Leibniz: Letzter Universalgelehrte starb vor 300 Jahren

Leibniz: Letzter Universalgelehrte starb vor 300 Jahren

Gottfried Wilhelm Leibniz war Entdecker, Erfinder und Weltverbesserer. Das Universalgenie galt schon zu Lebzeiten als einer der wichtigsten Gelehrten der Frühaufklärung. Als er am 14. November 1716 siebzigjährig in Hannover starb, hinerließ er eine Zettelwirtschaft von 200.000 Seiten. Dabei hatte "ich beim Erwachen schon so viele Einfälle, dass der Tag nicht ausreichte, um sie niederzuschreiben“. Sein Werken wirkt immer noch nach.

Schmackhaftes Nachwirken

Es ist schon etwas unfair: Gottfried Wilhelm Leibniz (1646-1716) beschäftigte sich mit vielfältigen Themen wie Gesundheitspolitik, Theologie und der Entstehung der Erde. Viele Menschen denken bei dem Namen trotzdem zunächst an Kekse. Das süße Nachwirken ist Hermann Bahlsen zu danken.

Foto: dpa/Holger HollemannDer Unternehmensgründer aus Hannover brachte 1891 die „Leibniz Cakes“ auf den Markt und benannte sie nach dem berühmten Hofrat, der Jahrzehnte in Hannover wirkte. Ende des 19. Jahrhunderts war es durchaus üblich, Lebensmittel nach bekannten Persönlichkeiten zu benennen. Populär wurden etwa auch Bismarckhering, Schillerlocke oder Mozartkugel.

Mehr als nur ein Keks

Gottfried Wilhelm Leibniz wurde am 21. Juni (nach gregorianischem Kalender am 1. Juli) 1646 in Leipzig als Sohn eines Professors geboren. Zeit Lebens sprudelte er über vor Ideen und suchte Lösungen für die großen Fragen der Menschheit. Unerschütterlich war er getrieben von Optimismus. Der Philosoph, Mathematiker und Fürstenberater war bis zu seinem Tod überzeugt davon, die Welt verbessern zu können. Er wollte die Spaltung der Kirche überwinden und entwickelte eine Universalsprache, um Missverständnisse zwischen den Völkern zu beenden.Foto: APA/EPA/HAUKE-CHRISTIAN DITTRICH

„Seine Visionen auf unterschiedlichsten Gebieten inspirieren Wissenschaftler bis heute“, sagt Michael Kempe, Leiter des Leibniz-Archivs in Hannover. So sei kürzlich ein Software-Entwickler aus den USA angereist, um in Leibniz' Schriften Anregungen für neue Algorithmen zu finden. Die niedersächsische Gottfried Wilhelm Leibniz Bibliothek verwahrt den Nachlass des großen Denkers. Etwa 200.000 handschriftlich beschriebene Seiten lagern hier, darunter der zum Unesco-Welterbe gehörende Briefwechsel mit 1300 Briefpartnern rund um den Erdball. Leibniz dachte global und suchte die Nähe zu Russland und China, um von anderen Kulturen zu lernen.

Foto: AP/KAI-UWE KNOTHBriefe von Leibniz sind UNESCO-Weltdokumentenerbe.In der Mathematik war der hochbegabte Wissenschaftler, der 40 Jahre lang als Hofrat und Bibliothekar des Welfenherzogs in Hannover wirkte, seiner Zeit weit voraus. Ohne das von Leibniz beschriebene Dualsystem gäbe es keine Computer. Seine Überlegung, dass Raum nichts Absolutes ist, verweist bereits auf Einsteins Relativitätstheorie. Mit Isaac Newton stritt er darüber, wer von beiden die Differential- und Integralrechnung erfunden hat.

Leibniz beschränkte sich nicht auf die Theorie, er war ein Forscher mit Hang zum Abenteuer. Der Hofrat kraxelte durch die Bergwerke des Harzes und konstruierte Windmühlen zum Antrieb von Pumpen. Weil ihm das Schreiben in der wackligen Postkutsche schwer fiel, entwarf er bequeme Reisesitze und Kabinen. Stets hatte der Frühaufklärer das große Ganze im Blick: Seine nie vollendete Geschichte der Welfen im Auftrag des Hofes begann Leibniz mit einer Abhandlung über die Entstehung der Erde. Grundlage waren auch eigene Fossilien wie ein versteinerter Mammutzahn.

