Science,English,and math 4th

【Thank you very much, Mayumichan】

I can start expressing myself in English again here because of Mayumichan who was a kind lady, so I`m grateful to her for it. 有難う、マユミちゃん😊

I`m going to start from the last response of the Science, math and English 3rd. Here we go.

【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

A book on math decided the life of Ramanujan. Its title was basic summary on pure appliance of math written by an English teacher on math. Formulas for taking exam were written in the book.

It`s just that some 6000 theorems and formulae were...

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【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

It`s just that some 6000 theorems and formulae were written in the book with every title. It was dull and boring, but Ramanujan was absorbed in thought whether or not the theorems and formulas were right, so he made sure for himself.

It`s just that simple comments were added to every theorem and formula, but there were none of demonstrations for them, so he had to think of original ways so as to make sure those theorems and formulas were right, but it was often some hints led to new discovery of new theorems.

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【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

Ramanujan learned to write down on a notebook when discovering new theorems and formulas for himself.

After that those notebook were put in order and collected three notebooks, and have been possessed at library in Madras university, but results of theorems and formulae were written in the notebooks but there were none of demonstrations for them.

He learned math with self education, so one-third of description on the notebook were already discovered before, but rest of the two-thirds were new discoveries.

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【A marvelous inspiration form the Indian magician】

《Coming across a book on math》

Its number are 3254 in total, and some of them are the ones without using the latest way no one can demonstrate.

77 years passed after his death, then all the theorems and formulae on the notebooks were demonstrated.

For example, pi. It`s expressed with addition of sum of numbers which continue infinitively. The author said he was going to express it in chapter 5, and this expression belongs to chapter 2.

The formula which Ramanujan thought of approaches the true value of the pi so quickly that we are...

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【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

The formula which Ramanujan thought of approaches a true value on the pie so quickly that we are forced to surprised at it.

When calculating first two numbers, its value is as same with that of the eighth decimal place on the true value of the pi.

To tell the truth, I`m not sure of the way of calculating, for I`ve never read the chapter 5, so I can`t have any comment so much, so I`m going to go on express as the book did.

Gottfried Leibniz is called a father on calculus of differential and integral.

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【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

He seems to be such a great mathematician that his ability on math was equal to Newton, though I have little knowledge on Newton, so I`m not sure how splendid both of them were on mathematical ability.

Leibniz also thought of formula on the pie. It seems to be well known and it was accurate to the third decimal place.

As a result, Ramanujan was better than Leibniz in relation to the way of calculation on the pi at least, but to my sorrow, I have little knowledge on the significance of the pi. I hate math.

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【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

It was not until 1987 after sixty years Ramanujan was dead that the accuracy on the formula of of pi by Ramanujan wasn’t demonstrated. After that the calculation on the pi has extended its number on digit by leaps and bounds.

No one seem to be able to think of the way like him, and when he was asked the resource of his imagination, he answered like the next.

I`m afraid that you can`t believe me, but the goddess who I pray for every day made me think of the way on the formula.

In addition he said an...

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【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

In addition he said an equation which doesn`t express any intention from the god is meaningless. Ramanujan discovered lots of theorems and formulas and they have influenced over lots of theories such as elementary particles, the cosmos, polymer chemistry, a study on cancer.

A famous physicist said studying Ramanujan has been to be significant, for his formulas aren’t only beautiful but have both something essential and something deep. It has turned out to be.

I have little knowledge on math, so my remark...

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【A marvelous inspiration from the Indian magician】

《Coming across a book on math》

As you know, I have little knowledge on math, so my remark, though it`s just my poor translation from the book, is superficial.

《Half dozen of new theorem every morning》

In 1913 Ramanujan learned to extract some theorems from his notebook and to send first rank of mathematicians letters in England which was a suzerain for India, and one of the mathematicians was Godfrey Hardy who was a central person in English mathematical world then.

Hardy read the letter written unknown formula carefully about for ...

