Logic

                Logic is not necessary for everyone.  The first rule of logic is that a logical system is consistent.  And since some people insist that consistency is just too much trouble, they consider logic a waste of time.

                We must live with these people.  We must allow them to make their own decisions about their lives.

                Most of the language of logic is found in the branch of mathematics called topology.  Topology is the study of sets and their interactions. 

SETS CONTAIN OBJECTS

                An object is a thing that can be considered to exist independently of oneself, with practical applications.  Everything else is subject.  We have names for objects.  In mathematics, we only consider the names (with their definitions) of the objects, not the objects themselves.  The actual objects are not a concern of the mathematician.  Only a physicist is concerned with actual objects.  He uses what is called applied mathematics.  Applied mathematics does two things that pure mathematics does not.  Applied mathematics first translates the physical world into names that can be dealt with mathematically.  Then, after the pure mathematics is finished, applied mathematics is used to translate the names back into a measurable prediction of the behavior of the objects.  Occasionally, applied mathematics uses rules relating to the objects to limit the pure mathematics that is used in between.

                When one identifies an object, one is identifying the object as a member of a set of objects that are alike in some way.  The task of truly identifying an object is difficult.  Objects are made of particles called atoms.  Truly identifying an object would have to include each one of the characteristics of the object, so it would have to include the locations of each one of the atoms of the object.  This alone would deter most people from tying to accurately identify an object.  Each one of these atoms exerts a gravitational attraction on each other particle in the Universe.  One must therefore identify the combined gravitational attraction of each other particle of the Universe on the atoms of the object, because these contribute to the description of the object.  In physics there is a problem called the multi-body problem.  Unfortunately, all the computers in the world are incapable of solving the multi-body problem for a system of more than two particles.  Solving for the practically (although not actually) infinitely many particles of the Universe is impossible.  Therefore, it is impossible to truly identify an object.  So, what we do is identify the object in some usefully approximate way.  This usefully approximate way of identifying objects has led to the development of topology.

                Topology is about as far removed from numerical mathematics as mathematicians get.  Topology does deal with numbers, but generally as members of sets.  Often, there are no operations defined for these numbers.  This makes the numbers look like mathematical names for sets of objects like cats.  You have cat number one, cat number two, cat number three, and so on.

                There are operations, but these usually have odd names to distinguish them from their numerical counterparts, and these operations may only work on the sets and not on the objects which the sets contain. 

IS

                A main topic in topology surrounds the English word IS.  The word IS is best avoided in mathematics due to its ambiguousness. 

                When we say a candy IS a piece of food, we mean that a candy IS A MEMBER OF THE SET OF pieces of food.  When we say, "An inch IS 2.54 centimeters."  IS means CAN BE SUBSTITUTED FOR. 

                Most of the time the word IS will mean IS A MEMBER OF THE SET OF.  When this is understood, written English becomes much more comprehensible.  In fact, English is difficult to comprehend unless it is translated into mathematical (usually topological) language. 

                Now you know.  If you want to participate in an intelligent discussion, learn topology. 

                If you want to respond to someone who has not learned topology, the following suggestions may be helpful.

1.  Come prepared with harmless topics and ways to change any conversation into a discussion of these.

2.  Find something which you can agree on, and let the other person do the talking. 

3.  Smile, be agreeable, and let the other say whatever he wants.  This tactic must only be used within the limits of one's own ethics.  Learn to keep quiet, but know how to avoid those who truly lack appreciation of higher things. 

                Everyone needs to be a certain distance away from that which is disturbing to him.  Any further than this, and the other person will feel bad, and you could miss their company.  Any closer, and you or the other person will feel violated. 

                Remember that people are generally more concerned about feelings than information.  It is often the emotions that words evoke that make the words meaningful.  This doesn't sound very mathematical, does it?  I say this is most mathematical.

                When discussing something, work diligently to identify what is being discussed, and then continually remind the other person of what is being discussed.  This needs to be emphasized.  Most of the fallacies people engage in can be dealt with efficiently and politely by simply reminding the other person of the topic being discussed. 

