奇しくもアインシュタインの誕生日の3月14日に亡くなった、偉大な理論物理学者スティーヴン・ホーキング博士。
(写真:現地時間2015年2月8日、イギリス・ロンドンで開催された英国アカデミー賞(British Academy Film Awards : BAFTA)にジェーン・ワイルド(写真右)と一緒にレッドカーペットに登場したスティーヴン・ホーキング博士)

紛れもなくアルベルト・アインシュタイン(Albert Einstein)以来最も偉大な理論物理学者で、数々の賞を授賞しているスティーヴン・ホーキング(Stephen Hawking)博士が、奇しくもアインシュタインの誕生日の3月14日に亡くなった。76歳だった。

ホーキング博士の発見は、生命そのものの全ての道理にかなった現代の論文を確証するものである。博士はNHS (国民保健サービス、英国の国営医療サービス事業)と福祉国家の信頼出来る支援者だったが、その生涯を通じて反戦運動家でもあった。また、ブラックユーモアの持ち主としても有名だった。彼はいくつかの宇宙の謎を解き明かした画期的な業績の一方、女性は「完全に謎だ」とよくジョークを飛ばしていた。

そして何よりも彼は、人々に影響を与える人間として輝きを放っていた。彼は半世紀以上も彼の平均余命を無視し、最も衰弱した状態で暮らしていたが、人生を大いに愛し、彼に与えられた全てのものに感謝した。体は諦めなければならなかったが、彼の精神は、私達が目がくらむような速度で飛び回っていた。彼の知識に対する渇望と生命構造を理解する為の緊急性は驚異的であった。

彼は「足元を見るのではなく、星を見上げて下さい。自分の目に見えるものを理解し、宇宙の存在を可能にしたものについて思い巡らしてみて下さい。好奇心を持って下さい」と人々に求めた。

ホーキング博士は、運動ニューロン疾患との勇敢かつ長い闘いの後、3月14日未明英・ケンブリッジ(Cambridge)の自宅で亡くなった。
彼の子供達、ルーシー(Lucy)、ロバート(Robert)、ティム(Tim)は、次の様な声明を発表した。
「私達は、愛する父親が今日亡くなった事を深く悲しんでいます。彼は偉大な科学者であり、その仕事と遺産が長年生き続けるであろう並外れた人物でした。彼の勇気と粘り強さは、その才気とユーモアと共に世界中の人々に影響を与えました。彼はかつて 『愛する人がいない宇宙なんて意味がない』と言いました。私達は、彼の事を永遠に寂しく思うことでしょう」

ホーキング博士の元妻のジェーン・ワイルド(Jane Wilde)の回想録を元にした2014年のヒット映画『博士と彼女のセオリー』(原題:The Theory Of Everything、イギリス映画、ジェームズ・マーシュ(James Marsh)監督)でホーキング博士を演じて、アカデミー賞主演男優賞を授賞したエディ・レッドメイン(Eddie Redmayne)は、「私が幸運にもお会いすることが出来た中で、本当に美しい心を持ち、驚異的な科学者であると同時に最も面白かった人を、私達は亡くしました。その比類ないご遺族の皆様にお悔やみ申し上げます」と述べた。

スティーヴン・ウィリアム・ホーキングは、第二次世界大戦のさなか学究肌の両親の4人の子供の長男として誕生した。父親は、医学の道を志す事を薦めたが、若き日のホーキング博士の関心は空へと向けられていた。彼の母親はかつて「スティーヴンは、常に不思議さに驚嘆する強い感性を持っていて、星が彼を引き寄せた事が分かりました」と振り返った。

17歳の時、彼は名門のオックスフォード大学(Oxford University)に入学して、物理学の学位課程に席をおいた。彼は自分でも認めている通り、決して真面目な学生ではなかった。「私は、大学にはあまり行きませんでした。人生を謳歌するのに忙しかったからです」と彼はかつて話している。しかし、それでも彼は自然科学の第一級優等学位を授与されて卒業する事が出来た。

