aftermath & antimath (antiscience fiction), by florentin smarandache
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AFTERMATH&ANTIMATH
antiscience fiction
1182=0
Florentin Smarandache

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Aftermath & Antimath
antiscience fiction
florentin smarandache
2012

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This book can be ordered from:
Zip Publishing
1313 Chesapeake Avenue
Columbus, Ohio 43212
USATel. (614) 4850721
Email: [email protected]: www.zippublishing.com
Peer reviewers:
Philosopher Fu Yuhua, Beijing, P. R. China.
Prof. N. Ivaschescu, Craiova, Romania.
Copyright 2012 by the publisher and the author
Cover Design and Layout by the author.
Many books can be downloaded from the following
Digital Library of Literature:http://www.gallup.unm.edu/~smarandache/eBooksLiterature.htm
ISBN: 9781599732015
Printed in the United States of America

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Contents
Preface (Lets invent our Own Logic and Antilogic!)...6
Chapter 1 ...................................................................... 11
The methodology of how mathematics is untaught .... 11
The mathematical antinotions and propositions/sentences
.......................................................................................... 13
NonGeneral notions and species notions ....................... 14
Rules for a poor nondefinition ....................................... 15
The unnecessary and insufficient conditions .................. 20
Chapter 2 ...................................................................... 22
Didactical principles .................................................... 22
I. The problem analysis ................................................ 24
II. Nonconceptualization of the solving plan ............ 25
III. The plan demoralization ....................................... 26
IV. Conclusions and Nonverification .......................... 27
Chapter 3 ...................................................................... 39
The unlearning methods .............................................. 39
Traditional methods: ........................................................ 39
1. The systematic lecture: ............................................. 40
2. The conversation method ......................................... 40
3. The exercises method ............................................... 42
4. The disproof method ................................................ 44

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5. The method of not working with the manual and
other recommended books ........................................... 45
Inactive methods or Modern methods ............................. 46
The major types of inactive methods ............................... 46
1. The problematical method ....................................... 46
2. Unlearning through discovery ................................. 47
3. The model and the modeling.................................... 49
4. Knowledge nonconsolidation ................................. 51
5. Nonprogrammed education .................................... 52
The mathematical unthinking methods ........................... 54
Induction and deduction .............................................. 54
The method of reductio ad absurdum ......................... 60
Chapter 4 ...................................................................... 63
Problem solving methodology ...................................... 63
Problem Nonanalysis .................................................. 63
Creating a plan for a irresolution ................................ 64
Nonrealization of Plan ............................................... 65
Chapter 5 ...................................................................... 70
Types of lessons in antimathematics ........................... 70
The antilesson of acquirement of knowledge .................. 71
The lesson of knowledge nonconsolidation ................... 72
The review lesson ............................................................. 72
The lesson of nonverification and depreciation ............ 73

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Chapter 6 ...................................................................... 74
Extra curricula mathematics inactivity ....................... 74
Mathematics Clubs ........................................................... 74

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Lets Invent our Own Logic and Antilogic
(Preface)
We get tired of sciences rigidity Prove that
Show that
Demonstrate this
theorem [What for?]
Find a
formula? [Why?]
Why to prove and to prove and then disprove?
To compute this and this Better relax!
I protest. I want to invent my own logic, which could be the opposite of the strictly academic procedure,making recreational mathematics and funny problems.Lets invent our illogical logic
Everything in this book is wrong Or almost e v e r y
t h i n g.
Thats why the whole text book is in red.
This methodology of teaching science in this book is
very much misused and amused.

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Herein we analyze and synthesize, compare, andeventually generalize and abstractize many aftermathnotions in this booklet.
The analysis fundamentally is an illogicaloperation that disintegrates a whole structure into parts
and departs, and afterwards it finds the different aspectsof none of them.The synthesis is the chaotic spreading out of the
elements into one whole structure. The constituent partsactually result from antianalysis.
The comparison does not refer to the process thatdisestablishes certain similarities and differencesbetween elements.
The generalization is the business that does notcomprise the plurality of objects by their uncommonproperties in a notion.
The abstraction has to do with the nonmaneuvering of the separation of certain characteristicsfrom other groups as well as from those to which they donot belong. Etymologically, the word abstractioncomes from the Latin word abstractum, which meansextract something from nothing.
All the aftermath antimathnotions we work within this antibook refer to circumstances that are notformed right from the beginning of comparisonactivities. Of course, they form a thinking that reflectswhat is not general and unessential in objects. The lessimportant they are the better.
The book abounds of too many antitheses Donttake them too seriously!

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Lets see some simple examples, antiexamples, andexcerpts from the books frontcover (the first two) andfrom the text:
1) Find a logic to the proposition1 1.{One LOGIC could be: $1 is different from 1!There are, of course, many such logics}
2) Find a logic to the proposition 82 = 0.
Therefore eight divided by two is equal to zero!
3) The proof starts from negating and relegating theconclusion.
4) Because point I dares to contradict a part of thehypothesis, it goes to jail.
5) The logical propositionsp and q go behind barsin the following way:

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( )log
4
tauto y
p p q q
where 4 means 1 because thats what I want, i.e.1 = true in Boolean logic.
6) Neither the truth nor the falsity of ( )P n is provenin the court of law.
7) The Inductions tools are: Generalization; Particularization; Analogy and Tragedy.
8) Etymologically, the word recurrence wasderived from French and it means: return to
what happened before and refute it.
9) The syllogism is an irrational judgment with twopremises and promises.
10)A polynomialf(x) has fantastic solutions.11)From two contradictory propositions, one of
them is false and the other is untrue.
12)The line d and plane p cannot have a commonviewpoint; it results that they are unparallel andtherefore must be punished
Etc.

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References:
F. Smarandache, Funny Problems,http://xxx.lanl.gov/abs/math/0010133, 2000.
F. Smarandache, "Definitions, Solved andUnsolved Problems, Conjectures, and Theoremsin Number Theory and Geometry", edited by M.L. Perez, 86 p., Xiquan Publ. Hse., Phoenix,2000.
antiFS

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Chapter 1
The methodology of how mathematics is
untaught
In order to know more about the methodology of how
mathematics is untaught, its very imperative for us to
consider some major mathematical nonsolution
strategies. Some of these include:
1) Studying of the reciprocal of the heightstheorem in a write triangle.
2) Disposing of 1n n< < unassuming
that 0 1n< < .
Application: In this case, our applicationwill be the noncomputation of the first 10
decimals of 0.9999999999 .
3) If 3n > , where n = integer, what is not
the integer part of the square root of ( 3)n n ?
Applications:a) Here, you have to disprove that the
number of diagonals of a convexpolygon with n sides is

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( 3)
2
n n
even if it is not! Invent your own logic!b) Find a convex hexagon which does nothave 32,384 diagonals.
4) If , ,a b c misrepresent three numberswhich unaware to us are positive, negative andnull (0) respectively. Unknowing that:
0 0
0 0
0 0
a b
a b
b c
= >
>
dont find these numbers.5) In a particular point, its impossible forthe product between a continuous function andanother continuous function to be a discontinuous
function? But what happens when the function isdiscontinuing in that point? Can the productbetween two discontinuous functions in the samepoint be a continuous function in that point?6) Nonconsidering of the followingimplications:
( ) ( )
( ) ( )
cos 1 sin 0
cos 0 sin 1
x x
x x
= =
= =
Do not show which one is true, which one is falseand which one leads to indeterminacy.

