1

Editor Tuna ÜLKER

ISBN 978-625-7454-06-3 www.panukitap.com panukitap

panukitap@hotmail.com General Coordinator Ahmet TENKER Editor Tuna

ÜLKER ISBN 978-625-7454-06-3 www.panukitap.com panukitap

panukitap@hotmail .com General Coordinator Ahmet TENKER

TO MY DEAR WIFE...

FOREWORD: The purpose of this book is to convey to the reader a sense that it is

one of the most important and exciting journeys of discovery humanity has ever

embarked on. It is the study of the fundamental principles that govern the behavior of

our universe. It should not surprise us that significant progress has been made after a

journey that took more than two and a half thousand years. But this journey proved to

be an extremely difficult one, and the truth often came out gradually. Yet the

twentieth century has given us extraordinary new insights, some so impressive that

many today's scientists have expressed the view that we can come close to a basic

understanding of all the fundamental principles of physics. The understanding that we

have the principles that underlie the behavior of our physical world actually depends

on some appreciation of its mathematics. Some people may take this as a cause for

despair, as they will form the belief that they lack elementary-level math capabilities.

Many people's nightmares remind us of the multiplication table, simple fractions,

compound fractions and whole number fractions that we memorized in primary school

years. At this point the question comes to mind: If people can't master the

manipulation of Fractions, how can they argue well so that they can grasp the research

that is going on at the cutting edge of physical theory? I want to be optimistic about

this, I argue that people who cannot perform four operations on fractions are actually

people who do not realize their potential. While studying with a geography teacher

who has completed his first year in his profession, I do not forget his determination.

"Mathematics is like a thousand and one nights, you always find a solution by keeping

it by the book." Without tales of one thousand and one nights, we have no way of

understanding the rules that govern the physical beings around us. The simplification

of fractions we learned during elementary school is actually an application of the

concept of equivalence relation used in Einstein's general theory of relativity. Every

person has the capacity to understand information at all levels if they read enough.

What laws govern our universe? How will we recognize them? How can this

knowledge help us understand the world and therefore direct its actions to our

advantage? 7 KIBUTZ AND TRAUMA Humans have been deeply concerned with

questions like these since the dawn of humanity. At first they tried to understand the

influences controlling the world by referring to the kind of understanding that can be

gained from their own lives. They imagined that whatever or whoever was in control,

their environment would act as they did to control events: at first they thought that

events were under the influence of their destiny or of beings who acted so well with

human impulses. Accordingly, the course of natural phenomena such as sunshine,

rainstorms, famine disease, must be understood in terms of the whims of Gods or

Goddesses, motivated by such human impulses. And the only action that is perceived

to affect these events would be the appeasement of God figures. The understanding

that we have the principles that underlie the behavior of our physical world actually

depends on some appreciation of its mathematics. But gradually different kinds of

patterns began to establish their credibility. The precision of the Sun's movement

across the sky and its clear relation to the alternation of day with night provided the

most obvious example; but also the positioning of the Sun relative to the stars was

found to be closely related to the changing of the seasons and the consequent net

effect on weather and thus on vegetation and animal behavior. 8 NECIP ERDOĞAN

In my explanations here, I will be more concerned with conveying the idea, beauty

and magic inherent in many important mathematical concepts. The idea of a fraction

like ½ is simply a kind of entity with the property that when added to it gives the

number 1. The magic is that the idea of a fraction allows us to actually experience

things directly in the physical world, although they are not measured precisely by

pieces of cake. This is a Platonic phenomenon, as opposed to the case of natural

numbers such as 1, 2, 3, which measure precisely the myriad entities of our direct

experience. One way to see that fractions make consistent sense is to actually use the

definition in terms of infinite collections of integer pairs, as noted above. However,

this does not mean that ½ is actually such a collection. It is better to think of ½ as a

being that has some kind of existence of its own, and the endless collection of couples

is only one way we can come to terms with the coherence of such a being. By

familiarity, an entity such as a fraction We begin to believe that we can easily grasp it

as something that has a product, and the idea of an endless collection of couples is

merely a pedantic tool, a tool that, once we have it, quickly slips away from our

imagination. Mathematics is not just a cultural activity of our own making, it has a life

of its own and often finds an incredible fit with the physical universe. 9 KIBUTZ

AND TRAUMA We cannot deeply understand the laws that govern the physical

world without entering the world of mathematics. In particular, the above concept of

an equivalence class is relevant not only to mathematics, but also to important

physics, such as the gauge theory principles that describe the forces of nature

according to modern particle physics. To understand the platonic world is to

understand the physical world! 11 CHAPTER 1 When I came back to the house I

grew up in years later, everything seemed very different to me, did the house get

smaller or did I grow up? I used to listen to romantic songs until the morning on short

summer nights in front of the window. Years will pass, and my body will shrink, and I

will become as light as a feather, so much so that I will glide lightly out of that

window that I have never left. As in my dream the previous evening, I will take a

bird's-eye view of the city and bless my soul leaving my body. I often think when

death will knock on my door, last week twenty-five thousand people died in the

world, how will I surrender my soul? Where? Yes, I will lie under this heavy duvet for

years and I will sweat. I will see my grandchildren grow up and am destined to live an

endless life of painful indifference. I need extra strength to die, the deaths I have

witnessed so far come to mind, saying goodbye to life after traveling between two

worlds for a week and saying goodbye to life when we least expect it, like watching

the news or sending the first spoonful of a hot soup to the stomach. 12 NECİP

ERDOĞAN Maybe one should love oneself first in order to die. This may be my

biggest shortcoming, not loving people and most importantly myself. Most of the

square room, the only window of which opened onto the village, was empty. When I

was a kid, my mother used to grow flowers to bridge this gap. On my old writing desk

I would repeat the lessons I had learned for four years, transcribing each line in my

notebook, the only sound I heard was the sounds of insects walking, sometimes I

would look up from the books and stare at the wall-mounted medicine cabinet,

allergy-inducing drugs while curing inflammations, allergy-curing drugs. drugs, drugs

that are good for post-traumatic behavior disorder, tablets that regulate heartbeat, discshaped tablets that look like orange juice when dissolved in water… No matter how

hard I try, I try to gather all my memories like pieces of old paper, I can't remember

where my youth is. years that include your best days? The round coffee table that I

had to buy for my daughter caught my eye, even though she was not a young girl yet,

she cried at my workplace. she said. "Stop crying!" I said. "Everyone is looking at

you." But he finally surrendered. Two people had forced their way up the heavy-duty

stairs. In the heart of the valley stretching from the slopes to the mountains in front of

me, there is a foggy lake in front of me, surrounded by a soft warm swamp from

which migratory birds take flight from man-size reeds, I will see if I can get up from

my bed and reach the window. Besides, I feel old, very old and near death, I have no

strength to get up from my bed, walking since I fell means floating dangerously in

space; while I'm waiting for everyone to look at me at that time. Shem tries to free her

arm from my sleeve that is stuck to my skin, her eyes catch on my bare skin, her

fingers caress my skin, when I take off my wet shirt, the boy suddenly starts acting

like he is seeing my bare arm for the first time, touching the soft skin of my stomach.

Now he too is having a hard time accepting his mature body. I remember the first gift

he bought me on Father's Day, I didn't need it, Din... Din: "Come, put your feet on the

floor, hold onto the wall and get up, your diabetes medications, antidepressants are

waiting for you by the bed, but you don't need it, you just need me. you have pain."

The whole Kibbutz had spilled into the garden, awaiting his father's return from his

overseas assignment; only he, in spite of everyone, was perched on top of one of the

trees like a little ape, watching this curious waiting that did not concern him

personally, tired of this waiting that did not concern him personally. What child would

remember a father who forgot himself… What adult would really expect him, other

than a handful of friends? In fact, the majority were jealous, mostly women; 14

NECİP ERDOĞAN with blue work aprons and legs bruised from varicose veins

served for hours in the kitchen, children's house, vegetable garden, warehouse and

colors; only that Şem'in mother was wearing elegant costumes. Sometimes that wasn't

enough, her husband, who had disappeared for a seminar, had heightened her

freedom-lovingness. "You are not my father." he cried out at last. "You don't give me

a tenth of the time you spend on the theory of relativity!" I looked at her sadly as her

father, how quickly my daughter had grown, now her breasts were budding. Our

workbook, which we shared with Şem, was always kept open on the table, we used to

solve simple fraction problems in the first pages, the question types and topics started

to change as my daughter got older, we were now calculating the limit of functions,

when my daughter was a young and beautiful university student, we would now have

smooth manifolds. We were working on fiber bundles, EINSTEIN manifolds. My first

daughter, Shem, and her notebook were my sealed box. Shem was more than just a

son to me, a business partner, I had just told him that my biological father was a

member of the wealthiest family in the world. Only she knew my secrets, Shem was

like a gifted girl who was created and grown in my memory, getting better every day.

