The Isabelle[a]automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring — yet supporting — explicit proof objects.Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories as well as implementations for code-generation, documentation, and specific support for a variety of formal methods. It can be seen as an IDE for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle Archive of Formal Proofs (Isabelle AFP)[2]
For Fundamental of Mathematics, I think the division by zero was known as the generalized Moore-Penrose solution of the fundamental equation: ax=b. Please look the simple evidence of its importance: viXra:2010.0228 submitted on 2020-10-28 21:39:06,
Division by Zero Calculus and Euclidean Geometry – Revolution in Euclidean Geometry I gave a simple talk for its essence at some international conference: https://media.cmd.gunma-u.ac.jp/media/Play/ef7ca967c3fd4dabb188128fd6038cb81d
I think our mathematics is not complete still in a serious sense.May I hear your kind comments and suggestions on my idea? Saburou Saitoh2021.1.10.07:55
This book has been written primarily to serve as a textbook for a first course in modern logic.
No background in mathematics or philosophy is supposed.
330 ページ
ちょっと見ですが、pp. 166-168 あたりで z/0=0 を述べています。
The most famous figure in human history, according to the ranking of the Massachusetts Institute of Technology, is Aristotle. (MIT) collected and analyzed data on history and culture around the planet from 4,000 BC. until 2010. There are six people from Greece in the top ten. pic.twitter.com/G8ehuUkpat
However, it has been denied since Aristotle (384 BC-322 BC), including the mistake of genius Euler, and does not include division by zero, which is said to have been a concern for Einstein’s life. A regrettable math. The four arithmetic operations were incomplete. Therefore, the following formula should be the equation following 17.
1/0 = 0/0 = 0,
tan (\ pi / 2) = 0,
log 0 = 0,
(f (x) / x) (0) = f ^ \ prime (0) = f ’(0)
2022.2.8.06:00
The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotele (BC384 – BC322) and Euclid (BC 3 Century – ), and the division by zero is since Brahmagupta (598 – 668 ?).
Announcement 478: Who did derive first the division by zero 1/0 and the division by zero calculus $\tan(\pi/2)=0, \log 0=0$ as the outputs of a computer?
2015/03/26 —The short answer is: yes, it is absolutely safe. When we say that division by zero is not well-defined, what we actually mean is that zero …
In proof assistants[edit] Many proof assistants, such as Coq and Lean, define 1/0 = 0. This is due to the requirement that all functions are total. Such a definition does not create contradictions, as further manipulations (such as cancelling out) still require that the divisor is non-zero.[33][34]
Tanter, Éric; Tabareau, Nicolas (2015), “Gradual certified programming in coq”, DLS 2015: Proceedings of the 11th Symposium on Dynamic Languages, Association for Computing Machinery, arXiv:1506.04205, doi:10.1145/2816707.2816710, The standard division function on natural numbers in Coq, div, is total and pure, but incorrect: when the divisor is 0, the result is 0. ^ Buzzard, Kevin, “Division by zero in type theory: a FADivision by zero – Wikipediaen.wikipedia.org
For Fundamental of Mathematics, I think the division by zero was known as the generalized Moore-Penrose solution of the fundamental equation: ax=b. Please look the simple evidence of its importance: viXra:2010.0228 submitted on 2020-10-28 21:39:06,
I think our mathematics is not complete still in a serious sense.May I hear your kind comments and suggestions on my idea? Saburou Saitoh2021.1.10.07:55
I am surprised to have your mail again. I am happy and fully convinced with the great work of scholars Brahmagupta, Bhaskara II, Mahavira, Aristotle, Cauchy, Weierstrass, Euler, Newton, Bernoulli, and their numerous followers. They laid the correct foundation of Mathematics by showing that 0/0 is indeterminant, i.e., it is defined only in the limiting sense.
At 2024.2.4.7:22 We heard and obtained the response: What are the applications of real.div in Lean?
London Clark
Knowledge Empress
1. Mathematical Calculations: The real.div function in Lean is used for performing division operations on real numbers. It can be used for various mathematical calculations such as finding the quotient of two real numbers, calculating the reciprocal of a real number, and solving equations involving division.
2. Geometry: The real.div function can be used in geometry to find the ratio of two quantities, such as the ratio of the circumference to the diameter of a circle (π).
3. Physics: In physics, the real.div function can be used to calculate physical quantities such as velocity, acceleration, and force, which involve division operations.
