Article #374Add/Reply to this discussion board posting
Subject: What Matters from a Robot's viewpoint
Author: DRAndrewWilliamHarrell
Posted: 8/31/2019 12:55:44 PM
WHAT MATTERS FROM A ROBOT’S VIEWPOINT
NUMBER WITH NUMBERS, NUMBERS WITH NUMBERS, MATHEMATICS,SCIENCE AND THEOLOGY, AND
ETHICS
Some recent tweets about Onenes numberically as uniqueness, firstness, ethics
“For a few of us, a certain Hope prevenes Faith in a God that answers prayers.
Then,afterwards, Faith,it,Love become a
happy 6-fold blessing.” @AndrewWHarrell
:
In order for ‘America First’ to happen as we trust divinely in things to happen right, the people
representing it, President Trump first, should be telling the truth, not just to the FBI and Congress, but
the to all of us and the press. Do I hear an Amen from God and us?
@AndrewWHarrell
What do you think? If you ever met God in person, would He be a realist or an Idealist? You could find
that first He is a realist, then an idealist, then He is inside of you the measure between the two. Finally,
He is neither, but emptiness inside of emptiness in you knowing Him. @AndrewWHarrell
28 Dec 09 Andrew W. Harrell @AndrewWHarrell
Our Father who art the One in Heaven! He is "our" Father, He is unique (there
is no one like Him). And,Christ Jesus makes Him known in us.
26 Dec 09 Andrew W. Harrell @AndrewWHarrell
Day after Christmas. "Meditating with Our Father who art the One in
Heaven." "Hallowed and Blessed be His Name." "His will be done."
31 Dec 09 @AndrewWHarrell
Day of the Holy Name of Jesus. May we all be One in God's name, even as I AM
that/who I AM One in it now. John 17:21
MOUNTAINTOP VIEW AND KNOWLEDGE LOOKING FROM THE CENTER OF THE PEAK see photos in the
2006 and 2007 pray postings at http://ourprayergroup.blogspot.com
DR, ANDREW WILLIAM HARRELL
Mathematics, UCBerkeley, 1974
LTC US Army Engineer Reserves(Ret.)
INTRODUCTION 4
DEFINITION OF SOME TERMS FROM THE THEORY OF KNOWLEDGE 6
DEFINITION OF GOD FOR THE ROBOT 51
DEFINITION OF SOME THEOLOGICAL IDEAS AND TERMS 63
SOME MORE HEAVENLY TERMS DEFINED MORE IMPLICITLY 73
DISCUSSION OF THEORIES OF CONSCIOUSNESS 92
DEFINITION OF SOME TERMS AND IDEAS IN LOGIC AND COMPUTER SCIENCE 94
DISCUSSION OF SOME DIFFERENT TYPES OF REALITIES 100
CONCEPTS AS EXPERT SYSTEMS 103
A SHORT HISTORY OF SOME OF THE MATHEMATICS RELATED TO THESE TOPICS 114
HOW NUMBERS ARE REAL 126
HOW THE OPTIMIZATION OF “THE BEST” AND “THE MOST” IN OUR THOUGHTS INSIDE OF
KNOWLEDGE NETWORKS IS RELATED TO THE THEORY OF NUMBERS AND ROBOTIC ETHICS 134
WE ARE LED BACK TO WHERE WE BEGAN TRYING TO UNDERSTAND CONCEPTS IN PLATO”S
DIALOGUE PARMENIDES 141
Zeno’s Paradox 148
APPENDIX A 158
REVIEW OF VOLUME I OF DEREK PARFITT’S “WHAT MATTERS” 158
DISCUSSIONS OF VOLUME II OF DEREK PARFITT”S “WHAT MATTERS” 17
SOME EPISTOLOGICAL MISTAKES IN DR. KITCHER”S REVIEW OF DR. PARFITT”S BOOK 28
APPENDIX B 32
USING A MECHANICAL DEVICE TO DEFINE THE CONCEPT OF THE NUMBER ONE 32
APPENDIX C 40
CAN WE PROGRAM ROBOTS WITH THE CAPABILITY TO REPENT WHEN A LOT OF US CANT EVEN
UNDERSTAND HOW TO DO THIS OURSELVES 40
APPENDIX D 43
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/******************************************************/ VAL*([_],LOOOO):- 0
/* IF PATH DOES NOT HAVE A FULL FORWARD EDGE MAY GO IN THIS DIRECTION*/ 0
/* IF PATH DOES NOT HAVE A EMPTY BACKWARD EDGE MAY GO IN THIS DIRECTION*/ 0
SIZE(A,B,S): 0
EXPRESS(C,D,F): 0
/* MINIMUM COST NETWORK PREDICATES*/ PREDICATES 0
DEPTHL**([J,_,[]). /*IF PRESENT NODE IS FINISH PT. THEN WE ARE DONE*/ 0
/***************************************************/ 0
DEPTH(ST,FI,L), 0
REVERSE(L,LL}, TIME(_,M,S,_), 0
BIBLIOGRAPHY 0
INDEX 3
INTRODUCTION
In order to understand how robots can have ethical behavior I think we humans will have to learn how to
understand what not only mathematicians and computer scientists, but also philosophers understand
the meaning of the word concept is. And, in order to understand concepts we must understand what
words and what their meanings are. And, in order to understand what meaning is we must understand
what the meaning of meaning, what the concept of a concept is,
Philosophically, mathematically, and in terms of computer science also. For me, after studying this
problem for 50 years or so I believe the key to solving it is to understand how numbers can be defined
to be real objects and have empirical meaning as well as logical spiritual existence.To do we will find we
are lead to the question of how to define the word God to robot’s, and how if we can’t prove He exists to
the robot to give them a set of facts and rules that allow Him, assuming He does exist to continue to
exist inside of them and help them with their ethical decisions. How can we argue philosophically, like
Aristotle and Dr. Mortimer Adler did for Him to continue to exist in all of us,,letting the robots and us
assume He already does.
In order to understand numbers with numbers and then mathematics, adding subtracting, multiplying,
dividing them, we must have first a way of representing them well, their structure. And, we must have a
way to operate on them well in order to understand what they can teach us. We can see them as 0s &
1s, in order to understand logic, decimal digits, for tax returns, real numbers for calculus, algebraic
roots of equations for geometry, complex numbers to help us understand airplane lift and electro-
magnetism, sur-real numbers for games, and to put to fill inside the gaps in the real continuum, etc.