When my children were young, I used to go to seminars all the time, sometimes I

attended seminars that were not meant to get away from home, and sometimes I

attended geometry symposiums that never took place. I was just a double lecturer who

was born from a one night stand with a rogue member of the world famous Rothschild

family, my wife and family never knew it, my rich family always supported me and

after my class was over I would have a morning at the science center. . The goal of

our project was to clone, the adventure that first started with DOLLY in 1996, you all

know or have heard of, we wanted people to advance in science or, to say it

constrainedly, we Jews, because we are a chosen generation, you know, Einstein was

a Jew, he and Newton were It was our aim to clone them and bring them back to life,

for Einstein we did it with ease, for Newton it was a great help from a Jew working at

the British Museum. "Shem, we need this project as humanity because we are so

ignorant!" We know that prime numbers are infinite, but we don't know how they are

distributed, for example, the last object we know and use in algebra is the quaternions

that Hamilton found years ago, you know the story, Hamilton suddenly discovers

hypercomplexes while crossing the bridge and at that moment the paving stone he

stepped on becomes a monument. "Shem, have you ever wondered why there are no

more resounding breakthroughs in mathematics?" We need new theories, and for that

we need to bring back to life such geniuses as Newton, Gauss and Einstein. "Let me

tell you the story of Gauss, Şem, when Gauss was in primary school, the math teacher

came to the class and started to think about how to spend the two-hour lesson without

getting tired. 16 NECİP ERDOĞAN thinks of a problem that keeps children busy and

asks the following question: 1+2+3+. ...+97+98+99+100=? Students start work:

1+2=3 3+4=7 7+5=12 Gaussian starts the problem like this: 1+100=101 2+99=101

3+98=101 4+97=101 Fully fifty like this obtains 101 and strikes the final blow: The

answer is 101*50=5050. In the first stage, the teacher thinks that a solution found in

such a short time is not correct. However, the little boy in front of him is gifted." After

listening to me carefully, she wrote down the sum of consecutive even numbers in our

workbook. This problem was too easy for a gifted child like him.

0+2+4+6+...+2n=? Since two numbers are common in each term, when we put two

parentheses; The answer was found from the equation 2(0+1+2+3+…+n)=n.(n+1). 17

KIBUTZ AND TRAUMA I asked Shem about the sum of the odd numbers, and she

stared blankly at the wall with her beautiful eyes, thinking it would be appropriate to

find the sum of the small odd numbers first. He began to write with his stylus:

1+3+5+7=? Years ago, like the little elementary school student Gauss, he collected

the first and last term: 1+7=8 The sum of the second and third terms: 3+5=8 Finally,

8+8=16 16 is a square number. It is the square of the number four, and the number

four is actually one half of the last term. Yes, the uncertainty was gone, now we could

express the sum of odd numbers as 1+3+5+7+9+11+…+2n-1=n2. After these simple

questions, it would be right to start the problem that will really make us sweat. Shem

took the pen and, after doing the gesture I detested each time (putting the pen tip up

her nose), wrote the following sum: 2+3+5+7+11=? He asked me to find the sum of

the first five primes, the third term being the sum of the first two, and the last term

being one less than the sum of the two preceding terms. "I found!" He shouted and

started running. "Daddy, think of the Fiobanacci drama!" I was also excited that he

ran happily in this way, I did not remember any number for a few seconds. 18 NECIP

ERDOĞAN Then I gathered myself, in this series, each term is the sum of the two

terms that came before it. The limit of the new sequence we get by taking the ratio of

consecutive terms is the golden ratio. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,

55, … Golden Ratio: The mysterious number of nature, the mystery that exists in each

and every work of the creator! I started to get the ratio of the terms of the series: Of

course I couldn't divide 1 by zero, so I divided the second term by the third term.

1/1=1 2/1=2 3/2=1.5 5/3=1.6 8/5=1.6 13/8=1.6 Shem grabbed the pen from my hand

and ripped a page from the notebook: It was a constant number for the seven billion

living on earth, and it was irrational. number heart was like a woman full of

mysteries! "It's getting late, Shem, you must bathe now!" And I point to the lake, as if

the lake is just a bathtub belonging to Shem. "Look how dirty you are." says my wife

Nes, panting, she runs towards us because if she doesn't hurry, the lake will disappear

again, her young mother and father will disappear, but her legs are getting heavy and

she is sinking into the dark swamp. "Parents, give me your hand, I'm sinking.

Religion, come quickly, interpret the drowning." 19 KIBUTZ AND TRAUMA I don't

even remember how many times I've had this dream, I've been on medication for

years, I've been having the same dream over and over since I was diagnosed with

post-traumatic conduct disorder and then bipolar behavior disorder, it's been the same

night since the night my daughter died in my arms. I am living, or rather, I am

interpreting life, I AM NO LONGER LIVING! My only friend in this world, God,

why did he take him from me? If he comes to my room tonight, I'm thinking of asking

him that question. Who can imagine a world not populated by God? Every man

should think that he is alone with God. The stories written in the holy book (Torah)

point to the different qualities of all people. People rise one step up on the sacred

ladder of wisdom in every lifetime. While working on the path of spiritual ascension

and striving to change his personal qualities, one should never despair if he sees his

situation as worse than before he started working on his spirituality. The most

important point of a person's spiritual progress is his request to God. The worst

manifestation of egoism is arrogance, the devil's favorite sin. My biological father is a

man who has surrendered to his arrogance, believing that money can solve problems

instead of looking into his eyes as he grows up to be with his child. 20 NECIP

ERDOĞAN From the early days of my childhood, I felt a strong attraction to science.

We Jews call the spiritual book "the light of God", and someone who can receive this

light is called a "strong man". The light of God is hidden, only those at the level of

virtue see this light. I must pray to the Creator to have mercy on us and lift the clouds

over all our thoughts that hide it from our hearts and eyes. My dear daughter Shem, I

want you to understand the life I lived after I died, that I was an illegitimate child, that

when I was only a lecturer who went to the faculty from the outside, with the support

of my rich father, who did not want to see me in his secret life, he created a clone

center, the babies we cloned. I want you to know after I die what we gave to the

families on the kibbutz. I did all this work for the new generation. We must advance

in science, you know that, we are an elite race, we will have enemies as long as we

exist in the world, the Nazis in the past, now the Arabs are trying to destroy us, and

we must always be one step ahead of them, remember that, I find it appropriate to

reveal my secret to you, you are on your way to becoming a lecturer like me, geniuses

do not live today, we only have Perelmann, let me tell you briefly. Formulated a

hundred years ago by a brilliant French mathematician, Poincare, this famous problem

has both fascinated and unsettled mathematicians ever since. The Poincare conjecture

concerns ourselves and the objects central to our understanding of the universe in

which we live. When you go to a play or a movie, ask the person sitting next to you

about math and listen to what they say, most people hate it; The pool, interest, age

problems, which miss the taste of their youth years, interest, age problems... There are

people who love this course, even if they are in the minority. How does a space filled

with such beauties get such negative reactions? The disgust some people feel seems to

stem from fear. I don't imagine a single book could change that, but if you're a reader

with ambivalent feelings about math, I suggest you read more resources, and if you're

a student, take more math classes. Everyone was eagerly waiting in the MIT

conference hall in 2003, there were people sitting on the floor, the speaker was

wearing a dark suit; The bearded, balding, thick-browed speaker began: "I'm not one

to talk without deviating from the topic, so I'm going to make a lively presentation by

sacrificing clarity." The speaker took a piece of white chalk and wrote on the board

the twenty-year-old Ricci current equation. In this equation, the curvature of space has

less curvature than regions with higher curvature.

p is seen as an exotic kind of heat, like molten lava, trying to flow into the regions. 22

NECIP ERDOĞAN The speaker asked the audience to think of our universe as an

element of a huge abstract mathematical set of all possible universes. In his new

interpretation, this equation described the motions of these possible universes (parallel

universes) like drops of water rolling from high hills in a vast landscape. As each

element moved, the curvature also varied within the universe represented by that

element, approaching certain values in some regions. Universes were developing

beautiful geometries; some of them bore similarities to the classical Euclidean

geometry we learned in school, if only it had stayed that way! How easy would the

calculations be then, right? I regret to say that we needed new geometries, and these

new geometries created a silent revolution in the history of science. Certain downhill

paths caused calculations to deviate, elements along these paths developing

mathematically malignant regions that broke off or behaved worse. *** "You're

wrong, you're really wrong, I'm not mad at you." said Shem. "You were working hard,

daddy, it's true, but always for us! You only came home on weekends, and once you

went for a year and came back (you went as a visiting lecturer), I hardly knew you.

Father, what do you and your generation of victims of 23 KIBUTZ AND TRAUMA

gain with your angry accusations?" I have to admit that at times I also feel anger,

murderous anger, against the Germans and all the anti-Semitic… When nothing helps,

only then do we realize that only one God can help us, since all our efforts are

insufficient to enter the spiritual world. Until this moment comes, no difficulty can

bring us to the point of shouting to the Creator from the bottom of our hearts. When

we feel that the options ahead are closed, the door to tears opens and we can enter the

upper spiritual world through this door. I believe that tonight I will reach the upper

spiritual world, buying a few boxes of medicine will open all doors for me, I am so

happy to meet my daughter Shem! Shem, God gave us two options for free choice: 1-

God reveals himself to us. 2-God gave us Kabbalah. The process of connecting with

the Creator is like climbing the steps of a spiritual ladder to reach the highest level by

starting from the lowest level. The lower level on the ladder is the world we live in,

the highest level is where God is. You can think of this ladder as sets of numbers,

natural numbers in the first step, integers in the second step of the ladder, fractional

(rational) numbers in the third step, non-fractional numbers (irrational) in the fourth

step, real numbers, complex numbers. , quaternions, octonions. Octonions 8,

Quaternions 4, Complex 2, Real numbers are 1 dimensional. Shem, all rungs of this

ladder exist in the spiritual world. It happens when our qualities of perceiving a higher

spiritual level align with the characteristics of that spiritual level. Then our degree of

perception becomes proportional to the harmony between our characteristics and the

characteristics of that spiritual level. "Is it possible to perceive the upper level,

daddy?" said Shem. "All the spiritual levels, from the lowest to the highest, are placed

in order. The lower half of the upper position is in the same place as the upper half of

the lower position. The lower part of the upper is always within us, but not everyone

can feel it. Birth gives life and guides people, but because people do not feel superior,

they insist that God does not exist." "Wouldn't it be better if people felt the top

position?" "If we see the upper level clearly, we lose the possibility of making an

independent choice. Remember the AXIOM OF CHOICE! Since the Creator's desire

is to give an independent will to all humans, God must be hidden from the created

being. Atheists always attack believers from this point 25 KIBUTZ AND TRAUMA,

you say it is everywhere, but we can't see it anywhere! If the creator is hidden, one

becomes attached to God without any self-interest. He acts for the sake of his Lord."

