「音楽はなぜ人に喜びを与えるのか」という長年続く議論は決着するのか？
「音楽がなぜ人に喜びを与えるのか？」ということは、はるか昔、それこそ古代ギリシャの哲学者・プラトンが既に考えていた疑問でした。音楽と喜びの関係は長い歴史の中で思想家たちが思いをめぐらせ、さまざまな考えが提示されてきた分野。中世から現代にかけて、どのように「音楽と喜びの関係」の考え方が変化してきたのか、哲学・歴史・政治を専門とするエディターのSam Dresser氏が記しています。
It’s hard to know why music gives pleasure: is that the point? | Aeon Essays
https://aeon.co/essays/its-hard-to-know-why-music-gives-pleasure-is-that-the-point
音楽と喜びの関係性について、中世ヨーロッパの音楽理論家たちは、アリストテレスの「旋律の音色が文章と一緒に働くことで、自然界を模倣する」という理論を支持していました。このような理論は詩の世界でも支持されており、「この世にある自然な感情」が作品の中で再現されることで、人は心地よさを感じるという考え方でした。
18世紀になると、このような考えは主流となっていき、「画家が自然の色合いや形を模倣するように、音楽家は音色、アクセント、吐息といった声の変化を模倣する。実際に、これらのサウンドは情緒や激情を発散する」と考える思想家もいたとのこと。
18世紀、理論家たちの多くは音楽の持つ「美の力」にますます興味を注いでいき、「音楽の形は感情の形を模倣しているのではないか？」「情熱を捉えることができる音色の変化があるのではないか？」という疑問が持たれるようになりました。外交官であり音楽理論家のヨハン・マッテゾンもその1人で「喜びは私たちの生命力から生まれるものなのだから、喜びを表現する音楽は開放的な旋律の跳躍を使うべきだ」と考えたとのこと。「絶望は下がり気味のメロディーラインで」「早いテンポは欲望」「ゆっくりしたテンポは悲嘆」といったことも、マッテゾンは提唱していきました。「音楽の特徴と人間の感情を関係させていく」というマッテゾンの考えは、現代でもみられ、「運動をするときはアップテンポの曲」「泣きたい時はしっとりした曲」といった選択が行われることも少なくありません。
by Alice Moore
しかし、18世紀の思想家にとって音楽表現の性質や真意を正確に突き止めることは簡単ではありませんでした。この難しさについて哲学者のドゥニ・ディドロは「絵画は対象物そのものを示してみせ、詩は対象物について説明します。しかし、音楽は対象物についての考えをかき立てるだけです。自然界を模倣する3つの芸術のうち、音楽は最も恣意的で、厳密さが欠けた力強い語りを魂に行います。対象物についてあまり示さず、イマジネーションだけを置いていくのです」と記しています。特に、テキストの含まれない音楽において、この特徴は顕著になるとのこと。
このように考えたのはディドロだけではありませんでした。18世紀終わりになると「詩と音楽が合わさることで自然界を模倣する」というアリストテレスの思想とは異なる考えが現れるようになります。学者のトーマス・トワイニングも同様に「音楽は正確性を欠いているからこそ喜びを生み出す」という考えの持ち主でした。これらの思想家たちは音楽の持つ「移ろいやすさ」「解釈に基づく演出」を強調したとのこと。音楽は人それぞれが自由に演奏でき、聞き手はその演奏を通して思いを巡らせ、新しい性質を発見し、意味を見いだし、喜びを感じるわけです。
しかし、18世紀の思想家にとって音楽表現の性質や真意を正確に突き止めることは簡単ではありませんでした。この難しさについて哲学者のドゥニ・ディドロは「絵画は対象物そのものを示してみせ、詩は対象物について説明します。しかし、音楽は対象物についての考えをかき立てるだけです。自然界を模倣する3つの芸術のうち、音楽は最も恣意的で、厳密さが欠けた力強い語りを魂に行います。対象物についてあまり示さず、イマジネーションだけを置いていくのです」と記しています。特に、テキストの含まれない音楽において、この特徴は顕著になるとのこと。
このように考えたのはディドロだけではありませんでした。18世紀終わりになると「詩と音楽が合わさることで自然界を模倣する」というアリストテレスの思想とは異なる考えが現れるようになります。学者のトーマス・トワイニングも同様に「音楽は正確性を欠いているからこそ喜びを生み出す」という考えの持ち主でした。これらの思想家たちは音楽の持つ「移ろいやすさ」「解釈に基づく演出」を強調したとのこと。音楽は人それぞれが自由に演奏でき、聞き手はその演奏を通して思いを巡らせ、新しい性質を発見し、意味を見いだし、喜びを感じるわけです。
by Jan Střecha
18世紀に活躍した音楽家の1人にヨハン・ゼバスティアン・バッハが存在しますが、バッハの複雑な音楽は、1つ1つの要素を聞いて解釈するのが難しいものでした。このため、思想家の中には「バッハの音楽を楽しむのは難しい」と考えたり、「聞き手が感情を抱くことができないのでは」と懸念する人もいました。弁護士・作曲家・音楽コメンテーターであったクリスチャン・ゴットフリード・クラウス氏はこの種の音楽を作り出す作曲家について「愛情を全て忘れてしまったのではないか」と記すほどでした。
しかし、別の文章でクラウスはバッハのような複雑な音楽について「全ての声がうまく働いた時、作曲かが言うような『壮大さ』『称賛』『熱意』『喜び』が表現されていることに気づいた。そして心は昇華されたより強い感情で満たされた」とも記しています。クラウスは最終的に、この種の複雑な音楽に「無限の体験」の可能性を見いだしました。
クラウスのような考えが広まる中で、19世紀に活躍した音楽評論家のエドゥアルト・ハンスリックは、マッテゾンが提唱した「音楽の特徴と人間の感情のつながり」という理論を批判するように。マッテゾンの音楽のとらえ方は、聞き手が1つの音楽を同じように聞くことを推奨する、というのがハンスリックの見方です。ハンスリックは、「音楽の喜びとは、作曲家が作品をどのようにデザインしたのかを理解しようとする試みによって得られる知的な満足である」と考えました。ハンスリックの考えは「音楽は喜びを与える」というこれまでの思想の真逆に位置するもの。ハンスリックにとって「ある種の複雑な音楽」は称賛に値するものでした。
18世紀に活躍した音楽家の1人にヨハン・ゼバスティアン・バッハが存在しますが、バッハの複雑な音楽は、1つ1つの要素を聞いて解釈するのが難しいものでした。このため、思想家の中には「バッハの音楽を楽しむのは難しい」と考えたり、「聞き手が感情を抱くことができないのでは」と懸念する人もいました。弁護士・作曲家・音楽コメンテーターであったクリスチャン・ゴットフリード・クラウス氏はこの種の音楽を作り出す作曲家について「愛情を全て忘れてしまったのではないか」と記すほどでした。