But, we must also have an algorithm for operating with them, and a control structure to give order to the
algorithm. In short we need a computer program with a data structure, a formula for computation, and
some kind of way of ordering the calculations of the formula to understand what numbers with
numbers mean.
And if these mathematical objects, numbers are to have application to science, we need to know how to
store them in reality, in a computer’s memory if we are using them in a robot, or in our minds in
ourselves and the natural world around us if we are using them for all of this.
Starting from these basic assumptions, do we have all we need to understand God and His ethics from
understanding these numbers and their laws that He has given us to obey? Probably not. But why not
see how far we can go toward understanding Him or Her with just this much and Him dwelling inside of
ourselves as a Holy Spirit to work with us and help us with the undertaking.
?
DEFINITION OF SOME TERMS FROM THE THEORY OF KNOWLEDGE
His first attempt at a definition of what a concept was to Plato was say basically:
Plato’s first definition of what a concept is (from the dialogue Meno)
,
---concepts form the meaning of meaningful words. ---concepts, smaller than a judgement, larger than
a sense impression are units of thought---well-defined relationships between concepts are themselves
concepts.
Plato’s second definition of what a concept is(from the dialogue Theatetus)
--- A concept is a rule that may be used to decide if an object falls in a certain group. It is an abstract
way of grouping thoughts. It deals with the information associated with the object of thought by asking
questions about it. This may be a simple process like the way we classify concrete objecs by the
“marks” of sense impressions: such as physical size or texture (this process is instinual in most
animals). Or it may be a more complicated process using a lot of other concepts.
But, in this dialogue, Plato was unable to clearly explain how one just asking a set of questions to use as
rules to define ideas and concepts is able to always know which questions to ask, when to stop asking
them, and how to present the final results of all the questioning…if, in face it is able to achieve a useable
‘final’ definition of anything if we follow this process.
The key question comes down to, “How do we use goals, help us understand the situation and define it
better in a classification type expert system.”? I hope to give some ideas of how we can do this in a
systematic procedure in a given situation later on. We will define some terms in the new relatively new
science of computer mathematics that will help. The dialogue where Plato did consider some ideas on
how to do this is his most difficult to understand, Parmenides. And, it turns out the most important goal
we will be able to understand better is the same one Plato talked about in this dialogue, “How do we
understand what ‘Oneness’ ‘is’?” How is it Truth, How is it Goodness. How is it ‘unification’ of other
ideas and thought?
Plato’s Theory of Knowledge was built around something he called “The Divided Line”
If you draw a diagram with four rows, two columns, and eight boxes:
Diagram: THE DIVIDED LINE
THOUGHT OBJECTS
REASON
(DIALECTIC) HIGHER FORMS
UNDERSTANDING
SCIENCE,
MATHEMATICS FORMS OF
SCIENCE AND
MATHEMATICS
BELIEF
(PERCEPTION) THINGS
OBJECTS
CONJECTURE
(IMAGINING) SHADOWS,
IMAGES,
REFLECTIONS
In the above diagram there are several terms which Plato used and we have defined yet. For Plato, what
didi a “form” mean… and what did the word “object” mean? Later on we will explain better what Plato
meant when he called something a “form”, for he did believe that it was a “something” and object. For
later empirical philosophers a form is just an object occurring and occupying a portion of space and
nothing more. But, Plato also believe that the meaning of the word form, its concept, was defined
differently if it referred to “ethical” ideas that if it referred to object occupying portions of space. He
believed that the concepts, the meanings of the words “goodness” and “truth” also were essences or
forms that were real, independtly existing things.
I
KNOWLEGE
OPINION
THEN EVERYTHING IN THE FIRST TWO ROWS IS CONSIDERED PART OF THE INTELLIGLE WORLD,
WHILE EVERYTHING IN THE SECOND TWO ROWS IS CONSIDERED PART OF THE VISIBLE WORLD.
By studying this diagram it is clear that Plato believed that mathematical entities were things not just
thoughts hence objects, hence real.
Now, some time later after the invention of digital computers and studying the theory of their operation
somewhat we can say that methods of looking at sense data in computers can be concepts that Plato
tried very hard his whole life to understand.
Concept of a Concept
Part I
1st DEFINITION OF A CONCEPT....
--concepts form the meaning of meaningful words;
--concepts, smaller than a judgement, larger than a sense impression are
units of thought;
--well-defined relationships between concepts are themselves concepts.
But beyond this, it is possible to list in more detail the most important
ways concepts work for us.
With development of appropriate practical backing they:
--Define object precisely, for our future reference and mutual communication;
--Abstract what different recurring experiences have in common, saving us
effort in the way we describe things;
--Make us able to imagine things, thinking about what isn't present;
--Spur problem solving, breaking us out of areas of mental confinement;
--Cue discoveries, the bright ideas we need to stimulate us forward;
--Help us learn how something works, and to remember what we have learned;
--Increase our understanding, making it easier to form still more concepts.
Stating a judgement involves conepts. Also, in making some decisions, we go
through a process of trials and errors performed mentally, before reaching a
decision. Concepts make this "idea testing" possible.
Even more fundamental than all this, is the way in which concepts develop
our first perceptions of the world. Some everyday examples of how concepts
do this include:
--the permanancy of concrete objects;
--the switchboard of three-dimensional space;
--which way is [up] and right-handedness;
--the rules of logical thought.
II
Here are some objections, straight from Plato's dialogues, to studying
concepts for their own sake:
The first comes from the dialogue of Socrates with Meno:
MENO. But how will you look for something when you don't in in the least
know what it is? How on earth are you going to set up something you don't
know as the object of your search? To put it another way, even if you come
up right against it, how will you know that what you have found is the thing
you didn't know?
Indeed. And, this can be followed up with a second, more specific, objection
to studying concepts and though processes: It is held that the study of the
method of thought for its own sake is of no use, except when almost
superfluous. In otherwords, isn't it ridiculous to separate the study of
thought from the day to day problems which we use if for?
Plato, stated this in his Socratic dialogue Charmides:
SOCRATES. And if a man know only, and has only knowledge of knowledge, and
no further knowledge of health and justice, the probability is that he will
only know that he knows something, and has a certain knowledge, whether
concerning himself or other men.
CRITIAS. True.
SOCRATES. Then, how will this knowledge or science teach him to know what he
knows? Say that he knows health; - not wisdom or temperance, but the art of
medicine has taught it to him; - and he has learned harmony from the art of
music, and building, from the art of building - neither, from wisdom or
temperance: and the same of other things.