„Er hat assoziativ gearbeitet und alles auf einmal gemacht“, erzählt Kempe. So kann sich auf einem einzigen Blatt eine technische Zeichnung, eine philosophische Idee, eine mathematische Formel und Klatsch vom Hofe finden. Später zerschnitt der Junggeselle, der gern seinen Wein mit Kirschsaft panschte, solche Zettel. Derzeit wird Leibniz' Werk „Mathematica“, das aus über 7000 Schnipseln besteht, mit modernster Computertechnik zusammengesetzt.Foto: APA/EPA/HAUKE-CHRISTIAN DITTRICHLeibniz bewohnte das Haus in Hannover von 1698 bis zu seinem Tod.

Das Gesamtwerk des Multi-Talents ist noch lange nicht erschlossen. Es ist vermutlich der größte Gelehrtennachlass der Weltgeschichte. Die bereits 1923 begonnene Gesamtedition der Schriften wird voraussichtlich bis zum Jahr 2055 dauern. Kempe glaubt, dass weitere Überraschungen in den Überlieferungen schlummern: „Leibniz hält Antworten bereit auf Fragen, die wir im Moment noch gar nicht stellen.“

Die vielen Veranstaltungen im Jubiläumsjahr haben das Universalgenie bekannter gemacht, ist der Leibniz-Experte Georg Ruppelt überzeugt. „Noch immer halten ihn mehr Menschen für einen Keksbäcker als für den Erfinder der Integral- und Differentialrechnung. Aber das ändert sich.“ Leibniz habe nicht nur Grundlagen für die digitale Welt gelegt. „Er war auch jemand, der den Ausgleich und den Frieden suchte.“https://kurier.at/wissen/gottfried-wilhelm-leibniz-letzter-universalgelehrte-starb-vor-300-jahren/230.440.413

記号の神様

\documentclass[12pt]{article}

\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}

\numberwithin{equation}{section}

\begin{document}

\title{\bf  Announcement 326:   The division by zero z/0=0 - its impact to human beings through education and research\\

(2016.10.17)}

\author{{\it Institute of Reproducing Kernels}\\

Kawauchi-cho, 5-1648-16,\\

Kiryu 376-0041, Japan\\

 }

\date{\today}

\maketitle

{\bf Abstract: } In this announcement, for its importance we would like to state the

situation on the division by zero and propose basic new challenges to education and research on our wrong world history.

\bigskip

\section{Introduction}

%\label{sect1}

By a {\bf natural extension} of the fractions

\begin{equation}

\frac{b}{a}

\end{equation}

for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$

\begin{equation}

\frac{b}{0}=0,

\end{equation}

incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the  case of real numbers.

 The division by zero has a long and mysterious story over the world (see, for example, Google site with the division by zero) with its physical viewpoints since the document of zero in India on AD 628,  however,

  Sin-Ei Takahasi (\cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing the extensions of fractions and by showing the complete characterization for the property (1.2):

 \bigskip

 {\bf  Proposition 1. }{\it Let F be a function from  ${\bf C }\times {\bf C }$  to ${\bf C }$ satisfying

$$

F (b, a)F (c, d)= F (bc, ad)

$$

for all

$$

a, b, c, d  \in {\bf C }

$$

and

$$

F (b, a) = \frac {b}{a },  \quad   a, b  \in  {\bf C }, a \ne 0.

$$

Then, we obtain, for any $b \in {\bf C } $

$$

F (b, 0) = 0.

$$

}

 Note that the complete proof of this proposition is simply given by  2 or 3 lines.

We should define $F(b,0)= b/0 =0$, in general.

\medskip

We thus should consider, for any complex number $b$, as  (1.2);

that is, for the mapping

\begin{equation}

W = \frac{1}{z},

\end{equation}

the image of $z=0$ is $W=0$ ({\bf should be defined}). This fact seems to be a curious one in connection with our well-established popular image for the  point at infinity on the Riemann sphere. Therefore, the division by zero will give great impact to complex analysis and to our ideas for the space and universe.

However, the division by zero (1.2) is now clear, indeed, for the introduction of (1.2), we have several independent approaches as in:

\medskip

1) by the generalization of the fractions by the Tikhonov regularization and by the Moore-Penrose generalized inverse,

\medskip

2) by the intuitive meaning of the fractions (division) by H. Michiwaki - repeated subtraction method,

\medskip

3) by the unique extension of the fractions by S. Takahasi,   as in the above,

\medskip

4) by the extension of the fundamental function $W = 1/z$ from ${\bf C} \setminus \{0\}$ into ${\bf C}$ such that $W =1/z$ is a one to one and onto mapping from $ {\bf C} \setminus \{0\} $ onto ${\bf C} \setminus \{0\}$ and the division by zero $1/0=0$ is a one to one and onto mapping extension of the function $W =1/z $ from  ${\bf C}$ onto ${\bf C}$,

\medskip

and

\medskip

5) by considering the values of functions with the mean values of functions.