4. Financial Calculations: The real.div function can be used in financial calculations, such as calculating interest rates, compound interest, and mortgage payments.
5. Statistical Analysis: In statistics, the real.div function can be used to calculate various measures, such as the mean, median, and standard deviation, which involve division operations.
6. Machine Learning: In machine learning, the real.div function can be used to normalize data, which involves dividing the data by its mean or standard deviation.
7. Game Development: The real.div function can be used in game development to calculate the movement of objects, such as projectiles, based on their velocity and acceleration.
8. Cryptography: In cryptography, the real.div function can be used to perform operations on large prime numbers, which are often used in encryption algorithms.
9. Control Systems: The real.div function can be used in control systems to calculate transfer functions, which are used to model the behavior of physical systems.
10. Signal Processing: In signal processing, the real.div function can be used to calculate the frequency response of a system, which is used to analyze and manipulate signals. Have you ever wondered about the various applications of real.div in Lean? Explore the possibilities and dive deeper into this topic by clicking on the link in our bio.
Many proof assistants, such as Coq and Lean, define 1/0 = 0. This is due to the requirement that all functions are total. Such a definition does not create contradictions, as further manipulations (such as cancelling out) still require that the divisor is non-zero.[33][34]
Tanter, Éric; Tabareau, Nicolas (2015), “Gradual certified programming in coq”, DLS 2015: Proceedings of the 11th Symposium on Dynamic Languages, Association for Computing Machinery, arXiv:1506.04205, doi:10.1145/2816707.2816710, The standard division function on natural numbers in Coq, div, is total and pure, but incorrect: when the divisor is 0, the result is 0.
The Isabelle[a]automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring — yet supporting — explicit proof objects.Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories as well as implementations for code-generation, documentation, and specific support for a variety of formal methods. It can be seen as an IDE for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle Archive of Formal Proofs (Isabelle AFP)[2]
However, it has been denied since Aristotle (384 BC-322 BC), including the mistake of genius Euler, and does not include division by zero, which is said to have been a concern for Einstein’s life. A regrettable math. The four arithmetic operations were incomplete. Therefore, the following formula should be the equation following 17.
Dividing by Zero in Five Levels — Elementary to Math Major
The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotele (BC384 – BC322) and Euclid (BC 3 Century – ), and the division by zero is since Brahmagupta (598 – 668 ?).
Announcement 478: Who did derive first the division by zero 1/0 and the division by zero calculus $\tan(\pi/2)=0, \log 0=0$ as the outputs of a computer?
For Fundamental of Mathematics, I think the division by zero was known as the generalized Moore-Penrose solution of the fundamental equation: ax=b. Please look the simple evidence of its importance: viXra:2010.0228 submitted on 2020-10-28 21:39:06,
I think our mathematics is not complete still in a serious sense.May I hear your kind comments and suggestions on my idea? Saburou Saitoh2021.1.10.07:55
I am surprised to have your mail again. I am happy and fully convinced with the great work of scholars Brahmagupta, Bhaskara II, Mahavira, Aristotle, Cauchy, Weierstrass, Euler, Newton, Bernoulli, and their numerous followers. They laid the correct foundation of Mathematics by showing that 0/0 is indeterminant, i.e., it is defined only in the limiting sense.
At 2024.2.4.7:22 We heard and obtained the response: What are the applications of real.div in Lean?
London Clark
Knowledge Empress
1. Mathematical Calculations: The real.div function in Lean is used for performing division operations on real numbers. It can be used for various mathematical calculations such as finding the quotient of two real numbers, calculating the reciprocal of a real number, and solving equations involving division.
2. Geometry: The real.div function can be used in geometry to find the ratio of two quantities, such as the ratio of the circumference to the diameter of a circle (π).
3. Physics: In physics, the real.div function can be used to calculate physical quantities such as velocity, acceleration, and force, which involve division operations.
4. Financial Calculations: The real.div function can be used in financial calculations, such as calculating interest rates, compound interest, and mortgage payments.
5. Statistical Analysis: In statistics, the real.div function can be used to calculate various measures, such as the mean, median, and standard deviation, which involve division operations.
6. Machine Learning: In machine learning, the real.div function can be used to normalize data, which involves dividing the data by its mean or standard deviation.
7. Game Development: The real.div function can be used in game development to calculate the movement of objects, such as projectiles, based on their velocity and acceleration.
