Point Counter Point(33) by Aldous Huxley

Point Counter Point(33) by Aldous Huxley


‘Yes, and probably a death’s head into the bargain,’ said Spandrell. ‘It’s a question of thickening arteries.’

‘But what makes the old such an Arab tea party is their ideas. I simply cannot believe that thick arteries will ever make me believe in God and morals and all the rest of it. I came out of the chrysalis during the War, when the bottom had been knocked out of everything. I don’t see how our grandchildren could possibly knock it out any more thoroughly than it was knocked then. So where would the misunderstanding come in?’

‘They might have put the bottom in again,’ suggested Spandrell.

She was silent for a moment. ‘I never thought of that.’

‘Or else you might have put it in yourself. Putting the bottom in again is one of the traditional occupations of the aged.’

The clock struck one and, like the cuckoo released by the bell, Simmons popped into the library, carrying a tray. Simmons was middleaged and had that statesman-like dignity of demeanour which the necessity of holding the tongue and keeping the temper, of never speaking one’s real mind and preserving appearances tends always to produce in diplomats, royal personages, high government officials and butlers. Noiselessly, he laid the table for two, and, announcing that his lordship’s supper was served, retired. The day had been Wednesday; two grilled mutton chops were revealed when Lord Edward lifted the silver cover. Mondays, Wednesdays and Fridays were chop days. On Tuesdays and Thursdays there was steak with chips. On Saturdays, as a treat, Simmons prepared a mixed grill. On Sundays he went out; Lord Edward had to be content with cold ham and tongue, and a salad.

‘Curious,’ said Lord Edward, as he handed Illidge his chop, ‘curious that the sheep population doesn’t rise. Not at the same rate as the human population.

One would have expected…seeing that the symbiosis is such a close…’ He chewed in silence.

‘Mutton must be going out of fashion,’ said Illidge. ‘Like God,’ he added provocatively, ‘ and the immortal soul.’ Lord Edward was not to be baited. ‘Not to mention the Victorian novelists,’ Illidge went on. He had slipped on the stairs; and the only literature Lord Edward ever read was Dickens and Thackeray. But the Old Man calmly masticated. ‘And innocent young girls.’ Lord Edward took a scientific interest in the sexual activities of axolotls and chickens, guinea-pigs and frogs; but any reference to the corresponding activities of humans made him painfully uncomfortable. ‘And purity,’ Illidge continued, looking sharply into the Old Man’s face,’ and virginities, and…’ He was interrupted and Lord Edward saved from further persecution by the ringing of the telephone bell.

‘I’ll deal with it,’ said Illidge jumping up from his place.

He put the receiver to his ear. ‘Hullo!’

‘Edward, is that you?’ said a deep voice, not unlike Lord Edward’s own. ‘This is me. Edward, I’ve just this moment discovered a most extraordinary mathematical proof of the existence of God, or rather of…’

‘But this isn’t Lord Edward,’ shouted Illidge. ‘Wait. I’ll ask him to come.’ He turned back to the Old Man. ‘It’s Lord Gattenden,’ he said. ‘He’s just discovered a new proof of the existence of God.’ He did not smile, his tone was grave. Gravity in the circumstances was the wildest derision. The statement made fun of itself. Laughing comment made it less, not more, ridiculous. Marvellous old imbecile! Illidge felt himself revenged for all the evening’s humiliations. ‘A mathematical proof,’ he added, more seriously than ever.

‘Oh dear!’ exclaimed Lord Edward, as though something deplorable had happened. Telephoning always made him nervous. He hurried to the instrument. ‘Charles, is that…’

‘Ah, Edward,’ cried the disembodied voice of the head of the family from forty miles away at Gattenden. ‘Such a really remarkable discovery. I wanted your opinion on it. About God. You know the formula, m over nought equals infinity, m being any positive number? Well, why not reduce the equation to a simpler form by multiplying both sides by nought? In which case you have m equals infinity times nought. That is to say that a positive number is the product of zero and infinity. Doesn’t that demonstrate the creation of the universe by an infinite power out of nothing? Doesn’t it?’ The diaphragm of the telephone receiver was infected by Lord Gattenden’s excitement, forty miles away. It talked with breathless speed; its questions were earnest and insistent. ‘Doesn’t it, Edward?’ All his life the fifth marquess had been looking for the absolute. It was the only sort of hunting possible to a cripple. For fifty years he had trundled in his wheeled chair at the heels of the elusive quarry. Could it be that he had now caught it, so easily, and in such an unlikely place as an elementary school-book on the theory of limits? It was something that justified excitement. ‘What’s your opinion, Edward?’