しかし、別の文章でクラウスはバッハのような複雑な音楽について「全ての声がうまく働いた時、作曲かが言うような『壮大さ』『称賛』『熱意』『喜び』が表現されていることに気づいた。そして心は昇華されたより強い感情で満たされた」とも記しています。クラウスは最終的に、この種の複雑な音楽に「無限の体験」の可能性を見いだしました。
クラウスのような考えが広まる中で、19世紀に活躍した音楽評論家のエドゥアルト・ハンスリックは、マッテゾンが提唱した「音楽の特徴と人間の感情のつながり」という理論を批判するように。マッテゾンの音楽のとらえ方は、聞き手が1つの音楽を同じように聞くことを推奨する、というのがハンスリックの見方です。ハンスリックは、「音楽の喜びとは、作曲家が作品をどのようにデザインしたのかを理解しようとする試みによって得られる知的な満足である」と考えました。ハンスリックの考えは「音楽は喜びを与える」というこれまでの思想の真逆に位置するもの。ハンスリックにとって「ある種の複雑な音楽」は称賛に値するものでした。
by Radek Grzybowski
20世紀、ハンスリックの考えに否定的な理論家もいました。哲学者のスザンヌ・ランガーはアートの解釈における「象徴性」を説いた人物。それまでの思想家と同じく、ランガーは音楽の「欠如」という部分に意味を見いだしました。ランガーにとって音楽は、ハンスリックが考えたような「決まった意味がある」ものではなく、「未完成の象徴を暗示的に示している」ものでした。象徴としての音楽を聴くことによって、人は自分自身にとっての感情的な物語を作ることができます。自分自身の感情に照らし合わせて意味を決定していく作業は知的なものであるとランガーは考えました。
2007年に亡くなった作曲家のレナード・B・メイヤーも全体論や構造を重視するゲシュタルト心理学の観点からランガーと同じ立場に立つ人物です。メイヤーは、音楽は抽象的で非指示的な要素から成り立ち、音楽を聴いた人は自分の「予想」から逸脱した部分に喜びを感じると考えました。
ランガーやメイヤーの思想は現代にも引き継がれています。特に現代の音楽教育でこのような理論が用いられることが多く、特定の聴き方が重視され、「音楽の喜びは、音楽を作り出した社会的現実性から何かを見つけ出すことにある」と考えられがちです。しかし、このような思想では、音楽を作った「個人」は軽視されるリスクもあります。
20世紀、ハンスリックの考えに否定的な理論家もいました。哲学者のスザンヌ・ランガーはアートの解釈における「象徴性」を説いた人物。それまでの思想家と同じく、ランガーは音楽の「欠如」という部分に意味を見いだしました。ランガーにとって音楽は、ハンスリックが考えたような「決まった意味がある」ものではなく、「未完成の象徴を暗示的に示している」ものでした。象徴としての音楽を聴くことによって、人は自分自身にとっての感情的な物語を作ることができます。自分自身の感情に照らし合わせて意味を決定していく作業は知的なものであるとランガーは考えました。
2007年に亡くなった作曲家のレナード・B・メイヤーも全体論や構造を重視するゲシュタルト心理学の観点からランガーと同じ立場に立つ人物です。メイヤーは、音楽は抽象的で非指示的な要素から成り立ち、音楽を聴いた人は自分の「予想」から逸脱した部分に喜びを感じると考えました。
ランガーやメイヤーの思想は現代にも引き継がれています。特に現代の音楽教育でこのような理論が用いられることが多く、特定の聴き方が重視され、「音楽の喜びは、音楽を作り出した社会的現実性から何かを見つけ出すことにある」と考えられがちです。しかし、このような思想では、音楽を作った「個人」は軽視されるリスクもあります。
by Xektop10
しかし、そもそも「音楽」において「何が喜びを生み出すか」という点ばかりが重視されているのではないか、という指摘もあります。音楽と喜びはいずれも主観的な認識に基づきます。両者とも具体的で明確な現象ですが、その性質上、詳しく説明することが難しいもの。長い歴史の中で音楽と喜びの関係について議論されてきた理由がこの性質にあるとするならば、両者の関係について決着を付けるのは非常に難しいことだといえそうです。https://gigazine.net/news/20180907-music-gives-pleasure/
しかし、そもそも「音楽」において「何が喜びを生み出すか」という点ばかりが重視されているのではないか、という指摘もあります。音楽と喜びはいずれも主観的な認識に基づきます。両者とも具体的で明確な現象ですが、その性質上、詳しく説明することが難しいもの。長い歴史の中で音楽と喜びの関係について議論されてきた理由がこの性質にあるとするならば、両者の関係について決着を付けるのは非常に難しいことだといえそうです。https://gigazine.net/news/20180907-music-gives-pleasure/
ゼロ除算の発見は日本です：
∞？？？
∞は定まった数ではない・
人工知能はゼロ除算ができるでしょうか：
とても興味深く読みました：2014年2月2日 4周年を超えました：
ゼロ除算の発見と重要性を指摘した：日本、再生核研究所
ゼロ除算関係論文・本
ソクラテス・プラトン・アリストテレス その他
テーマ：社会
The null set is conceptually similar to the role of the number ``zero'' as it is used in quantum field theory. In quantum field theory, one can take the empty set, the vacuum, and generate all possible physical configurations of the Universe being modelled by acting on it with creation operators, and one can similarly change from one thing to another by applying mixtures of creation and anihillation operators to suitably filled or empty states. The anihillation operator applied to the vacuum, however, yields zero.