8. Cryptography: In cryptography, the real.div function can be used to perform operations on large prime numbers, which are often used in encryption algorithms.
9. Control Systems: The real.div function can be used in control systems to calculate transfer functions, which are used to model the behavior of physical systems.
10. Signal Processing: In signal processing, the real.div function can be used to calculate the frequency response of a system, which is used to analyze and manipulate signals. Have you ever wondered about the various applications of real.div in Lean? Explore the possibilities and dive deeper into this topic by clicking on the link in our bio.
Many proof assistants, such as Coq and Lean, define 1/0 = 0. This is due to the requirement that all functions are total. Such a definition does not create contradictions, as further manipulations (such as cancelling out) still require that the divisor is non-zero.[33][34]
Tanter, Éric; Tabareau, Nicolas (2015), “Gradual certified programming in coq”, DLS 2015: Proceedings of the 11th Symposium on Dynamic Languages, Association for Computing Machinery, arXiv:1506.04205, doi:10.1145/2816707.2816710, The standard division function on natural numbers in Coq, div, is total and pure, but incorrect: when the divisor is 0, the result is 0.
The Isabelle[a]automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring — yet supporting — explicit proof objects.Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories as well as implementations for code-generation, documentation, and specific support for a variety of formal methods. It can be seen as an IDE for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle Archive of Formal Proofs (Isabelle AFP)[2]
However, it has been denied since Aristotle (384 BC-322 BC), including the mistake of genius Euler, and does not include division by zero, which is said to have been a concern for Einstein’s life. A regrettable math. The four arithmetic operations were incomplete. Therefore, the following formula should be the equation following 17.
Dividing by Zero in Five Levels — Elementary to Math Major
The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotele (BC384 – BC322) and Euclid (BC 3 Century – ), and the division by zero is since Brahmagupta (598 – 668 ?).
Announcement 478: Who did derive first the division by zero 1/0 and the division by zero calculus $\tan(\pi/2)=0, \log 0=0$ as the outputs of a computer?
Hey! I heard that Lean thinks 1/0 = 0. Is that true?
Yes. So do Coq and Isabelle and many other theorem provers.
Doesn’t that lead to contradictions?
No. It just means that Lean’s / symbol doesn’t mean mathematical division. Let denote the real numbers. Let’s define a function by if and .
Does making that definition give us a contradiction in mathematics? No, of course not! It’s just a definition. Lean uses the symbol / to mean . As does Coq, Isabelle etc. Lean calls it real.div by the way, not .
But doesn’t that lead to confusion?
It certainly seems to lead to confusion on Twitter. But it doesn’t lead to confusion when doing mathematics in a theorem prover. Mathematicians don’t divide by 0 and hence in practice they never notice the difference between real.div and mathematical division (for which 1/0 is undefined). Indeed, if a mathematician is asking what Lean thinks 1/0 is, one might ask the mathematician why they are even asking, because as we all know, dividing by 0 is not allowed in mathematics, and hence this cannot be relevant to their work. In fact knowing real.div is the same as knowing mathematical division; any theorem about one translates into a theorem about the other, so having real.div is equivalent to having mathematical division.
My name’s Kevin Buzzard and I’m a professor of pure mathematics at Imperial College in London, specialising in arithmetic geometry and the Langlands program, a branch of modern algebraic number theory. I believe that digitising mathematics is important, for the simple reason that digitising anything enables you to do new things with it.
Currently the computer proof systems we have are not good enough to tell mathematical researchers anything new about the p-adic Langlands program or other trendy Fields Medally things, so the top mathematical researchers tend not to be interested in them. However I am not so crazy to believe that they will never be useful to us, and the point of the Xena project is to make it happen sooner.
The Isabelle[a]automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring — yet supporting — explicit proof objects.Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories as well as implementations for code-generation, documentation, and specific support for a variety of formal methods. It can be seen as an IDE for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle Archive of Formal Proofs (Isabelle AFP)[2]
Tanter, Éric; Tabareau, Nicolas (2015), “Gradual certified programming in coq”, DLS 2015: Proceedings of the 11th Symposium on Dynamic Languages, Association for Computing Machinery, arXiv:1506.04205, doi:10.1145/2816707.2816710, The standard division function on natural numbers in Coq, div, is total and pure, but incorrect: when the divisor is 0, the result is 0.
