Science,English,and math 4th

【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is it true?》

Kurt Godel was a Czech.

Mathematic world used to be in a chaotic state at a time from the latter in the 19th century to the beginning in the 20th century.

Geometry which was born an interest from a figure and developed into technical skill for civil engineering and navigating ships.

Algebra which started to seek for something unknown and developed into theories on equations.

Differential and integral calculus which was necessary so as to ask for a value on an area of a figure and to solve...

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【A perfectionist who demonstrated an imperfect theorem】

《If saying I’m a liar, is it true?》

Differential and integral calculus which is necessary in order to ask for the value on an area and to solve physical phenomena.

Statistics which started to rule over a nation.

The thorny on probability which was born in the pursuit of benefits when gambling.

They were born one after another with no connection, and grew up respectively as if each field had lived together in a building named math.

Then a movement of trying to rebuild the mathematic world with the concept of set established by Cantor

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【A perfectionist who demonstrated an imperfect theorem】

《The paradox on a barber》

The theory on set is the origin from modern math, the feeling was heightened up then, and Bertrand Russel who was much talked as the greatest logician since Aristotle was aware that what is called the paradox by Russel on the set. Its example on the next paradox of a barber is well known.

There is a single barber shop in a town, and the barber shop is managed by a single barber. The barber was imposed a rule on himself.
If there are the ones who don’t shave for themselves in the town, he will shave for them.

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【A perfectionist who demonstrated an imperfect theorem】

《A paradox on a barber》

If there are the ones who shave for themselves in the town, the barber don’t shave for them.

By the way, who shaved for the barber?

As he said he won’t shave for the ones who shave for themselves in the town, if he shaves for himself, his remark is inconsistent.

On the other hand, as he said he will shave for the ones who don’t shave for themselves in the town, his action is contradictory to his remark.

The barber imposed the rule on himself, but the rule prevents himself from shaving, but he can’t choose...

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【A perfectionist who demonstrated an imperfect theorem】

《The paradox on a barber》

...but he can’t choose not to shave for himself.

So Russel wrote Principia Mathematica made up with three volumes with Alfred North Whitehead who was his teacher so as to avoid the paradox.

The book is based on the theory on set and tried to unify the whole math which we the humankind acquired until then and to demonstrate it with signs alone. It was written with the magnificent concept.

Almost all the part of the first volume was used so as to define 1 alone. Needless to say, after that its pace increased.

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【A perfectionist who demonstrated an imperfect theorem】

《The paradox on a barber》

Then high level mathematical concept appeared one after another, but its last part came to an end with a description that from this place we can seek a value with the same way. It was unfinished and the description seemed to be irresponsible and apathetic, the author said like that.

《if number of stamps are below five, its gift...》

Though having described that demonstrating with signs alone simply, it was one of great feats by Russell and his teacher. Its base is a truth table 真偽表.

For example, let’s...

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【A perfectionist who demonstrated an imperfect theorem】

《If the number of stamps are below five, its gift...》

Let’s suppose there was a card on which has several collected stamps, and there is a promise that if the stamps are more than five, its owner gets a present.

The condition of more than collected five stamps is P and other condition of being able to get the present is Q.

I’m going to think over whether or not P and Q are true, and if P stands up, then Q does, then what is its relation between true and false?

1 If the collected stamps are more than five, the owner can get a present.

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【A perfectionist who demonstrated an imperfect theorem】

《If the collected stamps are below five, its present...》

1 When the Q is true as the proposition which means the promise shows, the collected stamps are more than five, so P is true too, as a result P＝ Q stands up, so P ＝Q is true.

2 Though the collected stamps are more than five, the owner doesn’t get the present. Then P is true, but Q is against the proposition, so P ＝Q doesn’t stand up, so P ＝ Q isn’t true.

3 As the collected stamps are below five, the owner doesn’t get the present, but it is showed in the promise, so P ＝ Q is true

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【A perfectionist who demonstrated an imperfect theorem】

《If the collected stamps are below five, the present...》

3 So, P ＝ Q stands up.

These proposition struck no one as incongruous until now, but as to the last 4, everyone is hard to accept it at first.