Zero in this case is the null set - it stands, quite literally, for no physical state in the Universe. The important point is that it is not possible to act on zero with a creation operator to create something; creation operators only act on the vacuum which is empty but not zero. Physicists are consequently fairly comfortable with the existence of operations that result in ``nothing'' and don't even require that those operations be contradictions, only operationally non-invertible.
It is also far from unknown in mathematics. When considering the set of all real numbers as quantities and the operations of ordinary arithmetic, the ``empty set'' is algebraically the number zero (absence of any quantity, positive or negative). However, when one performs a division operation algebraically, one has to be careful to exclude division by zero from the set of permitted operations! The result of division by zero isn't zero, it is ``not a number'' or ``undefined'' and is not in the Universe of real numbers.
Just as one can easily ``prove'' that 1 = 2 if one does algebra on this set of numbers as if one can divide by zero legitimately3.34, so in logic one gets into trouble if one assumes that the set of all things that are in no set including the empty set is a set within the algebra, if one tries to form the set of all sets that do not include themselves, if one asserts a Universal Set of Men exists containing a set of men wherein a male barber shaves all men that do not shave themselves3.35.
It is not - it is the null set, not the empty set, as there can be no male barbers in a non-empty set of men (containing at least one barber) that shave all men in that set that do not shave themselves at a deeper level than a mere empty list. It is not an empty set that could be filled by some algebraic operation performed on Real Male Barbers Presumed to Need Shaving in trial Universes of Unshaven Males as you can very easily see by considering any particular barber, perhaps one named ``Socrates'', in any particular Universe of Men to see if any of the sets of that Universe fit this predicate criterion with Socrates as the barber. Take the empty set (no men at all). Well then there are no barbers, including Socrates, so this cannot be the set we are trying to specify as it clearly must contain at least one barber and we've agreed to call its relevant barber Socrates. (and if it contains more than one, the rest of them are out of work at the moment).
Suppose a trial set contains Socrates alone. In the classical rendition we ask, does he shave himself? If we answer ``no'', then he is a member of this class of men who do not shave themselves and therefore must shave himself. Oops. Well, fine, he must shave himself. However, if he does shave himself, according to the rules he can only shave men who don't shave themselves and so he doesn't shave himself. Oops again. Paradox. When we try to apply the rule to a potential Socrates to generate the set, we get into trouble, as we cannot decide whether or not Socrates should shave himself.
Note that there is no problem at all in the existential set theory being proposed. In that set theory either Socrates must shave himself as All Men Must Be Shaven and he's the only man around. Or perhaps he has a beard, and all men do not in fact need shaving. Either way the set with just Socrates does not contain a barber that shaves all men because Socrates either shaves himself or he doesn't, so we shrug and continue searching for a set that satisfies our description pulled from an actual Universe of males including barbers. We immediately discover that adding more men doesn't matter. As long as those men, barbers or not, either shave themselves or Socrates shaves them they are consistent with our set description (although in many possible sets we find that hey, other barbers exist and shave other men who do not shave themselves), but in no case can Socrates (as our proposed single barber that shaves all men that do not shave themselves) be such a barber because he either shaves himself (violating the rule) or he doesn't (violating the rule). Instead of concluding that there is a paradox, we observe that the criterion simply doesn't describe any subset of any possible Universal Set of Men with no barbers, including the empty set with no men at all, or any subset that contains at least Socrates for any possible permutation of shaving patterns including ones that leave at least some men unshaven altogether.
https://webhome.phy.duke.edu/.../axioms/axioms/Null_Set.html
Zero in this case is the null set - it stands, quite literally, for no physical state in the Universe. The important point is that it is not possible to act on zero with a creation operator to create something; creation operators only act on the vacuum which is empty but not zero. Physicists are consequently fairly comfortable with the existence of operations that result in ``nothing'' and don't even require that those operations be contradictions, only operationally non-invertible.
It is also far from unknown in mathematics. When considering the set of all real numbers as quantities and the operations of ordinary arithmetic, the ``empty set'' is algebraically the number zero (absence of any quantity, positive or negative). However, when one performs a division operation algebraically, one has to be careful to exclude division by zero from the set of permitted operations! The result of division by zero isn't zero, it is ``not a number'' or ``undefined'' and is not in the Universe of real numbers.
