Чтение на выходные«Ноль: Биография опасной идеи» Чарльза Сейфе
На русском выходит книга американского математика о тайне цифры «ноль». «Воздух» публикует отрывок из нее — о причинах фобии ноля.
Устрашающие свойства пустоты
В начале времен, когда жил Имир,
не было в мире ни песка, ни моря,
земли еще не было и небосвода,
бездна зияла, трава не росла.
не было в мире ни песка, ни моря,
земли еще не было и небосвода,
бездна зияла, трава не росла.
— Старшая Эдда —
Трудно представить себе, что можно бояться числа. Однако ноль был неразрывно связан с бездной — с пустотой. Людей преследовал изначальный страх перед бездной и хаосом, а также и перед нолем.
Древние народы верили, что до возникновения Вселенной существовали только пустота и хаос. Древние греки утверждали, что сначала матерью всего была тьма, а из тьмы возник хаос. Тьма и хаос породили остальную Вселенную. Еврейские предания о сотворении мира говорят, что земля была в состоянии хаоса и пустоты, пока Бог не пролил свет и не придал ему его свойства. На иврите это выражается так: tohu v’bohu. Роберт Грейвз связывает tohu с Техомотом, доисторическим семитским драконом, свидетелем рождения Вселенной, тело которого стало небом и землей. Bohu связывалось с Бегемотом, знаменитым чудовищем из еврейских легенд. Согласно древнеиндийской традиции, Создатель сбивал масло из хаоса и превратил его в землю, а северный миф рассказывает о том, как открытая бездна покрылась льдом, а из хаоса, порожденного смешением огня и льда, возник первобытный великан. Пустота и беспорядок были изначальным естественным состоянием космоса, и всегда существовал грызущий страх перед тем, что в конце времен беспорядок и бездна воцарятся снова. Ноль олицетворял собой эту бездну.
Однако страх перед нолем коренился глубже, чем страх бездны. Для древних математические свойства ноля были непостижимы, столь же окутаны тайной, как и рождение Вселенной. Причина этого крылась в том, что ноль отличается от всех остальных чисел. В отличие от других цифр в вавилонской системе, нолю никогда не позволялось стоять в одиночку — и не без основания. Оказавшись сам по себе, ноль ведет себя странно, по крайней мере не так, как остальные числа.
Если прибавить число к самому себе, оно изменится. Один плюс один — уже не один, а два. Два и два дают четыре. А вот ноль плюс ноль есть ноль. Это нарушает основной принцип счисления, называемый аксиомой Архимеда и говорящий, что если прибавлять число к самому себе достаточное количество раз, результат превзойдет по величине любое другое число. (Аксиома Архимеда была сформулирована в терминах площадей; число рассматривалось как разница между двумя неравными площадями.) Ноль же отказывается увеличиваться. Он также отказывается увеличивать любое другое число. Сложите два и ноль, и вы получите два; дело выглядит так, словно вы и не складывали ничего. То же самое происходит и при вычитании. Отнимите ноль от двух, и вы получите два. Ноль не имеет реальности. Однако это лишенное реальности число угрожает нарушить простейшие математические операции, такие как умножение и деление.
В области чисел умножение означает растяжение — в буквальном смысле слова. Представьте себе, что числовая ось — это резиновая лента с делениями на ней. Умножение на два может рассматриваться как растяжение резиновой ленты вдвое: то деление, которое приходилось на отметку «один», теперь переместилось на «два»; приходившееся на «три» — на «шесть». Аналогично умножение на одну вторую сходно с некоторым сжатием резиновой ленты: деление на «два» перемещается на «один», деление на «три» — на «полтора».
Но что происходит при умножении на ноль? Сколько бы раз ни взять ноль, все равно будет ноль, и все деления соберутся на ноле. Резиновая лента порвалась. Вся числовая ось нарушилась.