ダンスに熱心にいそしみ、大学のボート部では漕ぎ手を務めていたが、徐々に体の異変に気付き始めた。時々滑ったり、バランスを失ったり、言葉がつかえるようになったのである。1963年、彼がケンブリッジ大学大学院の宇宙学の博士課程1年生の時、父親に説得されて病院を訪れた。「彼らはベッドを傾けて、私の腕の筋肉サンプルを採取し、電極を私に刺して、X線不透過性の液体を私の脊椎に注射し、X線によってそれが上がり下がりするのを観察しました。その後、彼らは私の病名を教えてくれませんでした。ただ、私は多発性硬化症ではないが、典型的な症例だと知らされました」

彼は、運動ニューロン疾患と診断された。これは、筋肉を制御する神経が閉鎖している状態である。彼は余命2年と宣告された。逆説的に響くかもしれないが、余命宣告が彼の研究に拍車をかけた。博士課程が終了する前に人生が終わる可能性がある為、スティーヴンには無駄にする時間はなかった。不熱心な学生が突然熱心な学者に変身したのである。彼はかつて、「私の将来に暗雲が垂れ込めていましたが、驚いた事に、現在の生活を以前にも増して楽しんでいる事に気付きました。私の研究も進展し始めました」と語っている。

愛も彼が生きる希望を保ち続けるのに一役買ったようだ。診断の直前にスティーヴンは、大晦日のパーティーで仲間の学生のジェーン・ワイルドと出会い、翌年婚約し、更にその翌年結婚した。

当初 スティーヴンの病気は急速に進行した。まもなく彼は松葉杖に頼るようになり、1969年には車椅子生活を余儀なくされた。しかし彼は「私の障害について怒るのは時間の無駄です。私達は、人生と上手く付き合っていかなければなりません。私は、上手く付き合ってきたつもりです。あなたがいつも怒っていたり、文句ばかり言っていたら人々は、あなたの為に時間を使ってくれなくなってしまいます」とストイックに語っている。

幸運にも彼の病気の進行が遅くなり始めた。しかし介護者として、また主婦としての妻の負担が顕在化し始める。彼の処女作『ホーキング、宇宙を語る : ビッグバンからブラックホールまで』(A Brief History Of Time)の成功の後、ジェーンは3人の子供を養育し、今や重度障害者となった夫の通訳者を務める為に、自分の仕事で成功を収めるという夢を諦めざるを得なかった。この本の大成功とそれに伴うメディアからの注目、そして介護者の絶え間ない存在が相まって、夫婦の関係は、控え目に言っても緊張が高まっていた。30年間を共に過ごした後、彼らの結婚生活は崩壊し、スティーヴンは、彼の看護士の一人のエレイン・メイソン(Elaine Mason)
との再婚を発表した。

博士は、既に尊敬され、成功した学者であったが、彼の著作によって予期せぬ有名人となり、更なる出版の機会とその著作を通して彼の理論を講義する為のプラットフォームを提供した。彼は、「『ホーキング、宇宙を語る : ビッグバンからブラックホールまで』がベストセラーになることを全く予想していませんでした。私が最初に宇宙についての大衆向けの本を執筆する着想を得たのは、1982年の事でした。執筆のきっかけは、一つには娘の学費を支払う為にお金を稼ぐ事でした」と告白している。

宇宙と時間の謎を解明しようとした彼の献身は、多くの賞と賞賛とファンをもたらした。運動ニューロン疾患と診断された時、このかつてなく楽観的な科学者は、かつて「私は動くことも出来ませんし、コンピューターを通じて話さなければなりませんが、私の心の中では、私は自由です」と語った。

ゼロ除算の発見と重要性を指摘した：再生核研究所、日本

2017年11月15日(水)
テーマ：

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.

 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.

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

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
…/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

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.

再生核研究所声明 424(2018.3.29): 　レオナルド・ダ・ヴィンチとゼロ除算

次のダ・ヴィンチの言葉を発見して、驚かされた：

ダ・ヴィンチの名言 格言｜無こそ最も素晴らしい存在

我々の周りにある偉大なことの中でも、無の存在が最も素晴らしい。その基本は時間的には過去と未来の間にあり、現在の何ものをも所有しないというところにある。この無は、全体に等しい部分、部分に等しい全体を持つ。分割できないものと割り切ることができるし、割っても掛けても、足しても引いても、同じ量になるのだ。