Just as one can easily ``prove'' that 1 = 2 if one does algebra on this set of numbers as if one can divide by zero legitimately3.34, so in logic one gets into trouble if one assumes that the set of all things that are in no set including the empty set is a set within the algebra, if one tries to form the set of all sets that do not include themselves, if one asserts a Universal Set of Men exists containing a set of men wherein a male barber shaves all men that do not shave themselves3.35.
It is not - it is the null set, not the empty set, as there can be no male barbers in a non-empty set of men (containing at least one barber) that shave all men in that set that do not shave themselves at a deeper level than a mere empty list. It is not an empty set that could be filled by some algebraic operation performed on Real Male Barbers Presumed to Need Shaving in trial Universes of Unshaven Males as you can very easily see by considering any particular barber, perhaps one named ``Socrates'', in any particular Universe of Men to see if any of the sets of that Universe fit this predicate criterion with Socrates as the barber. Take the empty set (no men at all). Well then there are no barbers, including Socrates, so this cannot be the set we are trying to specify as it clearly must contain at least one barber and we've agreed to call its relevant barber Socrates. (and if it contains more than one, the rest of them are out of work at the moment).
Suppose a trial set contains Socrates alone. In the classical rendition we ask, does he shave himself? If we answer ``no'', then he is a member of this class of men who do not shave themselves and therefore must shave himself. Oops. Well, fine, he must shave himself. However, if he does shave himself, according to the rules he can only shave men who don't shave themselves and so he doesn't shave himself. Oops again. Paradox. When we try to apply the rule to a potential Socrates to generate the set, we get into trouble, as we cannot decide whether or not Socrates should shave himself.
Note that there is no problem at all in the existential set theory being proposed. In that set theory either Socrates must shave himself as All Men Must Be Shaven and he's the only man around. Or perhaps he has a beard, and all men do not in fact need shaving. Either way the set with just Socrates does not contain a barber that shaves all men because Socrates either shaves himself or he doesn't, so we shrug and continue searching for a set that satisfies our description pulled from an actual Universe of males including barbers. We immediately discover that adding more men doesn't matter. As long as those men, barbers or not, either shave themselves or Socrates shaves them they are consistent with our set description (although in many possible sets we find that hey, other barbers exist and shave other men who do not shave themselves), but in no case can Socrates (as our proposed single barber that shaves all men that do not shave themselves) be such a barber because he either shaves himself (violating the rule) or he doesn't (violating the rule). Instead of concluding that there is a paradox, we observe that the criterion simply doesn't describe any subset of any possible Universal Set of Men with no barbers, including the empty set with no men at all, or any subset that contains at least Socrates for any possible permutation of shaving patterns including ones that leave at least some men unshaven altogether.
https://webhome.phy.duke.edu/.../axioms/axioms/Null_Set.html
I understand your note as if you are saying the limit is infinity but nothing is equal to infinity, but you concluded corretly infinity is undefined. Your example of getting the denominator smaller and smalser the result of the division is a very large number that approches infinity. This is the intuitive mathematical argument that plunged philosophy into mathematics. at that level abstraction mathematics, as well as phyisics become the realm of philosophi. The notion of infinity is more a philosopy question than it is mathamatical. The reason we cannot devide by zero is simply axiomatic as Plato pointed out. The underlying reason for the axiom is because sero is nothing and deviding something by nothing is undefined. That axiom agrees with the notion of limit infinity, i.e. undefined. There are more phiplosphy books and thoughts about infinity in philosophy books than than there are discussions on infinity in math books.
http://mathhelpforum.com/algebra/223130-dividing-zero.html
http://mathhelpforum.com/algebra/223130-dividing-zero.html
ゼロ除算の歴史：ゼロ除算はゼロで割ることを考えるであるが、アリストテレス以来問題とされ、ゼロの記録がインドで初めて６２８年になされているが、既にそのとき、正解1/0が期待されていたと言う。しかし、理論づけられず、その後１３００年を超えて、不可能である、あるいは無限、無限大、無限遠点とされてきたものである。
An Early Reference to Division by Zero C. B. Boyer
http://www.fen.bilkent.edu.tr/~franz/M300/zero.pdf
An Early Reference to Division by Zero C. B. Boyer
http://www.fen.bilkent.edu.tr/~franz/M300/zero.pdf
OUR HUMANITY AND DIVISION BY ZERO
Lea esta bitácora en español
There is a mathematical concept that says that division by zero has no meaning, or is an undefined expression, because it is impossible to have a real number that could be multiplied by zero in order to obtain another number different from zero.
While this mathematical concept has been held as true for centuries, when it comes to the human level the present situation in global societies has, for a very long time, been contradicting it. It is true that we don’t all live in a mathematical world or with mathematical concepts in our heads all the time. However, we cannot deny that societies around the globe are trying to disprove this simple mathematical concept: that division by zero is an impossible equation to solve.
Yes! We are all being divided by zero tolerance, zero acceptance, zero love, zero compassion, zero willingness to learn more about the other and to find intelligent and fulfilling ways to adapt to new ideas, concepts, ways of doing things, people and cultures. We are allowing these ‘zero denominators’ to run our equations, our lives, our souls.
Each and every single day we get more divided and distanced from other people who are different from us. We let misinformation and biased concepts divide us, and we buy into these aberrant concepts in such a way, that we get swept into this division by zero without checking our consciences first.
I believe, however, that if we change the zeros in any of the “divisions by zero” that are running our lives, we will actually be able to solve the non-mathematical concept of this equation: the human concept.