4 Though the collected stamp are below five, the owner gets the present. Then the owner must have been indignant at it, and it would said it shouldn’t have gathered the stamp so hard, but the first proposition said nothing in relation to the time when the stamps were below five. It doesn’t say that the owner can’t get the present.

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【A perfectionist who demonstrated an imperfect theorem】

《If the collected stamps are below five, the present...》

4 As a result, P ＝ Q stands up, the author said like that, and continued.

It seems to be quibbling, but sometimes presents are left so much that the sponsor delivered presents for the guests of which number on stamps were below five, but the first promise doesn’t prohibit it.

If looking at the both of 3 and 4, the promise said nothing when the stamps were below five, so even if the owner of the stamps whether or not got the present, the proposition, it means the promise, is true

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【A perfectionist who demonstrated an imperfect theorem】

《Semantics and syntax》

Even if it’s true mathematically, if there are lots of quibbles like that, I’m afraid that a riot may have been caused, though the presents were delivered, so it seems to be all right.

A way of thinking over the meaning of each condition and judging whether or not the proposition is true. It’s called a semantics 意味論的方法.

But if adopting the semantics, a meaning of words which we usually everyday life apt to come into a conclusion when judging something. Sometimes it’s unclear and shows us equivocal situation...

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【A perfectionist who demonstrated an imperfect theorem】

《Semantics and syntax》

It’s sometimes unclear and shows us an equivocal situation we can’t judge whether or not it’s true, but as math aims at being perfect, so it isn’t very good.

Creating signs which we don’t use everyday life newly and showing a proposition with the sighs alone, the idea was born. The truth table is its first step. It seems to be a kind of a chart. What kind of chart? I’m sure I have to express it, but I feel sleepy, so I’m going to give it up tonight. Good night, everyone.

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【A perfectionist who demonstrated an imperfect theorem】

《Semantics and syntax》

If using the truth table, we can judge various propositions mechanically. Soon, relying on symbols alone, a way of making ahead with a demonstration was contrived as if we had calculated. We call it a syntax.

In the syntax, without thinking of its meaning at all, we demonstrate formally according to the fixed rule. Once having fixed its rule, as the demonstration proceeds automatically we should be careful of an axiom, it means a premise when starting.

If the axiom was in the wrong, an conclusion which we get...

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【A perfectionist who demonstrated an imperfect theorem】

《Semantics and syntax》

If the axiom includes something wrong, a conclusion from which we get will be in the wrong automatically.

The syntax on demonstration for math has been arranged by an English, George Boole, and a German Frege Gottlob and has been completed through the Principia Mathematica by Russell and Whitehead.

《The truth table》

I expressed that if the collected stamps are more than five, its owner can get a present before, so I’m going to show it with the truth table.

Then the truth table has twenty square. A vertical ...

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【A perfectionist who demonstrated an imperfect theorem】

《The truth table》

I’m afraid it’s just a recitation of my expression before. There should have been a better way, but it doesn’t occur to me. I’m sorry for it.

The vertical row has five. A place of the upper part on the rightmost is a blank column, and from the second place, 1,2,3, and 4.

1 is the condition that the collected stamp is more than five, the owner can get a present.

2 is the one that the collected stamp is more than five, but the owner can’t get the present.

3 is the one that the collected stamp is below five, so the...

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【A perfectionist who demonstrated an imperfect theorem】

《The truth table》

3 is the one that there is below five on the collected stamp, and the owner couldn’t get the present.

4 is the one that there is below five on the collected stamp, but the owner could get the present.

Each of those four conditions connects with two elements and we can reach a conclusion. Whether it’s the truth or not.

One of the elements is P that the collect stamp is more than five, and the second is Q that being able to get the present. The third element its conclusion.

For example, both of the P and Q are the...

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【A perfectionist who demonstrated an imperfect theorem】

《The truth table》

For example, both P and Q are the truth, so the conclusion is the truth. It means that the owner has more than five collected stamps, so it can get the present. Then three of the columns which 1 links to P, Q, and conclusion are all truth.

When making ahead with other connection, we can get each conclusion which I expressed before.