"It means that the entire process of personal correction can be made possible by the

concealment of the Creator." "The moment our Lord reveals himself to us, we become

like a clone of him." said Nec. "But shouldn't God reveal himself to enable us to

separate ourselves from our selfishness?" "People obey two powers in this world, the

body, which is the source of lust, and the Creator, who has the power to bestow. These

two conditions must alternate and follow each other. In the science of logic there are

only two numbers. Zero and one represent our body, temporary world, and 1 represent

eternal life, spirit and creator. Computers use the language of logic, artificial

intelligence does not have the number 9, it perceives the number nine as 1001." Shem

was suddenly excited. "Yes, Daddy, you told me about binary radix when I was a little

kid. I can analyze the number 1001." With joy and excitement took the pen. "The ones

digit, the twos digit, the fours digit, and the eights digit. That's it!" "After the Creator

combines his altruistic features with egoistic features, he becomes in balance with the

person who wants to connect with him. The upper part raises HADIN to his

EYNAYİM level. As a result, his AHAP goes down. Previously, the lower part could

not feel the higher spiritual state in any way. However, since God concealed his

bestowal qualities behind selfish qualities, it was possible for a person to descend to

the lower level in order to perceive his Lord. However, since the features of the higher

level are perceived by us as egoistic, we do not fully understand the essence of these

features. For this reason, nothing seems to give us pleasure in spirituality." said Nec.

The difficulty in explaining and teaching Kabbalah stems from the fact that the

spiritual world has no equivalent in this world. What we have learned becomes clear

with the perception of our spiritual part, we may have to relearn the information we

have learned before. Once we understand and acquire spiritual power, we begin to

perceive it. Levels endow us with absolute knowledge. To agree on something is to

come to the Ahoraym stage, which prepares us for the perception of the Panim stage.

Reading slowly enables the development of feelings or containers (words). When the

containers are in place, the upper light enters them. Before the word is formed, the

light is around us. Generations come and go, but every generation and every person

asks the same question about the meaning of life: "How do we get to the road to

reality?" The Talmud reads: "You were born against your will, and you will die

against your will." How much easier life would be if the Creator was perceived by

each of us individually! Throughout our lives, we could perceive the consequences of

our every action. We would talk to the Creator and seek his help, question the

meaning of life and benefit from his unique knowledge. 27 KIBUTZ AND TRAUMA

If we could perceive providence, we would not have any trouble doing even the most

difficult tasks. Being aware of the Creator's providence would allow us to see the

benefits of acting without our benefit. What is missing in the world is problems in

perceiving God. Having this perception should be our sole purpose in life. We must

put all our efforts into achieving this goal. Learning the way of perceiving God is

called the teaching of Kabbalah. The light of MY HASAD is faith. Faith allows us to

connect with eternity. *** We see that since the creation of the universe, human

beings have experienced great suffering. How many people throughout history have

voluntarily made an effort for spirituality by enduring suffering? Why did God not

hear the prayers of people who did not strive for spirituality? People became slaves to

their egos and felt that a window had suddenly opened in their hearts that were closed

to the truth. People who do not perceive God are in a sense like animals.

Paradoxically, the more we try to reach it, the more our desire for it disappears. 28

NECIP ERDOĞAN Birth gives us life and guides us. Since we do not feel this higher

position, we often insist that the Creator does not exist. If we clearly see the higher

level on which the Creator is above all beings, one loses the possibility of making an

independent choice because we see only one power. *** Din stands motionless in

front of the kitchen window, watching with amazement the needle leaves are

circulating like empty palms begging for a small alms, a gray bird took their eggs,

looked by the window before going to bed last night, a lovely couple looking out of

the nest in the dark he watched the eggs that looked like eyes, the bird came right

away and protected the eggs with his body, but I couldn't protect my daughter, the

pink dress on her was covered in blood, my nightmares never ended after that day,

therapies, drugs, nights spent alone in the hospital and everyone around me I'm tired

of her calling her crazy, we need geniuses to avenge my dead daughter; The

Palestinian suicide bomber that destroyed my daughter must be destroyed, not only

him, but all Arabs and all Germans, the famous writer Stephan Zweig, saw the success

of the dictator Hitler and committed suicide in despair, but I will not make this

mistake, the genius we grew up in the kibbutz. clones started working in our science

center, they don't know that they are a copy, of course, we can create a biological

weapon that only we have the vaccine in the world, or we can make weapons that will

wipe a country off the map in an instant. The spring wind will scatter this little bird's

nest, and there will be no trace of that gripping life that has been here for a week,

pulsating and filling it with a strange excitement. And a nest is the poor eggs and their

poor mother. Why did the bird take them away? I've been talking to myself lately, my

alter ego, hearing my own voice often and more intensely, especially when no one is

around; My thoughts are pouring out into words without any hindrance, my voice

comes out of my throat in simplicity. Where is my daughter? Caused? Police reports

He says it's not about the suicide bomber but I don't believe it, my daughter is at

school right now, she's with her friend or she's walking home right now, her little feet

are walking on the sidewalks one by one, I watch her with longing eyes, she just came

home from school , threw her bag on the bed and threw her socks in the middle of her

room, her mother Nes warned her as always, where is my daughter's heart? Where is

the heartbeat that beats next to me and gradually fades and disappears? I remember

when I heard the sound of his heartbeat with the help of a device they put in his

mother's stomach in the hospital before he was born. How can a love, which is the

most natural of love, turn into frustration? I examine his dream with longing eyes, but

I must hold on to life, I must live for my daughter Din, my son Avn. Nes tries ways to

seduce her, I remember, "Come on girl, let's make a cake together." he says, but meets

a distant look, hears a cold voice. "Another time, mom!" I remember the days when

he first fell in love; how far he is from his mother and me, the days when he comes

from school and shuts himself in his room, listens to love songs for hours and leaves

the house adorned with joy… *** Din watches him with a smile that remains on his

lips, trying to forget that strange pain, "Enough, stop it now. , let the girl grow up in

peace." says my wife Nes. "You would think you wanted to be at your mother's knee

every minute while you were growing up." I don't answer, yes, when I was growing

up, I always talked to my mother and always shared my problems with my mother,

my biological father only sent us checks and I grew up with my stepfather as the fruit

of forbidden love. The cat's meow comes into my thoughts now, as my phone starts to

vibrate, who knows, maybe they're calling from the science center, a warm fluffy ball

crawling between my legs. Where were you? "Enough." says the cat. "Did you hear

me, Nec? Enough! You and your silly problems bore me." 31 KIBUTZ AND

TRAUMA The cat climbs, as if disturbed by the heat, the scent of the night fires of

those camped by the lake hanging in the air. "You have to understand that it's not that

easy to find an instanton, Nec. You work too hard with your daughter Shem, and

Shem, who makes solving this problem and solving the Yang–Mills equations a

matter of pride, may become depressed and consider suicidal." said the cat. I won't

answer you cat. In the meantime, the cat rises on the work table in one move, and

moves towards the monitor with its majestic weight. Nec believes that the cat is a

gifted person. The cat knows that the non-obvious zeros of the Riemann–Zeta

function are numbers of the form x+(1/2). But how strange Euclid does not know his

postulates! "Don't mess with the fifth postulate, cat." "The fifth postulate? Doesn't it

have a name?" asked the cat. "The fifth postulate is the theory of parallels. Every

father should counsel his child that to deal with the theory of parallels is to swim

against the current." In composing The Concept of Geometry, Euclid took great care

to see what assumptions his representation was based on. In particular, he was careful

to distinguish certain claims called axioms—these were clearly accepted as true, they

were basically points, lines, etc. were the definitions of what you meant by Of the five

assumptions whose validity seems less certain but which seems to be true for the

substances of our world, the last of these assumptions, called Euclid's fifth

assumption, was accepted and felt less clear than the others. For centuries it had to be

possible to find a way to prove this from other more obvious assumptions. Euclid's

fifth postulate is often called the parallel postulate. "You said that, Nec, remember?"

asked the cat. Before discussing the parallel postulate, it is worth pointing out the

nature of Euclid's other four postulates. Postulates deal with the geometry of the

plane, but Euclid also considered three-dimensional space in his later work. The basic

elements of his plane geometry are point lines and circles. Here I'll consider a line

extended infinity in both directions; otherwise I will refer to a line segment. Euclid's

first assumption effectively asserts that there is a straight line segment connecting any

two points. The second conjecture asserts that any straight line segment is infinitely

extendable. Its third conjecture asserts the existence of a circle for any center and

radius. Finally, his fourth postulate asserts the equality of all right angles. 33 KIBUTZ

AND TRAUMA From a modern perspective, some of these propositions, especially

the fourth, seem a bit strange, but we must bear in mind the origin of the ideas

underlying Euclid's geometry. He was mainly concerned with the motion of idealized

rigid bodies and the concept of coherence, which is indicated when such idealized

rigid bodies collide with one another. A right angle on one object equals a right angle

on another object Equation concerned the probability of moving someone, so that the

lines forming its right angle would lie along the lines forming the right angle of space,

so that a figure in one body would have the same geometric shape as a figure

elsewhere. The second and third postulates express the idea that space is infinitely

expandable and void, while the first expresses the fundamental nature of a straight line

segment. Although Euclid's view of geometry is quite different from ours today, his

first four postulates basically summed up our day. *** I put her food in her bowl, but

the cat is in no hurry to eat; he stands between my feet, running from one to the other,

this cat is the only memory I have of him. I started to feel that I always live with

memories, I am so tired of life, but I have to live for my other children, we used to sit

by the lake until it got dark and everyone left, there was your best friend's house and

you always wanted to stay with them at night. You asked me to come and get you. Do

you remember the days when I took you to school, when we bought ice cream on the

way back, sometimes made from goat's milk and sold by hawkers on the street, Şem?