>I believe deep down that we all have a heart, a conscience, a brain to think with, and, above all, an immense desire to learn and evolve. And thanks to all these positive things that we do have within, I also believe that we can use them to learn how to solve our “division by zero” mathematical impossibility at the human level. I am convinced that the key is open communication and an open heart. Nothing more, nothing less.
Are we scared of, or do we feel baffled by the way another person from another culture or country looks in comparison to us? Are we bothered by how people from other cultures dress, eat, talk, walk, worship, think, etc.? Is this fear or bafflement so big that we much rather reject people and all the richness they bring within?
How about if instead of rejecting or retreating from that person—division of our humanity by zero tolerance or zero acceptance—we decided to give them and us a chance?
How about changing that zero tolerance into zero intolerance? Why not dare ask questions about the other person’s culture and way of life? Let us have the courage to let our guard down for a moment and open up enough for this person to ask us questions about our culture and way of life. How about if we learned to accept that while a person from another culture is living and breathing in our own culture, it is totally impossible for him/her to completely abandon his/her cultural values in order to become what we want her to become?
Let’s be totally honest with ourselves at least: Would any of us really renounce who we are and where we come from just to become what somebody else asks us to become?
If we are not willing to lose our identity, why should we ask somebody else to lose theirs?
I believe with all my heart that if we practiced positive feelings—zero intolerance, zero non-acceptance, zero indifference, zero cruelty—every day, the premise that states that division by zero is impossible would continue being true, not only in mathematics, but also at the human level. We would not be divided anymore; we would simply be building a better world for all of us.
Hoping to have touched your soul in a meaningful way,
Adriana Adarve, Asheville, NC
https://adarvetranslations.com/…/our-humanity-and-division…/
Lea esta bitácora en español
There is a mathematical concept that says that division by zero has no meaning, or is an undefined expression, because it is impossible to have a real number that could be multiplied by zero in order to obtain another number different from zero.
While this mathematical concept has been held as true for centuries, when it comes to the human level the present situation in global societies has, for a very long time, been contradicting it. It is true that we don’t all live in a mathematical world or with mathematical concepts in our heads all the time. However, we cannot deny that societies around the globe are trying to disprove this simple mathematical concept: that division by zero is an impossible equation to solve.
Yes! We are all being divided by zero tolerance, zero acceptance, zero love, zero compassion, zero willingness to learn more about the other and to find intelligent and fulfilling ways to adapt to new ideas, concepts, ways of doing things, people and cultures. We are allowing these ‘zero denominators’ to run our equations, our lives, our souls.
Each and every single day we get more divided and distanced from other people who are different from us. We let misinformation and biased concepts divide us, and we buy into these aberrant concepts in such a way, that we get swept into this division by zero without checking our consciences first.
I believe, however, that if we change the zeros in any of the “divisions by zero” that are running our lives, we will actually be able to solve the non-mathematical concept of this equation: the human concept.
>I believe deep down that we all have a heart, a conscience, a brain to think with, and, above all, an immense desire to learn and evolve. And thanks to all these positive things that we do have within, I also believe that we can use them to learn how to solve our “division by zero” mathematical impossibility at the human level. I am convinced that the key is open communication and an open heart. Nothing more, nothing less.
Are we scared of, or do we feel baffled by the way another person from another culture or country looks in comparison to us? Are we bothered by how people from other cultures dress, eat, talk, walk, worship, think, etc.? Is this fear or bafflement so big that we much rather reject people and all the richness they bring within?
How about if instead of rejecting or retreating from that person—division of our humanity by zero tolerance or zero acceptance—we decided to give them and us a chance?
How about changing that zero tolerance into zero intolerance? Why not dare ask questions about the other person’s culture and way of life? Let us have the courage to let our guard down for a moment and open up enough for this person to ask us questions about our culture and way of life. How about if we learned to accept that while a person from another culture is living and breathing in our own culture, it is totally impossible for him/her to completely abandon his/her cultural values in order to become what we want her to become?
Let’s be totally honest with ourselves at least: Would any of us really renounce who we are and where we come from just to become what somebody else asks us to become?
If we are not willing to lose our identity, why should we ask somebody else to lose theirs?
I believe with all my heart that if we practiced positive feelings—zero intolerance, zero non-acceptance, zero indifference, zero cruelty—every day, the premise that states that division by zero is impossible would continue being true, not only in mathematics, but also at the human level. We would not be divided anymore; we would simply be building a better world for all of us.
Hoping to have touched your soul in a meaningful way,
Adriana Adarve, Asheville, NC
https://adarvetranslations.com/…/our-humanity-and-division…/
5000年？？？？？
2017年09月01日(金)NEW !
