Copyright © James R Meyer 2012 - 2016 www.jamesrmeyer.com
As the morning burst open, gilded shafts of light were crashing everywhere, seeking new spaces as the sun rose inexorably into the sky. It was a signal that the past few days of incessant rain were over, allied by a light breeze that was chasing away the dampness that had seeped into the very substance of every material thing. In spite of this, Ralph McNeil was walking from his flat to Trinity College for the first lecture of the day without any particular enthusiasm. He didn’t notice the sunlight boldly splintering through the faded leaves that still clung to the trees. He was too busy thinking. In most aspects Ralph McNeil was average, almost non-descript. He was neither tall nor short, not particularly unattractive nor particularly good-looking.
But he had one significant characteristic that set him apart from his contemporaries, and that was his inclination to think a great deal, and to think in great detail. This inclination was an inescapable consequence of his complete inability to take anything for granted. Everything had to fit into his own personal scheme of understanding. Everything had to be understood totally, utterly, completely. There was no room for the inexplicable in Ralph’s scheme of things. There could be no mystery, only a failure to understand. Where others would be content to shrug their shoulders and say, Well, that’s just the way it is, Ralph would continue thinking about it, studying every aspect, chipping away at it until it would finally fit neatly into the space that he had reserved for it in his scheme of understanding. Ralph would acknowledge that there were several reserved spaces in his scheme of understanding. That did not dismay him at all; on the contrary, it meant that he always had something to think about. For him, thinking was a most enjoyable activity.
Today he was thinking that a year had passed since he had started his university course, reflecting that his engineering course, at least the first year of it, had been easy. He hadn’t worked that hard at his coursework, being too easily distracted by some detail that he would devote almost his entire attention to until he was satisfied that he understood it completely. Now that another year was starting he felt somehow underwhelmed. He was starting to wonder if it was all too easy. His success in the past year had only seemed to feed the doubts that had begun to creep into his mind. Doubts that would rise up and confront him, unannounced and uninvited. Not doubts about his academic ability, not doubts that he would be able to brush aside any difficulties that the second year of the engineering course might bring, but doubts that engineering was what he really wanted to do, doubts that it would be enough to sustain his interest for the rest of his life.
He envied those students that seemed to be content to slog through the course, doing just enough to pass their exams, spending the rest of their time doing what students are supposed to do: going out, drinking, socialising, pulling girls, and slowly recovering the next morning. He had tried that approach and had very quickly discovered that it didn’t really do anything for him. It all seemed so trite that it could provide nothing more for him than a quick burst of pleasure that soon faded into a distasteful awareness of the mediocrity of it all.
Now he was approaching one of the landmarks of old Dublin, the imposing facade of Trinity College. He stopped at the pavement, waiting for a break in the traffic. He looked around, watching people rushing about in all directions, going to work, or to lectures or appointments. What was it all for? He watched as a girl came up from behind him. She is rather beautiful, he thought. But she wasn’t looking at him; she was preoccupied, her eyes looking blankly ahead. He watched as she came up alongside him, hoping that she might glance at him. She didn’t. She walked right past him. With a start, Ralph saw that she was walking straight into the path of an oncoming car. Without thinking, he reached out after her, grabbed her arm, and pulled her forcefully towards him. She fell back towards him with a cry of alarm and crumpled into a heap at his feet as the car sped past them, its horn blaring.
“Are you all right?” he asked, looking down at her.
She looked up, her face drained of colour. A grimace flashed briefly across her face, replaced by an attempt at a wry smile.
“I think so. You must think I’m a total idiot.”
“No, no, not at all. It could happen to anyone. Can you get up?”
She took his proffered hand and he helped her to her feet.
“Well,” he said, “doesn’t look like you’ve broken any bones. Though I bet you’ll have a few bruises.”
She nodded, then closed her eyes and put her hand to her head.
“You need to sit down,” said Ralph. “There’s a café just up the street…” He put his hand out. “Let me take your backpack.”
When they were in the café, Ralph ordered two coffees and carried them over to the table where the girl was sitting. She was on the fine-featured side, fine-boned but well-proportioned, with dark, almost black shoulder length hair, tinged with a hint of auburn, almost straight but curled inwards at the ends. As Ralph sat down, he realised that he was staring at her and he suddenly looked down at his coffee. Momentarily flustered, he pulled his hand through his hair, a gesture that had become a habit whenever he felt awkward or embarrassed. As this happened rather frequently, he was fortunate to have hair with a gentle wave that fell back naturally into a soft sculpted shape. Fumbling for something to say, he said, “I didn’t catch your name.”