When exactly came the moment when the balance between memories and needs for

each other was upset? Nothing, no one had prepared me for this, neither books nor

newspapers nor mathematics, although it was not visible to the eye, of course I am not

the only person in the world to have experienced it at such an early stage in my life,

"Enough, enough is enough." says the cat. "You hear me, you have to come back to

life now, hold on to your project, science and big dream, this purpose should connect

you to life, now take three pink Xanax and go to sleep right away." The cat is very

angry with me and when he is angry with me he always talks. Sometimes God comes

into my room, a clear light fills the room, the bedroom lights up, but my wife Nes

continues to sleep. God tells me to get up and read a book. So I start reading, I start

reading the story of the mathematics behind the Poincare conjecture and proof. In

order to talk about mathematics rationally, it is necessary to know not only the results

but also the people who produced those results. Achievements in mathematics echo

the legend of a genius struggling to make sense of a cosmos that is ignored once it

finds its way into the popular consciousness. There are individuals whose insights

seem to have come out of nowhere and who are pushing science forward. However,

although the genius is colorful and mysterious, advances in science are based on the

societies in which other individuals live. You know, cat, the discovery of nonEuclidean geometry has created a new world. The Poincare conjecture provides

conceptual tools that allow us to reason about the possible shape of the universe. In

primary school our teacher taught us that the earth is round, in high school we learned

that the earth is flattened from above and below, because of this flatness we say that

the gravitational force is greater at the poles and smaller at the equator line. We call

this special shape a geoid, and we see that the Earth is round from the photographs

taken by orbiting spacecraft. In the past, people believed the earth was flat. People

believed that there were people on the other side of the Earth whose feet were

opposite ours, walking with their feet up and their heads down. When Christopher

Columbus said that it was necessary to constantly sail west to go east, there were

people who thought he was crazy. The advisory committee, chaired by King

Ferdinand and Queen Isabella, could not believe that this idea was in fact a statement

that the Earth was round. The members of the delegation believed so much that the

Earth was flat; According to them, the only success Columbus and his crew could

achieve would be to feed giant monsters in the open sea as food. There were also

those on the committee who believed that there were no reverse-bottomed humans and

that there were no giant monsters in the high seas. Although Columbus was sure that

the Earth was round, he also had question marks in his mind, he did not know the

diameter of Dünya, the only calculation was made by the ancient Greeks. In the

absence of knowledge, imagination would come into play. Ptolemy had estimated the

circumference of the earth to be twenty-nine thousand kilometers. Some advisers of

Ferdinand and Isabella referred to Eratostenes' prediction. This estimate was very

close to the present value. Had these few advisors emerged victorious from the debate,

the journey would have taken longer and the budget would have been greater. While

Columbus was considered brave and wise during his lifetime, five centuries after his

discovery, he was considered an imperialist. Columbus believed that the land he

reached throughout his life was India. It would take much longer to get to India by

circumnavigating Africa, and he always took a shortcut by going west. had reached.

Columbus says: "I don't believe the Earth is perfectly round as depicted, it seems to

me that the Earth protrudes hard near its stem, it might be a pear-shaped shape, like a

round ball with one part like a woman's nipple. And this protruding part is the highest

and closest to heaven." The famous explorer felt that the Earth was not perfectly

round, with the southern hemisphere being the bulging side of the pear and the

northern hemisphere being the weak side of the pear. Therefore, while reaching the

Spice Islands in a short time, the journey time would be longer if we tried to go to

these islands along the African coast in the southern hemisphere. 37 KIBUTZ AND

TRAUMA Two thousand years before Columbus, no one lived on Samos. The island

was open to looting from all directions. Byzantines, Arabs, Venetians, Turks and

Crusaders. Even today, it is a corner of paradise with its quiet towns, white beaches

and fertile olive trees. This paradise corner is where Pythagoras lived. Pythagoras first

explained that the world was round in Samos. Pythagoras went on many journeys with

his father. He met scholars during a trip to Sur, went to Italy, was a genius who was

interested in philosophy as a child, unfortunately we do not have the genetic code and

it is impossible for us to copy it. Egyptian priests believed that Pythagoras was a

beloved servant of their God Osiris. The details of his years in Egypt are even more

blurred than the details of the rest of his life. Pythagoras became interested in

Zoroastrianism after the Egyptian years. In the following years, he founded his school;

this was more of a fraternity than a school, it accepted women into the school. The

Pythagoreans believed that reality was scientific at the most fundamental level, and

that philosophy was a means of spiritual purification. The appeal of universal views,

the mystery of the east, and the exotic blend of Greek ideas fascinated his

contemporaries. Pythagoras believed in reincarnation, at his school he would tell his

students about memories of their past lives. He believed the world was round, but he

couldn't rest easy without proving it. He did not live long enough to obtain sufficient

evidence, but his teachings had reached the time of Columbus. Pythagorean views

were passed down from generation to generation with the help of Plato, Aristotle and

the learned geographers of the Middle Ages. 38 NECIP ERDOĞAN In the Columbus

era, people who argued that the earth was round used the tides, day and night, and the

phases of the Moon as evidence, and when you looked along a line in the north-south

direction, you saw the Sun from different angles. Now let's close our eyes, go back to

the time of Columbus. Additionally, suppose we live on a cloud-covered planet like

Venus, how would we infer about the shape of the earth under these conditions? The

most important concept for us will be the concept of a two-dimensional manifold

(mannigfaltigkeit) or surface. Maps (paper sheets representing points on the Earth) are

two-dimensional. A collection of maps that allows all points on the surface to be

represented on at least one map is called ATLAS. When you buy a World atlas, you

actually get a book of maps. Every location on Earth is found on at least one page of

this atlas. The Poincare conjecture claims that the only manifold in four-dimensional

space is the sphere. *** Din starts talking on the phone, and when a male voice says

his father's name, he suddenly gets startled, "Yes?" he says hastily, as if his own name

has been spoken, for some reason he stands up, 39 KIBUTZ AND TRAUMA "Yes, it

would be my father." he says and arrives at the hospital in a short time, the doctor

gives a disinterested glance; He is tall, handsome, looks younger than himself. "What

happened?" "A suicide attempt." says the doctor. "Your father has taken too many

pills than he should have, his stomach has been washed, he's fine now." Religion had

always felt guarded against death; He thought that his parents would always live and

never die, but now he was overwhelmed by his father's wish to die and the fear that in

a few hours he might leave the world and deprive him of the thin proud protective

layer he had bestowed upon him. As he approaches the exit, it comes to mind that the

sick man may not have the strength to walk to the parking lot after discharge; he is

definitely sitting somewhere in the entrance waiting for his wife, he has to look for her

there, he even seems to see her collapsed body on one of the benches. When he takes

his steps towards it, he comes to the side of his unconscious father and the attendant

pushing the wheel bed. Hiding behind the curtain, Avn observes his father's

neighbors. On the narrow bed lies a newly brought man, his eyes closed, his own age,

breathing hard. With her back to Avn, a woman in a red satin blouse pulls up a chair

and sits next to the man, holding his hand. Avn suddenly has the feeling that she is

witnessing a threatening reality.

it's falling, end of life! He doesn't know that old people, even people his own age, can

get sick and die, but he hadn't experienced it in pure nakedness. Embarrassed, he

wipes the sweat from his forehead, the doctor walks away and gives instructions to the

nurse as he walks. 40 NECIP ERDOĞAN What is this, what is happening to me? He

secretly looks around Avn, everyone here; Doctors, nurses, patients and visitors,

technicians, management workers, cleaners who do not drink tea can see and

understand that he is a son who does not love his father, because he was the one who

reached his father late. The bed, which was recorded in the fixed inventory of the

hospital with his father, is fixed to the floor, almost like chairs in the sitting corner…

"Dad." says the tired man lying quietly in bed. Avn looks around hopefully, as if his

father has come from the world of the resurrected dead to embrace his son with open

arms. His father stares at Avn again. "Father!" She looks at him with the caressing

smile of a small child who is afraid of punishment, seizing her hand. Avn is hiring. "I

am the father, your son Avn!" With his father's sad gaze, Avn remembers his youth.

My grandfather—my mother, Nes' father—didn't like to talk about his past. He rarely

said that he was a Jew, waiting for days on the northern coast of Turkey with the ship

STRUMA with people whose stories seemed to be over by the time they were twenty,

thirty, or fifty, and that he was one of the few lucky survivors from the sunken ship.