CRITIAS. That is evident.
Well, we won't try to hide it: in contrast to these carefully worded
sentences, the rest of this section of this book will list instances in
which the study of thought processes has value to us - instance in which it
has a very real value.
All along it must be kept in mind that steps forward in knowledge proceed
awkwardly. The initial conception of a new discovery always has many details
that are wrong. Socrates takes a critical position in those two quotations.
But, he later states that for his part the desire to know what is good (not
necessarily the actual knowledge itself) is the most important thing. The
sum of what we don't know is so vast -- we will never get anywhere without a
lot of pure desire to step beyond what we don't know with what we aren't
sure of. There is no way to obtain knowledge without making mistakes and
blunders. There is no roal road to any difficult art or skill.
WE SUMMARIZE: Our progress will depend on how important we conclude that it
is to try to do what we aren't sure of; what we don't seem capable of; what
we haven't been prepared for; and to what extent the theory that we do
construct being built on hard-learned experience, does not outdistance our
capability to produce results.
III
Not attempting to formulate any new principles, we will work with familiar
ideas, elaborating them, the method can be called the study of concepts
through concepts. Lets lay this technique out in part before hand:
.... We trace out the definition of a lessor known concept by recognizing in
it the same elements as are present in a better known concept. In literature
this method is used in the form of metaphors and similies. In ordinary
speech it is the method of analogies.
....Since we are actually hunting concepts, we can bring one out in the
open, so to speak, by asking questions about. These questions will expose
what dols the concept together.
.... By seeking out those elements in concepts that constantly reoccur, we
obtain the most economical description of them.
.... We wait for it (the work) to mature; repeatedly changing its
presentation, searching for the best presentation of what [seems] profound;
convinced that simple and elegant solutions are the shortest distance
between two points. The work will be the product of many months, even years,
of stubborn thinking and the subject of pride.
In the dialogue Theatetus, Plato first considers the claim of sense
impressions (by themselves) to be knowledge. A rational argument is then
given to discount this possibility. It is asked how false judgments are
possible under these circumstance. A model is given that compares this to
sense impressions not fitting the patterns in wax formed by objects. Then
examples of false judgments are given which don’t fit into this paradigm of
knowledge.
Then the possibility of true belief being knowledge is considered.
First “having” knowledge is compared to holding birds (ideas) in cages in
our minds. The cages are the truths of experience which constrain statements
we can make about the ideas. In this situation it is argued that it is
possible to make false judgments when we depend on these cages to
interpret our factual
experience. An example of a lawyer twisting the facts of a case to make the
jury come to the wrong conclusion is given.
Finally a definition for knowledge is given which is somewhat better.
Knowledge is said to be true belief along with an account or logos. This
account justifies our thoughts by making a sound (based on sense
impressions) rational argument about the experiences we have had. In order
words we are saying that in some sense knowledge is “assured belief”. Here
we see the importance of having a logos about God if we are to hope to be
able to know him.
However, the limitation of even this definition of knowledge is explained
further on in the dialogue. If an account is a rational logical
demonstration (two different methodological approachs to do this will be
explained later in this discussion), it must start with
certain “unknowable”, “first order facts”. This is because we have already
declared knowledge based on objects ( or sense impressions) not be knowledge
by themselves. But any argument will then just be a series of rational
exercises (tautologies) which come to no new conclusions about these
unknowable starting points. The conclusions cannot be new because they cannot
be any more knowable or true than what we started with.
It wasn’t until Immanuel Kant that human philosophy explained how it was
possible to have synthetic a priori truths. It we assume we humans have
certain abilities (what Jesus called talents) in our minds Then, by
postulating the truth of these God given abilities the truth of these
postulates themselves can
be determined and influenced by the way we perceive reality. The objects
that Plato called unknowable in themselves are this way because we don’t
have access to studying their true nature. This is because they are in a
sense outside of us. But our postulates about what is inside our selves are
accessible through our subjective intuitive and common sense minds. So these
talents can be good
starting points along with our experience about the World they can help form
knowledge when we have a logos (or account of the connection between what
has happened to us and what situation we started with). They then form an
innate nature common to all of humankind.
IV
THE MIND AS A SWITCHING YARD----PASCAL "ONE MUST
SUBSTITUTE DEFINITION FOR THE DEFINED."
Names are associated with objects; so are their meaning. But an objects's
name can be one the tip of the tongue when a corresponding meaning isn't:
how else do we know sounds when we hear them, or other things when we see
them. The answer is that our mind automatically deals with things by sorting
them into pre-established groups. We have, in our mind, a switching yard. If
we want to know what a song or anything else is, we must ask ourselves
question about the mental definition of it. It will be helpful to discuss the
nature of this mental switching yard, in order to understand how to ask
these questions.
Suppose that pieces of information start off in the mind like freight cars
start at the high point in a railroad switching yard:
-- They coast down the hill; switches guide each one into its proper siding.
There the cars wait, with others of the same destination to be sent to form
a train [ see classification type expert system which is defined below];
-- Each time we send a "new" object rolling down the hill, new switches are
added, and the network becomes better at sorting out information;
-- As we look at a scene of things in front of us, the mind couples thought
onto the corresponding parts of the scene -- and all of a sudden, what isn't
a blur, is a series of meaningful images moving through the mind [ see
backward oriented reasoning as defined below].
In a freight yard the starting points of the cars is on a hill. But, before
getting to the bottom of the tracks the cars may reach resting places. To
get out of these resting places often requires some extra help [see genetic
algorithm as defined below]. This is the case if there are substantial
depressions in the ground: or it may be that only a little push is needed;
if they, (the thoughts) are stuck on a rise in the ground level.
As images come through the mind to be sorted into concepts, they too reach
resting places. These are resting places in the switchboard of the mind,
speaking figuratively. If the route doesn't take us the full distance (does
not give us enough understanding), there may be valleys; and in this case
the help of other concepts needs to be introduced (more new switches added-
things we haven't classified yet). There may be rises. In this case we just
need to reflect on the situation. We must conciously give the cars a little
push in our mind.
It is possible to push this metaphor too far. But it does provide the means
to understand the quotation at the beginning of the section. The quotation
in French is "Substitutuer mentalement les definition a'la place des
definis". this means "Substitute mentally the defining facts for the
defining terms".