Thus we can understand its conclusion mechanically. It’s the truth table. As you know, I’m not good at expressing something, so I’m afraid whether or not I could make myself understood.

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【A perfectionist who demonstrated an imperfect theorem】

《The imperfect theorem by Godel》

The demonstration on the imperfect theorem by Godel seems to be very hard. I can express what the author said but it doesn’t always mean that I can understand what it means.

The author said he was going to omit its process and to introduce us the conclusion which Godel demonstrated in the imperfect theorem. Needless to say, I agree with the author.

Considering the formal syntax in which we treat natural numbers, a proposition exists in which no one can demonstrate, though when handling other things...

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【A perfectionist who demonstrated an imperfect theorem】

《The imperfect theorem by Godel》

...though when handling other things, the proposition may have been existed.

The author said a reader who is quick to pick up on things may be aware of it, it’s the same structure as the paradox on self reference which the author showed us, I’m a liar, so neither we can affirm nor deny it, though to my sorrow, I can’t be conscious of it.

As to the imperfect theorem by Godel, its title has been out of control on account of shocking its title, the title alone has been active, separating from its ....

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【A perfectionist who demonstrated the imperfect theorem】

《The imperfect theorem by Godel》

...separating from its original meaning, some people said that it was demonstrated that the math was in the wrong, and others said we could see the limit on us the humankind. It has been done frequently with a plausible way, but it’s a misunderstanding altogether.

The imperfection theorem means that there is a proposition which neither we can affirm nor deny, but it doesn’t mean that math has failed nor a thing which we thought to be right was in the wrong.

《The later years on the perfectionist》

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【The perfectionist who demonstrated the imperfect theorem】

《Later years of the perfectionist》

Godel showed an imperfection on a formal demonstration with the syntax, and after that, it prompted a remarkable progress on not only math, logic but computational science, for a computer understands an order and repeating judgement is formal, especially it is said that it has influenced on Alan Turing who changed the part of on the base of computer into a theory and was called a father of artificial father.

To my sorrow, it’s too hard to understand for me.

Godel defected to America with his...

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【The perfectionist who demonstrated the imperfect theorem】

《Later years of the perfectionist》

Godel defected to America with his wife so as to avoid a persecution on the Jews from Nazi in his mid thirties. Their guarantor was Einstein.

There seemed to be an oral examination on American constitution in order to get a citizenship in America.

On the day of the exam, Godel said like the next others around him.

When studying the constitution, it turns out that America has a possibility in itself that it may change into a despotic nation lawfully.

Einstein was forced to panic.

Godel was...

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【The perfectionist who demonstrated the imperfect theorem】

《Later years of the perfectionist》

Godel was a perfectionist and a nervous temperament, especially in his later years, its tendency became strong.

Except meals which his wife cooked, he tried not to eat anything, and he was afraid of being assassinated by poisonous gas that he left all the windows open even if in winter.

At last when his wife was in hospital, he starved to death because he didn’t eat anything. Then his weight was no more than 29.5 kg.

【Beauty in math exists in internal pleasure】

《If math isn’t beautiful...》

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【Beauty in math exists in its internal pleasure】

《If math isn’t beautiful...》

Tchaikovsky said if math isn’t beautiful, math itself wasn’t born perhaps. Except for something beautiful, is there other thing which the greatest geniuses have been in math attracted?

Do you think whether or not math is beautiful? Do you think you are sure of it? Or you don’t think so? The author said he’s thought math is beautiful, but I don’t think it at all.

When looking up the word on the beauty in 広辞苑, it says it stimulates perception, sensation, and feeling and internal pleasure is caused. What is ...?

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【Beauty in math exists in internal pleasure】

《If math isn’t beautiful...》

What is caused internal pleasure from math?

The author said his theory is that it depends greatly on the four next natures of math.

1 symmetry

2 rationality

3 elements of surprise

4 being concise

《Symmetry》

If Tokyo tower and Mt.Fuji are left-right asymmetric, lots of the people have never been fascinated like that.

In math, as to a figure, when folding from a point of line and lay two of them, two of the figures fit perfectly, we call it a line symmetry.