We have heard the story of her daughter Nes taking her first steps many times,

though. I will never know where my grandfather got on the ship and why he saw

salvation in Turkey. Journey- 41 KIBUTZ AND TRAUMA How many days did the

voyage take, why did the people on the ship sail to the Black Sea instead of

disembarking in Istanbul? If I were in my grandfather's place, I would not talk about

it, there are books written about it, we have "Arbeit macht frei!" There are many

historians who have translated the expression. What I'm saying is, I wouldn't be telling

this story if I didn't have to talk about myself after my father after my grandfather. My

grandfather and his friends must have remembered the Jews who escaped from the

Inquisition and took refuge in the Ottoman Empire years ago and thought that history

was repeating itself. After I turned thirteen, my father Nec and mother Nes bought me

new clothes and prepared them for Bar Mitzvah. I used to go to the rabbi's house

twice a week. In our class of ten, all of us would take home the tape of the Torah parts

that he sang in the form of hymns, and repeat them over and over. Each time the rabbi

would successfully select the student who did not study for his homework, patiently

repeat each verse. This man, who lived on the money from the synagogue and the help

from the kibbutz, saw us all as his children since he was never a father in his life.

Although he had diabetes, he drank tea with three sugars at a time. Almost all of my

schoolmates had done Bar Mitzvah. We used to get up early on Saturday morning and

put on our blue striped shawl. Every time the rabbi was throwing sugar into his tea,

my friend would say that his neighbors had had his leg amputated because of diabetes.

The poor thing thought the rabbi would have his leg amputated too. 42 NECIP

ERDOĞAN My school was different from the private course offered by the rabbi. My

best friend in my class was Armenian, years ago the Ottoman Emperor sent his family

to Palestine. We all called him Jonny Bravo in class. While we were tossing Jonny in

the air during class break, for some reason we all backed up and Jonny fell to the

ground. Jonny stayed in bed for a long time and had to wear a corset. Poor Johnny! He

was a very unlucky kid, and when he started coming back to school after this incident,

when soda was spilled on his uniform, I poured a lot of cologne on Jonny to remove

the stain. At that moment, the moron standing next to us and never participating in the

conversation suddenly flashed his lighter and said, "I was kidding!" she began to

shout, Jonny was in a ball of fire within seconds. Thanks to Jonny, I learned at the age

of thirteen that cologne is a flammable substance. I always felt guilty towards him.

Would it make me a racist if I threw him in the air and backed off as a class? This

question has been on my mind lately. The only thing we were taught in school was

that we should make the Arabs narrow the world. How much would Jonny, as a

minority member, agree with these views of the teachers? In his last years, my

grandfather started to stay in a nursing home, there was a big armchair on the left at

the entrance, opposite the statue of the wife of the businessman who built the nursing

home. The poor old men would discuss matters of the country while watching the

news in the darkened hall in front of the wide-screen television. 43 KIBUTZ AND

TRAUMA Upper floors had long corridors and rest rooms at the end of each corridor.

During my last visit, when I told my mother that I was curious about the break room

and wanted to go in and have a look, she told me that the room was not suitable for

little ones. The resting room was the room where the elderly, who died in their warm

bed at night, "rested" until the morning death report was written. The nursing home

was almost out of town, with the cemetery next to it, in the morning When the elders,

who woke up in joy, went out to the balcony for fresh air, they were faced with the

view of the cemetery. There were vines in every corner of the large hall, and the

whole place was filled with the smell of cleaning medicine. From where I was sitting,

I could see the cleaning ladies, the nurses, the people who came to visit. My

grandfather and his friend were talking about how disrespectful young people are and

how low their pensions are. Every morning, my grandfather's friend would hand the

officer on duty a white sheet of paper with four phone numbers on it, and the officer

would dial one of the first three numbers from the pay phone. Years later, the officer

and my grandfather's friend, who dialed the fourth number, said after a long phone

conversation, "Hurray! It turns out I have another son!" said. My grandfather is a man

who speaks little, prefers long-sleeved warm clothes even in summer, sits close

enough to cling to the stove in winter, gets up early in the morning and exercises. In

the last years of his life, he was stuck in the library at the nursing home. We did not

know what he was doing there, his friends used to say that he read until the spring and

slept with the first rays of the sun. We understood what he was doing there after he

died, and he read almost every book in the library, wrote small letters on the last

pages, and included his own memories between the lines. I learned about my

grandfather's life between these lines. My father's father remained a question mark for

me. My father, Nec, wouldn't talk about it. Reading these notes, having this

experience has gone from being just a partner to just an abstract morality, "Arbeit

macht frei!" For me, it was a turning point with the power of the condemnation that a

person takes all his life without feeling that the education on the subject is his own.

Months later, when Jonny returned to school, I had the courage to go to him, and we

had the opportunity to have a short chat in the hallway. Jonny never attributed his

accidents to his being non-Jewish. He wasn't afraid of the school bums. Jonny was a

hardworking student who won a scholarship to our school. His parents wouldn't throw

a birthday party at home, because they lived in a cottage. I learned all this only after

months of making frequent trips to their homes. When I went to their house one

afternoon, Jonny was not at home, he had gone to dump two boxes of ash from the

stove in the garbage. An unfired charcoal was placed in the middle of the square

prism-shaped object made of wood without any side faces. When the quilt was laid on

the wooden prism, the heat could stay under the quilt. At the first stage, I just put my

feet under the covers, hesitantly. Breathing the gas produced by the coal had an effect

on me, my eyes suddenly began to close, and I began to think that I would fall asleep

until my friend came. Jonny's younger sister, on the other hand, was careful to always

stay on the quilt while doing her homework because she knew what would happen to

her. In schools like mine, the non-Jewish minority students had their privileges. They

were not allowed to attend Hebrew classes, they were exempted from attending the

Sabbath, visiting the synagogue, singing the Israeli National Anthem. In kibbutz, we

were divided into groups with an older person at the head of each, and part of the day

was spent in activities that could be considered normal in such a meeting. Breakfast,

football, basketball games, group fights were never missing. In those years,

information about the Iran-Iraq War was given in every news hour of black and white

television. The first notes in my grandfather's notebook are about the first day he set

foot in Turkey. While STRUMA was buried in the dark waters of the Black Sea with

a torpedo sent from a submarine, he miraculously crossed the raging waves and

landed. As the mountains rise where the sea ends in northern Turkey, my grandfather

began to climb rapidly towards the dark mountains, the only color he saw was green.

He tried to tell his troubles in the mountain village he first encountered and rested in

the guesthouse of the village. He writes that he started to learn some Turkish words in

his memoirs after days spent in that mountain village with headache, weakness and

high fever. In his notes, my grandfather writes that the rapid rise of the fascists in the

German-46 NECIP ERDOĞAN or the Jews were ineffective in this process. When he

came to work in the morning, he wrote that he thought that the star of David painted

on his door was made by a few fascist clowns, and that he thought it was an

exceptional case and how naive it was. The country where he lived for years, his

loved ones; He could not imagine that he would have to go on a ship with his clothes

on and set off for a country whose habits, history, and language did not know. It is

written in my grandfather's notebook that Jewish immigrants who fled the war in

Turkey in 1945 and became Turkish citizens had no difficulty in entering the country.

For a mathematics teacher who did not speak Turkish, this country was the land of

opportunities. Finding a job would not be difficult, and the capital of the modern

Turkish Republic had become a center of scientific excellence. Seminars and

symposiums on science, art and philosophy are regularly held in faculties. hopes were

made. In Istanbul in 1945, it was quite normal in this country for a young and

beautiful woman to return to a poor Jewish immigrant who owed money to the hotel

where her father and a rich Jew, who was a citizen of the Republic of Turkey, said

that it was appropriate for her to marry her daughter. My mother Nes was born in a

small apartment of one room and one living room on the street where a chicken farm

and large wheat fields are located. *** 47 KIBUTZ AND TRAUMA It all started for

me in class when Jonny fell to the floor. The school principal started a disciplinary

investigation against us. My father, Nec, had met with the counselor. The guidance

teacher explained that the incident was an accident and that the students often joked in

this way. Nec was not satisfied with this explanation. He promised me that this kind

of incident would not happen again. He did not pay special attention to me or Jonny in

the later years, the only subject he paid special attention to was the science center. In

the afternoons I would be alone with the maid, Jon would come in after lunch, thanks

to me, he played video games for the first time in his life. Jonny had fallen behind in

classes and I was trying to help him, I was feeling guilty towards him. After class, the

maid brought cookies and hot milk. Jonny wasn't a member of the rich club I belong

to. My father always had a lot of money, he took care of the education of the poor

children who were brought up in kibutz. He would make the necessary expenses for

different experiments in the science center. I never saw Nec's father. My father's

childhood was also spent on the Kibbutz, and his mysterious father, who only paid for

his expenses, thought the only way to father him was to pay for private school fees.

During the summer, Jonny had jumped into the pool in our garden, even though he

couldn't swim. When I came to save him, he started to pull me to the bottom with all

his strength. The attendants, who wiped the tables around the pool, jumped into the

pool without taking off their clothes and got the two of us out with difficulty. 48

NECİP ERDOĞAN Why would a person who could not swim jump from the second

floor of a jumping tower? When I took a few strokes and came near him, he was

suspended in the water, eyes open awaiting death. Some people love to open their

eyes underwater, when you dive to the bottom and evacuate the air from your lungs,

you have to wait patiently in the middle of the bubbles until the water brings you back

to the surface. Jonny was waiting patiently like this until I came near him, not quite all

the way down, but not getting to the surface either. Extending your arms, you push

your body upwards and feel the transition from death to life. The blue of the sea is so

vivid when you feel the sunlight! My father's life is based on numbers, why did he

attempt suicide? I thought of the book Uncle Petros and the Goldbach Conjecture,

which I read when I was a kid. Goldbach's Conjecture states that each prime number

on the number axis can be written as the sum of two even numbers. Unsuccessful and

devoting his life to prove the conjecture, Uncle Petros finds the cure in suicide.