Faced with a problem, not sure how to solve it, there is a reason for much
of the uncertainity that occurs. We often fail to spell out the meaning of
the ideas involved in the problem. This is when we should go back [see
examples below] to the definitions of the ideas that we have in our minds:
-- Since the switching patterns branch out as they get more specific; our
questions go from the general to the particular, the identification process
quickly sorts one thing out of myriads of possibilities;
-- The "unknowns" in the problem are interpreted in terms of the conditions
and data that are given; new connections are formed mentally; we see things
clearer and in a conscious sense;
-- By varying the conditions of the problem, the mind searches for a related
problem that it has solved before; the general idea of a solution emerges;
patiently, the problem solver waits until all the troubling points are dealt
with, to insure that all the information is accounted for.
Second and third definition of a concept----
2nd DEFINITION OF A CONCEPT is a rule that may be used to decide if a object falls in a
certain group. It is an abstract way of grouping thoughts. It deals with the
information associated with the object of thought by asking questions about
it. This may be a simple process like the way we classify concrete objects
by the "marks" of sense impressions: such as physical size or texture (this
process is instinctual in simplier animals). Or it may be a more complicated
process using a lot of other concepts.
3rd DEFINITION a CONCEPT is a structure, existing in four dimensions in which meaning is stored, the
structure, or form is not composed only of objects existing in space and time in these dimensions but
can also be data or information stored as objects and functions in what mathematicians now call a
category, after Dr. Grothendieck, Dr. Maclane, and others defined it so in the 1950s.
It took a whole previous millennium and more than a few centuries in the last millennium for
philosophers, mathematicians and logicians to come up with the complication definition of what
mathematicians now call a STRUCTURE.
Kant was once oif the first philosphers to try and define what he called a STRUCTURE then, in terms of
indwelling intuitive wisdom related to his ideas about ethics. It is now something less empirical, but still
metaphysicial than what we had called an “OBJECT” or a “SET” before
A CATEGORY, as we now define it, is a mathematical structure consisting of the following DATA 1)
Objects, 2) Maps and 3) for each map an object as DOMAIN and also what is called RANGE or CO-
DOMAIN OF the function., 4) for each object A an identity map with DOMAIN A and CODOMAIN, 5) For
each pair of maps A,B a composite map C from the DOMAIN OF A to the CO-DOMAIN of B, 6) some
identity laws and associativity laws for the maps.
See the book “Conceptual Mathematics” by Lawvere and Schanuel for a more complete discussion and
explanation of this.
A PROPERTY in terms of what we now call CATEGORIES mathematically is defined as being a SUB-
OBJECT of a CATEGORY. This is different than what we
call a sub-object in set theory. In set theory, sets are made up of objects or elements, and sub-objects
of sets are what are defined as subsets….or parts of sets defined by group or elements in the larger
group or set of elements. This subset may be defined by the condition of satisfying a logical property,
and hence be a class.
These PROPERTIES would now be what Plato called “forms” having physicial reality in our mathematical
and also material universe.
But, a SUB-OBJECT S (also called a “SHAPE”) of a category, of which some of the maps in the
category are considered as being defined by the condition of satisfying logical proprositions.is defined
by INCLUSION MAPS IN THE CATEGORY.
An, INCLUSION MAP, i IN A CATEGORY from S to X is a MONOMORPHISM or MONIC MAP satisfying
“For each object T in the CATEGORY and each pair of maps, s1, s2, from T to another object S in the
CATEGORY is1=is2 implies s1 = s2. Lawvere and Schanuel page 335.
Frege when he wrote his book on numbers as concepts separated the parts of
Number into objects and functions. He called the functions from the numbers to the numbers the
conceptual part of his mathematical scheme of their reality. Associated with the objects he called
numbers were also their names This later developed into what he called the objects of numbers defined
as sets, like the empty set and the set of the set containing no elements, which also was called an
empty set.
Bertrand Russell his attempt to formalize completely mathematics and mathematical philosophy, called
the conceptual part of numbers, composed of objects and functions their names. This was because he
was philosophical what we called a “nominalist” or someone who believes the reality of concepts exists
only.
Formally and that formality is and only exists as a “name” not an empirical reality.
Russell came close to defining what we now can a category, with his theory of classes and the
structures which he called types.
Classes, or concepts, for Russell were functions like Frege had defined them but they also could create
reality through collecting the groups, heaps, or sets of elements that satisfied them.
For Russell the meaning of the objects he called numbers was also apart of the conceptual part of
them.
Meaning had an “extensional” or empirical part plus an “existential” or actual part and also an
“intensional” or “essential” part. Thus we have the “four-dimensional” part of the conceptual structure
that he believed in.
In order to define numbers in terms of categories and not sets we must set aside an extra metaphysical
dimension of meaning for what are called in defining real numbers for calculus “LIMITS”. In the set
theoretical approach to defining real numbers we define sequences of elements set theoretically by first
defining the ordered set theoretic product of two elements in the sequence, then putting an order on
the set theoretic elements and, calling on an axiom of infinity in order to postulate the existence of the
limit of the sequence of set theoretic elements. In using category theory to define limits of sequences
of functional numbers referred back to objects through mapping diagrams, we have to postulate a new
category of objects and functions with “CONE OBJECTS” and maps inside the category from functional
numbers to their regular number objects and also the cone object. This adds another dimension of
meaning, in addition to the extension, intension, existence, essence dimensions which go back to Plato
and Aristotle to what we now call the concept of the real number so categorically defined. See the book
“Category Theory for the Sciences by David I. Spivak in the references for more details on this.
Types were structures or groups of objects which form the domains and ranges, or co-domains of the
logical functions or propositions in his scheme of analytic logic.
When Godel proved his famous result about the existence of computable, non-analytic, logical
propositions he invalidated Russell’s key “axiom of reducibility” that postulated that any set formed by
already defined types of elements satisifyng the classes of logical propositions was itself another
“type” or “structure.” Made up of elements of the type already defined.
You might want to read his mathematical formalist book “Introduction to mathematical philosophy” and
also the criticism of it by the structuralist Mary Tiles in “The Philosophy of Set Theory” for a more
Complete explanantion of this.
Plato believed to the contrary that concepts themselves, have a reality apart from their names and
which exists both empirically and intuitively, first as an intuitive reality. Aristotle believed that the form
of the form of numbers exists itself as a real object and first as a empirical reality.
More modern “structuralists” like Yair Neumann referred to what they call “Mathematical structures of
natural intelligence” which they postulate exists completely in our individual minds as we learn them
there and become laid out
There as the Psychologist Jean Piaget has written and philsoophised about.
