The Shackles of Conviction

By James R Meyer

If you encountered something that you thought had to be wrong, what would you do? Would you try to prove that it was wrong? That’s what happens when Ralph McNeil encounters Kurt Gödel’s proof of Incompleteness. He sets out to prove it wrong. This is the story of how he tries to prove it wrong. It is also the story about the inner turmoil of the man who wrote the proof - Kurt Gödel.

• Language: English
• Publisher: James R Meyer
• Release date: Nov 15, 2010
• ISBN: 9781906706029

James R Meyer

I am interested in how little attention is paid to the limitations of the language when it is used to make statements that are supposedly logical. The consequences of this are particularly evident in mathematics, where there are theories that are based on the philosophy that numbers and other mathematical concepts are ‘actual’ things that exist independently of any physical reality. Such beliefs are commonly held on an almost subliminal level; most people have never taken the time to carefully examine the basis and the consequences of such beliefs. It is because of such beliefs that detailed considerations of language are ignored - the ‘actual’ non-physical reality is considered all-important - with the result that a detailed evaluation of the possibility of errors due to limitations of language is generally considered unnecessary.Every statement has to be stated in some language. If assumptions are made that ignore some aspects of the language of the statement, then how can we be sure that the statement is entirely logical? In particular, when a statement refers in some way, either implicitly or explicitly, to some language, whether it is the language of the statement itself or some other language, there is a significant possibility of confusion.Unless every aspect of such statements is very carefully analyzed, a statement that superficially appears to be logical may actually contain subtle errors of logic. In my work, I show how such errors can occur and how we can avoid such errors by careful analysis of language.

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Contents

Title Page

Introduction

Dedication

Preface

1. Incompleteness Of Understanding

2. Intimations Of Immortality

3. Intimations Of Obsession

4. Hearts And Minds

5. The Jester’s Gauntlet

6. A Passion For Life

7 .A Very Good Place To Start

8. The Cloud

9. Curiouser And Curiouser

10. The Awakening

11. First Steps

12. A Poisoned Letter

13. Cracking The Code

14. Confronting Reality

15. True Interpretations

16. All In The Mind

17. Relationships That Count

18. Incompletely Unsure

19. Of Machines And Men

20. Future Unforetold

21. No Substitute For Substitution

22. A Successful Operation

23. Simply Better

24. The Truth Will Set You Free

25. Intolerance Of Weakness

26. Deja Vu

27. A Certain Degree Of Truth

28. Charmed

29. Time Is Beckoning

30. Invitation To A Death

31. Year’s End

32. Marching Orders

33. Disclosure and Closure

34. Snow Is Only Water

Afterword

Index

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Introduction

There are many who say that Gödel’s Incompleteness proof is a unique and colossal achievement - that it will forever outshine every other proof that there can ever be.

There are many who say that Gödel’s Incompleteness proof shows that human intuition will always be able to prove more than any computer ever can.

There are many who say that Gödel’s Incompleteness proof demonstrates that to prove something is a weaker act than showing it to be true.

There are many who say that there are now so many versions of Gödel’s Incompleteness theorem that Gödel’s proof of incompleteness is now considered to be unassailable.

There are many who say that nobody loses sleep over Gödel’s Incompleteness proof any more because it is quite obvious that it is correct.

Surely they can’t all be wrong?

There are a few who insist that Gödel’s proof is a result of some logical trickery, some sleight-of-hand, or some misunderstanding.

Surely those few must be deluded?

Below is a list of some significant mathematicians, logicians and philosophers (in no particular order) who have concluded that Gödel’s Incompleteness proof is correct:

David Hilbert, Albert Einstein, John von Neumann, Alonso Church, Bertrand Russell, Alan Turing, Ernest Zermelo, Willard Quine, Alfred Tarski, Thoralf Skolem, Roger Penrose, Douglas Hofstadter, Andrei Kolmogorov, John Dawson, Gerhard Gentzen, Ernest Nagel, Gregory Chaitin, James Newman, Solomon Feferman, Harvey Friedman, Stephen Hawking, Verena Huber-Dyson, John Lucas, Raymond Smullyan, Torkel Franzén, Stephen Kleene, George Boolos, Jean van Heijenoort, John Rosser, Hilary Putnam, Juliet Floyd, Solomon Feferman, Rudolf Carnap, Emil Post

… in fact, the list includes almost all mathematicians, logicians and philosophers of significance that were alive at the time of publication of Gödel’s proof, or since.

* * *

Below is a list of significant mathematicians, logicians and philosophers who have stated that Gödel’s Incompleteness proof must be flawed:

Ludwig Wittgenstein

Surely it must be Wittgenstein who is wrong… or is he?

