１＋１＝２
『ヨビノリ』で「１＋１＝２の証明」というのを観ました。ペアノの公理を利用するものです（証明の具体的な内容は省略します）。
そこで、「１」とか「２」というのは、数というより記号だなと思いました。たしかに、漢字で書けば「壱」「弐」です。記号ですよね。だから「△」でも「×」でもいいわけです。
なぜか、０（ゼロ）だけは定義されているんですね（「０」は自然数です）。
実は０も記号です。「ゼロ」でも「オー」でも「丸」でもいいわけです。「★」でもいいのです。
ゼロってなんでしょう。「無」のような気もします。「私の家にはゼロ匹の猫がいます。」というのは「私の家には猫がいない」ということですよね。
ですから、ゼロというのは「否定」なわけです。ゼロが「存在する」というのはどういうことなのでしょうか。
そういえば、読もう読もうと思っていて読んでいない本がありました。本書『零の発見』です。
本書の構成
前半が「零の発見ーアラビア数字の由来ー」、後半が「直線を切るー連続の問題ー」です。
数学史ではありません。数学と文化、社会との関係を描いています。
そろばん
私たちに馴染みのある、そして次第に見なくなっているそろばんの話から始まります。つまり位取りの話です。ヨーロッパでもそろばんは使われていたんですね。そろばんで、玉（珠）が動いていないところが「０」ですね。
10進法なら10個の数字ですべての自然数を表すことができる
「南アフリカのブッシュマンお数詞が一とニとにとどまっていて、それからさきはただ「たくさん」という言葉しかない」（P.18）民俗学も人類学も変わってきているので、この知見が現在も正しいのかどうかはわかりません。必要のない知識は伝わりませんから、その社会のあり方が数え方を規定するでしょうね。
あなたは実際にいくらまでの数を数えたことがありますか。ひつじが１匹，ひつじが２匹，ひつじが３・・・。たとえば、１億円の札束を見たことがある人は多いと思いますが、実際に数えた人は少ないでしょう。１億円を持っている人だってとても少ないと思います。「１億」「１兆」という数字は誰がなんのために必要なのでしょうか。日常生活で必要な数（数えられる数）を考えてみてください。「４にん家族」とか、「ボールペンが５本」とかそんなに多くないんじゃないでしょうか。ポールペンが５本（あるいは１００本）あったとしても、実際に一回に使えるのは１本です。「体がいくつあっても足りない」と思っても実際には体は一つしかありませんから。
小麦粉の粒は数えることが難しいけど、ご飯粒なら数えることができそうです。でも、「今朝はご飯を〇〇粒食べた」とは言わないですよね。「朝ごはん、一膳（茶碗一杯）食べた」と言うわけです。
私たちが実感できる（使える）数は決して多くはないのです。私は６T（テラ）のハードディスクを３個もっています。１テラは1,000,000,000,000ですが、数えられる数ではないので、「６テラのハードディスク」という「もの」を「３個」もっているわけで、実際には「６」と「３」という数字しか使っていないことになります。[1][2]
数字の間に「,（コンマ）」がついていますが、これは欧米では「千」をまとまりとしているからです。そしてその「千」が「一まとまり」になって「千個」まとまって次の塊となります。１億円を１円玉でもって歩くことはできません。１万円札というのは、１万を一まとまりにしているわけです。中国や日本４桁ごとに点を打っていました。
筆算と紙
そろばんの衰退は、筆算が盛んになった（電卓ではありません）ことが原因で、筆算のためには大量の紙が必要です。紙は印刷技術に欠かせないものです。ルネサンス、宗教改革、大航海時代等との関係が述べられています。数学はそれ自体で発展（変化）するものではなくて、いろいろな条件が影響するのです。
これがこの本のすごいところで、数学を数学という枠組みでとらえるのではなく、歴史・社会的視点から叙述しています。
小数と分数
後半です。話は進んで、小数と分数の話になります。循環小数の話から、無限級数の話へ。
無限級数のカッコの位置を変えるだけで、０にも１にもなる。「これで万物ガ無からつくられた消息がわかる、といって、随喜に涙を流した数学者もあった、という話が伝わっているのである。（LF）今から考えると、ただ滑稽というよりほかはないが、当人は、それでも、大まじめであったのである。もっとも、こういう話はなにも数学の場合にかぎらずとも、他にもあることであろうし、また、昔のこととばかりもいい切れないように思われる。」（P.71）著者のユーモアと皮肉ですが、私のことを言われているみたいです（笑）。
