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GEOMETRY OF EXISTENCE

AND THE

KNOWLEDGE  UNIT

© Copyright 1993
By Jonathan Burch
All rights reserved

The geometry of existence is a method for defining, describing and comparing change in the universe.  The initial geometry is limited to the macro world above the level of the atom, where time is irreversible and an object in space cannot be two places at the same time.  It may be that this geometry, with some modifications to the axioms, or another geometry, will describe the micro world below the level of the atom, where time is reversible and an object can be at more than one place at the same time, and describe the effects of relativity.  The universe is conceived as a container (space-time) which contains a substance (energy) that may be organized in various ways (according to the rules of organization based on category structure).  The substance flows from higher concentrations to lower concentrations and thereby causes movement, which is characterized as change.

GEOMETRY  A geometry is a small set of assumptions or axioms from which a larger body of rules is developed logically.  If the assumptions are granted, then the rest of the description that follows is correct.  If the assumptions of a geometry fit a set of circumstances in reality, then the rest of the description in the geometry accurately or correctly describes the set of circumstances in reality.  Euclidean geometry accurately describes the set of circumstances of a plane.  Reimannian geometry does not correctly describe the set of circumstances on a plane, but does on the surface of a sphere.  In this geometry certain assumptions or axioms are given.  Then the description of the rest of the subject is developed logically from the axioms.  If it turns out that the axioms fit the reality of the universe, and the universe is logical in the way described in the axioms especially those of organization, then this description should describe the universe, and what is in it, and how change is caused in it.

 The container in which all change happens in the universe is space-time.  There are three mutually perpendicular dimensions of space and one dimension of time.  The four dimensional space-time geometry is built up step by step.  It begins with Euclid's axioms for a two-dimensional plane.  Then a particular type of plane, a squarefield, like a checkerboard, is defined.  Then axioms about time are given and time is added to the squarefield as the third dimension.  Finally the other spatial dimension is added to convert the squarefield into a cubefield plus time.  This completes the four dimensional description of space-time.

Euclid's Axioms.

 The five axioms of plane geometry according to Euclid are:

 1. Any two points may be joined by a line segment.

 2. Any line segment may be extended to form a line.

 3. A circle may be drawn with any given center and distance.

 4. Any two right angles are equal.

 5. If a line m intersects two lines p, q such that the sum of the interior angles on the same side of m is less than two right angles, then the lines p and q intersect on the side of m on which the sum of the interior angles is less than two right angles.

(Meserve, Bruce E. Fundamental Concepts of Geometry, Dover Publications, N.Y., 1983)

 FIRST LAW OF TIME

  Time occurs in an ordered sequence of events, and the sequence is irreversible.

 FIRST LAW OF SPACE

  An object cannot be at two different points in space at the same time.  Two objects cannot occupy the same space at the same time.
 

 The substance which is contained in space-time in the universe is energy.  Energy occurs in many forms.  The energy axioms are given, and transformations from one form to another are described, as well as the flow of energy from a higher energy concentration space-time location to a lower energy concentration space-time location.
 

THE ENERGY AXIOMS

 FIRST LAW OF ENERGY.

  Energy cannot be created or destroyed.
  (The Law Of Conservation of Energy.)

 SECOND LAW OF ENERGY.
 
  Energy in space-time moves from a higher energy state to a lower energy state, and ultimately tends toward entropy.
(The Law of Flow of Energy.)

 THIRD LAW OF ENERGY.

  Energy occurs in two phases, radiation and matter, with many forms of energy in each phase and any form of energy can be converted into any other form of energy. (The Law of Energy Transformation.)

 FOURTH LAW OF ENERGY.

  One form of energy can resist the movement of or contain another form of energy.
(The Law of Energy Resistance and Containment.)

  COROLLARY

  One form of energy can exert a force of resistance to the movement of the same or another form of energy.  Overcoming the force of resistance to accomplish the movement requires an equal and opposite amount of energy.

  COROLLARY
 
  The force of resistance of one form of energy can be so strong that it contains another form of energy, so that the other form of energy cannot move beyond the space-time boundaries of the first form of energy.

  COROLLARY

  One form of energy can exert a force to move another form of energy if the other form of energy resists or contains the movement.  The force resisting or preventing the movement from happening is the force of resistance.  The force attempting to cause the change is the force of movement (force of change or force of solution).  If the force of movement is equal to or greater than the force of resistance, the movement and the change will occur.  Then the force of movement is called the force of solution.  It is strong enough to solve the problem of causing the movement.

 FIFTH LAW OF ENERGY.

  Nature tends to develop energy efficient change.
  (The Law of Energy Efficient Change.)
 

ENERGY CONVERSION TABLE
 Any form of energy can be converted into any other form of energy (Third Law of Energy).  It requires energy to cause a conversion.  Some conversions produce a complete change to one new form, or phase of energy.  Most conversions produce more than one form of energy.  Some of it is available for useful work (movement) and some is not (waste energy or heat energy).  The conversion table gives several forms of energy, the other forms into which they may be converted, the energy cost of the conversion and method to make it start and the conversion products.  The point of the energy conversion table is to use the table to determine where all the energy comes from and where the energy goes in any change.  All of the energy that goes into the change should be accounted for after the change, since energy is neither created nor destroyed (First Law Of Energy).
 