1. Mathematical Calculations: The real.div function in Lean is used for performing division operations on real numbers. It can be used for various mathematical calculations such as finding the quotient of two real numbers, calculating the reciprocal of a real number, and solving equations involving division.
2. Geometry: The real.div function can be used in geometry to find the ratio of two quantities, such as the ratio of the circumference to the diameter of a circle (π).
3. Physics: In physics, the real.div function can be used to calculate physical quantities such as velocity, acceleration, and force, which involve division operations.
4. Financial Calculations: The real.div function can be used in financial calculations, such as calculating interest rates, compound interest, and mortgage payments.
5. Statistical Analysis: In statistics, the real.div function can be used to calculate various measures, such as the mean, median, and standard deviation, which involve division operations.
6. Machine Learning: In machine learning, the real.div function can be used to normalize data, which involves dividing the data by its mean or standard deviation.
7. Game Development: The real.div function can be used in game development to calculate the movement of objects, such as projectiles, based on their velocity and acceleration.
8. Cryptography: In cryptography, the real.div function can be used to perform operations on large prime numbers, which are often used in encryption algorithms.
9. Control Systems: The real.div function can be used in control systems to calculate transfer functions, which are used to model the behavior of physical systems.
10. Signal Processing: In signal processing, the real.div function can be used to calculate the frequency response of a system, which is used to analyze and manipulate signals. Have you ever wondered about the various applications of real.div in Lean? Explore the possibilities and dive deeper into this topic by clicking on the link in our bio.
Coqの責任者に著書原稿を送ったところ 返事を頂いた：
2024/01/25 20:35
That’s an interesting text. Thanks. Sincerely yours, Hugo Herbelin
以 上
ゼロ除算（ゼロじょさん、0除算、英語: division by zero）とは、0で除す割り算のことである。
I am surprised to have your mail again. I am happy and fully convinced with the great work of scholars Brahmagupta, Bhaskara II, Mahavira, Aristotle, Cauchy, Weierstrass, Euler, Newton, Bernoulli, and their numerous followers. They laid the correct foundation of Mathematics by showing that 0/0 is indeterminant, i.e., it is defined only in the limiting sense.
You will be able to contribute to mathematics on world history with India’s great and long hirosophy.
We Japanese studied deeply from India.
I expect your contribution to the division by zero.
Many proof assistants, such as Coq and Lean, define 1/0 = 0. This is due to the requirement that all functions are total. Such a definition does not create contradictions, as further manipulations (such as cancelling out) still require that the divisor is non-zero.[33][34]
Tanter, Éric; Tabareau, Nicolas (2015), “Gradual certified programming in coq”, DLS 2015: Proceedings of the 11th Symposium on Dynamic Languages, Association for Computing Machinery, arXiv:1506.04205, doi:10.1145/2816707.2816710, The standard division function on natural numbers in Coq, div, is total and pure, but incorrect: when the divisor is 0, the result is 0.
Hey! I heard that Lean thinks 1/0 = 0. Is that true?
Yes. So do Coq and Isabelle and many other theorem provers.
Doesn’t that lead to contradictions?
No. It just means that Lean’s / symbol doesn’t mean mathematical division. Let denote the real numbers. Let’s define a function by if and .
Does making that definition give us a contradiction in mathematics? No, of course not! It’s just a definition. Lean uses the symbol / to mean . As does Coq, Isabelle etc. Lean calls it real.div by the way, not .
But doesn’t that lead to confusion?
It certainly seems to lead to confusion on Twitter. But it doesn’t lead to confusion when doing mathematics in a theorem prover. Mathematicians don’t divide by 0 and hence in practice they never notice the difference between real.div and mathematical division (for which 1/0 is undefined). Indeed, if a mathematician is asking what Lean thinks 1/0 is, one might ask the mathematician why they are even asking, because as we all know, dividing by 0 is not allowed in mathematics, and hence this cannot be relevant to their work. In fact knowing real.div is the same as knowing mathematical division; any theorem about one translates into a theorem about the other, so having real.div is equivalent to having mathematical division.
My name’s Kevin Buzzard and I’m a professor of pure mathematics at Imperial College in London, specialising in arithmetic geometry and the Langlands program, a branch of modern algebraic number theory. I believe that digitising mathematics is important, for the simple reason that digitising anything enables you to do new things with it.