連続―無限
後半は有理数と無理数の話に移っていきます。そして、『ゼノンの逆理』の話へ。
「ちなみに、貨幣制度がはじまったのも、ギリシャ人の植民地的発展にともなって、地中海沿岸の各都市の間に大規模な商取引がおこなわれるようになって結果であって、この制度もまたイオニアをへて、しだいにギリシャ本土にまでひろまっていったものであった。」（P.94）ちょっと微妙かな。でも、貨幣制度が商取引からはじまったことは間違いないでしょう。
「ギリシャ人から見れば文字は単に記録の具であるにとどまり、思想や知識を伝達する手段としてはとうてい「語られる言葉」のような力を持たないものと考えられていたのである。すなわち「書かれた言葉は生きた命ある言葉の単なる模像、単なる影法師に過ぎない」、生き生きとした真の知識は、たがいに質問に答え、反駁にあい、誤解（FF）を正し、脱漏をおぎなうというふうに、口で語られる言葉をもってする談論の方法によって初めてこれを伝えることができる、というのである。（LF）学校教育と通信教育の優劣というよな問題を考えるとき、・・・」（P.95-96）
この説は、ソクラテスとプラトンの違いを考えるときに明瞭になってきますね。ソクラテスはまさしく問答（対話）に固執しました。でもプラトンは文字マニアでした。（『ソクラテスの弁明』）
プラトンはソクラテスと違っていたというより正反対の考え方を持っていたのかもしれません。「偉大な小心者プラトン」か「変わり者、あるいは変革者プラトン」か。
ここで著者はエス・エチ・ブチァー（1850~1910）の『ギリシャ天才の諸相』（『希臘天才の諸相』、のち『ギリシア精神の様相』）をとりあげます。
この本"Some Aspects of the Greek Genius"は、和辻哲郎・田中秀央の訳がありますが、旧字体のもののようです。読みたいですが・・・。
表語文字（表意文字）と表音文字、文字と芸術の乖離、法律に対する希臘精神の態度など、おもしろいことが満載のようです。
「ギリシャ人は、す学的事実―たとえば、ユークリッド幾何学における諸定理―は数学者がこれを発見するに先だって、すでにそれ自身存在しているものと考えていた、これに反して、現代では、数学的事実は、ポアンカレのいったように、「数学者が―時として数学者の気まぐれがこれを創造する」のであると考えられている、ということができるであろう。」（P.105）これは自然に対する見方そのものの変換ですね。（『いま自然をどうみるか』）
ピタゴラス教団の話。ソクラテスは「ソフィストに反して、客観的な善の概念を確立しようとつとめたのであった。ソクラテス自身は数学そのものには興味をもたなかったといわれるが、ソクラテスが概念に定義を与えること―すなわち、それぞれの概念が何であるかということ、その本質、その内容を述べること―を唱道したことは、以後の数学の方法に多大の影響を与えずにはいなかった。」（P.145）
この後の数学的説明はわかったようで、わからなったです(^_^;)。
零の発見
零の発見については、客観的な事項を述べているだけです。そういう叙述も好ましいと思います。
「ゼロがある」。無があるということは、「無いものは無い」「存在しないものは存在しない」ということです。当たり前のことと思ってしまうまえに、（混乱しそうですが）考える余地はありそうです。
「自然は真空を嫌う」と古代ギリシャ人は考えていたようです。それがヘレニズムを通してキリスト教に引き継がれます。[3]
「無はない」「すべてが有である」という考えは、「有るべきじゃないものを認める」と「有るべきじゃないものはなきものにする」という考えを生みます。どちらにしても、世の中に「有」あるいは「肯定」しかないのは窮屈ですよね。
別の形態や、別の社会を認めるということは大切なことだと思います。私は、「無いものは有る」ではなくて、「無いものは無いままで有る」「否定は有の否定ではなく、否定は有のあり方そのものである」と考えています。
読み終えて
この本が書かれたのが１９３９年。８０年以上も前です。歴史学（考古学）は以後、様々な発見ととともに変化しているようです。しかし、本書が貫いている考え方と方法論はいまも重要だと思います。名著です。
⟨impressions⟩
1 + 1 = 2
I saw "Proof of 1 + 1 = 2" in "Yobinori". It uses Peano's axioms (the specific content of the proof is omitted).