Perhaps these categorical natural mathematical structures can become for
Us mathematicians as basis for a more “humanist” philosophy of mathematics of the future in which
mathematicians are both “formalists” and”idealist” as different
Points in their reasoning.
Concept of a Concept
Part II
V
A short history of developments in the concepts of mathematical logic
A Concept is a data object (which is the result of a functional computation)
After the writings in the 1760’s of Immanual Kant it took some time to translate his metaphysical an
epistemological insights into scientific and theosophical application. In the 1890’s the English
mathematician/Anglican priest George Boole invented a way for algebraists to do logic. And, the
German mathematical philosopher Gottlieb Frege wrote his seminal paper on algorithmic computation
called “concept writing”. For them mathematics was the study of what was invariant logically and
conceptually under change of notation. Was it about the real world? Yes, in some sense, but as Kant
had pointed out it is impossible for us to know this “real world” completely while we are doing
mathematics about. So, in another sense it was also true that the mathematics they were doing was,
“not about the real world.” But about our mental constructs of it.
So, what is the new way of thinking about concepts as functions and computers? For more than a
thousand years Aristotle’s logical and definitional schemes of logic dominated the fields of study in both
philosophy, science, and mathematics. According to the traditional logic and philosophy of Aristotle
formal reasoning followed from four possible forms of judgment (the universal affirmative judgment, all
of A is B, which we will denote as an A, the particular affirmative judgment, some of A is B, which we will
denote as an I, the universal negative judgment, None of A is B, which we will denote as an E, and the
particular negative judgment, some of A is not B, which we will denote as an O). Along with these four
type of judgment there were four figures of syllogistic inference. If we break each syllogism into a
subject S, predicate P, and middle term M, the resulting four figures of syllogistic inference can be
denoted:
MP PM MP PM
SM SM MS MS
SP SP SP SP.
Based on purely combinatorial grounds this gives 256 different kinds of syllogisms .If we let the
unknown X mean is S, the unknown Y mean is M, and the unknown Z mean is P, and express the above
syllogistic forms in the predicate Boolean calculus of English mathematician George Boole the premises
in the first line above can be written:
| ‘Y v Z| ; |’Y v ‘Z| ; |’Z v Y| ; |’Z v ‘Y| where ‘Y means not Y, X v Y means X or Y, and |X|
means the truth value of the variable X. eg |’Y v Z| means the truth value of not Y or Z ( which is the
same using the normal definition of implication as the truth value of (Y implying Z).
The bottom two lines can be written:
| ‘Y v X| ; |’Y v ‘X| ; |’X v Y| ; |’X v ‘Y|
And
| ‘X v Y| ; |’X v Z| .
Using, this symbolic notation it is possible to that the truth or falsity of all 16 of the possible Aristotle
syllogisms may be completely represented and tested in this Boolean functional algebraic formalism .
But, now the question arises does the list of Aristotelian possibilities exhaust all the Boolean ones?
Take the sentence “If there is a son, then there is a father.” for a counter example to asserting this in all
situations. Let X stand for is a son and Y for is a father. Translating the sentence into our notation we
have:
‘| X | v | Y |
But, this expression can have many different truth values depending what X and Y stand for. And, there
is no way using this notation only to capture the relationship between what X and Y stand for into the
one argument Boolean algebra formalism. What is needed in terms of a formal to device which
expresses the meaning of the statements “is a son” and “is a father” adequately is what is called a
logical predicate with two arguments. We would then have predicates “is_son(X,Y)” and
“is_a_father(X,Y)” with the logical axiom connecting and defining the two predicates, “is_son(X,Y)” if
and only if “is_a_father(Y,X)”. X If and only Y being is a short hand notation for the assertion that X
implies Y and Y implies X. A relation is not a uniquely determined mapping from a domain to a range like
a function, it is just a set of order pairs (X,Y) where X and Y run through different sets. Given a value for
X there may be many different Y’s that the relation is defined for. But, given a value for X and Y the
relation has a value of true or false for that ordered pair.
Gottlieb Frege followed Pascal’s and Immanuel Kant’s suggestions and make the first attempt to try and
give us a precise logical definition of what had already been defined as a function by Rene Descartes.
Descartes defined a function as a mapping [correspondence] between metaphysical objects in one
realm and
another that could be shown to be well-defined [associating some object in the functions range to each
object in its domain] and one-to-one [not associating more than one to each object]. Once we have
defined what a function is conceptually we can study them abstractly. What we later came to call
computer programs are composed of algorithms [ordered sets of rules for the operation of functions on
spaces] + data structures [ways in which we store information symbolically in the program’s memory…
for more information on what I am talking about here see the influential 1976 book on computer
programming with this title by Niklaus Wirth]. For a more recent discussion of how the concept of a
computer program can be defined in terms of the categorical concepts we have introduced already see
the book in the references: Category Theory for the Applied Sciences. Concepts as we explained earlier
are the form in which we choose to transmit and work with our self-developed and God given
knowledge. So, when we
say that we can consider a concept as a computer programming we are claiming that this program is
not just a means or tool that we use to answer questions, search our memories but has value in itself
and is a form of our knowledge. So, with this way of looking at things [and our knowledge of them] I
think that we somewhat closer to considering what we are trying to understand about how we actually
understand sense information, our thoughts about this sense information and when we actually can
make warranted beliefs or judgements about whatever truth is in them. Some of these judgements will
be ethical and subjective, some objective and scientific, some philosophical and mathematical. But, in
all three cases we want to try and understand the functional processes in our minds and brains [which
may not necessarily be operating on/with the same data…realities or substances] which cause us to
come to reach these conclusions [judgements about knowledge and the creation of new knowledge for
us].
fourth definition of a concept----
A concept is a data structure. That is, it is a predefined set of object type. These types can be
frames with slots [classes], words, numbers, lists, streams, or variables. In certain situations theys can
be recursively defined. But, the final tree structure is usually limited to have only a finite number of
branchs. The information it contains is the values or attributes of the objects that the data structure
describe.
Some Definitions of Terminology Related to These Ideas
Attribute -- Defines the qualities or values contained in a class and the type of information that make up
a class. For example, the class car can have the attributes "type of engine" and "top speed".
Attribute value -- An actual number or confidence factor representing the degree of certainty with
which a factor is known.