 The rules of organization in the universe are derived from the category structure of the universe.  The category structure determines how attributes, entities with attributes, and relationships with attributes between entities are formed into categories.  A category has a membership definition.  Entities and relationships that meet the qualifications of the definition are members of the category.  All other things in the universe are non-members, outside the category.  Categories can have sub categories, and be members of larger, super categories.

 The category structure of the universe determines how the things in the universe may be organized.  There are certain limitations on how this organization can occur.  A proposed organization might meet the organization rules, but not the space-time rules or the energy rules.  If an entity does not meet all three sets of rules, it cannot exist in the physical universe.  One definition of existence in the physical universe, is meeting all three sets of rules.

Change in the physical universe can be described.   Knowledge is a description of change in the physical universe.  Change in terms of the space-time rules, the energy rules and the organizational rules may be expressed by a description in some physical symbol system.  This is called a description.  It is a description where the elements of the physical symbol system are symbols that stand for, or represent, the elements of the thing described.  A physical symbol system could be ink on paper, electronic signals in a computer or chemical-electrical signals in a brain.

 A physical symbol system has to have its symbols exist physically in the universe within the limits of all three sets of rules.  However, the description does not have to be limited by the space time rules or the energy rules or all of the organizational rules.  It is only a description, not an entity in the physical universe.  Therefore there can be descriptions that describe entities that can exist physically and there can be descriptions of entities that cannot exist physically, because the description is of something that violates one or more of the three sets of rules.  A description can violate the organizational rules, by using some of the organizational rules to make the description, but not conforming to all of the organizational rules that apply to that entity.

IMAGINATION

 It is possible to have a description of some organized entity in some physical symbol system.  It is possible to describe the limitations of the three sets of rules for any entity.  Therefore it is possible to describe any entity that can exist physically in the universe (within the limits of the three sets of rules).  In addition it is possible to describe an organized entity that does not conform to the all the limits of the space-time rules, or all the limits of the energy rules, or all the limits of the organizational rules.  Such a description may be called a partial description, because it does not include all of the relevant rules in all three systems for the entity described.  Such a partial description may also be called imaginary, because such imaginary entities do not have to conform to all the physical rules in how they are formed and what they do.  Even if an imaginary entity conforms to all of the rules (organizational, space-time and energy rules), except one, it cannot exist physically in the universe.

 It can exist as a concept or description in some physical symbol system, even if it cannot exist physically.  All such concepts (imaginary or real) are derived from the organizational rules.  Concepts are descriptions of organized entities according to the category structure and rules of the universe.  Such concepts, based on the category structure of the universe, form all of logic, set theory (finite and infinite sets), mathematics, thought, language, perception, stories which give meaning and explanation, religion, law, scientific theories and the myths that control social organization.

 Thought is structured.  Existence is structured.  Both structures are determined by the organizational rules based on the category structure of the universe.

NATURAL RELEVANCE

 Relevance is important, because of catagory signature.  Every entity has a category signature.  An entity's category signature is all the categories of which that entity is a member during any period of time.  All the categories in an entity's category signature are relevant to the entity during that time.  All other categories in the universe are irrelevant to the entity during that time.

 If the entity is changed during some period of time so that it becomes a member of some different categories, the attributes, entities, relationships and categories that are involved in the change are relevant to the entity duiring that period.  All other parts of the universe are irrelevant to that change in the entity.  This is natural relevance, which occurs due to the combination of the inclusive and exclusive aspects of the category structure of the universe.  This could be described as a natural relevance category which included all the categories and their parts that are relevant to the change in the entity, and an irrelevance category, which included all other categories and their parts in the universe.  For example, if a category membership in a category did not change during the time period, thre category would be irrelevant to the change.

NATURAL PURPOSE (Natural Teleology)

 If a natural process such as self organization or selection focuses on a certain type of change in an entity, it is possible to describe the process as occurring for the purpose of causing the change.  A process that causes a certain change may be selected because it causes that change, and the change is adaptive.  It is possible to describe that selection as purposeful.  As evolving creatures gain intelligence to perceive and manipulate energy in their environment, and become aware of obstacles and opportunities to survival and flourishing, the actions of such creatures do become purposeful.  None of this requires outside, supernatural intervention to "give" the purpose.  Purposeful behavior in the sense described here is completely natural.  In the past some scientists have felt threatened that purposeful behavior suggested a hint of the divine, which threatened the independence of their method of learning about the way reality worked (theory and confirming experiment).  Anything that smacked of teleology was denigrated as patently false.  However, to be so harsh a judge is to throw out all of nature and science.