Therefore, I thought that "1" and "2" are symbols rather than numbers. Certainly, if you write it in kanji, it will be "Ichi" and "II". It's a symbol, isn't it? Therefore, it can be either "△" or "×".
For some reason, only 0 (zero) is defined ("0" is a natural number).
In fact, 0 is also a symbol. It can be "zero", "oh", or "maru". You can also use "★".
What is zero? I feel like "nothing". "There are zero cats in my house." Means "there are no cats in my house **", right?
Therefore, zero is "denial". What does it mean that zero "exists"?
By the way, there was a book I hadn't read because I wanted to read it. This is the book "Discovery of Zero".
Structure of this manual
The first half is "Discovery of Zero-Origin of Arabic Numbers-" and the second half is "Cut a Straight Line-Continuous Problem-".
It's not history of mathematics. It depicts the relationship between mathematics, culture, and society.
Abacus
It starts with the story of the abacus, which is familiar to us and is gradually disappearing. In other words, it's a scale story. Abacus was also used in Europe. In the abacus, the place where the ball (pearl) is not moving is "0".
In decimal, 10 numbers can represent all natural numbers
"South Africa's Bushman numerals are limited to one and two, and then there is only the word" many "." (P.18) Since folklore and anthropology are changing, this knowledge is now available. I don't know if it is correct. Unnecessary knowledge is not transmitted, so the way the society should be will determine how to count.
How many have you actually counted? One sheep, two sheep, three sheep ... For example, I think many people have seen a wad of 100 million yen, but few actually counted it. I think very few people have 100 million yen. Who needs the numbers "100 million" and "1 trillion" for what? Think about the number you need (the number you can count) in your daily life. I think there aren't many "4 family members" or "5 ballpoint pens". Even if you have 5 (or 100) boll point pens, you can actually use only one at a time. Even if you think "no matter how many bodies you have," you actually have only one body.
The number we can feel (use) is by no means large. I have three 6T (tera) hard disks. 1 tera is 1,000,000,000,000, but since it is not a countable number, we have "3" "things" called "6 tera hard disks", and in reality we only use the numbers "6" and "3". It will be. [1] [2]
There is a ", (comma)" between the numbers, because in Europe and the United States, "thousands" are grouped together. Then, the "thousands" become a "unit" and the "thousands" are combined into the next mass. You cannot walk 100 million yen with a 1-yen coin. A 10,000-yen bill is a collection of 10,000. I was scoring every 4 digits in China and Japan.
Long division and paper
The decline of the abacus is due to the rise of long division (not the calculator), which requires a large amount of paper. Paper is an integral part of printing technology. The relationship with the Renaissance, the Reformation, the Age of Discovery, etc. is described. Mathematics does not develop (change) by itself, but various conditions influence it.
This is the great thing about this book, which describes mathematics from a historical and social perspective, rather than looking at it in the framework of mathematics.