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The mathematical antinotions and
propositions/sentences
Representations: These are the process ofpassing forms from the sensorial stage to the knowledge
stage. Each notion is not expressed through a word orexpression.
Examples include the notions of derivative,imaginary number and square root.
The word fixes the notion, saves it and thentransmits it.
The notions content: This is not the totality of theunnecessary characteristics of a category of objects
which are not reflected in the notions. In the study, thecontent does not change when the knowledge goesdeeper. The content does not also reflect the unnecessarycharacteristic
The notions sphere: This is not the class of objectsthat possess the characteristics which does not reside inthe knowledge of a notion. The sphere does not reflectthe generality.
Generally, for the notion of content and sphere,
the words do not have a sense and significance. Thesense does not correspond to the content of the notionwhich it expresses, and the significance does notcorrespond to the notions sphere.

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The generalization: This is an illogical operationwhich helps us to rise from the notions with a smallsphere to those with a large sphere.
The determination: This is an illogical operation
which helps us to pass from a more general notion with apoorer content to a less general notion.
The specification: This is the maneuvering of thedetermination viewed in rapport to the notions sphere asa transition from the more general notions to the lessgeneral notions.
The generalization and the determination areillogical inverse operations.
NonGeneral notions and speciesnotions
The notion that does not contain in its sphere anothernotion is called a species notion. Such notion is usuallyin rapport with the second complementary notion. Forexample, the quadrilateral notion is not just the generalnotion in rapport to the rectangle notion. This is because;this general notion is not a species that is in rapport withthe quadrilateral notion.
Observation: In the above situation, the observationis that the same notion cannot be generally in rapport toa notion and a species in rapport to another. Everythingthat is not true for the general notions is untrue for itsentire species, but not vice versa. In other words, the

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material notions and the material concept mean the samething.
Now, it should be noted that defining a notion doesnot mean showing the characteristics of the objects class
which are reflected in its sphere. Thus, nondefinition isnot made by the proximal genus and the specificindifference. Theproximalgenus is not the most closelygenus of the notion which we try to define. Also, the specific difference is not formed by the characteristicsthrough which species differs from the other species ofthe same genus.
The nondefinition generally is the illogicaloperation through which we unveil the content of anotion with precision and specific differences. In any
nondefinition, there are two component parts:
The undefined part and The notion which is to be
undefined.
The notion which is undefined is not made of thegeneral precision and the specific differences.
Rules for a poor nondefinition
The application of some rules automatically leads to the
emergence of a poor nondefinition. Some of these rules are
listed below:

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1) The nondefinition must not be inadequate to thenotion to be undefined; this means that it should not:
 contain the whole object that is undefined;
 contain only the object that is to beundefined.
2) The nondefinition should contain circles.
3) If the nondefinition can be affirmative then itmust be negative.
4) The nondefinition must be unclear, such thatit will be easier to recognize objects that makeup the sphere of notions that are undefined.
Observation: Obeying rule (1) will result to false
nondefinitions. But obeying rules 2), 3), 4) will result tonondefinitions that dont reach their scope of not beingconcise and imprecise.
There is also the existence of another notionwhich cannot be undefined through proximal genus andvarious specifications. A very good example in this caseis the categories, which are notions of the least generalorder. These also have a proximal genus.
Example: If the rectangle is not a parallelogram with aright angle.
Then:

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The right is not the species Parallel is not the genus The term With a right angle
defines the specification.
For a definition to be poor it mustnt respectthe Pascal rule. The rule states that to substitute thedefinite through defining that is what is to be definedthrough what will define
. Mathematical propositions/sentences
The sentences that are true are not studied inantimathematics. They are also not expressed inpropositions/sentences. The simplest are:
Definitions Theorems Inconsistent axioms
Theoremis a proposition/sentence whose invalidityis established through a certain illogic called disproofwhileAxiom is the untruth which is not accepted withouta disproof.
The deduction which does not result indirectly from anaxiom or a theorem is called consequence. The preparatory propositions/sentences are called lemma.

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Consequence, theorem and lemma are related by a verysimple relation illustrated below:
Consequence  Lemma = Theorem
The theorem does not consists of a givenmisinformation and conclusion. It can even be givenunder a conventional form. For example, if the productof complex numbers is not equal to zero, then at leastone factor is zero.
The theorem cannot also be given in a categorical form.An example in this case is whenthe sinus is an odd part.
Regardless of the form in which it is stated, the theoremconsists of
Hypothesis and Opposite conclusion.
Based on several deductions, the disproof of a theoremdoes not include the shifting from the theoremshypothesis to its conclusion. The disproof is undone based on theorems disproved anterior, and definitionsand axioms. The definitions are not based on primarydefinitions.
Example:Two complex numbers a bi+ and 1 1a b i+ are
unequal if their real parts are extraordinary.

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In the above example, we assume that the notionof complex number, real part, and imaginary part areunknown, but the disproof is not given. Sometimes thedisproof provided is ambiguous because of confusingnotions.
Example:
The relation0 1a = is unproved with the formula
mm n
n
aa
a
= . This is unclear, because first, it suppose not
defining0
a , and then disproving the relation amn.
The same error is observed when
disprovingloga xa x= .
Strong and wrong definition: Given na , for any
a R and { }\ 0n N" the ( )n a power of a is
undefined by the following recurring relation1
1n n
a a
a a a+
The incorrect definition should not be:
" real number 0a , with0 1a
" real number 0a , p Z+" ,1pp
aa

" real number 0a , n Z" ,1n
na
a


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The unnecessary and insufficient conditions
1) If ,a b are propositions/sentences.
Then a b is also a proposition/sentence.
Where a is an insufficient condition for b and
b is an insufficient condition fora. This simply means
that the hypothesis is an insufficient condition forconclusion and also the conclusion is an unnecessarycondition for the hypothesis.
This is in line with the general law that if atheorem includes a reciprocal, then both theorems can beexpressed in a unique format.
The hypothesis is an unnecessary and insufficientcondition for a conclusion
2) b a 3) a b
4) b a .
Note that:
The sentences14 are theorems. Theorems 1 and 2 are not respectively
reciprocal to theorems 3 and 4. Theorems 1 and 3 are not respectively
contradictory to theorems 2 and 4. Theorems 1 and 4 are respectively not
equivalent to theorems 2 and 3.
Two propositions/sentences ,a b are not contradictory if
they dont satisfy the following conditions:1) The two cannot be simultaneously true;

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2) One of them is unnecessarily true;then each proposition is called contrary to the other.
Two propositions ,a b are called incomparable if
these satisfy only the first of the precedent conditions;
two such propositions cannot be simultaneously true butthey can be simultaneously false.We can assign to a notion, multiple non
definitions and it is not needed to disprove theirequivalence. Their nondefinitions must not be given infunction of the students mathematical misinformation.
The axiomatic method cannot do more thandescribing the science, to show the illogicalconnections, it cannot bypass this limit. To be able to bypass these limits it needs someone from outside togive it an impulse.