К несчастью, нет способа обойти этот неприятный факт. Любое число ноль раз — ноль; это свойство нашей системы счисления. Чтобы в повседневно используемых числах был смысл, они должны обладать тем, что именуется свойством дистрибутивности, что лучше всего видно на примере. Представьте себе, что в магазине игрушек мячи продаются по две штуки, а кубики — по три. Соседний магазин игрушек торгует наборами из двух мячей и трех кубиков. Каждая упаковка из двух мячей и каждая упаковка из трех кубиков — такой же один предмет, как и упаковка с набором мячей и кубиков из соседнего магазина. Если быть последовательным, то покупка семи упаковок мячей и семи упаковок кубиков в первом магазине должна быть тем же самым, что и покупка семи наборов во втором. Это и есть свойство дистрибутивности. Используя математическую запись, мы выразили бы это так: 7 x 2 + 7 x 3 = 7 (2 + 3). Все получается правильно.
Если же применить это свойство к нолю, получается нечто странное. Мы знаем, что 0 + 0 = 0. Возьмем в качестве примера число 2. 2 + 0 = 2 + (0 + 0); согласно свойству дистрибутивности, мы также знаем, что 2 (0 + 0) — то же самое, что 2 x 0 + 2 x 0. Однако это означает, что 2 x 0 = 2 x 0 + 2 x 0. Чем бы ни было 2 x 0, когда вы прибавляете это число к самому себе, оно остается тем же самым, очень похожим на ноль. На самом деле это он и есть. Если вычесть 2 x 0 из обеих частей равенства, мы увидим, что 0 = 2 x 0. Таким образом, что бы вы ни делали, умножение числа на ноль дает ноль. Это зловредное число сжимает числовую ось в точку. Однако сколь бы досадным ни было это свойство, истинная сила ноля делается очевидной при делении, а не умножении.
Если умножение растягивает числовую ось, то деление сжимает ее. Умножьте какое-нибудь число на два, и вы растянете резиновую ленту — числовую ось — вдвое; разделите результат на два, и резиновая лента сожмется вдвое, произведя действие, обратное умножению. Производя деление, вы уничтожаете следствие умножения: метка на резиновой ленте, переместившаяся на новое место, возвращается в прежнее положение.
Мы видели, что произошло при умножении числа на ноль: числовая ось была уничтожена. Деление на ноль должно было быть противоположностью умножению на ноль — оно должно было бы восстановить числовую ось. К несчастью, этого не происходит.
В предыдущем примере мы видели, что 2 x 0 есть 0. Таким образом, чтобы совершить действие, обратное умножению, мы должны предположить, что (2 x 0) / 0 вернет нас к 2. Точно так же (3 x 0) / 0 должно вернуть нас к 3, (4 x 0) / 0 — к 4... Однако каждое из чисел 2 x 0, 3 x 0, 4 x 0, как мы видели, равно 0, так что (2 x 0) / 0 = 0 / 0, (3 x 0) / 0 = 0 / 0, (4 x 0) / 0 = 0 / 0. Увы, это означает, что 0 / 0 = 2, а также 0 / 0 = 3, 0 / 0 = 4... Это же бессмыслица!
Странные вещи происходят и в том случае, если мы посмотрим на 1 / 0 с другой точки зрения. Умножение на ноль должно произвести действие, обратное делению на ноль, так что 1 / 0 x 0 должно быть равно 1. Однако мы видели, что любое число, умноженное на ноль, дает ноль. Нет такого числа, которое, умноженное на ноль, давало бы 1, по крайней мере, среди чисел, с которыми мы встречались.
Хуже всего то, что если вы необдуманно разделите на ноль, вы можете разрушить все основы логики и математики. Достаточно всего один раз — один-единственный — разделить на ноль, и это позволит вам математически доказать все что угодно. Вы сможете доказать, что 1 + 1 = 42, а из этого вывести, что Эдгар Гувер был инопланетянином, Уильям Шекспир — узбеком, и даже что небо — в горошек.