Decimals and fractions
It's the second half. The story goes on, and it's about decimals and fractions. From the story of recurring decimals to the story of infinite series.
Just by changing the position of the parentheses of the infinite series, it becomes 0 or 1. "It is said that some mathematicians were happy to shed tears, saying that they could understand the news that was made from all things. (LF) From now on, it's just humorous. There is nothing else, but he was still serious, though there may be other stories like this, not just in mathematics, and in the olden days. It seems like I can't cut it. ”(P.71) The author's humor and irony, but he seems to be talking about me (laughs).
Continuous-infinite
In the second half, we will move on to the discussion of rational numbers and irrational numbers. And to the story of "Zeno of Elea".
"By the way, the monetary system started as a result of the colonial development of the Greeks and the large-scale commercial transactions between cities on the Mediterranean coast, and this system also made Ionia. It gradually spread to the mainland of Greece. ”(P.94) It's a little subtle. But there is no doubt that the monetary system started with commercial transactions.
"From the Greek point of view, letters were merely a tool of record, and were thought to have no power like" spoken words "as a means of communicating thoughts and knowledge. That is, "written words are merely images of living and living words, merely shadow masters", and the lively and true knowledge answers questions, refutes, and corrects misunderstandings (FF). It is said that this can only be conveyed by a method of discourse with spoken words, such as to prevent omissions. (LF) When considering the superiority and inferiority of school education and distance learning ... ”(P.95-96)
This theory becomes clear when considering the difference between Socrates and Plato. Socrates just stuck to the question and answer (dialogue). But Plato was a character enthusiast. ("Socrates's defense")
Plato may have had the opposite idea rather than being different from Socrates. Is it "Great Mind Plato" or "Weirdo or Transformer Plato"?
Here, the author takes up "Aspects of Greek Genius" ("Aspects of Rare Genius", later "Aspects of Greek Spirit") by S. Ech Butcher (1850-1910).
This book "Some Aspects of the Greek Genius" has a translation by Tetsuro Watsuji and Hidenaka Tanaka, but it seems to be in the old font. I want to read ...
It seems to be full of interesting things such as ideographic characters (ideographic characters) and phonetic characters, the divergence between characters and art, and the attitude of a rare spirit toward the law.
"The Greeks thought that mathematicians-for example, theorems in Euclidean geometry-already existed before mathematicians discovered them, whereas in modern times It can be said that the mathematical facts, as Poancare said, are believed to be "mathematicians-sometimes the whims of mathematicians create this." (P.105) This is a transformation of the view of nature itself. ("How do you see nature now?")
The story of the Pythagorean cult. Socrates "contrary to the Sophist, he sought to establish an objective concept of good. It is said that Socrates himself was not interested in mathematics itself, but Socrates gives a definition to the concept-that is, The advocacy of what each concept is, its essence, and its content-has had a great influence on the subsequent methods of mathematics. "(P.145)
It seems that I understood the mathematical explanation after this, and I did not understand (^ _ ^;).
Discovery of Zero
Regarding the discovery of Zero, it only states objective matters. I think that such a statement is also preferable.
"There is zero." The existence of nothing means that "there is nothing that does not exist" and "there is nothing that does not exist". There seems to be room to think (although it seems confusing) before taking it for granted.
It seems that the ancient Greeks thought that "nature hates vacuum". It is passed on to Christianity through Hellenistic period. [3]
The idea that "there is nothing" and "everything is present" gives rise to the ideas of "acknowledge what should not be" and "make what should not be without". Either way, it's cramped that there is only "yes" or "affirmation" in the world.
I think it is important to recognize different forms and different societies. I do not think that "there is something that is not there", but that "there is nothing that is not there" and "denial is not the denial of existence, but denial is the way it is."
Finish reading
This book was written in 1939. It's more than 80 years ago. History (archeology) seems to have changed with various discoveries since then. However, I think the ideas and methodologies that this book adheres to are still important. It is a famous book.