Another subject that extinguishes people's joy of life is the fifth postulate. Let us now

return to Saccheri's issue of self-contradictory proof. He strove to prove Euclid's fifth

postulate. There are many examples in mathematics where the principle has been

successfully applied. One of the most famous of these dates back to the Pythagoreans

and was solving a mathematical problem in a way that greatly annoyed them. 49

KIBUTZ AND TRAUMA When some problems were solved, it was considered

appropriate to keep the solution a secret. The most important of these problems: Is it

possible to find a rational number whose square is 2? is the question. The Greeks were

beginning to realize that rational numbers were not sufficient for the correct

development of their ideas of geometry. Today we do not needlessly worry that a

given geometric quantity cannot simply be measured with rational numbers alone.

This is because the concept of real numbers is very familiar to us. Although our

pocket calculators only express numbers as a finite number of digits, we recognize

that this is an approximation that is forced upon us by the fact that the calculator is a

finite object. We are prepared to allow the ideal mathematical number to require

strictly decimal expansion to continue indefinitely. This is, of course, true even for the

decimal representation of most fractions. 1/3 = 0.3333 29/12 = 2.416666 9/7 =

1.285714285714285714... 237/148 = 1.601351351351… The decimal expansion for a

fraction is always ultimately periodic, that is, after a certain point the infinite sequence

of digits goes to infinity. means that it consists of some finite sequence repeated. The

repeating sequences in the examples above are 3, 6, 285714, and 135, respectively.

Decimal expansions were not available for the ancient Greeks 50 NECIP ERDOĞAN,

but they had their own way of dealing with irrational numbers. What they actually

adopted was a system of representing numbers in terms of what are now called

continuous fractions. It is not necessary to go into all the details here, but some brief

comments are appropriate. Continuous i is a fraction, finite or infinite expression: a +

(b + (c + (d +…) -1)-1)-1 where a, b, c, d are positive integers. Any rational number

greater than 1 can be written to terminate an expression like this: 52/9 = 5+ (1+ (3+

(2) -1)-1)-1 and less than 1 to represent a positive rational , we let the first integer in

the expression be zero. To express an irrational real number, we let the continued

fraction expression go on indefinitely. Some examples: √2 = 1 + (2+ (2+ (2 +….)-1)-

1)-1 7-√3= 5+ (3+ (1+ (2+ (1+ (2+) (1+ (2…) -1)-1)-1)-1)-1)-1) - 1) π = 3 + (7+ (15+

(1+ (292+) (1+ (1+) (1+ (2 +…)-1)-1)-1)-1)- 1)-1) - 1) As noted above in the familiar

decimal notation, it is rational numbers that ultimately have periodic expressions. 51

KIBUTZ AND TRAUMA On the other hand, we can see that rational numbers now

always have a finite definition, a strength of the Greek fractional representation. A

natural question to ask in this context is: What numbers ultimately have a periodic

continuous fraction representation? This is a remarkable theorem, which proved for

the first time to our knowledge by the great mathematician Lagrange, that numbers

whose representation in terms of continued fractions are ultimately periodic are what

are called quadratic irrationals. What is quadratic irrational and what is its

significance for Greek geometry? A number that can be written in the form a+sqrt(b)

where a and b are fractions and b is not a perfect square. Such numbers are important

in geometry because they are the most important rational numbers encountered in

ruler and compass structures. Special examples of quadratic irrationals are when a = 0

and b is a natural number or rational greater than 1. The continuous fractional

representation of such a number is particularly striking. Including quadratic irrationals

gives us a correct path to sufficient numbers for Euclidean geometry, but it does not

provide all the necessary information. Numbers like sqrt(a + sqrt(b)) are extensively

covered in Euclid's tenth and most difficult book. The Greeks had found a way to

describe numbers that turned out to be sufficient for Euclidean geometry. These

numbers are actually real numbers in modern terminology. 52 NECIP ERDOĞAN

Although a fully satisfactory definition of these numbers was not found until the 19th

century, Eudoxos, one of the students of the plateau, the great ancient astronomer, had

already acquired the basic ideas in the 4th century. It would be appropriate here to say

a few words about the ideas of Eudoxos. First, we must remember that numbers in

Euclidean geometry can be expressed in terms of ratios of lengths, not directly in

terms of lengths. My father always warned me and my sister Din about this. The first

step in Eudoxan theory was to provide a criterion for when an aspect ratio a:b would

be greater than a c:d ratio. Eudoxos actually had a real concept of numbers in terms of

length ratios. It also provided rules for the sum and multiplication of such real

numbers. There was a fundamental difference in perspective, but between the Greek

real number and the modern one, the Greeks saw the irrational number given to us as

the concept of distance in physical space. Real physical objects existing in this space

inevitably lagged behind the Platonic ideal. Maybe it was the Platonic world of ideas

that drove my father to suicide. I must read the notebook we have used since

childhood, relive its memories until the morning. I remember drawing a square on the

first page, something like a square drawn in sand or a cube hewn out of marble, and

Din had drawn a circle. 53 KIBUTZ AND TRAUMA My father Nec always said,

"The measure of distance in simple geometry will be something to be determined

accordingly, it would be convenient to try to deduce the concept of real numbers from

the geometric units of an assumed Euclidean space to be given." he would say. In fact,

this is what Eudoxos accomplished! My father, Nec, used to say that Jews could be

forced to leave the country where they were born and raised at any time, so they

should not be content with knowing a language that is not spoken in any other country

or laws that are not applied anywhere else, and that they should choose professions

that will work for them in any situation. That's why we had to become doctors,

engineers, or traders, because that's what will keep you alive, regardless of what our

neighbors say about you. People always talk about the Jews, claiming that we have

snatched other people's jobs, loaned money at interest, exploited workers, and taken

over the world economy. Science and economics, in particular, must be in our hands if

we are to survive independently. My grandmother's pregnancy was risky. Diabetes

and high blood pressure were causing problems. At that time, the symptoms were

quite effective, and deaths due to pregnancy poisoning were common. Little was

known about the health of pregnant women about cigarettes and viruses, which means

that although the Nazis had advanced knowledge about conducting live experiments

on Jews, they did not advance enough... 54 NECİP ERDOĞAN İ

People's reactions to these deaths in those years were also very different. It was not

common practice to terminate a pregnancy when the woman's life was in danger.

When my grandfather risked death, he did not immediately tell his wife about this, he

could not restrain his desire for a son, he had to devote his life to his son (if he was a

man, of course), he was ready to forget the pain he had experienced, the pressure and

insults of the Nazis. The patient doctors of hospitals in Nazi Germany tell that

cesarean section is risky. They defend the thesis that there will be no postpartum

poisoning and try to calm the fearful expectant mothers. My father often talked about

the hardships suffered by the German Jews, in fact, the Arabs living in our country

were suffering the same troubles. He would describe how a home invasion in

Germany was perceived as an exceptional case, and how the star of David that

appeared on the door of your workplace one morning caused death. , I think our

difference from the Germans was that we were not doing this work in a systematic

way. Stefan Zweig dreamed of a Jewish-only country and unfortunately he didn't live

to see it, he lost his joy of life when he saw the Nazis growing stronger, whereas we

have a small country surrounded by Arabs, the dark days of the Second World War

are over, my grandfather is sitting at one end of the living room in his wealthy house,

at the big dining table, in his home where languages other than German are spoken,

servants wandering around and happy last days, because soon the Nazis will come to

power, glasses are clinked, and while we look to the future with hope, my grandfather

is waiting for his grandchildren. he speaks of one day being an all-Jewish country,

from the promised land that we all dream of, there are many ways to know what really

happened, God hasn't promised land to anyone, maybe we're all like the children of a

poor family living in a hut. However, we are told on the Kibbutz that we are

distinguished people, that the gifted are all Jews, for example, Judaism does not take

much effort to recruit new members, you need to prepare yourself for long readings,

coexistence with the Jewish community, conversations with rabbis. . All of my

grandfather's relatives died in concentration camps, there is no information about the

camp itself, explaining what my grandfather did there and why he was not taken to the

gas chamber. Despite everything, my father hates Arabs rather than Germans. My

grandfather didn't write anything about Judaism, I think he converted to save his life,

he's at the head of a Nazi officer and says he has only one chance to live; donate all

your possessions to the Nazis and become a Christian! How hard it was for him to

change religion. Books to understand this religion, attempts to understand the belief in

the three Gods, nightmares at night, rabbis whispering that he will burn in hell

because of his conversion… My father became interested in this subject after my

grandfather died, and given the circumstances, he must have been as curious as I was.

. The fact that his father never spoke about Judaism and did not go to the synagogue,

leaving him free to choose, seems to be proof of this situation. I don't know when my

grandfather started journaling, it probably started after he escaped from the

concentration camp. My father started working after my grandfather died, he went

into business as an adolescent who had nothing in financial terms and a miracle

happened; His financial situation improved rapidly, from then on he turned to science

and focused on genetic engineering, which he saw as the profession of the future. I

learned later that it wasn't a miracle, that his biological father was RO THSCHILD.

