. The outline is always a postulate.
In most cases the postulate contains postulates. It always
contains some instruction. The footnotes are secondary definitions,
criteria for definitions, and principles.
The purpose of peacemaking conversation is
peace. The conversation itself is the tool. The result of the
conversation is a list of commonly understood principled axioms and
principles with footnotes. The footnotes are secondary
definitions and criteria for definitions. As a result we maintain
the peace rather than our governments and our churches.