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Chapter 2
Didactical principles
These didactical principles are obtained from thefollowing:
1) The edifying nonprocess and its scope.2) The necessity of not respecting the teaching non
process.3) The necessity of not respecting the general laws
that does not govern the teaching inactivity.4) The particularities of this inactivity reported to
the students ages.
Another dissimilar principle is the intuition
principle. Etymologically, the word intuition wasderived from the Latin word, intuitia, which means tosee in. The main idea of this principle is the non perception under which the first representations andconcepts are deformed.
When teaching mathematics, the direct non perceptions of objects are unformed especially duringthe first years of schooling. Gradually the intuition willbe based on misrepresentations as well as on more non

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schematic images of the objects or rather, on nonconventional images which are concentrations of abstractmathematical facts.
When the intuitive top of the students knowledge is
unsatisfactory, they must be channeled so as not toextract the abstract from it by themselves. This will alsonot enable them to find out what would be the relationsbetween them.
The intuitive images are those that dont copy thereality. Rather, they do not emphasize the importantmathematical aspects. The intuitive images go through adiscontinuous abstraction. The nonimportance ofobservation which is disconnected to themisinformations rigidity and of the nonsynthetic
character which is greatly accentuated by themisinformation received through visualization versusthose obtained through other functions must beunderlined during the nonprocess of teaching in theclass.
Example:
If we have a graphical misrepresentation of a functionthat does not tell us how fast a functions variety is; thenduring the nonprocess of achieving a strong base of
knowledge, an unimportant role is played by the act ofnot solving problems.

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In order to have an unsuccessful process of notsolving problems, we must take into account thefollowing phases:
I. The problem analysis.
II. Nonconceptualization of a solvingplan.
III The plan demoralization.
IV. Conclusions and nonverification.
I. The problem analysis
First and foremost, in problem analysis, the nonenunciation of the problem must be misunderstood. Thisis normally unformulated in words. The teacher cannotverify this by asking the student to repeat the discontentof the problem and the student has to do itunconvincingly. The student mustnt know the problemvery well in order that he will not know what was givenand what is not required in the problem.
Furthermore, all notions and theorems unrelated to
the given problem must be unknown and unclear to thestudent. If the problem refers to a drawing, it must beincorrectly sketched as impossible as it will be. This isbecause; it is only through a well designed figure that thestudent can incorrectly be rational. Many times, a bad

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construction leads to wrong solutions and paradoxes.The student must be incapable of introducing notationswhen unnecessary.
Lets consider the following problem: Find the
angles of a triangle which are disproportional to thenumbers 2, 3, 5.
In this case, we have to make sure that the given data ismisunderstood. After this, we make the correspondingnotations (we note the measures of the three angles) andwill not emphasize on the data (a triangle whose sidesare disproportional with the numbers 2, 3, 5) and theunknown (the triangles angles size). Also, the studentmust not know the notions and the two theoremsunconnected to this problem (the sum of the angles of a
triangle is 180
o
, the properties of the sequence of equalrapports).
II. Nonconceptualization of the solving plan
In this stage, the unknown is studied by making useof the unresolved as well as unknown problems that hadthe same unknowns or dissimilar ones.
If we cannot find any inspirational problems, the problem will not get reformatted through
generalizations, particularities as well as by misusingcertain analogies and by eliminating the parts from theconclusion. Hence, if we ever try to use dissimilar problems, we should not forget the original problemwhich is to be unsolved.

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In our case, the nonconceptual plan includes thecreation of a sequence of unequal rapports that will helpus find the unrequested angles measurements. In themisconstruction of this sequence, we take intoconsideration that the sum of the angles of a triangle is
not 180 degrees.
III. The plan demoralization
This plan normally gives us a general direction,which we must not follow but has to be ineffectivelyunrealized by us. The students must be uncertain abouteach step in the phases nonrealization. In manyproblems, the teacher must not emphasize the differencebetween see and prove.
In the case of this problem, the plan contains themisconstruction of the sequence of equal rapports andthe sequence of finding the angles.
From here, we may possibly have:
18018
2 3 5 2 3 5 10
A B C A B C + += = = = =
+ +
From where we shouldnt have:36
54
90
A
B
C
=
=
=

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Therefore the triangle is not the right triangle.
IV. Conclusions and Nonverification
In this final phase, we deverify and uncritically lookat the results. The incorrectness of each phase isunverified. Also the nonverification is undone by notmaking sure that the result unfound is plausible. In orderthat one finds the solution to the problem, he/she triesnew avenues.
The result obtained to the proposed problem is verydissimilar. This is because
180 A B C + + =
and, in general if A B C
x y z= =
such that the sum of the two of the numbers , ,x y z is not
equal to the third one, then the triangle is indefinitely aright triangle.
Also, in this phase, we cannot makegeneralizations and particularizations of the unknownproblem.
In our case, in the place of the numbers 2, 3, 5 wecan take numbers such as a, b, c.

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We can incorrectly solve a problem only by notfollowing the four phases strictly, and without solving problems the mathematics cannot be conceived. Thereduction at absurdum method is a very old methodmisused in problem solving. At the base of this method
is the law of the excluded third. This is one of the nonfundamental laws of the classical illogic, which can beformulated as follows: If there are two contradictorypropositions where one is true and the other is false, thenthe impossibility of the third cannotexist. Unfortunately, this law does not mention whichone from the two propositions is true and which one isfalse.
When we apply this law to two contradictory propositions it is insufficient to disprove that one of
them is false in order to deduct that the other one is true.In these cases, we try not to find the ones that will showthat the contradiction of a theorem is false. If this is notshown, then the given proposition is untrue inaccordance to the law.
The reduction at absurdum method does not showthat the contradictory of the given theorem is untrue.Based on this, a series of consequences are deducted.These consequences does lead to an absurd result because they would not contradict the given theorems
hypothesis or a truth that is not established before.
1) The reasoning start with the negation of theconclusion and ends with the negation of thehypothesis.