テーマ：数学
Former algebraic approach was formally perfect, but it merely postulated existence of sets and morphisms [18] without showing methods to construct them. The primary concern of modern algebras is not how an operation can be performed, but whether it maps into or onto and the like abstract issues [19–23]. As important as this may be for proofs, the nature does not really care about all that. The PM’s concerns were not constructive, even though theoretically significant. We need thus an approach that is more relevant to operations performed in nature, which never complained about morphisms or the allegedly impossible division by zero, as far as I can tell. Abstract sets and morphisms should be de-emphasized as hardly operational. My decision to come up with a definite way to implement the feared division by zero was not really arbitrary, however. It has removed a hidden paradox from number theory and an obvious absurd from algebraic group theory. It was necessary step for full deployment of constructive, synthetic mathematics (SM) [2,3]. Problems hidden in PM implicitly affect all who use mathematics, even though we may not always be aware of their adverse impact on our thinking. Just take a look at the paradox that emerges from the usual prescription for multiplication of zeros that remained uncontested for some 5000 years 0 0 ¼ 0 ) 0 1=1 ¼ 0 ) 0 1 ¼ 0 1) 1ð? ¼ ?Þ1 ð0aÞ This ‘‘fact’’ was covered up by the infamous prohibition on division by zero [2]. How ingenious. If one is prohibited from dividing by zero one could not obtain this paradox. Yet the prohibition did not really make anything right. It silenced objections to irresponsible reasonings and prevented corrections to the PM’s flamboyant axiomatizations. The prohibition on treating infinity as invertible counterpart to zero did not do any good either. We use infinity in calculus for symbolic calculations of limits [24], for zero is the infinity’s twin [25], and also in projective geometry as well as in geometric mapping of complex numbers. Therein a sphere is cast onto the plane that is tangent to it and its free (opposite) pole in a point at infinity [26–28]. Yet infinity as an inverse to the natural zero removes the whole absurd (0a), for we obtain [2] 0 ¼ 1=1 ) 0 0 ¼ 1=12 > 0 0 ð0bÞ Stereographic projection of complex numbers tacitly contradicted the PM’s prescribed way to multiply zeros, yet it was never openly challenged. The old formula for multiplication of zeros (0a) is valid only as a practical approximation, but it is group-theoretically inadmissible in no-nonsense reasonings. The tiny distinction in formula (0b) makes profound theoretical difference for geometries and consequently also for physical applications. T
https://www.plover.com/misc/CSF/sdarticle.pdf
とても興味深く読みました：
2017年09月01日(金)NEW !
テーマ：数学
Former algebraic approach was formally perfect, but it merely postulated existence of sets and morphisms [18] without showing methods to construct them. The primary concern of modern algebras is not how an operation can be performed, but whether it maps into or onto and the like abstract issues [19–23]. As important as this may be for proofs, the nature does not really care about all that. The PM’s concerns were not constructive, even though theoretically significant. We need thus an approach that is more relevant to operations performed in nature, which never complained about morphisms or the allegedly impossible division by zero, as far as I can tell. Abstract sets and morphisms should be de-emphasized as hardly operational. My decision to come up with a definite way to implement the feared division by zero was not really arbitrary, however. It has removed a hidden paradox from number theory and an obvious absurd from algebraic group theory. It was necessary step for full deployment of constructive, synthetic mathematics (SM) [2,3]. Problems hidden in PM implicitly affect all who use mathematics, even though we may not always be aware of their adverse impact on our thinking. Just take a look at the paradox that emerges from the usual prescription for multiplication of zeros that remained uncontested for some 5000 years 0 0 ¼ 0 ) 0 1=1 ¼ 0 ) 0 1 ¼ 0 1) 1ð? ¼ ?Þ1 ð0aÞ This ‘‘fact’’ was covered up by the infamous prohibition on division by zero [2]. How ingenious. If one is prohibited from dividing by zero one could not obtain this paradox. Yet the prohibition did not really make anything right. It silenced objections to irresponsible reasonings and prevented corrections to the PM’s flamboyant axiomatizations. The prohibition on treating infinity as invertible counterpart to zero did not do any good either. We use infinity in calculus for symbolic calculations of limits [24], for zero is the infinity’s twin [25], and also in projective geometry as well as in geometric mapping of complex numbers. Therein a sphere is cast onto the plane that is tangent to it and its free (opposite) pole in a point at infinity [26–28]. Yet infinity as an inverse to the natural zero removes the whole absurd (0a), for we obtain [2] 0 ¼ 1=1 ) 0 0 ¼ 1=12 > 0 0 ð0bÞ Stereographic projection of complex numbers tacitly contradicted the PM’s prescribed way to multiply zeros, yet it was never openly challenged. The old formula for multiplication of zeros (0a) is valid only as a practical approximation, but it is group-theoretically inadmissible in no-nonsense reasonings. The tiny distinction in formula (0b) makes profound theoretical difference for geometries and consequently also for physical applications. T
https://www.plover.com/misc/CSF/sdarticle.pdf
とても興味深く読みました：
10,000 Year Clock
by Renny Pritikin
Conversation with Paolo Salvagione, lead engineer on the 10,000-year clock project, via e-mail in February 2010.
For an introduction to what we’re talking about here’s a short excerpt from a piece by Michael Chabon, published in 2006 in Details: ….Have you heard of this thing? It is going to be a kind of gigantic mechanical computer, slow, simple and ingenious, marking the hour, the day, the year, the century, the millennium, and the precession of the equinoxes, with a huge orrery to keep track of the immense ticking of the six naked-eye planets on their great orbital mainspring. The Clock of the Long Now will stand sixty feet tall, cost tens of millions of dollars, and when completed its designers and supporters plan to hide it in a cave in the Great Basin National Park in Nevada, a day’s hard walking from anywhere. Oh, and it’s going to run for ten thousand years. But even if the Clock of the Long Now fails to last ten thousand years, even if it breaks down after half or a quarter or a tenth that span, this mad contraption will already have long since fulfilled its purpose. Indeed the Clock may have accomplished its greatest task before it is ever finished, perhaps without ever being built at all. The point of the Clock of the Long Now is not to measure out the passage, into their unknown future, of the race of creatures that built it. The point of the Clock is to revive and restore the whole idea of the Future, to get us thinking about the Future again, to the degree if not in quite the way same way that we used to do, and to reintroduce the notion that we don’t just bequeath the future—though we do, whether we think about it or not. We also, in the very broadest sense of the first person plural pronoun, inherit it.