It is a more balanced approach to admit forms of purpose and teleology, but then ask where does the purpose come from.  If it is just a different form of description of natural selection or self organizing systems, it does not require divine intervention.  If it is due to the mind of an aware personality which functions by setting goals and solving problems, the source of the purpose is the individual personality, not divine intervention.  However, simply because one explanation is a natural explanation, it does not rule out the possibility of divine intervention, it merely give an alternative explanation.  Divine intervention is still logically also an alternative explanation.

The situation is similar to Heizenberg's Uncertainty Principle.  We cannot know for certain.  However, the uncertainty in the origin of purpose should no more stop scientific inquiry into nature than the uncertainty of location and momentum stopped studies in quantum physics.  Those who believe in religion and divine intervention may legitimately believe in divine purpose, without fear of ultimately being proven wrong.  Those who believe in natural processes, subject to human discovery and description, as the source of all purposefulness in the universe may legitimately believe in the comprehensiveness of nature and the lack of necessity or desirability for the concept of divine intervention.  Both views are legitimate and there is in principle no way to distinguish between them.  People will believe in what works for them.  That is the way the human personality has evolved, or was designed.  The most that scientists can do is tell a story that does describe and explain experience, and predicts accurately what will happen under certain conditions.  The best that religion can do is tell a story that does describe and explain experience, and predicts what will happen under certain conditions.

Each person must decide for himself or herself which story works best for him or her.  Actually the decision will be made unconsciously by the Automatic Belief System (ABS), whether the individual likes the result or not.  With the ABS, stories that produce decisions that result in success in a situation are given more association strength with that type of situation.  Stories that are unsuccessfull or less successful in the situation lose association strength. When that type of situation occurrs again, the ABS selects the story with the most association strength to use in that situation.  This asociation strength is experience consciously as belief.  This is an unconscious, automatic process. One can lose a cherished belief, or gain a despised belief, but have no conscious control over either belief.  Nature did not see fit to give us conscious control over the Automatic Belief System by which we choose our response stories.  In any case the various individuals, believing different stories, sometimes passionately, must learn to live adaptively together, recognizing that they each have to have stories to live by, and respecting each other's story as valuable for the person who believes it, and respecting the person as a valuable partner who functions with that story.

RULES OF ORGANIZATION

 An entity may meet the organizational rules, but not the space-time rules.  Forms of energy can be organized in various ways in space-time.  The axioms of organization are given.  Category structure and the rules of organization are developed.

 FIRST LAW OF ORGANIZATION.
 
Any aspect of space, time, energy or organization is a substance.  There are two types of substances, continuous and discrete.  Each substance forms a dimension.  A dimension is formed of any substance which is quantitative in the sense that there can be more or less of it (continuous, ordered dimension, such as length or temperature). A dimension is formed of a discrete substance where there are more or fewer of its unordered group of discrete components (resemblance dimension).  A resemblance dimension is ordered in the sense that an entity can possess more or fewer of the discrete components or prototype attributes (0, 1, 2, ..., n)  which define the resemblance dimension.
(The Dimension Law.)

 SECOND LAW OF ORGANIZATION.

  A range attribute is a limited range along an ordered dimension, such as a certain length or color.  A prototype attribute is one of an unordered group of attributes that together form the pool of attributes which define a resemblance dimension.
(The Attribute Law.)

 COROLLARY

 A range attribute is a range along an ordered dimension, which can include any contiguous range of or none of the ordered dimension.
 
 

 COROLLARY

 A prototype attribute can be either a range attribute or another prototype attribute.  The smallest prototype attributes are range attributes.

 COROLLARY

 A common attribute is a defining attribute of a common category in which all members have the common attribute.  A common attribute may be either a range attribute or a prototype attribute.

 A resemblance attribute has some minimum number, or more, of the prototype attributes (or discrete components) along a resemblance dimension.

 THIRD LAW OF ORGANIZATION

  Any collection of entities which are similar in that they all possess one or more attributes (range or prototype) in common, and are therefore different from all other entities which do not possess these particular attributes, are members of a common category with a category membership definition of being an entity which possess all of these particular attributes.  The defining attributes of a common category are called common attributes.  A collection of entities, which has a minimum or more of the group of attributes which are the prototype attributes of a resemblance dimension form the members of a resemblance category.  All other entities do not have this minimum number of these prototype attributes, and are not members of this resemblance category.  The category membership definition of a resemblance category is having a minimum or more of the prototype attributes, which form the resemblance dimension.  The more prototype attributes a member has, the more typical, or better example the member is of members of the category.  A prototype attribute is one of the collection of attributes (common or resemblance) which forms a resemblance dimension.  A resemblance attribute is the attribute of an entity that it is a member of a certain resemblance category.  A resemblance attribute is also a collection of particular prototype attributes which qualify it as a member of that resemblance category.  A resemblance category need have no member which has all the prototype attributes.  There may be no best member.  There may be no attribute which all members of a resemblance category have in common.
(The Category Law.)
 

 THE FOURTH LAW OF ORGANIZATION

  Every aspect of the universe is a member of one or more categories.  Every set of categories with only one member in common forms a category signature of that common member, which may be called an entity.  Every entity has a unique category signature at any instant in the universe. (How does relativity deal with this ).
(Category Signature Law.)