Class -- Defines the structure (in terms of its attributes) and behavior(in terms of its associated
methods and procedures) of an object. When it becomes an instance, it then holds the actual data
values of a particular realization of this type of object in the knowledge base. For example: a class
called human beings might have attributes related to the parts that differentiate our physical beings and
categories such as those related to its our mental and spiritual capacities. Some of the associated
methods and procedures of this class could be thinking, talking, walking. It can be considered as a
subclass of another class such as the class of living beings. The author and the reader are both specific
instances of a human being object.
Forward-Chaining -- Forward-chaining reasoning is an inferencing strategy in which the questions are
structured from the specific to the general. That is, it starts with user supplied or known facts or data
and concludes new facts about the situation based on the information found in the knowledge base.
This process will continue until no further conclusions can be reached from the user supplied or initial
data (using the rules and methods coknowledge base). (See the previous section for a more complete
explanation and an example).
Instance or Instantiation -- Specific occurrence of an object or a predicate. An object consists of its
class structure, which defines its attributes and behavior and its instances, which hold the actual values
of the object. Thus objects are instantiated (or defined) by a process of forward chained reasoning in
which the attribute slots are given values. An instance of the class human beings mentioned above
would refer to an individual person, such the reader of this report. Predicates are instantiated or
defined by a process of backward chained reasoing in which the arguments (some of which may be
recursively defined) are unified with previous facts or postulates.
List -- An ordered set of objects tied together one to another. Its length is not predefined, but it does
have a first and last element.
Number – In this beginning computing context, an integer or real number.
Object -- General term for a programming entity that has a record type data structure along with
attribute values and procedures or methods that enable it to represent something concrete or abstract.
It can be contrasted with other programming entities such as facts, rules, procedures, or methods. An
object's structure is defined by its class and attribute definitions. A class declaration is a data template
involved in representing knowledge which defines the structure of an object. For example, in the class
"human being" mentioned above some of the slots might be height and weight.
Recursion -- A process by which a data type, predicate or function is defined in terms of itself. This
situation of self relation allows the function or predicate to be computed in an orderly manner.
Stream -- An ordered set of objects tied together one to another. It has a first, but not necessarily a last
element.
Variable -- A name that represents the value of an unknown object.
Here is an example first application of many applications of how thinking about computing using this
3rd definition of a concept this way can be useful:.
OBJECT-ORIENTED ALGORITHM TO GROUP OR CLUSTER SETS
OF EXAMPLE DATA IN CATEGORIES
Start off with an initial set of clusters which
partition a set of sample data into groups.
For each example in a series of new data samples:
a) Compute the mean or centroid, or some other mapping
that quantizes or compresses the groups of data sample into
a small number of values.
b) Place each new example in the cluster which most
closely matches (resembles or is contiguous to) the initial
categories.
Recompute some measure of overall effectiveness of the
classification, such as:
Sum of squares of residual difference from each sample to
the cluster’s quantization value.
Iterate on all the new examples until this measure cannot
be improved further.
Quantization --- The process by which a set of data is
partitioned into parts of segments. This process proceeds from
the bottom up using distance measures to separate the whole
space into disjoint parts. Each part has a number or set of
values associated with it. So, when a new object is examined
it may be identified by the value of certain attributes
Aristotle considered sense information as the primary
set in creating our knowledge. He understood that this process
of quantization must occur before the algorithm of rule-based
conceptual identified (which was explained above) can occur.
It is an error to disregard this type of knowledge by
saying that things are quantized by their relation to causes.
The philosopher David Hume was the first to understand that
because this algorithm works with regard to resemblance and contiguity it is an entirely different
method than the earlier backward chaining algorithm.
But, also, in this midst of this learning process the opposite mistake can occur: Things are not
related to causes and definitions (essences) because the forward chaining algorithm above says they
are. But, the backward chaining rule-oriented algorithm to be explained below can be used
to learn how to separate out the parts which have already been
quantized from each other.
The philosopher Immanuel Kant has explained
to us that the way we as humans choose in the above manner
to group together categories of thought can itself influence the way we know things. Thus, knowledge,
although it is something which uses definitions (forms or essences) separate from us, is also something
which depends to some extend on structures or clusters of categories of thought that are inside of us.
And so, the sentence:
“We hold these truths to be self-evident,
all men are created equal”
can be interpreted to mean that we ourselves contain what Aristotle called the “potency” in our minds
for this statement to be true.
VII
A CONCEPT AS A LOGICAL RELATIONSHIP....
Given that we can compute with data structures and algorithms to get a conceptual model of a
functional and recursive program, how can the extent or scope of the form of this knowledge be
extended? Many logical problems can already be solved with the first four types of knowledge
representation that have been discussed:
1) The solution to logical problems such as: Jim is the Grandfather of Sue, Kathy is the mother of
Jim. What is the relationship of Sue to Kathy?
2) How do we find the river that flows through Missouri, Arkansas, Mississsippi and Lousianna?
These questions can be answered using the ideas already mentioned.
But, there is more to the language of what is called first and second order mathematical logic:
1) What do we mean when we say that a statement is false, or that the set of its solutions [variable
identifications that satisfy it] is empty? How can we determine if this is the case?
2) What do we mean when we say that a statement is always true no matter what values the
variables in its expression take?
3) What is a statement in predicative logic? How is it different from a mathematical function?
4) How does this difference effect the way we compute solutions that satisfy a set of statements?
How do we keep track of the partial solutions (store the data) so that we can explain how the conclusion
was reached?
5) How can we best compute the set of all ways of satisfying a predicative statement?
These problems require that we introduce quantifiers (there exists, for all) into statements. We also
have to better explain how the data sets in variable substitutions match into themselves and other
different pattern identifications.
5th Definition of a Concept ----- A concept is a logical relationship involving a predicative
statement (subset of n times Cartesian product of the domain values and variables, instead of just a
functional mapping). This logical relationship may also involve the question of the satisfaction of the
concept (truth in terms of a specific knowledge representation). It may also involve the notion of a set
of variable identifications in some model [data + algorithm] . And, it may also involve the notion of how
a method for determining truth searches through the space of variable identifications inside of a pre-
determined set of program search rules [logic + control] as a part of determining what the algorithm
used will be.
From an implementation standpoint a recursive function such as those defined in the previous
section means a function along with a stack [ array of data in memory ] to hold the variable
identifications, return point, and so forth. A predicate is equivalent to a recursive function along with its
stack along with another memory area to hold the goallist, partial solution, partial variable
identifications, binding array, backtrack points, and so forth.