Renny Pritikin: When we were talking the other day I said that this sounds like a cross between Borges and the vast underground special effects from Forbidden Planet. I imagine you hear lots of comparisons like that…
Paolo Salvagione: (laughs) I can’t say I’ve heard that comparison. A childhood friend once referred to the project as a cross between Tinguely and Fabergé. When talking about the clock, with people, there’s that divide-by-zero moment (in the early days of computers to divide by zero was a sure way to crash the computer) and I can understand why. Where does one place, in one’s memory, such a thing, such a concept? After the pause, one could liken it to a reboot, the questions just start streaming out.
RP: OK so I think the word for that is nonplussed. Which the thesaurus matches with flummoxed, bewildered, at a loss. So the question is why even (I assume) fairly sophisticated people like your friends react like that. Is it the physical scale of the plan, or the notion of thinking 10,000 years into the future—more than the length of human history?
PS: I’d say it’s all three and more. I continue to be amazed by the specificity of the questions asked. Anthropologists ask a completely different set of questions than say, a mechanical engineer or a hedge fund manager. Our disciplines tie us to our perspectives. More than once, a seemingly innocent question has made an impact on the design of the clock. It’s not that we didn’t know the answer, sometimes we did, it’s that we hadn’t thought about it from the perspective of the person asking the question. Back to your question. I think when sophisticated people, like you, thread this concept through their own personal narrative it tickles them. Keeping in mind some people hate to be tickled.
RP: Can you give an example of a question that redirected the plan? That’s really so interesting, that all you brainiacs slaving away on this project and some amateur blithely pinpoints a problem or inconsistency or insight that spins it off in a different direction. It’s like the butterfly effect.
PS: Recently a climatologist pointed out that our equation of time cam, (photo by Rolfe Horn) (a cam is a type of gear: link) a device that tracks the difference between solar noon and mundane noon as well as the precession of the equinoxes, did not account for the redistribution of water away from the earth’s poles. The equation-of-time cam is arguably one of the most aesthetically pleasing parts of the clock. It also happens to be one that is fairly easy to explain. It visually demonstrates two extremes. If you slice it, like a loaf of bread, into 10,000 slices each slice would represent a year. The outside edge of the slice, let’s call it the crust, represents any point in that year, 365 points, 365 days. You could, given the right amount of magnification, divide it into hours, minutes, even seconds. Stepping back and looking at the unsliced cam the bottom is the year 2000 and the top is the year 12000. The twist that you see is the precession of the equinoxes. Now here’s the fun part, there’s a slight taper to the twist, that’s the slowing of the earth on its axis. As the ice at the poles melts we have a redistribution of water, we’re all becoming part of the “slow earth” movement.
RP: Are you familiar with Charles Ray’s early work in which you saw a plate on a table, or an object on the wall, and they looked stable, but were actually spinning incredibly slowly, or incredibly fast, and you couldn’t tell in either case? Or, more to the point, Tim Hawkinson’s early works in which he had rows of clockwork gears that turned very very fast, and then down the line, slower and slower, until at the end it approached the slowness that you’re dealing with?
PS: The spinning pieces by Ray touches on something we’re trying to avoid. We want you to know just how fast or just how slow the various parts are moving. The beauty of the Ray piece is that you can’t tell, fast, slow, stationary, they all look the same. I’m not familiar with the Hawkinson clockwork piece. I’ve see the clock pieces where he hides the mechanism and uses unlikely objects as the hands, such as the brass clasp on the back of a manila envelope or the tab of a coke can.
RP: Spin Sink (1 Rev./100 Years) (1995), in contrast, is a 24-foot-long row of interlocking gears, the smallest of which is driven by a whirring toy motor that in turn drives each consecutively larger and more slowly turning gear up to the largest of all, which rotates approximately once every one hundred years.
PS: I don’t know how I missed it, it’s gorgeous. Linking the speed that we can barely see with one that we rarely have the patience to wait for.
RP: : So you say you’ve opted for the clock’s time scale to be transparent. How will the clock communicate how fast it’s going?
PS: By placing the clock in a mountain we have a reference to long time. The stratigraphy provides us with the slowest metric. The clock is a middle point between millennia and seconds. Looking back 10,000 years we find the beginnings of civilization. Looking at an earthenware vessel from that era we imagine its use, the contents, the craftsman. The images painted or inscribed on the outside provide some insight into the lives and the languages of the distant past. Often these interpretations are flawed, biased or over-reaching. What I’m most enchanted by is that we continue to construct possible pasts around these objects, that our curiosity is overwhelming. We line up to see the treasures of Tut, or the remains of frozen ancestors. With the clock we are asking you to create possible futures, long futures, and with them the narratives that made them happen.
https://openspace.sfmoma.org/2010/02/10000-year-clock/
by Renny Pritikin
Conversation with Paolo Salvagione, lead engineer on the 10,000-year clock project, via e-mail in February 2010.