\medskip

Furthermore, in (\cite{msy}) we gave the results in order to show the reality of the division by zero in our world:

\medskip

\medskip

A) a field structure  containing the division by zero --- the Yamada field ${\bf Y}$,

\medskip

B)  by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,

\medskip

C) by the reflection $W =1/\overline{z}$ of $W= z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero, not the point at infinity.

\medskip

and

\medskip

D) by considering rotation of a right circular cone having some very interesting

phenomenon  from some practical and physical problem.

\medskip

In (\cite{mos}),  many division by zero results in Euclidean spaces are given and  the basic idea at the point at infinity should be changed. In (\cite{ms}), we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.

\medskip

See  J. A. Bergstra, Y. Hirshfeld and J. V. Tucker \cite{bht} for the relationship between fields and the division by zero, and the importance of the division by zero for computer science. It seems that the relationship of the division by zero and field structures are abstract in their paper.

Meanwhile,  J. P.  Barukcic and I.  Barukcic (\cite{bb}) discussed recently the relation between the divisions $0/0$, $1/0$ and special relative theory of Einstein. However, their logic seems to be curious and their results contradict with ours.

 Furthermore,  T. S. Reis and J.A.D.W. Anderson (\cite{ra,ra2}) extend the system of the real numbers by introducing an ideal number for the division by zero $0/0$.

 Meanwhile, we should refer to up-to-date information:

{\it Riemann Hypothesis Addendum - Breakthrough

Kurt Arbenz

https://www.researchgate.net/publication/272022137 Riemann Hypothesis Addendum -   Breakthrough.}

\medskip

Here, we recall Albert Einstein's words on mathematics:

Blackholes are where God divided by zero.

I don't believe in mathematics.

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.

 Apparently, the division by zero is a great missing in our mathematics and the result (1.2) is definitely determined as our basic mathematics, as we see from Proposition 1.  Note  its very general assumptions and  many fundamental evidences in our world in (\cite{kmsy,msy,mos}). The results will give great impact  on Euclidean spaces, analytic geometry, calculus, differential equations, complex analysis and  physical problems.

The mysterious history of the division by zero over one thousand years is a great shame of  mathematicians and human race on the world history, like the Ptolemaic system (geocentric theory). The division by zero will become a typical  symbol of foolish human race with long and unceasing struggles. Future people will realize this fact as a definite common sense.

We should check and fill our mathematics, globally and beautifully, from the viewpoint of the division by zero. Our mathematics will be more perfect and beautiful,  and will give great impact to our basic ideas on the universe.

 For our ideas on the division by zero, see the survey style announcements.

\section{Basic Materials of Mathematics}

  (1): First, we should declare that the divison by zero is possible in the natural and uniquley determined sense and its importance.

  (2): In the elementary school, we should introduce the concept of division by the idea of repeated subtraction method by H. Michiwaki whoes method is applied in computer algorithmu and in old days for calculation of division. This method will give a simple and clear method for calculation of division and students will be happy to apply this simple method at the first stage. At this time, they will be able to understand that the division by zero is clear and trivial as $a/0=0$ for any $a$. Note that Michiwaki knows how to apply his method to the complex number field.

  (3): For the introduction of the elemetary function $y= 1/x$, we should give the definition of the function at the origin $x=0$ as $y = 0$ by the division by zero idea and we should apply this definition for the occasions of its appearences, step by step, following the curriculum and the results of the division by zero.

  (4): For the idea of the Euclidean space (plane), we should introduce, at the first stage, the concept of steleographic projection and the concept of the point at infinity  -

   one point compactification. Then, we will be able to see the whole Euclidean plane, however, by the division by zero, the point at infinity is represented by zero. We can teach  the very important fact with many geometric and analytic geometry methods. These topics will give great pleasant feelings to many students.

  Interesting topics are: parallel lines, what is a line? - a line contains the origin as an isolated

point for the case that the native line does not through the origin. All the lines pass the origin, our space is not the Eulcildean space and is not Aristoteles for the strong discontinuity at the point at infinity (at the origin). - Here note that an orthogonal coordinates should be fixed first for our all arguments.

(5): The inversion of the origin with respect to a circle with center the origin is the origin itself, not the point at infinity - the very classical result is wrong. We can also prove this elementary result by many elementary ways.