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【A marvelous inspiration from the Indian magician】

《Half dozen of theorems every morning》

Hardy read the unknown theorems in letters carefully about for three hours with his colleague and came to an conclusion that the one who they dealt with was a genius without mistake.

Next year, Ramanujan was invited to Cambridge University, and learned to study with Hardy. Later Hardy said Ramanujan arrived with about a half dozen of theorems every morning, and Hardy guided him to add a demonstration to each of new theorems over and over again.

But Ramanujan had never received an orthodox education...

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【A marvelous inspiration from the Indian magician】

《A half dozen of theorems every morning》

But Ramanujan had never received an orthodox education on math, so he hardly understood anything on the demonstration itself. Almost all of the theorems which Ramanuj submitted to was shown from the Goddess, and the ones which they looked at with his own eyes.

He was at a loss when asking to demonstrate the theorems.

For example, let’s suppose that there was a one who had never watched UFO, and asking to express what the UFO was. Then expressing the UFO seems to be hard. His situation resembles it.

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【Marvelous inspiration from the Indian magician】

《Half dozens of theorems every morning》

Hardy gave up requesting demonstrations to Ramanujan soon, and when he was received the theorems, he regarded them as oracles from the god and adding demonstrations to the theorem was his job, he learned to be businesslike then.

The one which is worthy of mention among their joint research is a formula of approximate value on division number. What is the division number?

It’s...oh, I`m so sleepy that I can’t continue today any more. I have to go to bed, good night.

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【Marvelous inspiration from the Indian magician】

《Half dozen of new theorems every morning》

The division number is a total number of the way on addition that a natural number is divided. It includes the number itself.

For example, 4 is expressed with 4, 3＋1、2＋2、2＋1＋1、1＋1＋1＋1, it is done with five ways, so the division number on 4 is five.

The bigger the original number is, the harder the calculation on division becomes, but the equation on an approximation value between Ramanujan and Hardy is proud of high precision.

Is it so marvelous? To tell the truth, I`m not sure of its significance.

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【Marvelous magician from the Indian magician】

《A formula which has cleared a modern math》

But the communal operation between Ramanujan and Hardy didn’t continue for a long time.

Ramanujan wasn’t only a vegetarian but didn’t eat anything except for any dish cooked by Brahman. In addition, he was absorbed in cooperative operation with Hardy so much that he went on studying 30 hours without taking a rest and went on sleeping 20 hours after that.

He ruined his health due to the irregular life after three years when he went to England. He returned to India in 1919, but he was dead in the...

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【Marvelous inspiration from the Indian magician】

《A formula which has cleared the modern math》

…he was dead in the next year. Then he was just the age of 32.

After returning to India, Ramanujan wrote down something significant on math on a notebook and its fragment happened to be discovered.

Mock theta function 擬テータ関数 and its formulas of which number were six thousand over were written down on the notebook. The discovery was so splendid that it was compared with the discovery on the tenth symphony by Beethoven.

To tell the truth, I have none of knowledge on Beethoven, besides...

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【Marvelous inspiration from the Indian magician】

《A formula which has cleared the modern math》

To tell the truth, I`ve never have listened to any symphony on Beethoven, so I`m sure the simile on the author wasn’t suitable.

When the fragment on the notebook was discovered, lots of people thought it had common points with theta function developed by a German mathematician, so it was named like that, but what does it mean on the lots of formulas of the Mock theta function? Large part of them remain mysterious at present.

Theta function has played an important role on the superstring ...

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【Marvelous inspiration from the Indian magician】

《A formula which has cleared the modern math》

The theta function has played an important role on superstring theory of modern physics.

As to the Mock theta function, we`ve hoped it`s related to the swelling energy on the universe and the grand unified theory in which trying to express all the power with unification, and lots of mathematicians and physicists have studied it enthusiastically at present.

As to the superstring theory, its easy way of thinking was on the internet, so I`m going to express it.