  COROLLARY

   Change occurs when an entity becomes a new member of a category or ceases to be a member of a category.

  COROLLARY

   When a change occurs to an entity, those categories in which membership of the entity changes are relevant to the change.  All other categories in the entity's category signature are irrelevant to the change.

  COROLLARY

   The duration of membership in categories in the category signature of an entity will often be different.  Membership in some categories will last a long time. Membership in other categories will be fleeting, or last a relatively short time.  This circumstance explains the saying of Heraclitus, "A man cannot twice step into the same river".  The truth or falsity of this statement depends on what categories are being considered to define the river.  If categories are used where membership will change between the first step and the second, then the statement is true.  However, if the categories used to define the river do not change during the time from the first step into the water and the second step, then the statement is false.  If the river is defined by categories of both durations, then the statement is true and this circumstance is probably closer to what Heraclitus had in mind.  He apparently saw at least some change occurring constantly in what he defined as the river.

 
 THE FIFTH LAW OF ORGANIZATION

  Any two or more parts that combine together to form a larger whole, with one or more attributes of the larger whole which do not exist when the parts are not so combined (emergent attributes), form two levels of organization, with the larger whole forming the higher level of organization (the whole level or base level, L-0), and the parts forming the lower level of organization (the parts level, L-1) .
(The Law of Levels of Organization.)

  COROLLARY

   The larger whole is a member of a category of which the emergent attribute (prime attribute) is in some manner an element of the category membership definition.  The larger whole is a member of that category, because it possess the prime attribute, but the parts which form the whole are not, because they alone, or in some other organizational structure, do not possess the emergent prime attribute which only exists as an attribute of the larger whole.

  COROLLARY

   Hierarchical organization must proceed level by level.  A middle level cannot be skipped.  Hierarchical organization cannot develop directly from low level organization to high level organization.  The intervening levels of organization must be present to create the emergent attributes of each level on which higher levels are built.

  COROLLARY

   An entity on one dimension may be used to measure the presence or amount of another entity on the same dimension.  A linear entity of known amount of length may (a ruler) be used to measure the length of another entity.  Where a change occurs in a dimension, an entity which changes in proportion to another entity may be used to measure the amount of change in the other entity, such as a thermometer measuring the amount of heat in a liquid.

 
CHANGE IN THE UNIVERSE UNDER THE RULES OF SPACE-TIME, ENERGY AND ORGANIZATION

 Change occurs in the universe.  Space-time changes, energy changes form and location and distribution in space-time and organization of space time and energy changes.  For ease of explanation, relativistic effects and quantum effects will be ignored.  This description of change in the macro world or Newtonian universe will be given.  Later the rules and conclusions can be extended to include relativistic effects and the changes in space-time encountered in the micro world.

 To begin to illustrate the geometry of existence, two dimensions of space are used to develop a field of operation of change based on Euclid's axioms of plane geometry, called a "squarefield", which looks similar to a checker board.  Then time is added to the squarefield in the form of a clock to keep track of the sequence and amount of change in a given number of steps.  Then energy is added to the squarefield to provide the motivation for change.  Later the third spatial dimension is added to form the cubefield which operates with four dimensional space-time, energy and organization according to the rules of each of the three areas.  The rest is the history of the universe.

SQUAREFIELD ILLUSTRATION

 The Euclidian axioms plus the undefined terms of "point" and "line" lead to the development of a two dimensional plane.  On this plane a point can be either round or square, or some other shape.  Here the square will be used as the definition of a point.  A series of points or squares, one after the other along one dimension of the plane forms a line ( a line of squares).  Another line perpendicular to the first is a line on the other dimension on the plane.  Many lines with equal distances between them and parallel to these two perpendicular lines, form a squarefield.

 Local area can be defined as any limited, finite contiguous part of space-time.  Any intersection of two perpendicular lines on a squarefield may be arbitrarily selected as an origin.  The horizontal axis may be called the x axis and extension along the x axis may be called "length".  The perpendicular or vertical axis is called the y axis and extension along the y axis is called "width".  Each side of a square is one unit of length.  The square at the origin may be designated by its x axis and y axis values (x,y) as (0,0).  This uniquely describes this square on the squarefield using this arbitrary origin.  The local area on this squarefield is some finite number of squares from the origin along both axes.  Many types and steps of change can occur within this local area on the squarefield.

 Shapes can be described on the squarefield by coloring in squares that are part of a shape.  The squarefield looks like a checkerboard or a piece of graph paper.  When shapes are drawn on it, they can be any shape that can be made up by adding colored-in squares together.

 The squarefield (the local area) can be expanded to any size.  The larger the squarefield, the more rows on the x and y axes.  As the squarefield gets bigger, the resolution of the point and line get smaller or finer.  Theoretically as fine a resolution as is desired can be accomplished.  In computer terms the pixels can be as small as one wants.  In this way estimates of curved lines to any resolution can be made on a squarefield.