Умножение на ноль уничтожает числовую ось. Однако деление на ноль разрушает всю систему математики. Это простое число обладает большим могуществом. Оно стало самым важным математическим инструментом. Однако благодаря своим странным математическим и философским свойствам ноль пришел в столкновение с фундаментальной западной философией.https://daily.afisha.ru/archive/vozduh/books/nol-biografiya-opasnoy-idei-charlza-seyfe/
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：
神の数式：
神の数式が解析関数でかけて居れば、 特異点でローラン展開して、正則部の第1項を取れば、 何時でも有限値を得るので、 形式的に無限が出ても 実は問題なく 意味を有します。
物理学者如何でしょうか。
カテゴリ：カテゴリ未分類
そこで、計算機は何時、1/0=0 ができるようになるでしょうか。 楽しみにしています。 もうできる進化した 計算機をお持ちの方は おられないですね。
これは凄い、面白い事件では？ 計算機が人間を超えている 例では？
面白いことを発見しました。 計算機は 正しい答え 0/0=0
を出したのに、 この方は 間違いだと 言っている、思っているようです。
0/0=0 は １３００年も前に 算術の発見者によって与えられたにも関わらず、世界史は間違いだと とんでもないことを言ってきた。 世界史の恥。 実は a/0=0 が 何時も成り立っていた。 しかし、ここで 分数の意味を きちんと定義する必要がある。 計算機は、その意味さえ知っているようですね。 計算機、人間より賢くなっている 様が 出て居て 実に 面白い。
https://steemkr.com/utopian-io/@faisalamin/bug-zero-divide-by-zero-answers-is-zero
２０１８．１０．１１．１１：２３
カテゴリ：カテゴリ未分類
面白いことを発見しました。 計算機は 正しい答え 0/0=0
を出したのに、 この方は 間違いだと 言っている、思っているようです。
0/0=0 は １３００年も前に 算術の発見者によって与えられたにも関わらず、世界史は間違いだと とんでもないことを言ってきた。 実は a/0=0 が 何時も成り立っていた。しかし、ここで 分数の意味を きちんと定義する必要がある。 計算機は、その意味さえ知っているようですね。 計算機、人間より賢くなっている様が 出て居て 実に面白い。
https://steemkr.com/utopian-io/@faisalamin/bug-zero-divide-by-zero-answers-is-zero
２０１８．１０．１１．１１：２３
ゼロ除算、ゼロで割る問題、分からない、正しいのかなど、 良く理解できない人が 未だに 多いようです。そこで、簡潔な一般的な 解説を思い付きました。 もちろん、学会などでも述べていますが、 予断で 良く聞けないようです。まず、分数、a/b は a 割る b のことで、これは 方程式 b x=a の解のことです。ところが、 b がゼロならば、 どんな xでも 0 x =0 ですから、a がゼロでなければ、解は存在せず、 従って 100/0 など、ゼロ除算は考えられない、できないとなってしまいます。 普通の意味では ゼロ除算は 不可能であるという、世界の常識、定説です。できない、不可能であると言われれば、いろいろ考えたくなるのが、人間らしい創造の精神です。 基本方程式 b x=a が b がゼロならば解けない、解が存在しないので、困るのですが、このようなとき、従来の結果が成り立つような意味で、解が考えられないかと、数学者は良く考えて来ました。 何と、 そのような方程式は 何時でも唯一つに 一般化された意味で解をもつと考える 方法があります。 Moore-Penrose 一般化逆の考え方です。 どんな行列の 逆行列を唯一つに定める 一般的な 素晴らしい、自然な考えです。その考えだと、 b がゼロの時、解はゼロが出るので、 a/0=0 と定義するのは 当然です。 すなわち、この意味で 方程式の解を考えて 分数を考えれば、ゼロ除算は ゼロとして定まる ということです。ただ一つに定まるのですから、 この考えは 自然で、その意味を知りたいと 考えるのは、当然ではないでしょうか？初等数学全般に影響を与える ユークリッド以来の新世界が 現れてきます。