The theory says ultimate element...

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【The theory on superstring】

The theory has said the ultimate element on material isn`t any particle but a string!

As to the scale on the superstring, it is said 10 ー35［m］, on the other hand, as to the scale on an atom is 10 — 10［m］, so the superstring is extremely minimum.

There are several hundreds of kind of elementary particles, and it is said we can express them with a single string.

When the string swings, waves of which number of swings are different happen, and each of waves is equal to each of the element particles, and the vacuum is filled with the strings.

Does it mean that...

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【The theory on superstring】

Does it mean that each of elementary particles isn`t independent each other but each of them is connected as they were a single string? As a result, they are influenced each other?

As to its size, I understand it`s so minimum, but I`m not sure its unit, though even if I`ve heard, I`m afraid it won`t ring a bell.
　
I`m wondering how it is demonstrated, but lots of the people tried to do it at present.

I`m going to express the grand unified theory in relation to easy way of thinking.

【The grand unified theory】

There...

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【The grand unified theory】

I`m going to express, being based on an expression in the internet, the elementary particles are expressed as a single string, everything is unified with a single one.

There are four kinds of power in the natural world such as gravitation, electromagnetism, a strong power which works inside an atomic nucleus, and a weak power among the elementary particles.

Except for the gravitation, a theory which unifies other three powers is unified theory which is the theory for everything, and it`s called an ultimate theory which physicists have sought for.

The theory...

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【The grand unified theory】

As to the theory which unifies the power, the law of universal gravitation by Newton is well-known. Other ones are the theory on magnetism by Maxwell and electroweak unified theory by Weinberg-Salam is noted.

The theory for everything which physics has sought for is unifying the four kinds of power which consist of the world, and an equation which expresses an origin on the world.

It is no exaggeration to say that the theory on physics has sought the theory for everything from various fields, and a way for thought on the theory is the superstring theory.

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【What I`ve thought】

Though I`ve wanted to continue to express, but the battery on my iPad is about to run out.

When it is charged, we can express, but it is harmful for my iPad, so I hate it.

It will take more than a half day to charge completely, so please wait until then. I`m sorry for it.

By the way, I hate math, but learning something is fun, so studying math is necessary. Without studying math, I can`t understand physics, I`m afraid.

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【The grand unified theory】

The theory said the world is made up with elementary particles and the elementary particles aren’t points but a thing like string.

As to the elementary particles, quarks like mesons, protons have been discovered. More than thirty years have passed since the theory was proposed, but lots of unclear parts remained.

If the theory is completed, we can understand not only the elementary particles themselves but the situation in which the universe is born and vanish. If trying to express it, unless the time and space is tenth dimension, there is a contradiction.

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【The grand unified theory】

Tenth dimension? I can`t imagine it with my brain, so I`m going to return to my topic.

【Darkness on mathematicians who have thought of infinity】

The author said he used to heard a conversation between children in elementary school like the next.

The one asked the other. Do you know what the biggest number is?

The other answered. I know a trillion, but I don`t know bigger number than it.

The one said. Don`t you know it? It`s infinity.

The author said it`s a typical misunderstanding that we are apt to think of the infinity as a very big number, for we can`t ...

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【Darkness on mathematicians who thought of infinity】

...for we can`t think of the infinity as the finite number like the trillion.

Other mathematician said like the next with regard to the Infinity.

If trying to substitute the infinity for a very big number, it resembles that there is the sky far away in the horizon, so substituting the sky for the sea which is far away. No one can admit it.

It was in the time of Ancient Greece when we the humankind learned to think on the infinity seriously, but philosophers who were typical mathematicians like Pythagoras, Plato, and Aristotle hated it.

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【Darkness on mathematicians who thought of the infinity】

They thought everything in the world was finite, if having brought something infinite into a debate, it caused a confusion, so they hated the infinity.