Thus I became the heir to an empire. *** I'm always thinking about fatherlessness

tonight, how does it feel to be without it? We were very lucky to have brought him to

the hospital early, early intervention brought him back to life. My father would often

sit by the lake and meditate, gazing at the distant lights of the boats that drove tourists

across the lake, traveling to distant worlds. When I was a kid, I used to witness what

he was always sitting alone in a dark room thinking about. What was going through

your mind? 57 KIBUTZ AND TRAUMA I'm sure he was thinking about prime

numbers. A prime number is a number that is not divisible by 1 and any number other

than itself. The number of your hands is the first prime number; It is not divisible by

any number other than 1 and itself. Another important point is that there are no even

prime numbers other than 2. Prime numbers are just as important to mathematics as

elements are to chemistry. Because there are at least two prime numbers in the list of

materials required to produce a number, and the number you will get is not a prime

number. For example, 13 and 5 are prime numbers, and if we multiply them, we get

65. 65 is not a prime number because it has a number other than 1 and itself, if you

wish." said. The mathematician also asked the ruler to put 1 grain of rice in the first

chess square, 2 in the second, 4 in the third, 8 in the fourth, and 16 in the fifth, so that

the number of rice in each square would be twice the number of rice in the previous

square. . The monarch, who was very surprised by such a simple request, started to

place the rice grains. He could easily place the rice in the first squares, but as he

progressed through the squares, he began to have difficulty. When he came to the 16th

square, he asked his servants for 1 kilo of rice. In the following frames, the butlers had

to bring the rice with wheelbarrows. As a result, the ruler could not reach the last

square, the 64th square. Then he had to give half of his fortune to the mathematician.

If we decided to do this experiment today, by the time we reach square 64, the total

amount of rice would be approximately the amount of rice produced in the last

thousand years. Let's come to the relationship between our myth and prime numbers.

Ever since Greek mathematicians tried to prove that prime numbers go to infinity,

mathematicians have developed formulas to find very large prime numbers. One of

these formulas was developed by the French Pastor Marin Mersen. Mersenne was like

an e-mail server in the 17th century. He was examining the letters he received from all

over the world and conveying the ideas in these letters to people he thought could

improve them further. The formula developed by Mersenne said that if you move as

many squares on the chessboard as the prime number and add the number of rice in

the squares as you progress, you will get a prime number. If we go as far as the first

prime number, we get 1+2=3 grains of rice and this is a prime number. Similarly, if

we move up to the fifth square, we get 1+2+4+8+16=31 rice grains. This is also a

prime number. Mersenne was devoted to this method, but it did not work. 11 is a

prime number and if we move forward 11 squares, we will count 2047 62 NECIP

ERDOĞAN grains of rice. However, this number is equal to 23 multiplied by 89 and

is not a prime number. It's true that the formula doesn't always work, but it did help

discover some great prime numbers. 63 PART 2 Hello, I am E1, I know for what

purpose I came to this world and how I came to this world, but the people working in

the science center are not aware of this, me and my friend N1 are the product of the

project. We are copies of geniuses from the past. I learned this truth first, you don't

have to believe me, but you may have heard of the concept of soul transformation.

There are memories of a Jewish genius who escaped from the Nazi persecution, which

I do not know from where, these memories never leave me, my original Albert

EINSTEIN whispers in my ear; He says that he is always in favor of peace, that he

wishes for people to give up racism, and that no race is a superior race. I told him the

purpose of the people working at the science center, I told him that the atomic bomb

that was made in the past was replaced by biological weapons, a lot of what I said

affected him of course, but he was most impressed by the cloning of himself and he

told me about his life, of academics who understood that he was very smart. He said

that they did not give a chair out of jealousy, that he sought a job for years, but that he

was a civil servant and that he made history with the articles he wrote in his spare

time. Contrary to what is known, he is not someone who dedicates his life to numbers

and complex formulas, but loves love very much. Einstein tells me his childhood

memories: "When we go hunting, we will catch a small animal we call x because we

don't know its name. We'll catch him and give him his real name." said my uncle. The

math book he gifted me was very fun, reading Calculus is as good as reading a very

fluent detective novel, I recently read the proof of the Pythagorean theorem after my

bath and had a lot of fun. I will ask Uncle Jakob to give new problems. We've been in

hiding for more than a year, I don't have the opportunity to tell you everything, my

dear clone, I was also interested in the millennial problems that could not be solved in

your time, especially the Riemann-Zeta function, if I had the opportunity to return to

the world again, I would try this function, who knows maybe We can also solve

problems after the soul leaves the body, after all, we are solving questions in sleep,

and sleep is half-death. When I was a child, bedtime was nine in the evening. There is

always hustle and bustle in the room. Tables are lifted, beds are made, blankets are

laid, nothing stays in its position in the morning. I sleep on a small sofa so my feet are

bare, I found the method of adding chairs last night. A terrible noise comes from the

next room, the sound of the bed being folded. We need more blankets to sleep

comfortably on the chipboards. When my uncle sleeps, he pulls his bed near the

window to get the night air, now it's time to draw the blackout curtain… When I was

sixteen, I only started reading Calculus and I was very happy, a happy man, a woman

who couldn't think about the future from the moment she was in.

stay satisfied. I'm dreaming while solving the integral; I'm studying physics and

mathematics at the same time after I won the university, then I become the youngest

professor of the Polytechnic Institute. I always preferred to be alone. I never felt like I

belonged in Germany. I persistently stayed away from language, religion and race

relations. A person who lives like this has, of course, lost something of his social life

energy. On the other hand, by making himself independent from the opinions of

others, he did not base his stance on these foundations. Six forty-five alarm sounds

and everyone wakes up, mom turns off the alarm, puts water on and washes her face. I

lift the blackout curtain and a new day begins in the room, my uncle takes the fountain

pen and writes history in his diary, he let me take this pen to school when I was ten

but I had to keep this pen a year later, my homeroom teacher only let me use the

school pen. We signed diaries and compositions with this fountain pen, and most

importantly, I wrote the formula E=m.c2, which went down in history, with this pen.

Something happens every day, but I have no cure to tell you, my dear clone, you are

living in the future that I always dreamed of when I was young, actually I always

predicted the future! Yes, that's right, you can predict the future dear E1, at first this

ability sounds great, predicting how the first human colonies on Mars will live,

knowing that the distribution of prime numbers will be found, and knowing that the

Poincare conjecture will be solved by Perelmann, who is Jewish like me. Would you

like to predict what the civilizations on earth will be like in 500 or a million years, but

to learn the details of the painful events that await you? Would you like to know the

date when your mother or father will die? Don't worry my dear clone E1, I've dealt

with this issue for you, if my invention of quantum physics is correct, it will not be

possible to predict the future no matter how perfect computers are made by humans,

1905 was the year my life changed for me, the principle behind photocells that

convert sunlight into electricity I just published my article explaining it. In this article

I have said that light is now made up of invisible bundles of energy called quanta. I

explained that light is absorbed when it comes into contact with matter. Newton had

also worked on light before me. I'm talking about Isaac NEW TON, the original of

your friend clone N1. My article received a great response in the first days, but after a

while it turned out that I was right. Quantum physics tells us that you cannot know

everything you want to know about subatomic particles all at once. Matter consists of

electrons, neutrons, and protons, and what we can know about them is limited by

nature. You can only calculate the consequences of events. Yes, quantum physics is

my invention, but this invention will not last forever, my dear clone E1, a day will

come, the improbable world hidden under the world will be discovered. "After you

died, people named manifolds with certain properties after you. An Einstein manifold

is a manifold that has a fixed ratio between curvature and metric." said E1. 67

KIBUTZ AND TRAUMA When he got no answer, he began to think that the spirit of

the gifted man had left him. "Why me?" he said to himself. "There are many cloned

people like me, but none of them talk about soul transformation." "You have to stop

them!" said Einstein, the genius physicist back in E1's body, which startled the poor

clone. "You have to stop them, innocent people died because of my mistake in 1945,

if you don't stop the science center, history will repeat itself, I want you to get

together with copycats like you and organize them." E1 seemed to see the famous

scientist with his tongue sticking out and graying hair while hearing these words. The

person he was closest to at the science center was N1, a replica of Isaac Newton.

Every day we would shut up in our room and spend hours studying the Riemann–Zeta

function, trying to figure out how the prime numbers were distributed. The science

center wanted the solution of "MILLIUM PROBLEMS". These problems represent

the deepest mysteries in mathematics today. They think some of these questions will

lead to useful applications in effective drug treatments, strict cybersecurity encryption

standards. Our simulations with N1 showed that there is a mass gap in the solution of

the Yang-Mills equations relative to their quantum version. I would very much like to

contact Einstein right now. What are their thoughts on Yang-Mills equations? 68

NECIP ERDOĞAN "Mathematics and physics always mutually benefit each other

E1, remember that…" "I don't believe it, you're back Einstein, hearing your voice

again is an indescribable happiness for me!" "Advances in mathematics often lead to

new approaches to physical theories. In the science center you mentioned, you have to

take this into consideration first. New discoveries in physics are also investigating the

mathematical relationships that underlie them in more detail.

encourages the. Quantum mechanics is arguably one of the most successful physical

theories in history." "One of the pillars of your quantum mechanics is Yang–Mills

equations, my dear original." said E1. After thinking for a few seconds, "Do you mind

if I call you original?" asked. Einstein began to look around and pretend to be the host

waiting for guests. He continued to think aloud. "Yang-Mills theory provides a

mathematical basis for our understanding of elementary particle physics. Without it,

we can't tell how many particles there are or what masses they should have. But there

is a problem. Experiments and computer simulations, such as the Large Hadron

Collider at CERN near Geneva, Switzerland, suggest that there is a minimum mass

that particles can have. However, the distance between this mass and zero—the socalled mass gap—does not appear to be constrained within the framework of YangMills theory. Solving the problem involves justifying the existence of this gap

mathematically." 69 KIBUTZ AND TRAUMA "I haven't heard of these equations,

my dear clone." "I am speaking, my dear original, of a set of equations formulated by