“My name’s Patricia. Patricia Danielli.”
“My name’s Ralph. Ralph McNeil.”
“Rafe?” she said, looking puzzled.
“Yes. It’s spelt R-A-L-P-H, but it’s pronounced Rafe, not Ralf.”
“Oh, I see,” said Patricia, “like Ralph Fiennes, the actor?”
Ralph smiled. “Exactly.”
“Well, Ralph,” she said, “It’s a pleasure to meet you. You’re a knight in shining armour, no less.”
“Think nothing of it.”
He felt like adding, ‘It’s the first time I’ve succeeded in pulling a girl in ages’, but he decided against it. As she drank her coffee, Ralph could see that she was starting to brighten, and the colour was returning to her face.
“I suppose you’re wondering what I was doing back there,” she said.
“Yeah, I suppose I was. But it’s none of my business.”
“It’s nice of you to say that. I was thinking about my next essay, that’s all.”
Ralph scratched his head. “You’re kidding me. You’re telling me that you get so absorbed into thinking about an essay that you walk straight in front of an car?”
She nodded. “Yes. Sad, isn’t it?”
“Oh, I don’t know. Either it means that you must be so incredibly clever that you become so lost in thought that you’re oblivious to the outside world, or else you’re…” His voice trailed off. “But I’m going to bet that you’re incredibly clever. Am I right?”
Patricia turned her dark brown eyes away. “Not really. I try hard, that’s all.”
Ralph chuckled. “So hard that you don’t watch where you’re going. Must be one hell of an essay.”
“Maybe. It’s an essay on the meaning of truth.”
“On the meaning of truth? That really is something else. What are you studying?”
“Philosophy. And English as well, but that’s only because I can’t take philosophy on its own.”
“I see,” said Ralph, his voice deadpan.
“You sound disappointed.”
Ralph shrugged his shoulders. “No, no. It’s just that I don’t know a lot about philosophy, that’s all.”
“You know, almost everyone says that. So, Ralph, what are you studying?”
“Oh, I’m doing engineering. Not much philosophy involved in that, is there?”
As Patricia’s eyes studied him, Ralph suddenly felt rather like a mouse about to be consumed by a cat grown tired of playing.
“You’re just assuming that,” she said. “If you open your eyes and your mind, you’ll see philosophy everywhere. You shouldn’t write it off just like that.”
Ralph looked around the themed café, decorated in chic French style, the walls adorned with pictures of Paris, strings of garlic hanging from steel hooks, and he wondered what philosophy he was supposed to be seeing.
“You say that there’s philosophy everywhere. Where’s the philosophy in this café?”
“Can’t you see?” said Patricia. “Don’t just look, think. Look at these people. What are they thinking? Why are they here? Are they here just because they want some caffeine? Or do they feel an inner warmth because they feel comfortable here? Because they see it as a place where they can relax? Or are they trying to escape from reality?” She looked across at Ralph. “Do you see what I mean?”
Ralph pursed his lips. “Maybe. But that’s just this café. I was saying that there isn’t much philosophy involved in engineering…
It is now late autumn and the leaves lie cold and broken on the damp streets of the Austrian capital. For the comfortably off in Vienna, the approach of winter does nothing to dampen their sprit of optimism. For them the hardships endured during the Great War are a fading memory, and times are good for the middle classes. Vienna is now a city teeming not only with new ideas, but also with a new enthusiasm for life itself. Here, life is not to be approached tentatively - it is to be grasped and lived to the full. When darkness falls the city becomes alive with music, dancing and laughter as people throw themselves with thoughtless abandon into the delights of sensual gratification.
But not all. It is early evening, and in the bar The Moth there are just a few customers. One of them is Kurt Gödel. Now twenty-one years old and finished his course in mathematics and philosophy at the University of Vienna, he is studying for his doctorate. He stays with Rudolf, his older brother, in a town house owned by their father, just ten minutes from the university. Kurt revels in the freedom that comes with living away from home, especially when he has a generous allowance from his father that enables him to frequent The Moth. In the evening The Moth turns from a quiet bar into a lively nightclub. Kurt would not dare to be seen in a nightclub in his small hometown, but Vienna in the 1920s is not a place where such conventions hold any sway for the new wave of intellectuals. It is a nexus for new ideas, a hub where intellectuals of all philosophies and beliefs gather together to bounce their ideas off each other. While Kurt relishes the intellectual freedom of this vibrant city, he is at the same time intrigued and appalled by some of the ideas he encounters. He has always considered himself a free thinker, but he carries with him the unshakeable belief in a God that has a direct influence over the trajectory of every human life. When others talk excitedly of new philosophies that deny the need for a God of any form, he quells his deep irritation, nods sagely, and says nothing.