To tell the truth, if trying to think of the infinity with a way of thinking something finite, there are lots of something mysterious and unreasonable, the author said like that.

For example, let`s suppose there were two group of natural numbers. The one is positive integral numbers alone, we call it A and the other is the positive integral numbers are squared alone, we call it B.

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【Darkness on mathematicians who thought of the infinity】

Which have more elements, A or B?

Judging from the way of thinking in the finite world, needless to say it`s the A, for the natural numbers continue like 1,2,3 in the A…, on the other hand, the B is 1,4,9 …, the B has lots of space among the numbers, so the B is a part of the A. It`s an ordinary way of thinking.

But to my surprise, both groups of element on numbers are the same, for they are link to each other one by one like 1 and 1, 2 and 4, 3 and 9… we can classify them as a pair one by one. We call it one to one correspondence.

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【Darkness on mathematicians who thought of the infinity】

Galileo who was an Italian pointed out that it`s strange, for it`s just a part, but both of numbers are the same. It`s difference between finite and infinite. It is said it`s a study in which we the humankind have placed essence on infinite for the first time.

Gauss who was a German said if handling the infinity as real numbers, something unreasonable is caused as Galileo said. When using the word of infinite, we should use it, as adverb, changing it into something big infinitively.

Even Galileo and Gauss were not sure what do do...

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【Darkness on mathematicians who thought of the infinity】

Even Galileo and Gauss were unsure what to do in regard to the infinity, but it was Georg Cantor who settled down his research on the infinity outright for the first time and created an idea of a set 集合.

The set is gatherings, but in math it means something clear which we can judge whether or not it belongs to the group with some definition like the gathering on positive integral numbers, or the ones on hands when doing janken.

For example, gathering on something beautiful or the one on something delicious, we can`t judge an ...

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【Darkness on mathematicians who have thought of the infinity】

...we can`t judge whether or not it includes a group, so we don`t call it the set in math.

The set like the positive integral numbers of squared can link to other set on natural numbers one by one, so we call it a countable set.

The element on number of the countable set is as same as the set on natural numbers, so Cantor adopted a way of saying that they are the same cardinality. The author said the cardinality 濃度 in Japanese, but it`s different from the one like salty water. It`s different from concentration.

For example...

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【Darkness on mathematicians who thought of the infinity】

For example, there are two kinds of set like 1.2.3 and a.b.c. Each element on number is three, so two sets on the cardinality is the same.

In short, the cardinality is its number. Why wasn’t adopted a Japanese, 個数 when translating？ if the set is large infinitively, when counting and saying 何個, it struck us as incongruous, the author said.

A set is same the in the natural numbers on cardinality means that each element on the set is linked to that of the set on natural numbers each other. All of them can make a pair.

There is an...

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【Darkness on mathematician who thought of the infinity】

When there is an infinitive set and lining up its elements on a line number at a place of integral number like 1.2.3...one by one, there are the ones which we can`t put there exactly, so we are forced to put them on other space except for the natural numbers like 0.5, or route 2.

Then the situation on the number line of the infinite set is different from the one of natural numbers. It seems that the one on the infinitive set is more crowded, so the cardinality was translated into 濃度 in Japanese, the author said.

A set which we can...

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【Darkness in which mathematicians have thought the infinity】

《We`ve seen but we don`t believe it》

The set like the squared natural numbers which we can count on the cardinality is named aleph 0. Its element on the number of squared and the one on natural number is the same.

Aleph is the first alphabet in Hebrew.

There is the whole integral numbers including negative integral numbers and rational numbers which we can express with fractional numbers of which denominator and numerator can be expressed with integral numbers. Canton showed a degree on crowded of the two set are also aleph 1.

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【Darkness on mathematicians who thought of the infinity】

《We`ve seen but don’t believe it》

Oh! I made a mistake. Cantor showed the two of set were aleph 0.

By the way, I adopted the word of cardinality as the degree of crowded on the set, but it’s hard to understand, so I make it rule to say the degree of crowded from now on.