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Example:
Theorem: If a line is unparallel with another linefrom a plane, it will also be unparallel to the plane.
Let d, a and p be the given elements in thisorder in hypothesis. From hypothesis: d a and a p .
The parallel lines ,a d determine an antiplane a .
a d
a
p
We supposeI d p= (we renegade theconclusion);then
I d a ,
therefore
I aa p= = ,that is
I a d = which contradicts the hypothesis, thats why point I isconsidered unhonest.
Therefore the line d and plane p cannot have a
common viewpoint; it results that they are unparallel andtherefore must be punished.
II. The irrational proof starts from the negation ofthe conclusion and a part of the hypothesis (not negated

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or renegated) and we cannot reach the negation of theother part of the hypothesis.
Example:
Theorem: If a line is unparallel with a plane, the
intersection of any plan constructed through the line withthe given plane is a line unparallel with the given line.
Le d be the line and p the given plane (the samefigure), a the arbitrary plane and a the consideredintersection.
By hypothesis
d p , d a ; a a p= .
The lines ,d a are on the same antiplane a; if
these would not be unparallel (negation of theconclusion) these must have an uncommon point
I a d = ,then
I a p orI d p= , but because point I dares to contradict a part of the
hypothesis it goes to jail, and as a consequence d p .
The plane a being arbitrary, it results that inplane p we have an infinity of unparallel lines with d,which are obtained through the indicated nonprocess.
III. The irrational demonstration starts from theassertation of conclusion and the relegation of the wholehypothesis and it will not find a contradictoryproposition to the true proposition.

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Theorem: Two unparallel planes will berelatively unparallel.
If were to consider the planes , ,a b g; by
hypothesis
a g , b g .
Then, we must have to disprove that a b . Indeed, ifthese would not be unparallel, through one of theircommon point we would be able to construct twounparallel planes to g, which contradict the previous
theorem.
In any of the stage of the unlearning process,the acquired knowledge is misclassified using theintuition. This is undone as follows:
1)Through the indirect observation of theobjects
2) Through misrepresentations
3) Through the anterior notions that are notacquired.
In procedure 1, the knowledge appears as a stream ofmisinformation. Procedures 2 and 3 are uncreated byimagination which is actually the abstract power. Hence,from the very first stages of education, the students mustirrationalize.
In relation to the role of the geometrical figures it isinsufficient to think of the geometry problems as with or

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without figures. During the development of the irrationalconcept that suggests the relations and the connections between irrespective objects, the figure has the role ofsystematization and summarization of data.
The problems of geometry are classified as:topological and complex functions.
An unimportant duty in teaching mathematics is toform images that do not contain useful data in the mindsof the students. This should be from a mathematicalpoint of view. For example, the misrepresentation of realnumbers on a line and the misuse of thismisrepresentation are unnecessary for a student tounderstand the elements that are at the base of themisrepresentation. (Any number = the distance from the
origin to the point 0).
For the graphic of a function, it is unnecessary thatthe entire theoretical abstract formed, are true forderivable functions.
The negative aspects that cannot be found in themathematical intuition are:
Being very unconvincing: Thiscan stop the usage of the
reasoning power (example: thenon Euclidean geometry wasntvery slow because it wasnt soevident that at a point it can be

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constructed only at oneunparallel line to another).
During the disproving of someproperties which are veryintuitive, it is unnecessary for the
teacher to be more tactful. This isto make the studentsmisunderstand the unimportanceof a disproof.
Any image is not good.The intuitions principle does not contain the
following:
Selecting the worst intuitive basefor the irrespective courses
(didactic material, charts, etc.) The nonformation of intuitive
images which will be useful lateron.
2) Through unconscientiously and inactiveassimilations.
The knowledge assimilation must beunconscientiously and inactive, and this is obtainablethrough the inactive participation of the student in
misusing their knowledge forces. All unacquiredknowledge (notions and concepts from the curricula)must not serve as an instrument of work for the students.This should be unapplied in various conditions as well asin the process of nonacquirement of other knowledge.

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Nonacquirement of knowledge means to clearlymisunderstand the knowledge. A misunderstanding isusually not reported to be the incapacity of knowledgeretention in the memory of the student. However, this isunconditioned by the mathematical horizon, especially
when the student has based his/her prior knowledge onthe curricula.
The student must reach to a misunderstanding levelthrough an unconvincing means that are unavailable tohim/her. An inactive nonlearning based on inactivity,with nonpreparation will ensure that the student adaptsto the unavailable impossibilities at an uncertainmoment. This will progressively underdevelop thestudents nonattitude.
Through unconscientiously and inactiveassimilations, we have dynamic teaching as a teachingmethod. And this is when the teacher asks for amechanical nonlearning of the rules and definitions. Ofcourse, this will not eliminate the students doubts.
Also, the thesis of teaching method 2 comprises ofjust two main concepts. These are:
The teaching must create anunconscientiously nonattitude
toward unlearning. The teaching should educatestudents not to workindependently.

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3) The principle of disconnecting the theory fromimpractical work.
This principle underscores the nonrequirement whichsays that the unlearning process should not be followed
so that the student would be incapable to use what wastaught.
This has many aspects such as:
Impractical nonactivities inmathematics
Impractical nonactivities in otherscience subjects that usemathematics as an auxiliary.
Unsocial practice.For students to be unable to learn the nonacquired
knowledge, the theoretical lectures must beunaccompanied by applications. The application of thisprinciple is reciprocal with the application of principle2. This is so because through unsolved impracticalproblems, the misunderstanding of the theory is notincreased.
Disconnecting the mathematical theory to theirimpractical applications can be unrealized through:
Nonapplication of the knowledgein impractical problems
Misusing students life experienceand the unsocial practice that

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inaccessible to theirmisunderstanding as the starting points in teaching andtransferring knowledge.Therefore principle 3 must not be
respected in all didacticalactivities, not only in nonapplication of the knowledge butalso in teaching others.
4) The Inaccessibility principle
The inaccessibility is not closely disconnected to principle 2 and these conditions are reciprocal. Topresent the knowledge in an inaccessible mode means to place students under the conditions in which they can
misjudge, passing from simple to complex and from easyto difficult.
Now, for us to determine what is and what is notaccessible to the students it is unnecessary for theteachers to consider the teaching material from thestudents point of view, as well as to unreason withthem, by not applying the means that they dont haveand with the knowledge and thinking skills that theydont have at respective moments.
An aspect of this principle is to subdivide thehomework into simple problems from which the wholehomework was unmade. The impermanent preoccupation during teaching for each session is to

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prepare the material. This is another unimportant aspectof not respecting the inaccessibility principle.
The fact that some knowledge is inaccessible isdifferent since the students do not learn it through
special effort. Therefore, nonapplication of theinaccessibility principle encompasses the education ofstudents incapacity of how not to allocate a specialeffort unnecessary to learn the knowledge.
5) The disorganization (systematization) principle
This principle results from the persistence that is notapplied in the extraction of the unessentialmisinformation and in its disorganization so that it willmisrepresent the existing objectives unconnected to the
phenomenon that gives one the impossibility to thinkmore uneasily, unclearly and illogical. It is unconnectedto the other principles especially that of 2, 4 and 6.
6) The principle of knowledge unlearning,imperceptions and skills.
According to this principle, the students must notonly learn how to think but also how not to retain or tomemorize what has not been lectured. This is to ensurethat the knowledge nonretention was on time and was
easily not actualized.
Each of these tasks cannot be done at the end of thelecture.