From a hardware requirement standpoint, the knowledge representation required for this definition
of a concept is already covered in the idea of a stored memory programmable computer [deterministic
Turing machine] and explained in the 4th defintion of a concept. However, if we believe that ideas and
the way we organize these ideas in our thoughts have a reality in themselves, then these new ways of
representing are something different. If this is so, then there such a thing as a concept of a concept.
It is useful at this point to try and explain some of the details of these questions which may not have
been clearly understood over the last several thousand years. Fairly recently, it has become clear, that
objects and predicates are defined using quite different data structure and computational procedures:
Some more definitions and terminology that form examples of the observations concerning the
second definition of a concept and its limits
Warning -- there are two parts to every definition;
-- a rule to identify the object;
-- the assertion that the rule is adequate.
The second part is a "hidden assumption" that all definitions contain. We agree beforehand
that we know what we are talking about.
-- there is no guarantee that the [rules] [marks] won't need to be changed later.
-- the terminology "if and only if" exists only in the imagination.
When a computer goal-search search program is designed using a set of rules we can try to satisfy the
rules using either a bottom-up, forward chaining strategy, a top-down, backward chaining strategy, or a
combination of both bottom-up and top-down goal search using both forward and backward chaining. If
in the rules that set up the definition of terms a negative possibility is not allowed in one of the clauses
then we won’t ever be able to know whether the search was not satisfied…and hence the only if part of
the definition of the term is either. When the rule based classification system asks its series of
questions and dynamically enters information into its object-oriented database as a result of the answer
satisfying the conditions and hypotheses of the rules it may reach the end of the rule set before all the
questions that need to be asked have been (see the paper at the US Army Research Office’s 1994
Conference on Computing by Harrell which has an example of how this happens in a 41 rule expert
system to classify river bar creation and stream,river bed erosion ). This happens because the logical
system is what is called non-monotonic ( that is what logical assertions and theorems are proved in the
system may depend on the order in which the data and rules are instantiated inside of its logical
predicates). In this case the rule set needs to be sorted in terms of the order in which the conclusion
nodes and predecessor nodes are entered in the systems knowledge tree. The algorithm below does
this:
Here is another practical application, it uses the first definition of a concept, as a rule to define words
using uestions about their meaning:
ALGORITHM TO TOPOLOGICALLY SORT RULES IN AN EXPERT SYSTEM
Start) For the whole set of nodes of conclusions in the rules:
a. If every conclusion node has a predecessor, then stop. The rule-based system has a cycle
infeasible (that is, a partial order cannot be defined on it).
b. Pick a node V which has no predecessor.
c. Place V on a list of ordered nodes
i. When the nodes conclusion is assessed, if a terminal goal node is reached, print out the list of
rules used on the way to reach that goal/conclusion.
ii. Delete all edges leading out from V to other nodes in the knowledge tree.
d. Go to the start .
A short list of some basic logic and knowledge oriented terminology is listed below :
Backward-Chaining -- An inferencing strategy that is structured from the general to the specific. That
is, it starts with a desired goal or objective and proceeds backwards along a series of deductive
reasonings while it attempts to collect the hypotheses required to be able to conclude the goal. This
process continues until the goal is reached and it then displays its conclusion. (See following sections
for a more complete explanation and an example.
Algorithm -- A procedure that conducts a calculation in an ordered manner for the purpose of solving a
problem.
Antecedent -- The IF part of a conditional statement.
Clause – A formula to be included in conditional statement which contain a goal. Clauses do not contain
the terms which are displayed functional or predicate arguments in the more standard way of writing
logical statements. If the clauses in the consequents are restricted to always being positive the clause
is called a Horn clause.
Consequent -- The THEN part of a conditional statement.
Continuous function -- Rougly speaking, a function that has no jumps or breaks in its graph.
Differentiable function -- Roughly speaking, a function that has one and only one tangent line close to
function curve (graph) at each of the points on it.
Domain -- A set of objects which form the elements from which a function maps.
Expert System -- A computer program that represents and uses expert human knowledge to attain high
levels of performance in a problem area. An expert system has two basic components: a knowledge
base which contains the information (facts, rules, and methods) found in the problem area being
represented, and an inference engine or mechanism that make use of the knowledge base (by
scheduling and interpreting the facts, rules, and methods) to make conclusions and decisions that solve
problems that would normally take a human expert more effort.
Fact -- A collection of logical relations between objects
Forward-Chaining -- Forward-chaining reasoning is an inferencing strategy in which the questions are
structured from the specific to the general. That is, it starts with user supplied or known facts or data
and concludes new facts about the situation based on the information found in the knowledge base.
This process will continue until no further conclusions can be reached from the user supplied or initial
data (using the rules and methods coknowledge base). (See the previous section for a more complete
explanation and an example).
Function -- A mathematical mapping from one set to another which associates at most one value in the
range to each value in its range.
Genetic Algorithm -- A mathematical procedure designed to provide computational searches through
the combinatorial possibilities that might led to the goal of program. Data formats are created that
represent the intermediate values of attributes involved in the search. Then some of the values are
mutated at randomly selected points in the data format in order to see whether the goal can be
reached. This approach works better normally works better than trying to represent intermediate
attributes as differentiable functions and search in the direction of the tangents to the function curves.
The reason is that partial solutions have less tendencies to get caught in valleys or depressions.
Goal --- A top-level consequent of the rules in the knowledge base toward which Backward-Chaining
may be directed. (It is a hypothesis that the program will try to determine if some group of rules can be
instantiated together to satisfy)
Graph of a function -- The set of ordered couples or pairs (x and y coordinates) of the values in the
domain and range of the function, displayed or plotted in a two dimensional representation.
Knowledge Base -- The sum total of all the facts and rules through which inferences, conclusions, and
goals may be reached. This may change as new facts and rules are added or subtracted from the
overall system.
Knowledge Tree --- A graph showing the logic and data flow connections between rules and facts in the
knowledge base. A knowledge tree presents a graphical representation of the complete structure of the
knowledge base.
Mapping -- A set of ordered couples of objects. Thus, ((1,2),(2,3),(3,4)) is a mapping from the integers
to the integers.
Range -- A set of objects which form the elements into which a function maps.
Backward-Chaining -- An inferencing strategy that is structured from the general to the specific. That
is, it starts with a desired goal or objective and proceeds backwards along a series of deductive
reasonings while it attempts to collect the hypotheses required to be able to conclude the goal. This
process continues until the goal is reached and it then displays its conclusion. (See following sections
for a more complete explanation and an example.