レオナルド・ダ・ヴィンチ。ルネッサンス期を代表する芸術家、画家、彫刻家、建築技師、設計士、兵器開発者、科学者、哲学者、解剖学者、動物学者、ファッションデザイナーその他広い分野で活躍し「万能の人（uomo universale：ウォモ・ウニヴェルサーレ）」と称えられる人物

そもそも西欧諸国が、アリストテレス以来、無や真空、ゼロを嫌い、ゼロの西欧諸国への導入は相当に遅れ、西欧へのアラビヤ数字の導入は　レオナルド・フィボナッチ（1179年頃～1250年頃）によるとされているから、その遅れの大きさに驚かされる：

フィボナッチはイタリアのピサの数学者です。正確には「レオナルド・フィリオ・ボナッチ」といいますが、これがなまって「フィボナッチ」と呼ばれるようになったとされています。
彼は少年時代に父親について現在のアルジェリアに渡り、そこでアラビア数字を学びました。当時の�
�聖ローマ皇帝・フリードリヒ2世は科学と数学を重んじていて、フィボナッチは宮殿に呼ばれ皇帝にも謁見しました。後にはピサ共和国から表彰もされました。

ローマ数字では「I, II, III, X, XV」のように文字を並べて記すため大きな数を扱うのには不便でした。対してアラビア数字はローマ数字に比べてとても分かりやすく、効率的で便利だったのです。そこでフィボナッチはアラビア数字を「算術の書」という書物にまとめ、母国に紹介しました。アラビア数字では0から9までの数字と位取り記数法が使われていますが、計算に使うにはとても便利だったために、ヨーロッパで広く受け入れられることになりました。（

historicalmathematicians.blogspot.com/2012/03/blog-post.html　　02/03/2012 -）

ゼロや無に対する恐怖心、嫌疑観は現在でも欧米諸国の自然な心情と考えられる。ところが上記ダ・ヴィンチの言葉は　如何であろう。無について好ましいものとして真正面から捉えていることが分かる。ゼロ除算の研究をここ４年間して来て、驚嘆すべきこととして驚かされた。ゼロの意味、ゼロ除算の心を知っていたかのような言明である。

まず、上記で、無を、時間的に未来と過去の間に存在すると言っているので、無とはゼロのことであると解釈できる。ゼロとの捉え方は四則演算を考えているので、その解釈の適切性を述べている。足しても引いても変わらない。これはゼロの本質ではないか。さらに、凄いこと、掛けても割っても、ゼロと言っていると解釈でき、それはゼロ除算の最近の発見を意味している:  0/1 =1/0=0。－　ゼロ除算を感覚的に捉えていたと解釈できる。ところが更に、凄いことを述べている。

この無は、全体に等しい部分、部分に等しい全体を持つ。これはゼロ除算の著書DIVISION BY ZERO CALCULUS（原案）に真正面から書いている我々の得た、達したゼロに対する認識そのものである：

{\bf Fruitful world}\index{fruitful world}

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For example, in very and very general partial differential equations, if the coefficients or terms are zero, we have some simple differential equations and the extreme case is all the terms are zero; that is, we have trivial equations $0=0$; then its solution is zero. When we see the converse, we see that the zero world is a fruitful one and it means some vanishing world. Recall \index{Yamane phenomena}Yamane phenomena, the vanishing result is very simple zero, however, it is the result from some fruitful world. Sometimes, zero means void or nothing world, however, it will show some changes as in the Yamane phenomena.

\medskip

{\bf From $0$ to $0$; $0$ means all and all are $0$}

\medskip

As we see from our life figure, a story starts from the zero and ends to the zero. This will mean that $0$ means all and all are $0$, in a sense. The zero is a mother of all.

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その意味は深い。我々はゼロの意味をいろいろと捉え考え、ゼロとはさらに　基準を表すとか、不可能性を示すとか、無限遠点の反映であるとか、ゼロの2重性とかを述べている。ゼロと無限の関係をも述べている。ダ・ヴィンチの鋭い世界観に対する境地に驚嘆している。

以　上

*057 Pinelas,S./Caraballo,T./Kloeden,P./Graef,J.(eds.): Differential and Difference Equations with Applications: ICDDEA, Amadora, 2017. (Springer Proceedings in Mathematics and Statistics, Vol. 230) May 2018 587 pp. 

*Less than human　TYRELL/col.9610*

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