Most times he goes to The Moth early in the evening before it starts to get busy. He is a regular customer; the dark panelled walls and secluded corners offer him a tranquil retreat, while also giving him an opportunity to sit back and observe the rest of the world. He rarely stays late, preferring the subdued atmosphere of the bar before it gets noisy, which is when he slips out unnoticed. Usually he sits alone, and often can be seen with a distant look on his face. When he is like this, he is lost in his own world, deep in thought, contemplating, attempting to explore the outer reaches of the limits of human thought.
However, he is not so far removed from reality that he fails to notice when there is a pretty girl in the vicinity. There is a sound of laughter as a group of girls come into the bar. Kurt watches the girls as they cross the floor. He knows them; they are dancers at the nightclub. He has talked to them on other evenings; when they are not dancing, the dancers are expected to serve in the bar and talk to the customers. As the girls look around, they whisper to each other. Kurt notices that there is a new girl tonight, a pretty blonde. She looks around to familiarise herself with the surroundings, and as she does so, Kurt observes that she has a small purple birthmark on the left side of her face. But, he decides, it does not detract from her looks, and rather makes her seem more intriguing. As her glance lands on him, one of the other girls nudges her and whispers to her, “He’s eying you up. Seeing as you’re the new girl, you can talk to that one. He’s so creepy.”
The new girl looks bemusedly at the man sitting stiffly behind his table in a dark suit, white shirt and black tie, with neatly combed black hair and matching thick black glasses.
“Oh, surely he’s not that bad?” she says, “But why does he wear those awful black glasses?”
As they move on in, the other girl says, “Because he’s so serious. And I mean seriously serious.”
The girls go off to the changing rooms. When they emerge in their dancing gear, Kurt sees one of them elbowing the new girl. Then the new girl looks in Kurt’s direction and walks tentatively over to him.
“Hi,” she says brightly, “Can I get you anything?”
“No, I am quite all right. I have not finished this coffee yet.”
“Oh. Do you mind if I sit down here?”
He nods slowly. “Not at all,” he replies. “You are new here.”
“That’s right,” she says. “It’s my first night here, and I am feeling very nervous. I don’t know if they’ll like me here.”
“I am sure that they will like you,” he says. “Why on earth would they not?”
“Oh, I don’t know. Maybe they won’t like my dancing.”
“I see. Have you not done this sort of dancing before?” he asks.
“Oh, I have. Quite a lot. But you do get used to being in one place, and when you go to a new place, it’s like starting all over again.”
“You will be fine, I can tell,” he says.
“You can?” she laughs. “How can you tell?”
He does not reply immediately but sits back and contemplates. “You are modest, and you are pretty,” he says. “Men like girls who are modest and pretty.”
He pauses, then leans towards her, and lowers his voice. “You know, it does not really matter if you are not a great dancer. As long as you are pretty. And you are.”
She giggles. “You’re such a flatterer.”
“No. Just honest,” he says, without a glimmer of a smile. Then he asks her in a serious tone, “Are you a good dancer?”
Noticing that he is gazing at her, she looks away. “I think so. At my last place, plenty of men told me I was. But,” she says, tilting her head and shaking back her hair, “you can’t always tell by that.”
“Indeed. Such praise might be part of some underlying strategy.”
She raises her eyebrows and chuckles. “A strategy? Surely not? I’m sure you wouldn’t, would you?”
He looks at her for a moment. She is amusing, he thinks. Then he says, “Whether I would use such a strategy would depend on two things. One, whether I had a hidden objective. Two, whether I thought that the undeserved praise would be believed by the recipient.”
She laughs. “Aren’t you funny? Well, funny man, will you tell me if I dance well or not?”
“Of course,” he says.
“And will you be honest?
“If you want me to be…” This time his face betrays the faintest of smiles.
Then the band starts to tune up, and she tells him she has to go. The band play jazz arrangements of popular tunes, and the girls dance the familiar routines that Kurt has seen many times before. After a while the band stops for a break, and the girl comes back over to him.
“Well, what do you think? How’s my dancing?”
He says, “Well, being honest, I thought you were…” He pauses, then continues, “Good. No. Not good…
The Shackles of Conviction is published by james R meyer Publishing
There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book There’s something about Gödel comes under scrutiny.
I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).
I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.
There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.
For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem
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