1 + 1 = 2
I saw "Proof of 1 + 1 = 2" in "Yobinori". It uses Peano's axioms (the specific content of the proof is omitted).
Therefore, I thought that "1" and "2" are symbols rather than numbers. Certainly, if you write it in kanji, it will be "Ichi" and "II". It's a symbol, isn't it? Therefore, it can be either "△" or "×".
For some reason, only 0 (zero) is defined ("0" is a natural number).
In fact, 0 is also a symbol. It can be "zero", "oh", or "maru". You can also use "★".
What is zero? I feel like "nothing". "There are zero cats in my house." Means "there are no cats in my house **", right?
Therefore, zero is "denial". What does it mean that zero "exists"?
By the way, there was a book I hadn't read because I wanted to read it. This is the book "Discovery of Zero".
Structure of this manual
The first half is "Discovery of Zero-Origin of Arabic Numbers-" and the second half is "Cut a Straight Line-Continuous Problem-".
It's not history of mathematics. It depicts the relationship between mathematics, culture, and society.
Abacus
It starts with the story of the abacus, which is familiar to us and is gradually disappearing. In other words, it's a scale story. Abacus was also used in Europe. In the abacus, the place where the ball (pearl) is not moving is "0".
In decimal, 10 numbers can represent all natural numbers
"South Africa's Bushman numerals are limited to one and two, and then there is only the word" many "." (P.18) Since folklore and anthropology are changing, this knowledge is now available. I don't know if it is correct. Unnecessary knowledge is not transmitted, so the way the society should be will determine how to count.
How many have you actually counted? One sheep, two sheep, three sheep ... For example, I think many people have seen a wad of 100 million yen, but few actually counted it. I think very few people have 100 million yen. Who needs the numbers "100 million" and "1 trillion" for what? Think about the number you need (the number you can count) in your daily life. I think there aren't many "4 family members" or "5 ballpoint pens". Even if you have 5 (or 100) boll point pens, you can actually use only one at a time. Even if you think "no matter how many bodies you have," you actually have only one body.
The number we can feel (use) is by no means large. I have three 6T (tera) hard disks. 1 tera is 1,000,000,000,000, but since it is not a countable number, we have "3" "things" called "6 tera hard disks", and in reality we only use the numbers "6" and "3". It will be. [1] [2]
There is a ", (comma)" between the numbers, because in Europe and the United States, "thousands" are grouped together. Then, the "thousands" become a "unit" and the "thousands" are combined into the next mass. You cannot walk 100 million yen with a 1-yen coin. A 10,000-yen bill is a collection of 10,000. I was scoring every 4 digits in China and Japan.
Long division and paper
The decline of the abacus is due to the rise of long division (not the calculator), which requires a large amount of paper. Paper is an integral part of printing technology. The relationship with the Renaissance, the Reformation, the Age of Discovery, etc. is described. Mathematics does not develop (change) by itself, but various conditions influence it.
This is the great thing about this book, which describes mathematics from a historical and social perspective, rather than looking at it in the framework of mathematics.
Decimals and fractions
It's the second half. The story goes on, and it's about decimals and fractions. From the story of recurring decimals to the story of infinite series.
Just by changing the position of the parentheses of the infinite series, it becomes 0 or 1. "It is said that some mathematicians were happy to shed tears, saying that they could understand the news that was made from all things. (LF) From now on, it's just humorous. There is nothing else, but he was still serious, though there may be other stories like this, not just in mathematics, and in the olden days. It seems like I can't cut it. ”(P.71) The author's humor and irony, but he seems to be talking about me (laughs).
Continuous-infinite
In the second half, we will move on to the discussion of rational numbers and irrational numbers. And to the story of "Zeno of Elea".