(6): We should change the concept of gradients; on the usual orthogonal coordinates $(x,y)$,

 the gradient of the $y$ axis is zero; this is given and proved by the fundamental result

 $\tan (\pi/2) =0$. The result is trivial in the definition of the Yamada field. This result is derived also from  the {\bf division by zero calculus}:

\medskip

 For any formal Laurent expansion around $z=a$,

\begin{equation}

f(z) = \sum_{n=-\infty}^{\infty} C_n (z - a)^n,

\end{equation}

we obtain the identity, by the division by zero

\begin{equation}

f(a) =  C_0.

\end{equation}

\medskip

This fundamental result leads to the important new definition:

From the viewpoint of the division by zero, when there exists the limit, at $ x$

 \begin{equation}

 f^\prime(x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h}  =\infty

 \end{equation}

 or

 \begin{equation}

 f^\prime(x) =  -\infty,

 \end{equation}

 both cases, we can write them as follows:

 \begin{equation}

  f^\prime(x) =  0.

 \end{equation}

 \medskip

 For the elementary ordinary differential equation

 \begin{equation}

 y^\prime = \frac{dy}{dx} =\frac{1}{x}, \quad x > 0,

 \end{equation}

 how will be the case at the point $x = 0$? From its general solution, with a general constant $C$

 \begin{equation}

 y = \log x + C,

 \end{equation}

 we see that, by the division by zero,

 \begin{equation}

 y^\prime (0)= \left[ \frac{1}{x}\right]_{x=0} = 0,

 \end{equation}

 that will mean that the division by zero (1.2) is very natural.

 In addition, note that the function $y = \log x$ has infinite order derivatives and all the values are zero at the origin, in the sense of the division by zero.

 However, for the derivative of the function $y = \log x$, we have to fix the sense at the origin, clearly, because the function is not differentiable, but it has a singularity at the origin. For $x >0$, there is no problem for (2.6) and (2.7). At  $x = 0$, we  see that we can not consider the limit in the sense (2.3).  However,  for $x >0$ we have (2.6) and

 \begin{equation}

 \lim_{x \to +0} \left(\log x \right)^\prime = +\infty.

 \end{equation}

 In the usual sense, the limit is $+\infty$,  but in the present case, in the sense of the division by zero, we have:

 \begin{equation}

 \left[ \left(\log x \right)^\prime \right]_{x=0}= 0

 \end{equation}

  and we will be able to understand its sense graphycally.

 By the new interpretation for the derivative, we can arrange many formulas for derivatives, by the division by zero. We can modify many formulas and statements in calculus and we can apply our concept to the differential equation theory and the universe in connetion with derivatives.

(7): We shall introduce the typical division by zero calculus.

  For the integral

\begin{equation}

\int x(x^{2}+1)^{a}dx=\frac{(x^{2}+1)^{a+1}}{2(a+1)}\quad(a\ne-1),

\end{equation}

we obtain, by the division by zero,

\begin{equation}

\int x(x^{2}+1)^{-1}dx=\frac{\log(x^{2}+1)}{2}.

\end{equation}

We will consider the fundamental ordinary differential equations

\begin{equation}

x^{\prime \prime}(t) =g -kx^{\prime}(t)

\end{equation}

with the initial conditions

\begin{equation}

x(0)  = -h, x^{\prime}(0) =0.

\end{equation}

Then we have the solution

\begin{equation}

x(t) = \frac{g}{k}t + \frac{g(e^{-kt}- 1)}{k^2} - h.

\end{equation}

Then, for $k=0$, we obtain, immediately, by the division by zero

\begin{equation}

x(t) = \frac{1}{2}g t^2 -h.

\end{equation}

In those examples, we were able to give valuable functions for denominator zero cases. The division by zero calculus may be applied to many cases as a new fundamental calculus over l'Hôpital's rule.

(8):  When we apply the division by zero to functions, we can consider, in general, many ways.  For example,

for the function $z/(z-1)$, when we insert $z=1$  in numerator and denominator, we have

\begin{equation}

\left[\frac{z}{z-1}\right]_{z = 1} = \frac{1}{0} =0.

\end{equation}

However,

from the identity --

 the Laurent expansion around $z=1$,

\begin{equation}

\frac{z}{z-1} = \frac{1}{z-1} + 1,

\end{equation}

we have

\begin{equation}

 \left[\frac{z}{z-1}\right]_{z = 1} = 1.