                 There are two ways to keep a discussion sensible.  One is to keep the topic clear.  The other is to know what is being said. 

                This may sound simple, but it requires practice or else it is very difficult. 

                It is often so much easier to just not pay attention to what someone says rather than expend the effort it takes to know what the person is saying....NOT!!!

                IT IS ALWAYS WORTHWHILE TO KNOW WHAT YOU ARE LISTENING TO!

                If you want to listen, listen!  If not, then do something else! 

MATHEMATICAL OBJECTS

                A mathematical object is a pure abstraction.  Mathematical objects do not exist anywhere else except in the mind. 

                This concept of mathematics causes a lot of trouble.

                Aw, gee, that's too bad.  Oh, boo, hoo, hoo.  Live with it, baby.

THE DEFINITION OF DEFINITION

                A definition is a phrase or collection of phrases which can be substituted for the word they define. 

A definition describes or names the set of objects to which the thing being defined belongs.  A definition also distinguishes the thing being defined from all other members of the set.

 An adequate definition is complete and consistent.  A definition is complete when it can be substituted for its word anywhere that the word is correctly used.  A consistent definition cannot be used for any incorrect use of its word.  A good definition is terse, and uses small words.

                Work is needed to develop a system of definitions which are good.  Usually such definitions will not be given to you, so you have to find them yourself.  This is very difficult.  The purpose of this book is to help you find them, help you recognize them, and give you some.

BACK TO SETS

                There are two sets of particular interest to mathematicians.  These are the universal set and the empty set.  Both are arguably impossible.  If there is a set that is truly empty, then there is another set which contains everything.  Does the set which contains everything contain itself?  Does it contain the set of all sets which don't contain themselves?  Because these questions are extremely difficult to answer, the universal set is defined in a somewhat vague way.  What we do to live with these difficulties is define the universal set to be the set of all objects under consideration.  Similarly the empty set is the set which contains none of the objects under consideration.  By simply not considering the set of all sets which don't contain themselves, we are able to develop the system of mathematics called topology. 

INFINITY

                Another hot topic in topology is the topic of infinity.  There are two types of infinities.  There are countable infinities and uncountable infinities.  The set of counting numbers is an example of a countably infinite set.  The set of irrational numbers is an example of an uncountably infinite set.

Logic Errors

                A fallacy is a method of using a distraction or an invalid argument or false premises or vague terminology to seemingly support a contention.

Distractions

When someone makes a statement that does not directly refer to a contention under consideration then it is probably a distraction and needs to be forcefully removed from consideration. This removal must be complete.

Invalid Argument

One must pay careful attention to the logic used in an argument. There are certain invalid logic rules that are commonly used by people to support conclusions. A familiarity with these is extremely important.

False Premises

False premises are sometimes very difficult to identify. Usually false premises use vague terminology. Often false premises use ideas that are commonly accepted yet untrue.

Vague Terminology

The main source of vague terminology is the misuse of pronouns. This is very difficult to identify and requires careful observation of all that is said or written. Certain terms in standard spoken English are routinely misused so that the vagueness they imply can be used to apparently support a contention. One needs to be aware of this and needs to be ready to offer definitions of such terms that can be readily agreed upon.

The following is a list of logical errors that might be found in an argument. This list is designed to include the many different fallacies which comprise the many lists of logical fallacies.

A reference to an author rather than an idea. (Author of an idea could be many individuals.)

Reference to something not under consideration (off topic).

A reference to a belief or popularity.

Circular reasoning.

Statistical error (inadequate sample size (dramatic instance), bias in sample, bias in author, bias in questioner, lack of double blind, leading questions, environmental factors, statistics do not predict an individual event.)

Burden of proof (who is more responsible for proving their contention) rests on the affirmative, or according to situation.

It precedes, therefore it causes.

Inadequate inclusion of data.

Systems tend toward or away from equilibrium (not necessarily).

Ignoring randomness.

Guilty by association.

Introducing somewhat related ideas that do not pertain to the contention.

Parts of a thing do not necessarily behave the same as the thing.

Differences of scale. A paper airplane will fly if it is made from a range of paper sheets, but not beyond certain limits because problems of scale enter.