In the next, Cantor showed the set on irrational number is more crowded than aleph 0, and indicated it’s aleph 1.

In fact, both the number of points included on a number line, the ones on a plane and the ones in the space are the same, the author said like that.

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【Darkness on mathematicians who have thought of infinity】

《I’ve seen but I don’t believe》

Cantor also reached the same conclusion, and he was very surprised at it and wrote to his friend and he said he’d seen it but he didn’t believe it. His phrase of this is very well known.

There are simple drawing on a number line and coordinates on the book. Three points like 0.12, 0.3456, and 0.789012 both on the number line and coordinates.

0,12 on the number line is able to show on the coordinates. We can show it at the place 0.1 and 0.2 on the coordinates. If line up the number after the...

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【Darkness on mathematicians who thought of the infinity】

《I’ve seen it but don’t believe it》

It means that 0•1 links to the x axis, the horizontal one, and 0•2 links to the y axis, the vertical axis.

When lining up odd number after the decimal point and linking to the x axis and even numbers to the y axis. Then the number of 0.3456 on the number line links 0.35 and 0.46 on the coordinates.

Those links between the one on the number line and other on the plain is made up. Though both of the points on the number line and the plain are infinitive, but both number of the elements are the same.

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【Darkness to mathematicians who thought of the infinity】

《I’ve seen it, but don’t believe it》

I’m not good at expressing something, so I’m afraid whether or not I can make myself understand, but it can’t be helped. I can’t do anything more than that. Please pardon me for the poor expression.

The author continued like the next.

He’d had a recognition that points gather and they become a line, and the line gathers and they become the plain before, so he couldn’t believe that the number of points on a straight line and those on a plain were the same easily, but it was clear that he...

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【Darkness to mathematicians who have thought of the infinity】

《I’ve seen it but don’t believe it》

...but it was clear that he couldn’t refute. Then he was aware that common knowledge in the finite world isn’t accepted in the infinite world.

Besides, the degree on crowded isn’t limited to aleph 0 and aleph 1 alone. It’s limitless. Infinity isn’t a very big number but the general term on numberless numbers which have boundless expansion we can’t imagine at all in the finite world.

The mathematician said he saw it but didn’t believe it. I’ve also sympathized with him. Math is mysterious.

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【Darkness to mathematicians who have thought of the infinity】

《Confrontation between the great mathematician and his disciple》

As the concept on the set was born the infinity became an object for an argument scientifically for the first time then.

Cantor was a pioneer who opened the gate for a mysterious land like the Garden of Eden where theologians alone were permitted to go into. He opened the gate of the mystery land on the infinity for other mathematicians.

But his great achievement by the marvelous genius was so advanced that it wasn’t evaluated respectably when he was alive.

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【Darkness to mathematicians who have thought of the infinity】

《Confrontation between a great mathematician and his disciple》

Far from it! The idea which Cantor thought of was often criticized and attacked from other mathematicians, especially the one who criticized him severely was Leopoldo Kronecker who trained Cantor.

Kronecker was a great mathematician in Germany and appeared in the book several times, the author said like that, but to my sorrow, I don’t remember him.

He used to treat Cantor with affection before and when finding his job at an university, he lent a hand then, but ...

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【Darkness to mathematicians who thought of the infinity】

《Confrontation between a great mathematician and his disciple》

...but when Cantor started to study irrational number and the infinity, Kronecker criticized Cantor who used to be his favorite pupil. He called his disciple the one who was harmful to others. He began to oppose his pupil.

He thought numbers which weren’t able to express with integral number and the ones which weren’t finite weren’t worthy of thinking. He asserted that the one which wasn’t expressed with fraction of integral number and irregular number continued after...

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【Darkness to mathematicians who thought of the infinity】

《Confrontation between a great mathematician and his disciple》

Kronecker asserted that the one which wasn’t able to express with fraction of integral number and a number continued after the decimal point irregularly was nonsense.