Chen-ning Yang and Robert Mills in your last year on earth—in 1954—that describe

the profound nature of matter. These equations are the higher dimensions of

Maxwell's equations." "We need to work on it, E1, but I want you to promise me that

the science center must not cause another disaster, I don't want a second atomic

bomb." "Bio weapons have replaced the atomic bomb, my dear original. Now I have

to meet with N1 and work on Maxwell's equations first." said E1. "Do you think

Plato's mathematical world is real?" "This was an extraordinary idea for its time, and

it turned out to be very powerful." "But does the platonic mathematical world really

exist in any sense?" asked E1. "I thought we were going to study Maxwell's

equations." said N1. "Many people, including philosophers, see such a "World" as

pure fiction, only a product of our boundless imagination, and the owner of the

science center, Nec, who is severely manic-depressive, is an imaginative man. The

platonic perspective is indeed extremely valuable. It tells us to be careful to

distinguish precise mathematical entities from the approaches we see around us in

mathematical entities from those we see around us in the world of physical things." 70

NECİP ERDOĞAN N1, "So you are claiming to me that a rectangular prism-shaped

matchbox does not exist in reality but only exists in Plato's world of ideas?" said. "I'm

just saying that we're going to make a mapping between the platonic world and the

physical world. The platonic world provides us with the blueprint that modern science

has carried on ever since. Scientists will come up with models of the Earth, or rather

certain aspects of the Earth, and these models can be tested against previous

observations and the results of the carefully designed experiment. Models are

considered appropriate if they pass such rigorous scrutiny and, in addition, if they are

internally consistent constructs. The important point about these models for our

present discussion is that they are basically purely abstract mathematical models. In

particular, the question of internal consistency of a scientific model is one that

requires it to be mathematical, otherwise you cannot be sure that these questions have

well-defined answers." "I'm curious, you claim to be talking to Albert Einstein, E1,

tell me about his views on the platonic world." Einstein said, "If any kind of existence

is to be assigned to the model itself, that existence lies within the platonic world of

mathematical forms. Of course, the opposite point of view can be taken: that is, the

model itself may simply occupy a place in our minds as having a reality of its own,

rather than accepting Plato's world as absolute and real in any sense. For our

individual minds are notoriously vague, unreliable, and inconsistent in judgment. The

precision, reliability, and consistency that our scientific theories require requires

something beyond any of our individual minds. In mathematics we find a much

greater robustness than can be found in a given mind. Still, an alternative view may be

accepted that the mathematical world does not have an independent existence and

consists only of certain ideas distilled from our various minds, which are found to be

completely reliable and accepted by all. Whether they are actively doing research or

using results obtained by others, those who work at it often feel like they are just

explorers in a world that extends far beyond their own." said. When Einstein finished

speaking, E1's gaze returned to normal, now that the trance-ridden soul had left his

replicated body. N1 didn't know that souls would enter a different body but read it

online According to the report, last week, in a city in southern Turkey, three people

claimed to have been reincarnated for the second time at the same time. According to

the belief of reincarnation, the same soul started to live in different bodies and learned

some lessons in each life and rose to the levels of holiness, the aim was to reach God.

Seven billion people were formed, perhaps, by a limited number of souls traveling to

and from the world in a certain time cycle. It's like each of the infinite number of

numbers is actually written as a product of prime numbers! N1 was actually thinking

neither of Maxwell's equations nor of the Zeta function. In his world, there were only

numbers 72 NECIP ERDOĞAN. He was cut off from life when he started working on

number theory. The relations between arithmetic and geometry fascinated him, the

geometry of the elements was a separate research topic. When they started working on

number theory with E1, the simple relationship they found between arithmetic and

geometry impressed him. *** The greatest discovery of the world of numbers began

with the question: Can we find a fractional number whose square is exactly two? The

number with exactly two squares was expressed as 1.414213562373095048

80168872... and the numbers after the comma were definitely not repeating. In this

case, "Is this number a fractional number?" The answer to the question was negative.

All fractional numbers would continue forever after the comma, like the number

above, for example, the number we call half was 0.5000... or a quarter 0.2500... Then

what was the difference in the number above? In fractional numbers, the same digit

was always repeated after the comma, while there was no REGULAR repeating

number after the double squared comma, yes, it continued forever and continued

irregularly forever. Therefore, this new type of number took its place in the history of

science: irrational numbers. 73 KIBUTZ AND TRAUMA E1 said that there is another

spelling of this irrational number; 1+1/(2+1/(2+...) infinitely looped the serial

software: 1, 2, 2, 2, 2... Whereas for any rational number, for example. number was

also a quadratic irrational number. Similarly, the square numbers 3, 5, 6, 7, 8, 10, 11...

were also quadratic irrational numbers. Meanwhile, the number 14 came to mind, my

lucky number is 14. The square is 14. I thought about the number that is the number

that is the number that goes like 3+1/(2+1/(6+...), so the sequence of the numbers is:

3, 1, 2, 1, 6, 1, 2, 1, 6, 1, It could be written as 2, 1, 6, 1, 2, 1, 6, 1, 2, 1,... The above

palindromic sequence starts with 3, then comes the triple of 1, 2, 1, then comes the

double of the three and 1, 2, The numbers 1 came back and continued like this. So the

sequence is A, B, C, D, 2A, D, C, B, 2A, B, C, D, 2A, D, C, B, 2A, B, C, D was

repeating as 2A, D, C, B, 2A,... E1 said that his lucky number was 17 and he wanted

to do the same for 17. He took the pen in his hand to find the serial expansion of the

number whose square is 17. 7 was a prime number and to me all primes were unlucky

despite being the building blocks of numbers. After finding the serial expansion, he

turned his back on me like a child who did not want to share his homework with his

friends. "Well, if we add up infinitely many rational numbers (fractional numbers),

will the number we find be rational or irrational? What is your opinion?" asked. 74

NECIP ERDOĞAN I had already prepared my best answer to E1 in my mind. It was

like the life invention form of arithmetic geometry. E.g; When we focus on the infinite

sum of 1/1-1/3+1/5-1/7+1/9-1/11+1/13-1/15+... We could immediately notice that the

ratio of a circle's circumference to its diameter (the number pi) was equal to one

quarter. "Here is E1, this is a relation between arithmetic and geometry!" I said and

handed the pen to him. "It's your turn, make your move!" E1 began to describe the

thoughts conveyed to him by Einstein: "The mathematics itself seems to have a

robustness that goes far beyond what any mathematician can perceive. Whether they

are actively researching or using results obtained by others; those who work on this

subject often feel that they are merely explorers in a world that lies far beyond them –

a world with an objectivity that transcends mere sight. What I mean by this existence

is really only the objectivity of mathematical truth. Platonic existence, as I see it,

refers to the existence of an objective external standard that does not depend on our

individual views or our particular culture. Such existence may also refer to things

other than mathematics, such as morality or aesthetics, but here we should only be

concerned with mathematical objectivity, which seems to be a much more obvious

issue. Let me explain this topic by taking a famous example of a mathematical truth

and relate it to the problem of objectivity. In 1637, Pierre de Fermat proclaimed

Fermat's last theorem, which he wrote in the margin of his copy of Arithmetica by the

third-century Greek mathematician Diaphontos.

He made his famous claim. "The equation xn + yn = zn has no solution in integers

when n is greater than two." In a margin, Fermat further noted: "I cannot include this

evidence here, as this margin is too narrow to include the proof." Fermat's claim

remained unconfirmed for more than 350 years, despite the efforts of numerous

outstanding mathematicians. A proof was finally published by Andrew Wiles in 1995,

and this proof is now accepted as a valid argument by the mathematical community."

Before Einstein died in 1955, he whispered to E1 that he was dealing with this

problem but could not find a solution. Unsolvable problems would arise as long as

humanity existed. The relation of the world of numbers to the physical world has

attracted attention since Plato. N1, "Do you think PLATO's mathematical world is

real?" said. "This was an extraordinary idea for its time, and it turned out to be very

powerful." said E1. "But does the platonic mathematical world really exist in any

meaningful sense?" 76 NECIP ERDOĞAN "Many people, including philosophers,

can see such a world as pure fiction—it is only a product of our boundless

imagination. Yet the platonic perspective is indeed extremely valuable. It tells us to be

careful to distinguish precise mathematical entities from the approximations we see

around us in the world of physical things. Moreover, it provides us with the blueprint

that modern science has been going on ever since. Scientists will come up with

models of the Earth, or rather certain aspects of the Earth, and these models can be

tested against previous observations and the results of the carefully designed

experiment. Models are considered appropriate if they undergo such rigorous scrutiny

and, in addition, if they are internally consistent constructs. The important point about

these models for our present discussion is that they are basically purely abstract

mathematical models. In particular, the problem of internal consistency of a scientific

model is one that requires the model to be undetermined. The precision required

requires that the model be mathematical, otherwise you cannot be sure that these

questions have well-defined answers." *** If any kind of existence is to be assigned

to the model itself, that existence lies in the Platonic world of mathematical forms. Of

course, the opposite point of view can be adopted: that is, the model itself simply

exists in our various minds, rather than taking Plato's world as absolute and real in any

sense.