‘Well,’ began Lord Edward, and at the other end of the electrified wire, forty miles away, his brother knew, from the tone in which that single word was spoken, that it was no good. The Absolute’s tail was still unsalted.

‘Talking about elders,’ said Lucy,’did I ever tell either of you that really marvellous story about my father?’

‘Which story? ‘

‘The one about the conservatories.’ The mere thought of the story made her smile.

‘No, I never remember hearing about the conservatories,’ said Spandrell, and Walter also shook his head.

‘It was during the War,’ Lucy began. ‘I was getting on for eighteen, I suppose. Just launched. And by the way, somebody did almost literally break a bottle of champagne over me. Parties were rather feverish in those days, if you remember.’

Spandrell nodded and, though as a matter of fact he had been at school during the War, Walter also nodded, knowingly.

‘One day,’ Lucy continued, ‘I got a message: Would I go upstairs and see his Lordship? It was unprecedented. I was rather alarmed. You know how the old imagine one lives. And how upset they are when they discover they’ve been wrong. The usual Arab tea party.’ She laughed and, for Walter, her laughter laid waste to all the years before he had known her. To elaborate the history of their young and innocent loves had been one of his standing consolations. She had laughed; and now not even fancy could take pleasure in that comforting romance.

Spandrell nodded. ‘So you went upstairs, feeling as though you were climbing a scaffold…’

‘And found my father in his library, pretending to read. My arrival really terrified him. Poor man! I never saw anyone so horribly embarrassed and distressed You can imagine how his terrors increased mine. Such strong feelings must surely have an adequate cause What could it be? Meanwhile, he suffered agonies. If his sense of duty hadn’t been so strong, I believe he would have told me to go away again at once. You should have seen his face!’ The comic memories were too much for her. She laughed.

His elbow on the table, his head in his hand, Walter stared into his wine-glass. The bright little bubbles came rushing to the surface one by one, purposively, as though determined at all costs to be free and happy. He did not dare to raise his eyes. The sight of Lucy’s laughter-distorted face, he was afraid, might make him do something stupid—cry aloud, or burst into tears.

‘Poor man!’ repeated Lucy, and the words came out on a puff of explosive mirth. ‘He could hardly speak for terror.’ Suddenly changing her tone, she mimicked Lord Edward’s deep blurred voice bidding her sit down, telling her (stammeringly and with painful hesitations) that he had something to talk to her about. The mimicry was admirable. Lord Edward’s embarrassed phantom was sitting at their table.

‘Admirable!’ Spandrell applauded. And even Walter had to laugh; but the depths of his unhappiness remained undisturbed.

‘It must have taken him a good five minutes,’ Lucy went on, ‘to screw himself up to the talking point. I was in an agony, as you can imagine. But guess what it was he wanted to say.

‘What?’

‘Guess.’ And all at once Lucy began to laugh again, uncontrollably. She covered her face with her hands, her whole body shook, as though she were passionately weeping. ‘It’s too good,’ she gasped, dropping her hands and leaning back in her chair. Her face still worked with laughter; there were tears on her cheeks. ‘Too good.’ She opened the little beaded bag that lay on the table in front of her and taking out a handkerchief, began to wipe her eyes. A gust of perfume came out with the handkerchief, reinforcing those faint memories of gardenias that surrounded her, that moved with her wherever she went like a second ghostly personality. Walter looked up; the strong gardenia perfume was in his nostrils; he was breathing what was for him the very essence of her being, the symbol of her power, of his own insane desires. He looked at her with a kind of terror.

‘He told me,’ Lucy went on, still laughing spasmodically, still dabbing at her eyes,’ he told me that he had heard that I sometimes allowed young men to kiss me at dances, in conservatories. Conservatories!’ she repeated. ‘What a wonderful touch! So marvellously in period. The ‘eighties. The old Prince of Wales. Zola’s novels. Conservatories! Poor dear man! He said he hoped I wouldn’t let it happen again. My mother’d be so dreadfully distressed if she knew. Oh dear, oh dear!’ She drew a deep breath. The laughter finally died down.http://www.obooksbooks.com/2015/4472_33.html

\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 300: New challenges on the division by zero z/0=0\\
(2016.05.22)}
\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.

\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.

\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 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 $W =1/\overline{z}$ 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,
\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.

For our ideas on the division by zero, 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,ann293}).

\section{On mathematics}
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 impacts on Euclidean spaces, analytic geometry, calculus, differential equations, complex analysis and physical problems. See our announcements for the details.

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 impacts to our basic ideas on the universe.

\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.

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

\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.

\bigskip

We are standing on a new generation and in front of the new world, as in the discovery of the Americas.

\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 $z/0=0$,
Linear Algebra \& Matrix Theory (ALAMT)(to appear).

\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.

\end{thebibliography}

\end{document}