"By the way, the monetary system started as a result of the colonial development of the Greeks and the large-scale commercial transactions between cities on the Mediterranean coast, and this system also made Ionia. It gradually spread to the mainland of Greece. ”(P.94) It's a little subtle. But there is no doubt that the monetary system started with commercial transactions.
"From the Greek point of view, letters were merely a tool of record, and were thought to have no power like" spoken words "as a means of communicating thoughts and knowledge. That is, "written words are merely images of living and living words, merely shadow masters", and the lively and true knowledge answers questions, refutes, and corrects misunderstandings (FF). It is said that this can only be conveyed by a method of discourse with spoken words, such as to prevent omissions. (LF) When considering the superiority and inferiority of school education and distance learning ... ”(P.95-96)
This theory becomes clear when considering the difference between Socrates and Plato. Socrates just stuck to the question and answer (dialogue). But Plato was a character enthusiast. ("Socrates's defense")
Plato may have had the opposite idea rather than being different from Socrates. Is it "Great Mind Plato" or "Weirdo or Transformer Plato"?
Here, the author takes up "Aspects of Greek Genius" ("Aspects of Rare Genius", later "Aspects of Greek Spirit") by S. Ech Butcher (1850-1910).
This book "Some Aspects of the Greek Genius" has a translation by Tetsuro Watsuji and Hidenaka Tanaka, but it seems to be in the old font. I want to read ...
It seems to be full of interesting things such as ideographic characters (ideographic characters) and phonetic characters, the divergence between characters and art, and the attitude of a rare spirit toward the law.
"The Greeks thought that mathematicians-for example, theorems in Euclidean geometry-already existed before mathematicians discovered them, whereas in modern times It can be said that the mathematical facts, as Poancare said, are believed to be "mathematicians-sometimes the whims of mathematicians create this." (P.105) This is a transformation of the view of nature itself. ("How do you see nature now?")
The story of the Pythagorean cult. Socrates "contrary to the Sophist, he sought to establish an objective concept of good. It is said that Socrates himself was not interested in mathematics itself, but Socrates gives a definition to the concept-that is, The advocacy of what each concept is, its essence, and its content-has had a great influence on the subsequent methods of mathematics. "(P.145)
It seems that I understood the mathematical explanation after this, and I did not understand (^ _ ^;).
Discovery of Zero
Regarding the discovery of Zero, it only states objective matters. I think that such a statement is also preferable.
"There is zero." The existence of nothing means that "there is nothing that does not exist" and "there is nothing that does not exist". There seems to be room to think (although it seems confusing) before taking it for granted.
It seems that the ancient Greeks thought that "nature hates vacuum". It is passed on to Christianity through Hellenistic period. [3]
The idea that "there is nothing" and "everything is present" gives rise to the ideas of "acknowledge what should not be" and "make what should not be without". Either way, it's cramped that there is only "yes" or "affirmation" in the world.
I think it is important to recognize different forms and different societies. I do not think that "there is something that is not there", but that "there is nothing that is not there" and "denial is not the denial of existence, but denial is the way it is."
Finish reading
This book was written in 1939. It's more than 80 years ago. History (archeology) seems to have changed with various discoveries since then. However, I think the ideas and methodologies that this book adheres to are still important. It is a famous book.
＜注＞
[1] 私は、PCマニア的なところがあるので、そういうハードディスクをもっていますが、１台「数百万円~数千万円」の車を「数十台」もっていていても、実際に乗れるのは１台だけです。
[2] １テラコンピュータ用語で実際には十進数で1,099,511,627,776です。
[3] 「エーテル」の存在も必然的に導き出されます。それが否定された後、また現代物理学で形を変えて復活したのはおもしろいです。
[4] 『ゼノンの逆理』はよくわかりません。連続・非連続の話というより、『運動』の話だと思っています。つまり二律背反原則の否定です。「そこにあって、そこにない」、「AでありかつAではない」ということです。これが、「連続の定義」でとらえることができたら素晴らしいのですが。