When being in a point in its center, and it turns...

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【Something beautiful in math exists in internal pleasure】

《Symmetry》

When being a point in the center and rotating 180 degree, if the figure fits perfectly with the same position before it rotated, we call the figure a point symmetry.

In a numerical formula, if exchanging an order of the number, it reaches the same value, we call it a symmetric expression.

When being able to recognize something symmetric in the figure or in the numeral expression, I find it beautiful and it natural. The author said like that.

《Rationality》

Have you ever heard of saying that when a swallow flies low...

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【Something beautiful in math exists in internal pleasure】

《Rationality》

Have you ever heard a saying that when a swallow flies low, it will rain?

Foreseeing a weather from a natural phenomenon close to us beforehand like that is called a weather lore.

It seems it’s caused from an experience, but it has a ground.

When a low pressure zone which make it rain approaches, atmosphere which contains lots of moisture flows into near the ground, so insects which are prey to the swallow can’t fly high because their wings are heavy with humid atmosphere.

As a result, the swallows which try to...

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【Something beautiful in math exists in internal pleasure】

《Rationality》

As a result, the swallow which tries to eat the insects also learns to fly in a low altitude.

When listening to the rational explanation, I find it convincing, at the same time I frequently find it pleasant. The author said and continued.

I understand that it doesn’t always mean that everyone has the same sensation, but a way of logical thought which was formulated systematically through the Elements of Euclid in Ancient Greece has been accepted and developed ceaselessly until the modern era, for it means that not...

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【Something beautiful in math exists in internal pleasure】

《Rationality》

...for it means that not a few people have found that the rationality is something pleasant.

Except for the conviction, there is other reason why I’m fond of the rationality. It’s that even if its course is different, it reaches the same conclusion.

For example, there are two right angle triangles. They are the same figure altogether. The right angle is between two sides. The one’s length is 4 and the other is 3, and the oblique side was 5. There is a vertical line from the oblique side and it reaches the triangle...

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【Something beautiful in math exists in internal pleasure】

《Rationality》

A question is asking the value on the length of the oblique side.

I’ve said there are two of the same right angle triangles. The one is the side of which length is 4 is on the ground if there is the ground there.

On the other hand, as to the other right angle, its oblique side of which length is 5 is on the ground if there is the ground there.

Needless to say, two of the value on the area of the right triangles are the same, but two ways of asking the value are expressed.

The one is 4×3÷2, and the other is 5 × the...

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【Something beautiful in math exists in internal pleasure】

《Rationality》　　　　　　　　　　　　　　　

3×4÷2＝5 which is the length of the oblique side ×...

Oh! I’ve forgotten to express one more thing. There is the vertical line from the oblique line to the right angle.

When the base is the oblique line, the height of the right angle triangle is equal the straight line from the oblique line to the right triangle, so I call its length Z.

As a result, as to the area on the value of the two right angle triangles is the next one.

3×4÷2＝5×Z÷2, so Z ＝two and two-fifths. We paid attention for its area this...

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【Something beautiful in math exists in internal pleasure】

《Rationality》

We paid attention for its area this time, but in the next we do the similar two triangles.

When the length of the base is 4 on the ground if there is the ground, the triangle is expressed with A,B,C, and D.

The base is AB, the oblique side is AC, and the last one is CB. There is a vertical line from the oblique side to the right angle. As to its starting point from the oblique side, it’s D.

Then the two of the triangles, △ ABC and △ADB are similar, so AC：CB＝AB：BD, 5：3＝4：Z, so 5Z ＝12, Z＝two and two-fifths.

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【Something beautiful in math exists in internal pleasure】

《Rationality》

By the way, as you know, I hate math and I’ve never studied math so much, as a result, to my sorrow, I have little knowledge on math, so I was forced to be worried about. Two of the triangles, △ ABC and △ABD are similar, why?

But two of them are right angle triangles, and ∠ DCB and ∠ACB are the same angle. It means each of the two angles are the same, so the last angles of each of them are also the same, as a result, two of the triangles are similar.

The author continued.

In other words, being rational means that...