List -- An ordered set of objects tied together one to another. Its length is not predefined, but it does
have a first and last element.
Method --- A procedure stored in an object's class structure that can determine an attribute's value
when it is needed in the program , referenced in its class, or required to execute a series of procedures
because another value in the program changes. "When needed methods" are executed during
backward chaining to determine an attribute's value.
Node --- A vertex or point in the knowledge tree connecting the antecedents and consequents of rules
in the knowledge base. In most conventions the nodes are the rules and the antecedents and
consequents are the edges between the nodes or vertices.
Number -- An integer or real number.
Pattern Expression -- An expression containing variables an involving objects and their attributes.
These patterns in the expression contain combinations of symbols denoting constant and variable
objects. They will not normally containing predicates which have the ability to reference themselves in
their arguments.
Pattern Matching -- The process of matching a general pattern expression to an instantiation or specific
instance of an object or to another pattern expression. The process proceeds in a forward-oriented or
bottom up reasoning process.
Predicate -- A logical relation that affects one of more objects or variables. A predicate specifying a
relation between n types of arguments is usually written as a mapping (which must also be a function)
having n arguments. Predicates, as opposed, to relations may have one argument. Predicates are
defined by giving a series of logical rules which specify an algorithm for computing the value of the
function which specifies its name. Objects, as explained above, are defined by giving values to the
attributes that make up their structures or by computing these values using methods (which are usually
not recursive).
Procedure -- same as method.
Relation -- We speak of relations as holding between two things or among several things. Thus the
relation of being married holds
between a man and a woman. A relation between n types of objects is written in terms of a mapping
with n arguments.
Recursion -- A process by which a predicate or function is defined in terms of itself. This situation of
self relation allows the function or predicate to be computed in an orderly manner.
Stream -- An ordered set of objects tied together one to another. It has a first, but not necessarily a last
element.
Subgoal -- A relation, possibly involving objects of variables, which is necessary for the satisfaction of
another goal.
Unification -- The process by which a theorem proving machine (i.e. a logic programming compiler)
tries to math a goal against facts or already instantiated predicates on the left hand side of rules in
order to satisy that goal, or to determine one or more further sub-goals necessary to evaluate teh
original subgoal. The process uses backward oriented reasoning and proceeds in a top-down manner.
Variable -- A name which represents the value of an unknown object.
Word -- A name.
VI
CONCEPTS AS RELATIONS.
HOW CONCEPTS CHANG,
WHILE REMAINING THE SAME,
INSIDE OF THE STREAM OF CONCIOUSNESS
IN OUR MINDS
This way of looking at concepts assumes that
objects are appearing to us and the intellectual faculties in
our minds are able to identify and unify how the objects fit
together as time progresses.
METHODS AS CONCEPTS ----
While rules are used for backward directed goal-oriented reasoning, objects and recursively defined
data types are appropriate for building up forward directed production systems, models are appropriate
for procedural oriented, cased-based reasoning.
A model deals with some topic, a pattern of behavior, a procedure for accomplishing a taks, an overall
type of reality (World view).
A paradigm or case is:
1) a way of looking at a body of facts
2) an example, a particularly good example
3) a pattern, an all encompassing pattern.
One can mistake a paradigm for a theory - in the same way one can mistake a series of examples for a
definition. A good example ( a paradigm) can serve as a model for the interpretation of a body of facts.
However, when it becomes a model it becomes capable of being displace. Remaining a paradigm, an
example of a way of looking at things, it stands for what it is.
A conceptual model (as opposed to physical one) is something that exists in the mind. It is envisaged
and/or specified without actually being all that it represents. It may be required to simulate what one is
interested in, i.e. a topic, pattern of behavior, tasks, or World view. The better it simulates reality the
better a model it is.
6th Definition of a Concept--- A concept is a model (involving the essential parts of a series of cases or
examples) along with a program to learn, retrieve, identify, the concept (knowledge). The program has a
data structure part + an algorithm (interpretive procedure) part. The algorithm may consist of a set of
rules, as in an classification type expert system. It may be all the statements which are derivable from a
set of axioms involving predicates with variable terms and ground instances of facts. Or, it may be a
pattern identification routine, such as in a neural network.
--- A model can be of a set of logical relationships. In this case it is an interpretation of a set of
sentences which it satisfies A concept is a logical interconnection between facts which can be
interpreted as a model of thought,[see next section].
--- A model can be of the way something works. It can use data structures to construct a program to
simulate what something does.
It can be used to test whether facts fit into this viewpoint. And, in this sense it has a type of semantic
truth (A. Tarski) associated with it. It has a mapping (mapping is used here in the mathematical sense)
of truth tables which enables us to test whether any statement about the World can be true in some
interpretation of the model.
--- A model can be a way of looking at the World [a physical theory, ethical theory, philosophic theory,
religious theory]. In this case there is a interpretation [mapping] from the objects in the World to a set
of facts, constants, variables. There is a translation of physical laws, ethical, philosophic, religious
beliefs and postulates into rules connecting those facts, constants and variables.
IMPORTANT NOTE: The more we want to talk about knowledge related to the World and the
less about knowledge related to ethics, philosophy, theology, the more we need to introduce numbers
and their language mathematics into the propositions we write. For example, once we have numeric
data types, we can go from simple verbal propositions (qualitative judgements) to statements involving
numeric values (quantitative judgements). We can then make a philosophic classification type expert
system of the type given in the discussion of the second definition of a concept, into a quantitative
case-based reasoning tool.
--- In a classification type expert system, the model involves a data structure as in the 3rd definition of
a concept. It also involves some means of retrieving the information, along with a way of creating the
rules as in the 2nd definition of a concept.
How do we create the knowledge tree that we use in such a quantitative case-based reasoning expert
system? For the philosophic expert system we asked a series of questions from the general to the
specific about something we believe that we already have in the mind. Now, for this, we need a series of
examples or cases in order to develop cutoff values of object attributes in order to branch into the
knowledge tree. Again the questions go from the general to the specific. But how do we know which
attributes are general. Answer: we can construct statistical summaries and tables that analyze the
examples in the data to determine which attributes are most associated which the particular results we
are interested in. These attributes are then said to be the most general, because they determine the
first questions we need to ask in constructing a classification tree.
ALGORITHM TO CLASSIFY OR IDENTIFY AN OBJECT BY THE
EXPERIENCE WE HAVE OF IT
1) Construct a neural net consisting of a layer of input nodes or neurons, a hidden layer of
interconnecting nodes, an ouput node (on-o