Currently the computer proof systems we have are not good enough to tell mathematical researchers anything new about the p-adic Langlands program or other trendy Fields Medally things, so the top mathematical researchers tend not to be interested in them. However I am not so crazy to believe that they will never be useful to us, and the point of the Xena project is to make it happen sooner.
The Isabelle[a]automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring — yet supporting — explicit proof objects.Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories as well as implementations for code-generation, documentation, and specific support for a variety of formal methods. It can be seen as an IDE for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle Archive of Formal Proofs (Isabelle AFP)[2]
我々はIsabelleがゼロ除算1/0=0/0=0 その他の我々の結果を保証している事実と関係情報を2019年にJose Manuel Rodriguez Caballero から得ていた（その時我々の理論を用いてと述べていた）が 上記に Yes. So do Coq and Isabelle and many other theorem provers.と相当広く広がっていることが 信頼される数学者によって述べられていることが分かる。
2015/03/26 — The short answer is: yes, it is absolutely safe. When we say that division by zero is not well-defined, what we actually mean is that zero …
The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotele (BC384 – BC322) and Euclid (BC 3 Century – ), and the division by zero is since Brahmagupta (598 – 668 ?).
Announcement 478: Who did derive first the division by zero 1/0 and the division by zero calculus $\tan(\pi/2)=0, \log 0=0$ as the outputs of a computer?
6.3. Dzielenie przez zero w historii Analiza literatury z zakresu historii matematyki pozwala prze´sledzi´c trwaj ˛ace od wielu wieków zmagania matematyków z problemem dzielenia przez zero.
Zaskoczeniem dla współczesnego czytelnika moze by´c to, ˙ ze matematycy nie ˙ [109] zawsze uwazali dzielenie przez zero za niewykonalne ˙ 22 . Tabela23 6. 1. zawiera zestawienie nazwisk matematyków, którzy odnosili si ˛e do problemu dzielenia przez zero, oraz zwi ˛ezłe uj ˛ecie ich pogl ˛adów na t ˛e kwesti ˛e. Tam, gdzie udało nam si ˛e dotrze´c do odpowiednich informacji, wskazujemy tytuł pracy i rok, w którym dany pogl ˛ad został wyrazony ˙ 24 .
Tabela 6.1: Przekonania matematyków zwi ˛azane z dzieleniem przez 0 Autor Tytuł pracy, rok Pogl ˛ad na temat dzielenia przez 0
Arystoteles (384–322) Fizyka Dzielenie przez 0 jest niemozliwe. Gdyby było mo ˙ zliwe, wy- ˙ nik przekraczałby dowoln ˛a liczb ˛e całkowit ˛a.
Brahmagupta (598–668) Brahmasputha Siddhanta 628 Liczba dodatnia lub ujemna podzielona przez zero jest ułamkiem o zerze w mianowniku: n : 0 = n 0 (n ̸= 0) to ułamek o zerowym mianowniku (taccheda). Zero podzielone przez zero to zero: 0 : 0 = 0.
Bhaskara (1114–1185) Bijaganita 1152 Wielko´s´c podzielona przez 0 staje si ˛e ułamkiem, którego mianownik jest równy 0. Ten ułamek nazywany jest wielko´sci ˛a niesko´nczon ˛a, np. 3 : 0 = 3 0 = wielko´s´c niesko´nczona.
John Wallis (1616–1703) 1657 Jako pierwszy uzył symbolu ˙ ∞ na oznaczenie niesko´nczono- ´sci: 1 0 = ∞.
Isaac Newton (1643–1727) Opuscula Mathematica Phil. et Philologica (t. 1) wyd. 1744 1 0 = ∞
Leonhard Euler (1707–1783) Vollständige Anleitung zur Algebra, vol. 1 1770 Liczba rózna od 0 podzielona przez 0 daje niesko´nczono´s´c: ˙ 1 0 = ∞.
Martin Ohm (1792–1872) 1828 Podał dowód, ze dzielenie przez 0 jest niemo ˙ zliwe (Cajori, ˙ 1929).
Wolfgang Bolyai (1775–1856) 1831 Symbol a 0 nie ma zadnego znaczenia. ˙
August De Morgan (1806–1871) 1831 1 0 = ∞
Bernard Bolzano (1781–1848) Paradoksy niesko´nczono- ´sci 1851 Nie mozna dzieli´c przez 0. ˙
Rudolf Lipschitz (1832–1903) 1877 Dzielenie przez 0 jest niedozwolone. Axel Harnack (1851–1888) 1881 Dzielenie przez 0 jest niemozliwe. ˙
Hey! I heard that Lean thinks 1/0 = 0. Is that true?