ゼロ除算の誤解は深刻：
最近、3つの事が在りました。
私の簡単な講演、相当な数学者が信じられないような誤解をして、全然理解できなく、目が回っているいるような印象を受けたこと、
相当ゼロ除算の研究をされている方が、基本を誤解されていたこと、1/0 の定義を誤解されていた。
相当な才能の持ち主が、連続性や順序に拘って、４年以上もゼロ除算の研究を避けていたこと。
これらのことは、人間如何に予断と偏見にハマった存在であるかを教えている。
まずは ゼロ除算は不可能であるの 思いが強すぎで、初めからダメ、考えない、無視の気持ちが、強い。 ゼロ除算を従来の 掛け算の逆と考えると、不可能であるが 証明されてしまうので、割り算の意味を拡張しないと、考えられない。それで、 1/0,0/0,z/0 などの意味を発見する必要がある。 それらの意味は、普通の意味ではないことの 初めの考えを飛ばして ダメ、ダメの感情が 突っ走ている。 非ユークリッド幾何学の出現や天動説が地動説に変わった世界史の事件のような 形相と言える。
最近、3つの事が在りました。
私の簡単な講演、相当な数学者が信じられないような誤解をして、全然理解できなく、目が回っているいるような印象を受けたこと、
相当ゼロ除算の研究をされている方が、基本を誤解されていたこと、1/0 の定義を誤解されていた。
相当な才能の持ち主が、連続性や順序に拘って、４年以上もゼロ除算の研究を避けていたこと。
これらのことは、人間如何に予断と偏見にハマった存在であるかを教えている。
まずは ゼロ除算は不可能であるの 思いが強すぎで、初めからダメ、考えない、無視の気持ちが、強い。 ゼロ除算を従来の 掛け算の逆と考えると、不可能であるが 証明されてしまうので、割り算の意味を拡張しないと、考えられない。それで、 1/0,0/0,z/0 などの意味を発見する必要がある。 それらの意味は、普通の意味ではないことの 初めの考えを飛ばして ダメ、ダメの感情が 突っ走ている。 非ユークリッド幾何学の出現や天動説が地動説に変わった世界史の事件のような 形相と言える。
２０１８．９．２２．６：４１
ゼロ除算の４つの誤解：
ゼロ除算の４つの誤解：
１． ゼロでは割れない、ゼロ除算は 不可能である との考え方に拘って、思考停止している。 普通、不可能であるは、考え方や意味を拡張して 可能にできないかと考えるのが 数学の伝統であるが、それができない。
２． 可能にする考え方が 紹介されても ゼロ除算の意味を誤解して、繰り返し間違えている。可能にする理論を 素直に理解しない、 強い従来の考えに縛られている。拘っている。
３． ゼロ除算を関数に適用すると 強力な不連続性を示すが、連続性のアリストテレス以来の 連続性の考えに囚われていて 強力な不連続性を受け入れられない。数学では、不連続性の概念を明確に持っているのに、不連続性の凄い現象に、ゼロ除算の場合には 理解できない。
４． 深刻な誤解は、ゼロ除算は本質的に定義であり、仮定に基づいているので 疑いの気持ちがぬぐえず、ダメ、怪しいと誤解している。数学が公理系に基づいた理論体系のように、ゼロ除算は 新しい仮定に基づいていること。 定義に基づいていることの認識が良く理解できず、誤解している。
George Gamow (1904-1968) Russian-born American nuclear physicist and cosmologist remarked that "it is well known to students of high school algebra" that division by zero is not valid; and Einstein admitted it as {\bf the biggest blunder of his life} [1]：1. Gamow, G., My World Line (Viking, New York). p 44, 1970.
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