 \end{equation}

 For analytic functions we can give uniquely determined values at isolated singular points by the values by means of the Laurent expansions as the division by zero calculus, however, the values by means of the Laurent expansions are not always reasonable. We will need to consider many interpretations for reasonable values. In many formulas in mathematics and physics, however, we can see that the division by zero calculus is reasonably valid. See \cite{kmsy,msy}.

\section{Albert Einstein's biggest blunder}

The division by zero is directly related to the Einstein's theory and various

physical problems

containing the division by zero.  Now we should check the theory and the problems by the concept of the RIGHT and DEFINITE division by zero. Now is the best time since 100 years from Albert Einstein. It seems that the background knowledge is timely fruitful.

Note that the Big Bang also may be related to the division by zero like the blackholes.

\section{Computer systems}

The above Professors listed are wishing the contributions in order to avoid the division by zero trouble in computers. Now,  we should arrange  new computer systems in order not to meet the division by zero trouble in computer systems.

 By the division by zero calculus, we will be able to overcome troubles in Maple for specialization problems.

\section{General  ideas on the universe}

The division by zero may be related to religion,  philosophy and the ideas on the universe, and it will creat a new world. Look the new world introduced.

\bigskip

We are standing on a new  generation and in front of the new world, as in the discovery of the Americas.  Should we push the research and education on the division by zero?

 \bigskip

\bibliographystyle{plain}

\begin{thebibliography}{10}

\bibitem{bb}

J. P.  Barukcic and I.  Barukcic, Anti Aristotle—The Division of Zero by Zero. Journal of Applied Mathematics and Physics,  {\bf 4}(2016), 749-761.

doi: 10.4236/jamp.2016.44085.

\bibitem{bht}

J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,

Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).

\bibitem{cs}

L. P.  Castro and S. Saitoh,  Fractional functions and their representations,  Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.

\bibitem{kmsy}

M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,

New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,

Int. J. Appl. Math.  {\bf 27} (2014), no 2, pp. 191-198,  DOI: 10.12732/ijam.v27i2.9.

\bibitem{ms}

T. Matsuura and S. Saitoh,

Matrices and division by zero $z/0=0$, Advances in Linear Algebra

\& Matrix Theory, 6, 51-58. http://dx.doi.org/10.4236/alamt.2016.62007 http://www.scirp.org/journal/alamt　

\bibitem{msy}

H. Michiwaki, S. Saitoh,  and  M.Yamada,

Reality of the division by zero $z/0=0$.  IJAPM  International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html

\bibitem{mos}

H.  Michiwaki, H. Okumura, and S. Saitoh,

Division by Zero $z/0 = 0$ in Euclidean Spaces.

 International Journal of Mathematics and Computation

 (in press).

\bibitem{ra}

T. S. Reis and J.A.D.W. Anderson,

Transdifferential and Transintegral Calculus,

Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I

WCECS 2014, 22-24 October, 2014, San Francisco, USA

\bibitem{ra2}

T. S. Reis and J.A.D.W. Anderson,

Transreal Calculus,

IAENG  International J. of Applied Math., {\bf 45}(2015):  IJAM 45 1 06.

\bibitem{s}

S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices,  Advances in Linear Algebra \& Matrix Theory.  {\bf 4}  (2014), no. 2,  87--95. http://www.scirp.org/journal/ALAMT/

\bibitem{ttk}

S.-E. Takahasi, M. Tsukada and Y. Kobayashi,  Classification of continuous fractional binary operations on the real and complex fields,  Tokyo Journal of Mathematics,   {\bf 38}(2015), no. 2, 369-380.

\bibitem{ann179}

Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.

\bibitem{ann185}

Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.

\bibitem{ann237}

Announcement 237 (2015.6.18):  A reality of the division by zero $z/0=0$ by  geometrical optics.

\bibitem{ann246}

Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.

\bibitem{ann247}

Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.

\bibitem{ann250}

Announcement 250 (2015.10.20): What are numbers? -  the Yamada field containing the division by zero $z/0=0$.

\bibitem{ann252}

Announcement 252 (2015.11.1): Circles and

curvature - an interpretation by Mr.

Hiroshi Michiwaki of the division by

zero $r/0 = 0$.

\bibitem{ann281}

Announcement 281 (2016.2.1): The importance of the division by zero $z/0=0$.

\bibitem{ann282}

Announcement 282 (2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.

\bibitem{ann293}

Announcement 293 (2016.3.27):  Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.

\bibitem{ann300}

Announcement 300 (2016.05.22): New challenges on the division by zero z/0=0.

\end{thebibliography}

\end{document}