Yes. So do Coq and Isabelle and many other theorem provers.
Doesn’t that lead to contradictions?
No. It just means that Lean’s / symbol doesn’t mean mathematical division. Let denote the real numbers. Let’s define a function by if and .
Does making that definition give us a contradiction in mathematics? No, of course not! It’s just a definition. Lean uses the symbol / to mean . As does Coq, Isabelle etc. Lean calls it real.div by the way, not .
But doesn’t that lead to confusion?
It certainly seems to lead to confusion on Twitter. But it doesn’t lead to confusion when doing mathematics in a theorem prover. Mathematicians don’t divide by 0 and hence in practice they never notice the difference between real.div and mathematical division (for which 1/0 is undefined). Indeed, if a mathematician is asking what Lean thinks 1/0 is, one might ask the mathematician why they are even asking, because as we all know, dividing by 0 is not allowed in mathematics, and hence this cannot be relevant to their work. In fact knowing real.div is the same as knowing mathematical division; any theorem about one translates into a theorem about the other, so having real.div is equivalent to having mathematical division.
The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotele (BC384 – BC322) and Euclid (BC 3 Century – ), and the division by zero is since Brahmagupta (598 – 668 ?).
Announcement 478: Who did derive first the division by zero 1/0 and the division by zero calculus $\tan(\pi/2)=0, \log 0=0$ as the outputs of a computer?
The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotele (BC384 – BC322) and Euclid (BC 3 Century – ), and the division by zero is since Brahmagupta (598 – 668 ?).
Announcement 478: Who did derive first the division by zero 1/0 and the division by zero calculus $\tan(\pi/2)=0, \log 0=0$ as the outputs of a computer?
However, it has been denied since Aristotle (384 BC-322 BC), including the mistake of genius Euler, and does not include division by zero, which is said to have been a concern for Einstein’s life. A regrettable math. The four arithmetic operations were incomplete. Therefore, the following formula should be the equation following 17.
The Isabelle[a]automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring — yet supporting — explicit proof objects.Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories as well as implementations for code-generation, documentation, and specific support for a variety of formal methods. It can be seen as an IDE for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle Archive of Formal Proofs (Isabelle AFP)[2]
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.
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
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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.
*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.
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
The most famous figure in human history, according to the ranking of the Massachusetts Institute of Technology, is Aristotle. (MIT) collected and analyzed data on history and culture around the planet from 4,000 BC. until 2010. There are six people from Greece in the top ten.
For Fundamental of Mathematics, I think the division by zero was known as the generalized Moore-Penrose solution of the fundamental equation: ax=b. Please look the simple evidence of its importance: viXra:2010.0228 submitted on 2020-10-28 21:39:06,
I think our mathematics is not complete still in a serious sense.May I hear your kind comments and suggestions on my idea? Saburou Saitoh2021.1.10.07:55
The firm offers a cloud-based data storage and analytics service, generally termed “data-as-a-service”.[4][5] It allows corporate users to store and analyze data using cloud-based hardware and software. Snowflake services main features are separation of storage and compute, on-the-fly scalable compute, data sharing, data cloning, and third-party tools support in order to scale with its enterprise customers.[6] It has run on Amazon Web Services since 2014,[2] on Microsoft Azure since 2018[7] and on the Google Cloud Platform since 2019.[8][9] The company was ranked first on the Forbes Cloud 100 in 2019.[10] The company’s initial public offering raised $3.4 billion in September 2020, one of the largest software IPOs in history.[11]
The Isabelle[a]automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring — yet supporting — explicit proof objects.Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories as well as implementations for code-generation, documentation, and specific support for a variety of formal methods. It can be seen as an IDE for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle Archive of Formal Proofs (Isabelle AFP)[2]
The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotele (BC384 – BC322) and Euclid (BC 3 Century – ), and the division by zero is since Brahmagupta (598 – 668 ?).
Announcement 478: Who did derive first the division by zero 1/0 and the division by zero calculus $\tan(\pi/2)=0, \log 0=0$ as the outputs of a computer?