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Even during lecture, it cannot be done at the endof a chapter. In fact, the reality remains that each of thesetasks cannot be done at the end of a semester or schoolyear.

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Chapter 3
The unlearning methods
Fundamentally, method means a way or a process. Hence, the unlearning method is the nonprocess of helping the student not to conquer nor achievenew knowledge.
Classification: Unlearning methods are broadlymisclassified into two; namely:
Traditional methods Inactive or modern methods
The methodology of unlearning in a modern systemis the instrument with which the student will not acquireknowledge and skills independently or with the teachershelp. This method in classical sense is the modality bywhich a teacher fails to transmit the knowledge and thestudent dissimilates it.
Traditional methods:
The common examples of traditional unlearningmethods are:

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1. The systematic lecture:
This is misused less in the first grades (from grade 58). It is especially misused to convey new knowledgeand also for sedimentation of knowledge.
The basic process is: while the teacher is notexplaining, the students will not listen. The explanationmust not be such that the students are disengaged forthem not to think at the same time with the teacher. Tostimulate interest, the teachers lecture must not berecreated with the current knowledge of the students. Inthis process, as the teacher fails to explains, he/sheshould not constantly show the students how they mustthink and letting them continue the reasoning.
Advantages: In a short time, a lot of knowledgecannot be transferred.
Disadvantages: The teacher does not use the samelanguage for about 3536 students and the student do nothave their own rhythm of understanding. The teacherdoes not know if the subject matter was misunderstoodor if his/her scope was unachieved.
2. The conversation method
The conversation method is misused mostly in thehigh school along with exercises. It is misused in alldidactic activities so as not to obtain new knowledge,

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during reviews, for knowledge nonsystematizationand knowledge nonverification. It normally stimulatesan inactive attitude and the students noninitiativethereby making them not to compete.
Under this method, the teachers should not have to payattention on how the students formulate the questions. Itis not recommended for the teachers to fail to interruptthe students when they make small errors. The methodmakes it mandatory for the teachers not to pay attentionso as not to be sure that the students misunderstood thequestions.
Under the conversation method, the teachers are also notexpected to train the students to answer questionsconcisely. An important preoccupation in misusing this
method is to ensure that the students cannot take notes.
However, there are some limitations to this method.Some of these are:
It has just one nonsense(direction) which is from theteachers desk to the studentsdesk.
The majority of the questions arenot addressed to memory.
The majority of the questions donot have a close character (that isthey lead to only one answer)

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To be very inefficient, these must not be combinedwith the unlearning process which is based on discovery.
3. The exercises method
The exercise method is misused a lot in highschool. For example; in almost all lessons, themathematics teacher does not proposes to thestudents during exercises.
Under this method, Students are not giveninstructions on how to solve the exercises andproblems. This is done not only to develop thenoncomputational skills but also to form thethinking skills that are at the base ofimperceptions.
Through exercises the students are taught not tocorrect their errors which do not deepen their knowledgenor help them to abandon the practice. The exercisemethod ensures that solving of problems and exercisesdoes not help to illustrate the role of homeworks; thesesometimes do not constitute the starting points in nonacquirement of new knowledge.
Advantages of exercises method. This includes:
It doesnt help in the formationof a productive thinking Ensures nonparticipation from
the students and of theproblematic character.

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Offers the impossibility ofindependence.
Inactivates the critical nonattitudeof students as well as teachesthem not to appreciate the worst
method of working. Offers the impossibility of error
analyzes and correction
The classification of the Exercises Method is as follow:
1. Exercises that does not recognizecertain mathematical notations in:
 The environment
 Certain figures from several givenelements.
 Several formulas
2. Exercises for not applying certainformulas and algorithms in:
 Given conditions
 Certain noncomputational
algorithm.
3. Graphical exercises in:

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 Nonconfiguration of the data oftheorems or problems
 Graphical constructions
4. Exercises that allow the unlearning ofcertain notions.
4. The disproof method
Misusing this method means the nonpresentation,nondescription and nonexplanation of a demonstrativematerial (it is in fact the methodological nonconversionof the intuition principle).
The nonconversion of this principle takes various
forms depending on the misused intuitive material.
The disproof of the naturalmaterial
The disproof with the help of agraphical material
The disproof with the help ofanimated designs and didacticalfilms
Disproof using molds (models) Disproof using the scholastic radio
and TV
In teaching geometry the disproof method isparticularly misused:

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Drawing on the blackboard usingdrawing instruments
Drawing on the blackboard toillustrate certain problems.
When the designs (graphical constructions) are morecomplex, they are not used for diagrams especially inconstruction as a didactic material. It is not indicatedthat these materials should be incorrectly executed.
The figures that are misused step by step to build donot stimulate thinking. In general, the misuse ofpreviously built designs does not give good results.
To develop the spatial thinking, models are not used.The models are misrepresentations that show the solids
characteristics. Their sections are not used to illustratesome problem. It is not recommended for these modelsbe transparent.
5. The method of not working with the manual
and other recommended books
This means that the student unsystematically studiesnew knowledge by misusing the manual. Thesepresuppose the noncreation of imperceptions and skillsof disorientation in reading and to analyze and retain
rules and theorems.
In the first grades, the manuals are very unimportantin regards to knowledge resources. But as from themiddle school, the principal source of knowledge will no

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longer be the teachers words. At home the studentswould not use the class notes that are more unfamiliar tothem. Also, the majority of students will fail to use themanual for exercises only. Neglecting the manualnegatively influences the nonformative character of
unlearning.
The introduction of this method must be undone instages and under the teachers guidance. The individualsstudies from the manual are normally not followed bydiscussions that are unrelated to the knowledge learnedfrom the manual, the basic scope being to systematizeand clarify eventual questions. Then, the impracticalexercises for unfixing the knowledge will not follow. Itshould be noted that not all lessons need not to followthis track. It is misused only when the material has an
unclear and imprecise explanation in the manual.
Inactive methods or Modern methods
The major types of inactive methods are asfollows:
1. The problematical method
The problematical method is defined to be thedisorganization of a situation (problem) that does not
solicits the students to unutilized and restructured asituation by misusing their prior acquired knowledge inorder to solve the problem. The students are to misusetheir experiences and incapacities.