 Time is added as a dimension to the squarefield by adding a clock to measure the number of events in a series of changes on the squarefield.  The clock is a set of several loops of ten squares.  Each loop represents units, tens, hundreds, etc., of time.  Each time a change occurs on the squarefield, the units square moves one square around its loop.  When the units square has completed one loop (10 squares), the tens square moves one position in its loop.  After a series of changes is completed, the clock will tell how many steps it took to accomplish the change.  Different changes can be compared as to the time they take to complete, or when in the sequence of steps they occur.

 Figure 1 shows a squarefield with the clock in the lower left hand corner beneath the origin, which is marked "0,0".  A square shape has moved from the origin (0,0) one position to the right on the x axis to position (1,0).  The clock shows that this took one unit of time, which will arbitrarily be called seconds.  The units square is on the one position.  The squares of all the other loops of the clock are on zero.
 
 

 Shapes can be shown on the squarefield, and their attributes such as length, width, perimeter and area can be given by theorems derived from the axioms.  Figure 1 also shows a square and a rectangle at different locations on the squarefield.  The description of the square is given by L=2, W=2.  The position of the shape is given by the square that is a part of it which is closest to the origin.  The closest square is at (2,4).  The description of the rectangle is L=2,W=3 and its location is (4,1).  In this manner rectangular shapes can be defined and located on the squarefield.

 Change is represented by movement of squares from one location to another.  It takes one second on the clock to move one square a distance of one square, that is to move one shaded square one square position on the squarefield.  If a square is moved two squares it takes two seconds.  If two squares (a rectangle) are moved together one square each, it takes one second on the clock.

 Concept Theory Notation

 The notation for the organization of shapes on the squarefield is given by the notation developed by the author for Concept Theory.  There are two kinds of concepts, a molar concept, called an "arcon", and a molecular concept, called a "molecon" (pronounced moll eh con, similar to molecule).  An arcon is a concept which has only attributes as a whole.  For example a watch has many parts, but as a whole it has the attribute of weight, and of telling a certain time at a given moment.  Arcons are concepts that only have attributes as a whole, or molar attributes.

 A molecon is a molecular concept which has parts that have attributes and relationships among them.  The parts are arcons with their molar attributes and the relationships among the parts are called rules.  For convenience of designating which attributes of arcons and rules are relevant and irrelevant to something about the concept, these types of attributes are defined:

 Rel  (rell)  Relevant attribute of an arcon
 Ire  (eye er)  Irrelevant attribute of an arcon
 Rue  (roo)  Relevant attribute of a rule
 Irue  (eye roo)  Irrelevant attribute of a rule

 Figure 2 shows the basic structure of a concept made up of two arcons related by a rule to form a molecon.  The molecon is surrounded by a larger circle which represents a larger arcon that describes the molecon as a whole, without parts.  This process is called arconizing a molecon, describing parts as an organized whole with attributes as a whole.
 
 

 The notation for an arcon is a circle, symbolizing the concept as a whole.  Beneath the circle is a vertical attribute line with spokes sticking out to the right.  At the end of each spoke is a designation for a particular attribute.

 The notation for a rule is a set of armed parentheses.  The arms join the two arcons that are related by the rule, and a vertical upward line with spokes to the right designates the attributes of the rule.  For simplicity, usually only relevant attributes of arcons and rules (rels and rues) are shown.  Relevance will be defined below.

 The notation for a molecon is a combination of arcons and rules.  The simplest molecon is shown in Figure 2 with two arcons with one rel each, are joined by a rule which has one rue.

 The notion of levels of organization can be described in terms of concept theory.  The level of the molecon can be the base level or whole level designated L(0).  The level of the parts or arcons, which make up the molecon are the next lower level, theb part level designated L(-1).  There is an awkward phrase which describes turning a molecon (a concept with parts) into an arcon (a concept as a whole without parts).  This is called "arconizing a molecon".  It can be thought of as drawing a circle around a molecon, dropping an attribute line from the circle and describing attributes of the molecon as a whole.  Such a larger arcon, which describes attributes of the molecon as a whole, becomes a part in a larger molecon at the next level of organization up from the original molecon. Arconizing a molecon creates a new larger arcon which, when related to other arcons at its level creates a new level of organization, L(+1).  Organization occurs when two or more arcons are related by a rule.  Organization occurs with the creation of a molecon.  Organization at a level occurs with creation of a molecon at that level.  Sole or unrelated arcons at that level are not organized.

 One of the values of concept theory notation is that the concept diagrams help keep the organizational structure accurate.  The diagrams follow the organizational rules.  Attempted violations of those rules become readily apparent when a description that violated the organization rules cannot be diagramed.  Another benefit is that concepts can be seen and understood.  So long as only a few levels and a few relevant attributes are included, a coherent, understandable concept of any object of any size, complexity or level of abstraction may be described, for example from tiny loops of cosmic string to the whole universe.