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Dedication

This book is dedicated to my three children, Gráinne, Ralph, and Alexander, from whom I beg forgiveness for the times that I have neglected them as a father in order that I might work on this book. My hope is that perhaps it may teach them that joy does not come like the sand trickling through an hourglass. Joy is the great wash of the sea; sometimes calm, sometimes gathering up into great waves. It is not easy to learn to read it, to know whether it is time to stay or time to move on. But learning is one of the joys of life, and so it is entirely appropriate that learning about joy should itself be a great joy.

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Preface

Since most people never look at the introductory pages I have deliberately kept this as brief as possible. If you are really interested in the story of why and how I came to write this book, please look up my website:

http://www.jamesrmeyer.com

In this book, while it is a work of fiction, I have tried as far as possible to work to a framework of real events that occurred during Gödel’s lifetime. The dates of such matters as his graduation, publication of his Incompleteness Theorem, his marriage, mental breakdowns, lecture tours, the deaths of his acquaintances and other similar events are all approximately correct.

I felt impelled to write this book for two principal reasons. The first reason was that for me, the life of Kurt Gödel was a fascinating jumble of facts that never seemed to fit together. I found either a reverential portrayal of an intellect of superhuman proportions, or else a portrayal of a pathetic figure incapable of normal human existence. I thought I could perhaps give Gödel a human face that would in some way fit in with the salient facts of his life. The second reason was that since no-one has, up to now, been able to understand Gödel’s Incompleteness proof properly, there has been a plethora of misleading information written about it; I wanted to give an analysis of that proof in terms that are as accessible as possible, in order to help people to fully understand what is really going on in Gödel’s Incompleteness proof. Any liberties that I have taken with definitions and technical terms do not affect the principles involved.

I sincerely hope you enjoy this book.

James R Meyer

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Incompleteness Of Understanding

Dublin, October 2006

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 façade 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.

Okay. We’ll talk about that then. Just tell me what interests you about engineering.

Everything. I suppose I like precision, I like everything to be clear-cut, without any waffle.

Typical engineering student, then. What’s your favourite subject?

Mathematics. In fact, I’d thought of doing pure mathematics, but I decided against it. I didn’t want to get stuck in too narrow a field – not yet, anyway. I can always take a pure math course later on if I want.

I see. So you’d be interested in the basic foundations of mathematics?

I don’t know. I haven’t given it much thought, to be honest. I’m more interested in practical mathematics – mathematics that has some relevance to the real world.

Ah, yes, she said breezily. Practical mathematics. But don’t you think that all practical mathematics must be based on something? My opinion, and it’s only my opinion, is that the study of foundational mathematics must be more interesting than ordinary mathematics. Foundational mathematics opens up a whole world of higher mathematics that touches on important philosophical questions.

You call it higher mathematics. To me, it’s not. I think it’s airy-fairy mathematics. It hasn’t any relevance to the real world.

You’re wrong. There’s things that you can prove in what you call airy-fairy mathematics that you can’t prove in ordinary mathematics.

Ralph had to force himself against a derisory rolling of his eyes as he replied, Yeah, I know all about that sort of stuff. I know that there are people who’ll tell you that there are infinitely many numbers of one sort, and there are infinitely many numbers of another sort, but that even though there are infinitely many numbers of each sort, they tell you that there are more of one sort of number than the other sort. That’s what I call airy-fairy mathematics. That sort of stuff’s based on crazy theories that have no basis in reality and have no practical use at all. Only weirdos are interested in that kind of thing.

Patricia flashed him a look of annoyance as she said sharply, Like me?

Ralph bit his lower lip. "I’m sorry. I didn’t mean it like that. I just meant that this ‘higher’ math that you talk about has no real value."

She replied with an edge to her voice, That’s where you’re wrong. Using this higher math, you can prove that there are things that must be true, but can’t be proved using ordinary math.

What does that mean? Is this part of your essay on the meaning of truth?

It means just what I said. There are things where it can be shown that they have to be true, but ordinary mathematics can’t prove them. Ever. And yes, it’s a key part of my essay on the meaning of truth. It’s called the incompleteness of ordinary mathematics, and the incompleteness of ordinary mathematics can only be proved by a higher form of mathematics.

Wait a minute. What exactly do you mean by ordinary math and higher math?

Well, in order for me to explain that, you need to know some of the terminology. Do you know what a formal language is?

She paused for a moment, and then added quickly, A formal mathematical language, that is?

Sort of, but you’d better explain what you mean by it. Otherwise we might be talking at cross-purposes.

"All right. A formal language is a mathematical language where you don’t use any English words at all. You’ve got a strictly defined alphabet of symbols, such as the symbol for zero, the symbol for ‘or’, the symbol for ‘not’, and so on.