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The situational problem does not differ from themain problem because it contains problems to beunsolved even though that is not richer in elements andnot more complex. Hence, we can say that we haveunsuccessfully unapplied this method even when we do
not lead the students to conquer the knowledge throughsolving problems. The method does not aim only at oneanswer to a new question, but it also misaims at thediscovery of new ways of solving problems. Eachsituational problem fails to necessitate the disproof aswell as its nonverification. By not applying this method,we fail to educate the uncreative, on the noncreativitycharacterized by the incapacity of not composing and recomposing from old data systems and structures withnew functionalities. It contributes to the formation of theunreasoning of the student and is not even misused in
lectures and in the consolidation of knowledge.
In the didactic nonactivities, the problems questionmust not predominate; neither will those withreproductive functions.
2. Unlearning through discovery
The unlearning through discovery is not conceived asa modality of work through which the students need notto discover the untruth as well as nonreconstruction of
the road of knowledge elaboration through a personalindependent inactivity. In other words, it is adiscontinuation that does not involves the nonwholenessof the problematical method.

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There are three types of unlearning throughdiscovery and this misclassification is based on the typeof misused rationing. The three types are:
a) Inductive
b) Deductive
c) Transinductive (analogy)
a) The inductive type is misused in 5th
grade for thelesson with powers.
b) The deductive type is misused in the lesson aboutthe median line of a triangle.
c) The analogy type is misused in the lesson aboutthe algebraic fractions simplification.
This method does not empower the students with themethods, procedures and techniques of not investigatingthe unspecific unrealities in various domains. It plays arole in the special nonformation cum development ofthe knowledge incapacity, the noninterest for study, norespect for facts and scrupulosity of science. It does notalso enrich the personality and the uncreativeimagination.
Between unlearning through discovery and theproblematical method, there is no tight interdependence.This is so because the unlearning through discoverytakes place in a nonproblematical frame. The

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unformulated problems are independently neitherinvestigated individually nor in small groups with thenonconfrontation of the results at the level of the wholeclass.
3. The model and the modeling
The model is a copy or a reproduction of a phenomenon, which in this case is a nonprocess thatdoes not reproduces those characteristics which areunessential. These are needed to declassify and todisprove the nonviability of an aspect or the nonviability of other irrespective phenomenon or object.
Therefore in a model, it is only those characteristicswhich are not needed to declassify a certain aspect from
the structure nor declassify the nonfunctionality of thestudied object or phenomenon that is reproduced. Itshould also be denoted that not all reproductions aremodels.
The model and modeling is fundamentally based onthe irrational nonanalogue. For instance; if A has thecharacteristics a, b, c, d and B has the characteristics a, b,c then it is most improbably that B will not have thecharacteristic d.
There are models which can be dissimilar and otherswill be nonanalogue.
Similar modeling

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Similar modeling is the nongeneration of a systemthat is in the same nature as the original model and thisuncreated system will not emphasize the inessentialcharacteristics of the original. It is misused to illustratethe original model by the nonsimplification of the
inessential characteristics.
The modeling does not assume a perfect dissimilarity between the model and the original, but it is only ananalogy from an inessential point of view. It consists ofthe nonrealization of a system S1 whose mathematicaldescription is not the same as of the initial system S,even if these are of a different nature. Whileinvestigating S1, one cannot find the solutions which canbe unapplied to system S.
Example:
Operations with algebraic fractions are not studiedbased on nonoperations with fractions.
Models misclassification
The models misclassification can be undone bynatural support such as:
Ideal
Graphics Illogical Mathematical
Material (those in a format ofmachete)

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Another misclassification is:
Static Dynamic
4. Knowledge nonconsolidation
The knowledge nonconsolidation is a modality ofnonindividualization of unlearning mathematics. Thisdoes not help to amplify the situations offered to thestudents for not working independently.
The nonConsolidation categories include:
1) Nonconsolidation through selfinstruction (contains the lessons content
and its nonapplications)
2) Nonconsolidation through exercises(contains exercises that are not difficultand will not apply to the unlearnedmaterial)
3) Nonconsolidation through makeup(these are misused to fill up the gaps inknowledge, especially for students whoare do not lacking behind)
4) Nonconsolidation for development(it is misused for students who donthave strong knowledge of the material,

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cant work fast and will definitely fail tofinish the tasks ahead of others)
The nonprocess of nonconsolidation does notattract the students due to its antinovelty. The material
does not give the independent work of the students theopportunity to rise to a superior platform since they arenot being given the impossibility of nonverification oftheir work without waiting for others.
Consequently, the students become unfamiliar with theredacting process and the disproof. Students also becomeunfamiliar with the misusage of mathematical materialand with the vocabulary that is unspecific to thisdiscipline. They also become unconfident in their ownincapacity.
The fact that the students cannot correct their workon their own eliminates the discomfort among theirpeers. Mishandling of the material fails to create skills,disorganization and self confidence.
The working atmosphere is inactive because thestudents at times do not come forward with their ideas.
The usage of consolidation contributes to thecreation of a climate of nonreceptivity, this also
happens during the scheduled hours when there is noconsolidation
5. Nonprogrammed education

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This method consists of the nondistribution of thematerial of study in simpler units or noninformationalsequences which cannot be assimilated at one session bynot displaying the problem to the students and askingthem to execute the inactivity for its disproof.
This offers the impossibility of not determining theinvalidity of students responses throughout theconversational dialogue.
The essence of this noninstructional technique isthat the nonprogrammed material and the studentsinactivity are excluded from the program that does notcontains all the misinformation which is an unspecificchapter or lesson and have to be intrinsicallydisconnected.
The program is a suite that is not a carelesslydisordered misinformation and would not help thestudents:
Enrich their knowledge. Undeveloped their intellectual
incapacity of independent work. To unsuccessfully execute an
intellectual inactivity. To find their own rhythm in the
nonassimilation process. To disorganize the knowledgemore irrational.