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【Something beautiful in math exists in internal pleasure】

《Rationality》

In other words, being rational means everyone can reach the same conclusion as long as each of them adopts a logical thought, and the way of choosing is free as long as it’s logical, so I’m happy for it. The author said like that.

For example, let’s suppose that you learned to go to a cooking school and its instructor was irrational one, and the instructor forced you to adopt everything as the way the instructor was pleased.

As to a way of washing vegetable, the way of cutting it, measuring way of amount of the food...

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【Something beautiful in math exists in internal pleasure】

《Rationality》

...and an order of putting in a seasoning. The instructor showed you each way minutely. It didn’t allow to adopt other way at all. Even if your way was different a little, the instructor was enraged at it.

In addition, even if the same cooking, in the next time, the contents of its direction was different. All of the students attending the cooking school would be intolerable for it. They had always to be sensitive to the mood of the instructor, so the cooking school was rigid and stressful. No one can enjoy there.

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【Something beautiful in math exists in internal pleasure】

《Rationality》

If the instructor in the cooking school is rational, it will allow you to adopt various ways. In fact, a course of cooking delicious dish shouldn’t be the only one alone. If by any chance, you the students may come up with a good idea which makes the dish better than a recipe which the teacher prepared for.

If the instructor is rational, it’ll be willing to accept the device, and praise you. If you can go to the cooking school like that, you’ll have a fun. Every time you go to the cooking school, you will look ...

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【Something beautiful in math exists in internal pleasure】

《Rationality》

Every time you go to the cooking school, you will look forward to the class.

Being rational is connected with freedom on thinking, so the internal pleasure is caused.

《Elements of surprising》

When learning math, we often discover unexpected fact.

For example, if going on adding odd number, 1 ＋ 3 ＋ 5 ＋ 7...wherever stopping it, its value is a square measure. It seems that few people find it natural at once.

It’s expressed with an illustration and simple equations, and I want to express in English, but I find it hard.

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【Something beautiful in math exists in internal pleasure】

《Elements of surprising》

How is the square measure made up? The book said like the next.

It’s made up with L, but the direction of the L is turned. When adding the odd number, the size of the L is bigger a little.

The first odd number is 1, and 3 is added. The L is turned and is made up with three elements, and the square measure is made up with the four elements.

After that, 5 is added, and other L is added like the same way, but the size of L is bigger a little this time. It’s made up with five elements, and the other square...

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【Something beautiful in math exists in internal pleasure】

《Elements of surprise》

...then the other square measure made up with nine elements.

As a result, the figure is always the square measure, and its elements are square measure as well.

At first, when we can’t find a thing by intuition, but we’re convinced with a logical explanation, then we’re surprised and impressed with it. It’s one of internal pleasure.

On the other hand, if a thing which we find it natural is harped from the first to the end, we find it boring, at least we can’t find it anything pleasant there.

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【Something beautiful in math exists in internal pleasure】

《Surprise of elements》

When a mathematical thought made us find a fact unexpectedly and it caused the internal pleasure for us, so we may find something beautiful. It may be natural.

《Simplicity》

The biggest reason why we find the math is beautiful may be something simple.

It seems that there is a phrase that less is more. The phrase has been used from a long ago in the world of design. Its origin is an English poet used it in his work.

It means that when designing simple is better than decorating too much. It resembles a...

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【Something beautiful in math exists in internal pleasure】

《Simplicity》

It resembles a phrase, simple is best.

Leonardo da Vinci also said being simple is a refinement extremely.

When we need to follow a transient fashion, we sometimes should add various elements and decorate, but if seeking common beauty beyond the times, simplicity is indispensable.

For example, Golden Gate Bridge in San Francisco was built about eighty years ago, but we call it the most beautiful bridge in the world even at present. It's so simple that if something is removed from the bridge, it doesn’t exist as bridge.

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【Something beautiful in math exists in internal pleasure】

《Simplicity》

We’re forced to feel like that, for kinds of decoration are removed from the bridge.

A Japanese famous designer said it’s aesthetics made up with subtraction.

By the way, I’ve never seen the Golden Gate Bridge, so I’m not sure whether or not it’s beautiful.