For an introduction to what we’re talking about here’s a short excerpt from a piece by Michael Chabon, published in 2006 in Details: ….Have you heard of this thing? It is going to be a kind of gigantic mechanical computer, slow, simple and ingenious, marking the hour, the day, the year, the century, the millennium, and the precession of the equinoxes, with a huge orrery to keep track of the immense ticking of the six naked-eye planets on their great orbital mainspring. The Clock of the Long Now will stand sixty feet tall, cost tens of millions of dollars, and when completed its designers and supporters plan to hide it in a cave in the Great Basin National Park in Nevada, a day’s hard walking from anywhere. Oh, and it’s going to run for ten thousand years. But even if the Clock of the Long Now fails to last ten thousand years, even if it breaks down after half or a quarter or a tenth that span, this mad contraption will already have long since fulfilled its purpose. Indeed the Clock may have accomplished its greatest task before it is ever finished, perhaps without ever being built at all. The point of the Clock of the Long Now is not to measure out the passage, into their unknown future, of the race of creatures that built it. The point of the Clock is to revive and restore the whole idea of the Future, to get us thinking about the Future again, to the degree if not in quite the way same way that we used to do, and to reintroduce the notion that we don’t just bequeath the future—though we do, whether we think about it or not. We also, in the very broadest sense of the first person plural pronoun, inherit it.
Renny Pritikin: When we were talking the other day I said that this sounds like a cross between Borges and the vast underground special effects from Forbidden Planet. I imagine you hear lots of comparisons like that…
Paolo Salvagione: (laughs) I can’t say I’ve heard that comparison. A childhood friend once referred to the project as a cross between Tinguely and Fabergé. When talking about the clock, with people, there’s that divide-by-zero moment (in the early days of computers to divide by zero was a sure way to crash the computer) and I can understand why. Where does one place, in one’s memory, such a thing, such a concept? After the pause, one could liken it to a reboot, the questions just start streaming out.
RP: OK so I think the word for that is nonplussed. Which the thesaurus matches with flummoxed, bewildered, at a loss. So the question is why even (I assume) fairly sophisticated people like your friends react like that. Is it the physical scale of the plan, or the notion of thinking 10,000 years into the future—more than the length of human history?
PS: I’d say it’s all three and more. I continue to be amazed by the specificity of the questions asked. Anthropologists ask a completely different set of questions than say, a mechanical engineer or a hedge fund manager. Our disciplines tie us to our perspectives. More than once, a seemingly innocent question has made an impact on the design of the clock. It’s not that we didn’t know the answer, sometimes we did, it’s that we hadn’t thought about it from the perspective of the person asking the question. Back to your question. I think when sophisticated people, like you, thread this concept through their own personal narrative it tickles them. Keeping in mind some people hate to be tickled.
RP: Can you give an example of a question that redirected the plan? That’s really so interesting, that all you brainiacs slaving away on this project and some amateur blithely pinpoints a problem or inconsistency or insight that spins it off in a different direction. It’s like the butterfly effect.
PS: Recently a climatologist pointed out that our equation of time cam, (photo by Rolfe Horn) (a cam is a type of gear: link) a device that tracks the difference between solar noon and mundane noon as well as the precession of the equinoxes, did not account for the redistribution of water away from the earth’s poles. The equation-of-time cam is arguably one of the most aesthetically pleasing parts of the clock. It also happens to be one that is fairly easy to explain. It visually demonstrates two extremes. If you slice it, like a loaf of bread, into 10,000 slices each slice would represent a year. The outside edge of the slice, let’s call it the crust, represents any point in that year, 365 points, 365 days. You could, given the right amount of magnification, divide it into hours, minutes, even seconds. Stepping back and looking at the unsliced cam the bottom is the year 2000 and the top is the year 12000. The twist that you see is the precession of the equinoxes. Now here’s the fun part, there’s a slight taper to the twist, that’s the slowing of the earth on its axis. As the ice at the poles melts we have a redistribution of water, we’re all becoming part of the “slow earth” movement.
RP: Are you familiar with Charles Ray’s early work in which you saw a plate on a table, or an object on the wall, and they looked stable, but were actually spinning incredibly slowly, or incredibly fast, and you couldn’t tell in either case? Or, more to the point, Tim Hawkinson’s early works in which he had rows of clockwork gears that turned very very fast, and then down the line, slower and slower, until at the end it approached the slowness that you’re dealing with?
PS: The spinning pieces by Ray touches on something we’re trying to avoid. We want you to know just how fast or just how slow the various parts are moving. The beauty of the Ray piece is that you can’t tell, fast, slow, stationary, they all look the same. I’m not familiar with the Hawkinson clockwork piece. I’ve see the clock pieces where he hides the mechanism and uses unlikely objects as the hands, such as the brass clasp on the back of a manila envelope or the tab of a coke can.
RP: Spin Sink (1 Rev./100 Years) (1995), in contrast, is a 24-foot-long row of interlocking gears, the smallest of which is driven by a whirring toy motor that in turn drives each consecutively larger and more slowly turning gear up to the largest of all, which rotates approximately once every one hundred years.
PS: I don’t know how I missed it, it’s gorgeous. Linking the speed that we can barely see with one that we rarely have the patience to wait for.
RP: : So you say you’ve opted for the clock’s time scale to be transparent. How will the clock communicate how fast it’s going?
PS: By placing the clock in a mountain we have a reference to long time. The stratigraphy provides us with the slowest metric. The clock is a middle point between millennia and seconds. Looking back 10,000 years we find the beginnings of civilization. Looking at an earthenware vessel from that era we imagine its use, the contents, the craftsman. The images painted or inscribed on the outside provide some insight into the lives and the languages of the distant past. Often these interpretations are flawed, biased or over-reaching. What I’m most enchanted by is that we continue to construct possible pasts around these objects, that our curiosity is overwhelming. We line up to see the treasures of Tut, or the remains of frozen ancestors. With the clock we are asking you to create possible futures, long futures, and with them the narratives that made them happen.
https://openspace.sfmoma.org/2010/02/10000-year-clock/
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