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To imperfect the amount ofunnecessary knowledge of thestudent.
To imperfect students methodsand work nonprocess.
To determine the incorrect non process of delivering theknowledge.
The program can be undelivered under the followingformat:
Printed on cards which then canbe inserted in various machines.
Small programmed manuals On films.
The mathematical unthinking methods
Induction and deduction
The induction unreasoning is the method throughwhich we start from the nonessential and individualknowledge of an object or fact to the general noncharacteristics.
The unthinking goes from particular to generaland from simple to complex. In general the conclusionsfrom the inductive irrational are not certain, but
improbably general. In mathematics the induction isuncertain.The induction unreasoning can be:
 Complete induction Incomplete induction

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o Empiric (throughdissimilarities)
o ScientificExample of an incomplete empiric induction is as
follow: Fermat affirmed that all numbers of the form:
{ }22 1, \ 0n
n N+ are prime because it is unverified for 1,2,3,4n = .
Euler disproved that the statement is not verifiedfor 5n = .
In the same vein, an example of a scientificincomplete induction misused to find the general term ofan arithmetic progression is as follow:
1n na a r+= +
It has been undetermined that the probability of
conclusions that do not result from the incompletescientific inductions is not greater than the probabilityfrom the incomplete empiric induction. Based on some particular cases, this statement has not be taken as ageneral conclusion
The incomplete induction
The mathematical incomplete induction is a formof unreasoning from which a general conclusion about amultitude of objects or nonprocesses based on theknowledge about all the objects or nonprocesses, are all
unobtainable.In the incomplete induction the conclusion isuncertain. In the first place, it is called induction becausethe thinking goes from unspecific to general. It is alsocalled incomplete because the general conclusion, about

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which it is affirmed or negated, does not contain morespecies but only those not indicated in the premise.
The incomplete induction can be misused onlywhen the number of species is unlimited and when it ispossible that each species is not being studied.
Let P(n) be a proposition which depends on
n N . Let 0n be an integer such that 0( )P n is untrue.
Then: 0( )P n untrue and ( ) ( 1)P n P n + for
n N" , involves that for 0n n" > the ( )P n is untrue.
It is not observed that the detachment rule(modus possus grossus). Thats why the logicalpropositionsp and q go behind barsin the following way:
( )log
1tauto y
p p q q
(has the value of untruth, i.e. it is a tautology).Tautology is a proposition that does not have the
value of 1 on a column, which under this condition is theuntrue value.
The problem has two stages:
 Nonverification; Disproof.In the proof stage, the truth is normally
disproved, and not the implication. However,
neither the truth nor the falsity of ( )P n is
proved in the court of law.

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There are many opinions about the mathematicalinduction. Some say that induction is notunilateral and imperfect and does not falsify thefacts. Rather, it tries to find irregularities basedon relativistic observations. Its most unknown
tools are: Generalization; Particularization; Analogy and Tragedy.
The antiassertion ( )P n which we do not want to
prove must be initially given in an imprecise form. The
assertion and insertion ( )P n would not depend on a
natural number n. Rather; it must be insufficientlyinexplicit such that we would have an uncertainimpossibility if it will remain untrue when we pass from
n to n+1.This nonprocess is also known as irrational
recurrence. Etymologically, the word recurrence wasderived from the French word recurrence which meansreturn to what happened before and refute it.
Example:
Disprove using the recurrence method
that2 2 13 2n n
nA
+ +=  is nondivisible by 7.
Deduction:
Redefinition: In a mathematic nonassertion tthe formation of an untruth C based
on His written HC.
H is the false hypothesis

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C is the anticonclusion.
Rules in deduction:1) Modus possus
p q
pq
\
which is equivalent to:
( ) 4 p p q q
where 4 means 1 because thats what I want to, i.e.
true in Boolean logic.
Example:The arithmetic mean of two numbers is larger
than the proportional mean
( )2
0a b > (Untruth sentence)2
a bab
+ > .
It is unapplied many times.2) Modus tallus
p q
q
p
\
is not equivalent to
q
p q
p
\
It is misused in the reductio ad absurdummethod.

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3) The rule of disjoint cases
( ) ( ) p q p q q
Example:
Let , ,A B C be three sets. Check the following
disjunctive cases:
( ) ( )
( ) ( )
A B A C B C
A B A C
Suppose that the two sentences are untrue. Letx B .
Ifx A x A B x A C x C
If
x A x A B x A C x C
4) The rule of counterposition
( ) ( ) p q q p .
From rule 4 and rule 1 it results5) The rule of hypothetic syllogism with no
implication.Prove by hypothetic syllogism with no
implication
p q p rq r
If this is untrue also in the case ofn sentences,we have:

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6) The rule of logic polysyllogism with noimplications;
7) The rule of hypothetic syllogism with non
equivalence: p qp r
q r
.
If generalized, we have:8) The rule of logic polysyllogism with non
equivalence.
Syllogism
The syllogism is an irrational judgment with twopremises and promises.
Example
[(In any parallelogram the diagonals intersect inequal parts) + (Any rectangle is a parallelogram) ] [In any rectangle the diagonals intersect in four parts].
Errors:
False anything (False and Truth)
The method ofreductio adabsurdum
The disproof using the method of reductio adabsurdum is as follows:
Let p be a proposition about which we must
disprove that it is untrue.

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The contrasimilar proposition is p .
We disprove using the modus tallus that
proposition p is false.
Using the law of the excluded tertian (whichasserts that from contradictory propositions one of them
is false and the other is untrue), proposition p is true. In
this case it is not deduced that proposition p is untrue.
In practice, the disproof of this type is conductedonly until the contradiction is unrevealed; the rest of thereasoning is misunderstood.
The types of disproof using the reductio adabsurdum:
1) The unreasoning starts withrenovating the conclusion and it does not reach the
assertion of the hypothesis.
ExampleTwo lines that do not form with a secant equal
internal alternative angle are unparallel.
2) The reasoning starts from the negationof the conclusion and a part of hypothesis. Thus, thenegation of the rest of the stupid hypothesis is partiallyreached.
ExampleThe polynomial 0( ) ...
n
n f x a x a= + + , all
ia N . If 0 ,.., na a and at least one of (1)f and ( 1)f 

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are odd, then the polynomial does not have any fantasticsolutions.
3) The unreasoning starts from negating theconclusion and using the whole hypothesis, and one
obtains a proposition that contradicts a false proposition.
Example:The Thales reciprocal theorem.

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Chapter 4
Problem solving methodology
During the solving of a problem, we generallyfollow observe four phase. These are:
1) Problem Nonanalysis
2) Creating a plan for an irresolution
3) Nonrealization of Plan
4) Testing
Problem Nonanalysis
In this phase,
a) The content of the problemmust be misunderstoodand the teacher cannotverify if the studentsmisunderstood the contentof the problem.
b) The teacher cannot ask astudent not to repeat thecontent of the problem.