 Energy is represented on the squarefield by a half-moon shaped carrier of energy.  Energy moves square shapes on the squarefield.  The flat side of the energy carrier pushes against an adjacent shaded square and moves it one or more squares depending on how much energy the carrier has.  The amount of energy remaining in the carrier is shown by a number in the half moon.

 Initially, the three sets of rules are put together on a squarefield plus time to illustrate how and why change occurs in the universe.  First certain corollaries of the three sets of rules are developed.

COROLLARIES

RESOLVER.  A resolver is a method of ignoring the infinities in any situation.  Often in nature there are variables or extensions of something that are very numerous or great, some even infinite.  These are called here the infinities.  For example all the events that ever occurred in the universe form a continuous stream of change in the universe.  If one is to analyze any of this, the problem needs to be broken down into a smaller, more manageable size.  This can be thought of as creating categories with fewer and fewer members, so that operations can be done precisely on the few members that are left in the smaller category, and all non-members may be ignored.  One needs to have a way of defining only a portion of this infinite subject, study the portion and ignore the rest.  This is called "ignoring the infinities".  A resolver is a method of concentrating on the small, manageable part, and ignoring the infinities.  A resolver may be thought of as a category definition of a smaller category that allows one to ignore all the non-members.

 Geometry provides an example.  Consider a set of points on a line that form a one dimensional line which extends in either direction to infinity.  The usual procedure in geometry is to select some finite segment of the line and do geometry with it.  For example two points (A and C), are arbitrarily selected on the line, to form a line segment.  A third point (B), is selected between the first two.  Now various operations are done with the parts of the line between A and C, and the infinities on either side of A and C are ignored.  All that is considered is the line segment.  The rest is still there, but it is ignored.

 The method of selection is the resolver that resolves the problem of having too much line to deal with practically.

 Resolvers are methods for defining a small manageable amount of something large and unmanageable, so that progress can be made with what is manageable.  Richard Feyneman, for example used renormalization as a resolver.  It is a mathematical technique that let him ignore the infinities he had encountered, so that he could go on to develop quantum electro dynamics (QED), a theory that merges the electromagnetic force with the weak and the strong forces.

 If two dimensions are considered, then the infinities need to be controlled on both dimensions to have a manageable problem.  If a second spatial dimension, perpendicular to the first has a line along it, then the same selection method can be used as a resolver.

 The resolvers put left and right boundaries and top and bottom boundaries on the infinite lines, so that the infinities may be ignored and work can be done with the smaller, more manageable figure that is left.

 The concept of a resolver, which reduces a problem to manageable proportions is found throughout nature.  The size of the universe and the many infinities make this necessary when limited local resources are being used to cause change.

KNOWLEDGE
 
 Knowledge is a description of how to do something, specifically, how to cause or accomplish a particular change in the universe.  There are two kinds of knowledge, direct knowledge and indirect knowledge.

 A description of how something changes is the thing itself as it changes.  This is direct knowledge.  The most accurate and complete description of how the Earth changes from moment to moment is the Earth itself.  The Earth is a direct knowledge description of how the Earth changes.

 Indirect knowledge is a description in some physical symbol system of how something else changes.  The entities and relationships of the changing parts of the thing can be represented by other physical parts of the universe, such as ink lines on paper, or clay or computer models, or chemical-electrical impulses in the brain.  These are the physical symbols which represent the parts of the changing entity and the changing relationships among those parts.

THE KNOWLEDGE UNIT

 The knowledge unit describes change from one state to another.  It is usually in the form of a direct description in nature and an indirect description when humans use symbolic knowledge to describe the change.  A knowledge unit is like arconiozing a molecon.  A knowledge unit has three parts, an initial or base condition, a solution and a result.  The base condition is the disorganized parts at L-1.  The result is the new organized whole at L-0, the prime whole or prime entity, with a prime attribute as a whole.  The solution is the process of change from the disorganized parts to the orgasnized whole with its prime attribute.

 A knowledge unit has a set of initial conditions, called the "base condition", and a completed change called the "result".  In between is the process by which the base condition is changed into the result.  This process is called the "solution".  It is the solution to the problem of how to cause the change.  The base condition, solution and result form the structure of one unit of knowledge, or how to make that particular change happen.  A knowledge unit only describes changes that have actually occurred in the physical universe, or would occur if the right initial conditions occurred.  For example, if the energy is present at the right time and place in the right form and amount to make the change happen, the change will happen, absent some intervening factor not described in the knowledge unit.

 All the change that has ever occurred in the universe may be considered one continuous process of change, similar to an infinite line.  All the change that occurs in the universe from one moment to the next may be considered one change, for example how every particle in the universe changes in position with respect to all the other particles during that time.  This is too much change to describe in a manageable form.  Therefore some resolvers need to be used to reduce the description of a knowledge unit to manageable size, so that the infinities that surround it can be ignored.

 Different types of resolvers are needed to provide the boundaries for this problem than in the case of the infinite perpendicular lines, but the process is similar.  First the end points, a beginning and an end (such as A and C) are selected, and then the top and bottom points, a ceiling and a floor (such as D and F) are selected.