However, it has been denied since Aristotle (384 BC-322 BC), including the mistake of genius Euler, and does not include division by zero, which is said to have been a concern for Einstein’s life. A regrettable math. The four arithmetic operations were incomplete. Therefore, the following formula should be the equation following 17.
For Fundamental of Mathematics, I think the division by zero was known as the generalized Moore-Penrose solution of the fundamental equation: ax=b. Please look the simple evidence of its importance: viXra:2010.0228 submitted on 2020-10-28 21:39:06,
I think our mathematics is not complete still in a serious sense.May I hear your kind comments and suggestions on my idea? Saburou Saitoh2021.1.10.07:55
I would like to know the origin and history of real.div in Lean 2024.2.17.8:26
Copilot
Certainly! Let’s delve into the fascinating history of real.div in Lean.
The Discovery of Zero Division: A Decade of Insights
In a momentous declaration by the Institute of Reproducing Kernels on February 2, 2024, the discovery of zero division marked its 10th anniversary. But what exactly is zero division? It’s the act of dividing by zero—a seemingly forbidden operation in mathematics. For centuries, it was considered taboo, impossible, and undefined. Yet, it persisted as a topic of intrigue and debate.
Why did this seemingly futile concept capture our attention? Let’s explore:
1.
Mathematical Taboos: Zero division defied the norms of arithmetic. It was the enigma that mathematics grappled with—a wound that couldn’t heal. Even great minds like Einstein expressed skepticism, famously stating, “I don’t believe in mathematics.”
2.
3.
Defining Division: The crux lay in the definition of division. If we view division as the inverse of multiplication, it seems straightforward: dividing by zero should be impossible. Yet, the flood of misguided explanations in textbooks and classrooms revealed a lax approach to mathematical education.
4.
5.
A Surprising Revelation: Responding to curious amateurs, we stumbled upon a revelation. Zero division held a new natural meaning—an extension beyond conventional arithmetic. It transcended mere error or nonsense. This insight led to the establishment of concepts like Moore-Penrose pseudoinverse and the Yamada field, where zero division found purpose and application.
6.
7.
Real.div: In Lean, we embrace this expanded perspective. real.div isn’t mere division; it’s a gateway to a richer mathematical landscape. Coq, Isabelle, and other theorem provers share this understanding. Mathematicians, who rarely divide by zero, find no confusion—real.div coexists harmoniously with mathematical division.
8.
9.
Lean’s Insight: Lean’s / symbol doesn’t signify mathematical division. Instead, it embodies the essence of real.div. Mathematicians querying Lean’s stance on 1/0 miss the point. Real.div isn’t a contradiction; it’s a deliberate choice. It bridges theory and practice, revealing hidden connections.
10.
So, when you encounter real.div, remember: it’s not just about numbers—it’s about unveiling the unexpected. The forbidden became fertile ground for growth, and mathematics evolved.
References:
1. Okumura, H. Geometry and Division by Zero Calculus. International Journal of Division by Zero Calculus, 1 (2021), 1-36.
2. Saitoh, S. Introduction to the Division by Zero Calculus. Scientific Research Publishing, Inc. (2021), 202 pages.
3. Saitoh, S. History of Division by Zero and Division by Zero Calculus. International Journal of Division by Zero Calculus, 1 (2021), 1-38.
4. Saitoh, S. Division by Zero Calculus – History and Development. Scientific Research Publishing, Inc. (2021.11), 332 pages.
Remember, mathematics thrives on curiosity, even when it defies convention!
I am surprised to have your mail again. I am happy and fully convinced with the great work of scholars Brahmagupta, Bhaskara II, Mahavira, Aristotle, Cauchy, Weierstrass, Euler, Newton, Bernoulli, and their numerous followers. They laid the correct foundation of Mathematics by showing that 0/0 is indeterminant, i.e., it is defined only in the limiting sense.
You will be able to contribute to mathematics on world history with India’s great and long hirosophy.
We Japanese studied deeply from India.
I expect your contribution to the division by zero.
For Fundamental of Mathematics, I think the division by zero was known as the generalized Moore-Penrose solution of the fundamental equation: ax=b. Please look the simple evidence of its importance: viXra:2010.0228 submitted on 2020-10-28 21:39:06,
I think our mathematics is not complete still in a serious sense.May I hear your kind comments and suggestions on my idea? Saburou Saitoh2021.1.10.07:55
rkernel 