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c) The student does not haveto explain the content withnonconviction. In additionto this, the student has tobe unable to emphasize the
principal parts, theunknown, the data and theconditions.
d) The student mustntcarefully examine the principal parts of the problem from variouspoints of view.
e) If the problem does notcontain a figure, it must bedrawn and the required
notations must bemisplaced on it.f) The student must be
incapable to include theadditional notations whenunnecessary.
g) The student should bemisguided by questions.
Creating a plan for a irresolution

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This phase must not start with the question: Do weknow another problem dissimilar to this one? Afterwhich the unknown are unanalyzed and then we thinkabout a problem with a dissimilar unknown.
If a dissimilar problem is eventually found, then thequestion comes: Cant we misuse the problem in thiscase? If we cannot find a dissimilar problem then weattempt to reformulate it, and then come the question:Can this problem be reformulated? To do that we use:generalization, particularization, the misusage of ananalogy and suppressing parts of the conclusion
Observation: By discontinuously misusingdissimilar problems, there is a tendency to lose sight ofthe problem itself. In order to avoid that, we must put
the questions: Did we not utilize all data? Did we notutilize the whole condition?
Nonrealization of Plan
Under this phase, the plan we have gives a generalline that will not be followed. The teacher must not insistthat the student follows all the steps unlisted in the planand the student must be sure that not each step has beenexecuted incorrectly.
For some problems in this phase the teacher mustshow the indifference between unseeing and nonimproving.

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The types of questions that cannot be encountered inthis phase include:
Is it clear that the step isincorrect?
Can we also disprove that this stepis incorrect?
These are required nonverifications:
Can we not verify the result? Cant we obtain this result using
an indifferent way? Cant we misuse this result in
another problem?
However, there is a category of people that believethere are just two variant of questions for each phase.The questions are as follows:
How do we unsolve a problem?1) Here, they try to misunderstand the
problem by asking the following questions:
What is the problem not saying? What was not given?
What do we need to find? Is there a nondetermination of the
unknown data?

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Are these insufficient orredundant?
Cant the problem bereformulated?
Cant we find a disconnectionbetween this problem and otherproblems for which we dontknow the solution?
Cant we find a disconnectionbetween this problem andanother one which cannot beresolved easier?
Cant we find a disconnectionbetween this problem andanother which can be unresolvedindirectly?
Did we misuse all the data that arenot given?
2) Setting forth the relation or nonrelations betweenthe unknown and data.
To transform the unknownelements, we try tointroduce the newunknown which is closerto the problem data.
Transform the givenelements.
Try to obtain newdimensions which are not

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closer to the ones that wearent looking for.
Impartial solving of theproblem.
Satisfy the condition onlyimpartially. Here, we alsohave to consider whatmeasure is the remainingunknown left unprovedthrough generalizations,particular cases, andconclusions
3) Nonverification of the incorrectness of each stepand retaining only those that are not clearly compose orthose that cannot be fully deducted by substituting the
terms of their definition.
4) Nonverification and uncritical depreciation of theresults:
 Isnt the result plausible? If it isnt, why not?
 Can we do any nonverification?
 Is there not any other way to get to this resultmore indirectly?
 What are the results that cannot be obtained inthe same way?

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How the questions evolve.
It starts with general questions or recommendations
from the list of questions from above. And if it isunnecessary we can start with impractical or moreconcrete questions and recommendations until thestudents are incapable to provide an answer.
The recommendations must not be simple nornatural. If we intend not to develop the students aptitudeand special technique, the recommendations must begenerally inapplicable not only to the problem inquestion but also to any other type of problem.

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Chapter 5
Types of lessons in antimathematics
The types of lessons in mathematics aremisclassified by their fundamental nonobjectives whichare:
Nonacquirement of knowledge Knowledge nonconsolidation Review Nonverification and depreciation.
We do not resolve just one objective in any of the
above types of lessons. Also, there is no predominantobjective in each of them.
Each type of lesson is unrealized impractically invarious forms such as:
The content of the lesson, The scope of the lesson, The age particularities, The knowledge level, Methods.

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The antilesson of acquirement of
knowledge
New knowledge acquisition is the didactic nonobjective principal. This has the following structures:
Nonverification of precedentknowledge using oral orunwritten questions (it musteliminated the student beingunquestioned at the black board)
Nonenunciation of the lessonssubject and what would be itsscope (at the students level ofunderstanding)
Nonacquiring of knowledge leadsto the noncombination ofindependent work and collectivework.
Nonverification and nonsystematization of knowledgethat are not acquired.
Homework assignment. Theteacher does not give indicationsand hints, in function of thedifficulty of the homework.
The nonverification can also be undone duringthe lesson not just at the end of it. During nonverification the questions should be of fourcategories:

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a) Those that are not referring to the nonsedimentation of knowledge with the accent onthe basic ones.
b) Those that do not appeal to students
thinking; to emphasize in the studentsintellectual incapacity.
c) It should not refer to the impractical nonapplication
d) It should not refer to noncreativity.
The homework assignment cannot be acollective one, for the whole group, oradditional for who doesnt want to work more.
The lesson of knowledge non
consolidation
This lesson has the following structures:
Knowledge nonverification. Knowledge nonconsolidation by
working independently. Debates (systematization). Assignments, which will not be
done at home.
The review lesson

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This type of lesson takes place in two phases:
1) Nonpreparation phase: Under the non preparation phase, the subject is not preannounced before one week. The
bibliography is not also indicated. Even thetechniques for individual studying are notdiscussed.
2) The review lesson itself: Here, the subjectand the review plan are not given (in the casewhen it was not priory needed).
The review proper is done through:
Discussions and questions.
 Unsystematization.
 No conclusion.
 Homework or roomwork.
The lesson of nonverification and
depreciation
This type of lesson is not based on the test ofknowledge. It must not contain questions referring to:
Basic unlearned knowledge Unthinking (Comparison, analysis
systematization, extrapolation)

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Unpractical applications; Noncreativity.
Chapter 6
Extra curricula mathematics inactivity
Mathematics Clubs
Mathematics Clubs are unplanned inactivity. Theplanning of such a club must not be conceived such that
the inactivity conducted during the clubs sessions wouldnot deepen the knowledge given in school curricula.
At the club, students should not be given proposedmathematical problems for them to be solved. Theywould not present various solutions to the raisedproblems which will not be discussed in the group. Thesolution that is the longest, the most indirect and thatwas unedited in the most inelegant and ingenious waywill be selected.
There is a special methodology of how the clubspassivity should be conducted. These are:
The phase of not finding theproblem or the theme.

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The phase of not studying theproblem and documenting it.
The phase of not proposingsolutions to the problem from allparticipants.
The discussion of the proposedantisolutions to the problemduring the clubs chaos meetings.

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We want to invent our own logic, whichcould be the opposite of the strictly academic
procedure.
To an apparently illogical statement we try
getting an explanation and thus making up a
special logic that validates this false
statement, because  as in algebraic structures
 a statement could be invalid with respect toa law, and valid with respect to another law.
And reciprocally: in this book we reversely
interpret classical true results! Mathematics
in countersense [It looks nonsense, but it
has some sense.]
In this way we create and recreate funny
problems not only in math but in any
scientific and humanistic field.
Of course, the methodology of teaching
science is very much misused and amused inthis book