 How much change should there be in one knowledge unit?  That is like asking how long should a line segment be.  The basis for the selection is arbitrary, but sometimes is helped by other considerations.  Change is a continuous process (similar to the infinite line) in the sense that one thing changes into another, which eventually changes into the next in the chain of change.  Where does one begin in the continuous process of change to define a particular knowledge unit?  How does one make a brick out of mud?  How does one select point A?  First one begins at the end and selects point C.

 The initial step is to select an end point, similar to point C.  This is the final condition or resulting change which is to be accomplished or for which the description is to account.  This is the goal of the change.  This is the result in the knowledge unit.
Everything else to be resolved and defined will be involved in producing this one end result.

This is helpful, but still the problem is too big.  What initial conditions should go into the base condition?  What should be included in the result?  How can the problem be further simplified?

 RELEVANCE.  Relevance is a resolver used to further simplify the problem.  One aspect of the problem is selected, and then only those parts of the base condition, solution and result that are relevant to that part are selected.  All other parts are ignored.  The technique for selecting that part is to select the key or focus of the knowledge unit.  This key is the change that the knowledge unit describes how to accomplish.  "Relevant to what?" is the question.  To make it as simple as possible for the smallest possible unit of knowledge, only one item is selected as the relevance key for the whole knowledge unit.  Everything in the knowledge unit is relevant to the accomplishment of the change that produces that one item.  The item is not even an entity.  It is smaller than that.  It is only one attribute of an entity.

 The relevance key or focus of the knowledge unit is one attribute of one entity in the result.  We begin with the end (the one changed attribute) and work backwards to the beginning (the base condition from which the changed attribute arose).

 We begin by asking, "What attribute is of interest?  This attribute is called the "prime attribute".  It is an attribute of some entity called the "prime entity".  The prime attribute is a molar attribute or an attribute of the prime entity as a whole, such as the weight (prime attribute) of a ball (prime entity).  The prime attribute is the relevance key or focus of the knowledge unit, not the prime entity.  A prime entity may have many attributes as a whole.  If there is ever a conflict in relevance between the prime entity and the prime attribute, it is what is relevant to the production of the prime attribute that is used to determine what are relevant parts of the knowledge unit, so that everything else in the universe may be ignored.  A description of the production of the same prime entity, but with a different prime attribute forms a differenmt knowledge unit.  Sometimes it is convenient to determine relevance of parts of the unit to the prime entity.  This is acceptable, so long as the aspect of the prime entity being used is relevant to production of the prime attribute.

 Once the prime attribute and prime entity have been defined, then the end point, the result, which is similar to point C on the line has been selected.  All of the rest of the conditions in the universe which exist at the time of the result may be ignored.  Only those aspects of energy in space-time which are directly involved in producing the prime attribute and prime entity are relevant.  All others in the result may be ignored.

 The next step is to select the beginning, the base condition, which is similar to point A.  This is done by determining how much change is to be described in this knowledge unit.  How far back in the process of change should the unit go?  This selection is arbitrary, because in theory no matter how far back one goes in the change process to define the base condition, a solution process describing all the steps of change from the base condition to the result can be given.  Therefore the decision is based on other factors, such as convenience, prior knowledge or the beginning of the point of ignorance in the change process.  If step A - 1 is known, then the next step is step A.  If one is ignorant of how to do step A, then A is the beginning point of ignorance in the change process.  Depending on what one wants to accomplish with the knowledge unit, one could begin with the initial conditions just prior to step A at step A-1.  The selection may be a function of measurement.  If one can measure events at step A, but not step A-1, then begin at step A.

FRAME OF REFERENCE. The beginning point of the knowledge unit is the base condition.  When the beginning point has been selected, then the location in space and time of the beginning point can be determined.  Once the location in space and time of the base condition and the result have been determined, then a reference system can be selected.  A reference system is some part of the local environment that does not change in space and time during the change of the solution, against which the changes in the solution may be plotted.  The frame of reference can be though of as a box that contains the change process.  When the change occurs, things inside the box change, but the box does not change.  All of the changes in the knowledge unit can be referred to this rigid, stable, unchanging frame of reference.  It becomes the space-time framework to which all change in the knowledge unit is tied or compared.  Spatially it is like a Cartesian coordinate system.  Temporarily it is like a clock.  In Figure 1 the squarefield with its clock is the space-time reference system for shapes and changes of shapes on the squarefield.

 If the prime entity exists in the base condition, as well as the result, the prime entity can be the place to begin to ask questions of relevance.  If the prime entity does not exist in the base condition, but is created as a part of the solution process, the constituent parts which are joined together or taken apart to make the prime entity are places to start asking about relevance.  What parts of the prime entity or the constituent parts that will make it are involved in the change process that actually produces the prime attribute?  These are the relevant parts to the prime attribute, and therefore to the knowledge unit.

