書評：数学, キリスト教, ゼロ, 歴史
ゼロをめぐる衝突は、哲学、科学、数学、宗教の土台を揺るがす争いだった
数の概念
太陽暦と幾何学を発明したエジプト人
ピュタゴラスにとって音楽を奏でるのは数学的な行為だった
ゼロから無限が生まれた
『宇宙を復号（デコード）する 量子情報理論が解読する、宇宙という驚くべき暗号』チャールズ・サイフェ
表紙の色がよくない。これだけで売れ行きが悪くなっていることだろう。広く読まれるべき作品であるにもかかわらず。頭からケツに至るまでゼロについての話である。
こんな損害をもたらすことのできる数は他にない。（※軍艦）ヨークタウンを襲ったようなコンピューターの故障はゼロのもつ力のほんの一端でしかない。さまざまな文化がゼロに対して身構え、さまざまな哲学がゼロの影響のもとで崩れさった。ゼロは他の数と違うからだ。ゼロは、言語に絶するもの、無限なるものを垣間見させてくれる。だからこそ、恐れられ、嫌われてきた――また、禁止されてきたのだ。
本書は、ゼロの物語である。ゼロが古代に生まれ、東洋で成長し、ヨーロッパで受け入れられるために苦闘して、西洋で台頭し、現代物理学にとって常なる脅威となるまでの物語だ。ゼロを理解しようとし、この神秘的な数の意味をめぐって争った人々――学者と神秘主義者、科学者と聖職者――の物語である。西洋世界が、東洋からきたある概念から身を守ろうと（時として暴力的に）試み、失敗した物語だ。そして、一見無害に見える数が突きつけるパラドクスに、20世紀最高の知性さえうろたえ、科学的思考の枠組みが崩壊しそうになったという歴史である。
ゼロが強力なのは、無限と双子の兄弟だからだ。二つは対等にして正反対、陰と陽である。等しく逆説的で厄介だ。科学と宗教で最大の問題は、無と永遠、空虚と無限なるもの、ゼロと無限大をめぐるものである。ゼロをめぐる衝突は、哲学、科学、数学、宗教の土台を揺るがす争いだった。あらゆる革命の根底にゼロ――そして無限大――が横たわっていた。
ゼロは東洋と西洋との争いの核心にあった。ゼロは宗教と科学の闘いの中心にあった。ゼロは自然の言葉、数学でもっとも重要な道具となった。そして、物理学でもっとも深刻な問題――ブラックホールの暗黒のコアとビッグバンのまばゆい閃光――はゼロを打ち負かす闘いなのだ。
だが、ゼロは、その歴史を通じて、排斥され追放されながらも、それに立ち向かうものを常に打ち負かしてきた。人間は力ずくでゼロを自らの哲学に適合させることはできなかった。かえって、ゼロは宇宙に対する――そして神に対する――人類の見方を形づくってきたのだ。
【『異端の数ゼロ 数学・物理学が恐れるもっとも危険な概念』チャールズ・サイフェ／林大訳（早川書房、2003年）】
結局、キリスト教と仏教という思想的バックボーンが明暗を分けたといってよい。キリスト教（ギリシアを中心とした西洋）は神以外の無限を嫌ってゼロを否定し、仏教（インド）はゼロを空の概念から止揚した。ゼロは教義を揺るがす代物だった。ちなみに、マイナスもインドで誕生したそうだよ。西洋はこれも拒絶。
こなれた文章で一気に読ませる。最近読んだ数学本の中ではぴか一。ゼロは無であり、無限だった。量子論の世界でも元始の宇宙における真空は、物質と反物質が充満していた状態と考えられている。つまり、プラスマイナスゼロというわけ。http://d.hatena.ne.jp/sessendo/20090204/p1http://ameblo.jp/syoshinoris/entry-12089827553.html
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 293: Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0}
\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 declare that any parallel lines have the common point $(0,0) $ in the sense of the division by zero. From this fact we have to change our basic idea for the Euclidean plane and we will see a new world for not only mathematics, but also the universe.
\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 result is a very special case for general fractional functions in \cite{cs}.
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{taka}) (see also \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 } $
$$
F (b, 0) = 0.
$$
}
\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 (\cite{ahlfors}). Therefore, the division by zero will give great impacts 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 or by the Moore-Penrose generalized inverse,
\medskip
2) by the intuitive meaning of the fractions (division) by H. Michiwaki,
\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 $1/\overline{z}$ of $z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero,
\medskip
and
\medskip
D) by considering rotation of a right circular cone having some very interesting
phenomenon from some practical and physical problem --- EM radius.
\medskip
See also \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 division $0/0$ and special relative theory of Einstein.
Furthermore, Reis and Anderson (\cite{ra,ra2}) extends the system of the real numbers by defining division by zero.
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.
For our results, see the survey style announcements 179,185,237,246, 247,250 and 252 of the Institute of Reproducing Kernels (\cite{ann179,ann185,ann237,ann246,ann247,ann250,ann252}).
At this moment, the following theorem may be looked as the fundamental theorem of the division by zero:
\bigskip
{\bf Theorem (\cite{mst}).} {\it Any analytic function takes a definite value at an isolated singular point }{\bf with a natural meaning.}
\bigskip
The following corollary shows how to determine the value of an analytic function at the singular point; that is, the value is determined from the regular part of the Laurent expansion:
\bigskip
{\bf Corollary.} {\it For an isolated singular point $a$ of an analytic function $f(z)$, we have the Cauchy integral formula
$$
f(a) = \frac{1}{2\pi i} \int_{\gamma} f(z) \frac{dz}{z - a},
$$
where the $\gamma$ is a rectifiable simple Jordan closed curve that surrounds one time the point $a$
on a regular region of the function $f(z)$.