Students in junior high school learn the irrational number at present, but the great mathematician who led the world on math didn’t recognize it 150 years before. We are forced to be surprised, but handling something endless and the one which we can’t see its end mathematically is hard and courageous.

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【Darkness to mathematicians who thought of the infinity】

《After being sick spiritually and in his later years...》

Cantor was grieved over being lacking of understanding and obstinate personal remarks from his Kronecker who used to be his teacher.

There was one more thing from which Cantor suffered in the latter half of his life as if he had been pursed and attacked. It was a demonstration on hypothesis of continuum 連続体.

It means that there aren’t any other degree on crowded number between aleph 0 and aleph 1.

The continuum is a set on whole real numbers including rational and ...

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【Darkness to mathematicians who thought of the infinity】

《As being sick spiritually in his late years...》

Continuum is a set of the whole real number including rational number and irrational number, and a number line is filled with the real number.

The hypothesis on the continuum is a hypothesis that an infinite set which has an element that it is more than natural number and is less than real number doesn’t exist.

No one can prove nor show any evidence against the hypothesis. It has been demonstrated clearly at present, but Cantor believed he could and challenged over and over again...

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【Darkness to mathematicians who thought of the infinity】

《As being sick spiritually in his late years...》

...but needless to say he failed to do it, it’s natural, for no one can demonstrate it, but Cantor lost his confidence in his own ability as mathematician because of the setback.

Repeated criticism from Kronecker and the failure on demonstration of the hypothesis on the continuum. Two of the agony spread dark shadow over his heart and he fell sick spiritually at last.

In addition he suddenly started to be absorbed in a study on English history and English literature in his late years.

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【Darkness to mathematicians who thought of the infinity】

《As being sick spiritually in his late years...》

His theme was demonstrating a theory that the true author on dramas written by Shakespeare was Francis Bacon.

A German mathematician said no one can drive out of us from the paradise which Cantor opened for us, but the infinite world where Cantor reached with wings made of intellect and imagination may not have been any paradise but may have been a hell where devils live for him.

【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is it true?》

Do you...

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【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is is true?》

Do you know a paradox on self-reliance?

There is a premise or theory which look right, but a conclusion which we can’t accept easily is caused from the premise and theory, we call the conclusion the paradox.

As an example on the paradox of self-reliance, an well known is a remark that I’m a liar. It seems an ordinary one, but thinking of it carefully, the remark is a contradiction.

If saying I’m a liar, it means that the remark that I’m a liar itself is a lie, so in short I’m honest, but ...

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【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is it true?》

...but the premise started from the remark that I’m a liar, but there is the conclusion that I’m honest. It’s a contradiction. Then let’s suppose that the remark was a lie. Then what’s happened?

I’m honest, so the remark that I’m a liar is also true, so I’m a liar. As a result, it’s also a contradiction.

In short, as to the remark that I’m a liar, neither we can say it’s true nor a lie. In general, if a sentence has a construction that it’s false, we can’t judge whether it’s true or not.

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【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is it true?》

We call it the paradox on self-reference.

A rule saying there are no rules without exceptions, or a bill on the wall saying to prohibit to put any bill on the war are well known as the paradox on self-reliance.

By the way how about math?

There is a proposition which is the matter we can judge whether it’s true or not objectively in math. Is there any proposition which we can’t judge whether or not it’s true?

There are two of things alone in math. The one which is demonstrated it’s true, and...

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【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is it true?》

...or the one which is demonstrated it’s false. Lots of people recognize it in common.

Needless to say, there are some propositions which we can’t judge whether or not it’s true, but it’s because of us the humankind’s inability, so they will be classified either of truth or falsehood. Plenty of people seems to have thought like that, don’t they?

But there is a person who made it clear that there is a proposition in which neither we can demonstrate whether or not it’s true in math. He is Kurt Godel.

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