 Similarly with the solution steps.  The process of change from the base condition to the result may be capable of being broken down into definite steps which occur in order or in parallel.  U causes V, V causes W and W causes the prime attribute to occur.  Each of these solution steps should be considered separately for relevance.  Only those parts of each solution step which are involved in producing the change during that step that leads to the production of the prime attribute are relevant.  All other parts of each solution step can be ignored.

LEVELS OF ORGANIZATION.

 Now the base condition, solution and result have been identified, honed down by relevance and initially defined.  This is similar to selecting points A, B and C on the infinite horizontal line.  However, how far up or down should the unit go?  How detailed should the description be?  Should it extend from the micro level to the macro level, from quarks to quasars?  This is similar to the infinite vertical line.  Some method is needed to select only useful parts of the vertical line, and ignore the rest.  This is done by use of the concept of levels from organization theory.

 In this case one begins in the middle.  The prime attribute and the prime entity are assigned the basic level, designated L(0).  The next smaller parts of the prime entity which are relevant to the prime attribute and which make up the prime entity, are the next level down.  The level for these parts is designated L(-1).  The level of the next smaller parts which make up the parts of the prime entity are at organizational level L(-2).  An entity which is organizationally above the prime entity, is a super entity of which the prime entity is a part.  The level of the super entity is designated L(+1).

 The bottom of the knowledge unit, similar to point F on the vertical line, is selected by determining how many levels of organization down from the level of the prime entity, L(0), are relevant to the change from the base condition as defined, to the prime attribute.  This may be a matter of convenience or measurement, or interest.  One may only be interested in describing the solution at the level of human hand tools, and not at the level of molecules, or atoms.  One's measurement devices may have a limited resolution which limits the relevant level to changes larger than millimeter size, or light years for example.  One can arbitrarily select one or two levels down for the analysis of the change.

 The top of the knowledge unit is either the prime entity, L(0), or one or a few levels above it, L(+1, or ...).  Normally the top wouild be the prime entity at L(0).  Therfore a knowledge unit would have a top at L(0) and a bottom at L(-1).

SUMMARY OF METHOD FOR DESCRIBING A KNOWLEDGE UNIT.

 1. Select the prime attribute.

 2. Select the prime entity which psssesses the prime attribute.

 3. Define the result (relevance limitation).

 4. Select the base condition.

 5. Select the solution steps.

 6. Select the frame of reference  (something in the local area which does not change during the solution process).  Create a coordinate system based on the frame of reference.

 7. Select the levels of organization to be included in the unit.  The prime entity with its prime attribute are at level L(0).

 8. Do a relevance reduction on all of the parts of the unit.  Only those parts of the base condition, solution and result, which are relevant to the production of the prime attribute are retained in the description of the knowledge unit.

 9. Do a surface reduction.  Instead of describing a whole entity, just describe the surface which interacts with another surface in the change process.  If all mass is concentrated at a point, define the point and the amount of mass associated with it.  However, be able to handle any deformations of surfaces, and changes in shape or form of energy, and of course any changes in amount of energy.  Also, once the knowledge unit is described in its stripped down form, it must be possible to redescribe its parts in "normal", solid condition for clarity of understanding for those who want to see it in its solid (not surface) form.

 10. The knowledge unit may be further reduced by various methods of shorthand notation, such as a point representing a spherical mass, or an equation symbol representing some mathematically describable part of the knowledge unit.  Here too reconstruction to a "normal" description must be possible.

 11. The base condition, solution and result, in their reduced form, should now be defined in the coordinate system based on the frame of reference.

 12. A complete description of the space-time, organizational and energy changes from base condition to result and prime attrubute should now be described to complete the knowledge unit.

 The completed knowledge unit is a description of the change from a specifically defined base condition to a specific result producing one prime attribute by a specific solution process.  All of this is described in terms of space-time descriptions of all the relevant parts, their organization and changes in organization, and the forms and amounts of energy involved in the change from the base condition to the result.

 One who wants to know how the change happens may inspect the description of the change in the knowledge unit.  One who wants to make the change happen in the physical world need only recreate the base condition, and the change will occur, barring some other factor not included in the knowledge unit.  One who wants to find this solution and learn how to make this change, need only search for a description of this base condition and this result.  The "name", "signature", "address" or what ever the  unique identifying mark of the knowledge unit is called is the base condition plus the result. Finding this knowledge unit, one that has this base condition and this result, will produce a solution that works.  One who wants to compare various solutions to the same problem, may search for all knowledge units with the same base condition and result, but different solution steps.  Then the efficiency of the different solutions may be compared.  Efficiency is defined as the energy multiplied by the time required to cause the change.  The solution with the smaller product is more efficient.  It takes less time, or less energy or both.

 Knowledge units may be linked together to produce larger chains of solutions.  If a chain of solution steps is known, except for one step in the middle of the chain, then research can be done on a solution to the knowledge unit which has as its base condition the result of the last known knowledge unit in the chain before the unknown step, and which has as its result the base condition of the next known knowledge unit after the unknown step.  Knowledge units, once described, are useful tools for many uses in discovering, describing, understanding and controlling change in parts of the physical universe.