}
\bigskip
The essential meaning of this theorem and corollary is given by that: the values of functions may be understood in the sense of the mean values of analytic functions.
\medskip
In this announcement, we will state the basic property of parallel lines by the division by zero on the Euclidean plane and we will be able to see that the division by zero introduces a new world and fundamental mathematics.
In particular, note that the concept of parallel lines is very important in the Euclidean plane and non-Euclidean geometry. The essential results may be stated as known since the discovery of the division by zero $z/0=0$. However, for importance, we would like to state clearly the details.
\section{The point at infinity}
We will be able to see the whole Euclidean plane by the stereographic projection into the Riemann sphere --- {\it We think that in the Euclidean plane, there does not exist the point at infinity}.
However, we can consider it as a limit like $\infty$. Recall the definition of $z \to \infty$ by $\epsilon$-$\delta$ logic; that is, $\lim_{z \to \infty} z = \infty$ if and only if for any large $M>0$, there exists a number $L>0$ such that for any z satisfying $L <|z|$, $M<|z|$. In this definition, the infinity $\infty$ does not appear.
{\it The infinity is not a number, but it is an ideal space point.}
The behavior of the space around the point at infinity may be considered by that around the origin by the linear transform $W = 1/z$(\cite{ahlfors}). We thus see that
\begin{equation}
\lim_{z \to \infty} z = \infty,
\end{equation}
however,
\begin{equation}
[z]_{z =\infty} =0,
\end{equation}
by the division by zero. The difference of (2.1) and (2.2) is very important as we see clearly by the function $1/z$ and the behavior at the origin. The limiting value to the origin and the value at the origin are different. For surprising results, we will state the property in the real space as follows:
\begin{equation}
\lim_{x\to +\infty} x =+\infty , \quad \lim_{x\to -\infty} x = -\infty,
\end{equation}
however,
\begin{equation}
[x]_{ +\infty } =0, \quad [x]_{ -\infty } =0.
\end{equation}
\section{Interpretation by analytic geometry}
We write lines by
\begin{equation}
L_k: a_k x + b_k y + c_k = 0, k=1,2.
\end{equation}
The common point is given by, if $a_1 b_2 - a_2 b_1 \ne 0$; that is, the lines are not parallel
\begin{equation}
\left(\frac{b_1 c_2 - b_2 c_1}{a_1 b_2 - a_2 b_1}, \frac{a_2 c_1 - a_1 c_2}{a_1 b_2 - a_2 b_1}\right).
\end{equation}
By the division by zero, we can understand that if $a_1 b_2 - a_2 b_1 = 0$, then the commom point is always given by
\begin{equation}
(0,0),
\end{equation}
even the two lines are the same. This fact shows that the image of the Euclidean space in Section 2 is right.
\section{Remarks}
For a function
\begin{equation}
S(x,y) = a(x^2+y^2) + 2gx + 2fy + c,
\end{equation}
the radius $R$ of the circle $S(x,y) = 0$ is given by
\begin{equation}
R = \sqrt{\frac{g^2 +f^2 -ac}{a^2}}.
\end{equation}
If $a = 0$, then the area $\pi R^2$ of the circle is zero, by the division by zero; that is, the circle is line
(degenerate).
Here, note that by the Theorem, $R^2$ is zero for $a = 0$, but for (4.2) itself
\begin{equation}
R = \frac{-c}{2} \frac{1}{\sqrt{g^2 + f^2}}
\end{equation}
for $a=0$. However, this result will be nonsense, and so, in this case, we should consider $R$
as zero as $ 0^2 =0$. 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, in the sense of the Theorem,
from the identity
\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}
By the Theorem, for analytic functions we can give uniquely determined values at isolated singular points, however, the values by means of the Laurent expansion are not always reasonable. We will need to consider many interpretations for reasonable values.
In addition, the center of the circle (4.3) is given by
\begin{equation}
\left( - \frac{g}{a},- \frac{f}{a}\right).
\end{equation}
Therefore, the center of a general line
\begin{equation}
2gx + 2fy + c=0
\end{equation}
may be considered as the origin $(0,0)$, by the division by zero.
We can see similarly the 3 dimensional versions.
\medskip
We consider the functions
\begin{equation}
S_j(x,y) = a_j(x^2+y^2) + 2g_jx + 2f_jy + c_j.
\end{equation}
The distance $d$ of the centers of the circles $S_1(x,y) =0$ and $S_2(x,y) =0$ is given by
\begin{equation}
d^2= \frac{g_1^2 + f_1^2}{a_1^2} - 2 \frac{g_1 g_2 + f_1 f_2}{a_1 a_2} + \frac{g_2^2 + f_2^2}{a_2^2}.
\end{equation}
If $a_1 =0$, then by the division by zero
\begin{equation}
d^2= \frac{g_2^2 + f_2^2}{a_2^2}.
\end{equation}
Then, $S_1(x,y) =0$ is a line and its center is the origin $(0,0)$.
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{bb}
J. P. Barukcic and I. Barukcic, Anti Aristotle - The Division Of Zero By Zero,
ViXra.org (Friday, June 5, 2015)
© Ilija Barukčić, Jever, Germany. All rights reserved. Friday, June 5, 2015 20:44:59.
\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{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. 6(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{mst}
H. Michiwaki, S. Saitoh and M. Takagi,
A new concept for the point at infinity and the division by zero z/0=0
(manuscript).
\bibitem{ra}
T. S. Reis and James 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 James A.D.W. Anderson,
Transreal Calculus,
IAENG International J. of Applied Math., 45: 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{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$.}
(note)
\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.
\end{thebibliography}
\end{document}
0 件のコメント:
コメントを投稿