At first even if someone famous said something splendid, it doesn’t always mean that I can sympathy with it, but if saying like that, this thread can’t go ahead, but I’m forced to say it.

Not only mathematicians but scientists have wanted to give a clear...

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【Something beautiful in math exists in internal pleasure】

《Simplicity》

Not only mathematicians but scientists have wanted to give clear explanation to the common truth on the cosmos, and it’s the most original motivation, and they have thought that the common truth is simple and beautiful.

In fact lots of the theorems and formulas which the mathematicians found in the past are simple. The author said he was going to show one of them.

When counting the number on vertex, edge, and face of a cube, taking from its initial and each of numbers is expressed with V, E, F, then the next ..

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【Something beautiful in math exists in internal pleasure】

《Simplicity》

...then the next simple equation stands up. V— E＋F＝2. We call it the theorem on polyhedron by Euler.

There are illustration of a tetrahedron, a rectangular, and a pentagonal prism on the book, and everyone can see each number of vertex, edge, and face, so the equation holds water.

Thus it’s seemingly complicated, but simple in its essence. It frequently happens in math. The simplicity and the common truth are influenced each other, we find it beautiful when seeing it.

The author continued.

A feeling of wishing to...

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【Something beautiful in math exists in internal pleasure】

《Simplicity》

A feeling of wishing to be mathematic resembles the one of wishing to be beautiful. If wishing to be beautiful, we need a heart in which we feel something beautiful, so wishing to be mathematic, we need to polish a sensitivity in which we can feel that being mathematic is fantastic and beautiful.

It doesn’t always mean that I want to be mathematic, but I find math is interesting at least.

【Pythagoras and a secret art on numbers】

《Numbers have each character?》

Everyone has number which each of them feel a sense of ...

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【Pythagoras and a secret art on numbers】

《Numbers have each character》

Everyone has numbers which each of them has a feeling of closeness. For example, it’s the number of birthday, the other which we’re fond of from the long ago, or another of the uniform number on a baseball player who you like.

The author said he likes 8. When the number is expressed with a Chinese character, it fans our downward, so it’s auspicious, but he was absorbed in playing baseball in childhood, his favorite professional baseball player was Tatsunori Hara, and his uniform number was 8, so he learned to like 8.

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【Pythagoras and a secret art on numbers】

《Numbers have each characters?》

7 which positions in front of 8, and is frequently said lucky and is popular among us, but the author said he has an image on 7 that it keeps itself above all the vulgarity around it, and gives off aura which doesn’t let anyone get close to it, so he doesn’t have a feeling of closeness to 7.

9 which is after 8, and doesn’t always make friends with everyone, but is very reliable among its pals, so if being in difficulty, the author feel like being helped from the number.

The author said he has an imagination on...

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【Pythagoras and a secret art on numbers】

《Numbers have each of characters》

The author said he has an image on various numbers like that. He doesn’t established it by force but it was done automatically.

He said he was afraid that everyone may think he was eccentric, but he said the ones who are good at numbers have a natural tendency in common like that, though it doesn’t always mean that he listened to each of them and made sure of it.

Even if each of them has an original image for each number, each of them has different image to 7,8,9.

The ones who love music can distinguish between...

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【Pythagoras and a secret art on numbers】

《Numbers have each characters》

The ones who love music can distinguish difference between good and bad musical performance, and the ones who good at cooking recognize difference the right amount of salt between good and bad one, or difference the right heat between good one and right one.

The ones who love math are sensitive to difference between characters on numbers like that.

《A discovery by Pythagoras》

The prosperity on Pythagoras and his disciples in the Ancient Greece started from the time when walking,Pythagoras was aware something strange.

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【Pythagoras and a secret art on numbers】

《A discovery by Pythagoras》

When a smith stroke the iron, there was sound of iron, but the ones reverberated good and the others didn’t like that. Pythagoras and his disciples visited the smith and examined why there was difference between the sound.

It turned out that difference between the weight on the hammer used by each of smith caused the phenomenon, and after minute research, unexpected fact was discovered.

When the sound was reverberated good, the proportion on the weight of the hammer was simple proportion of the integral number, 2：1 or 4：3

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