Then you’ve got very strict rules of grammar, so that if a sentence doesn’t follow all the rules it’s not a valid sentence of the language. If a sentence doesn’t follow the rules, you’re not allowed to guess at what it might be meant to be. There can’t be any ambiguity at all, not like English – which is full of ambiguity."

She went on, It turns out that all of ordinary maths can be written in a formal language. So you might expect that every possible mathematical proof could be written in a formal language.

He interrupted her. Hang on. If it’s the case that all of ordinary maths can be written in a formal language, why don’t we actually use formal language when we’re actually doing ordinary maths?

Well, most of the time in ordinary mathematics, what you’re doing is using a semi-formal language. The problem with a completely formal language is that it’s very cumbersome to use. A simple proof could take hundreds of pages whereas a semi-formal proof might do the proof in a few pages.

What’s the use of a formal language then?

Well… it’s true that you wouldn’t want to use a formal language all the time. But the concept of a formal language is a very useful tool, a reference point, as it were. The way I see it, when mathematicians came up with the idea of a formal language, they wanted to clarify exactly what they were doing with their mathematics. They reckoned that if in principle that they couldn’t put the mathematics they were using into a formal language, then that meant they didn’t fully understand the mathematics they were using. If they could put it into a formal language, then they’d say they must have a full understanding of what they were doing with their mathematics. So even if they never actually used the formal language in practice, it was always a useful reference base point. It was to be the foundation rock on which the entire edifice of mathematics was to be built on.

She continued, When people had learned how to construct formal languages, they realised that with a formal language, you didn’t need to have any great insight to be able to produce a proof in a formal language. Start off with the basic sentences of the formal language, apply the proof rules over and over, and you end up with another sentence of the formal language. That sentence has been made following all the rules of the formal language, so that sentence has been proved. What you’ve done is that you’ve made a mathematical proof. In fact, it’s so straightforward that a machine could do it. Of course, when people first worked on formal languages, computers hadn’t been invented. But we do have computers now. So you can put a formal language into a computer as a program that will produce a proof. Before there was any computers, people used to think that some day there would be a machine that could be programmed with all the rules of mathematics, so that it would start churning out mathematical proofs – and there would be no mathematical proof that it couldn’t produce as long as the machine was powerful enough, and was given enough time.

Ralph watched as she spoke. Now all traces of fragility after her fall to the pavement had disappeared; her eyes were sparkling as she became more and more immersed in what she was saying.

But, she said, about 75 years ago, a man called Kurt Gödel came up with a theorem that proved that wasn’t the case. He proved that there are statements that can be made in a formal mathematical language but which can never be proved using the formal language – but despite that, they must be true. It was a turning point in the development of mathematics and it meant that mathematicians had to completely revise their notions of what truth in mathematics actually meant. It meant that the idea that it’s possible to translate every mathematical concept into a formal language was wrong. There are mathematical concepts that can be stated in a higher language that can never be stated in a formal language. Ever.

As she waited for a response, Ralph sat and thought. Then he said, "I hope you don’t mind me saying this, but this theorem you’re talking about sounds like the biggest load of nonsense I’ve ever heard. If the formal language can’t provide a proof, but your ‘higher’ math can provide a proof, surely all you’re saying’s that you’ve left out something from your formal language that’s in your ‘higher’ language?"

It might seem that way, but it’s not. Gödel’s proof can’t be broken down to a simple formal language. It operates on a completely different level, like being on a higher plane.

I’m sorry, but really… said Ralph, shaking his head, …that has to be rubbish. If no one can analyse that proof and put it into a fully defined mathematical language, then either they’re not smart enough, or there’s something wrong with the proof. It’s as simple as that.

She looked at him scornfully. It’s not as simple as that. Hundreds of logicians have looked at Gödel’s theorem and have found nothing wrong with it.

But haven’t you just said to me that the whole reason why people started using formal language was so they could make it precisely clear what they were trying to say?

She hesitated, and then said, Yes, but the whole point of Gödel’s theorem is that it proves that there can be no formal language that can say everything that we can say in higher math.

"Let me get this right. You’re saying that there can be no ambiguity and no errors in a formal language. That in a formal language everything’s completely defined and nothing’s left undefined. And then you tell me that you can talk about the formal language in your ‘higher’ language. Now, since this ‘higher’ language isn’t a formal language, that means there can be ambiguity and errors in this ‘higher’ language."

Ralph shrugged his shoulders and said, "And then you tell me that there’s a proof in this ‘higher’ language, this fine language that can have ambiguity and errors. And you tell me that this proof proves that there are statements that can be proved to be true in this ‘higher’ language, but which can’t be proved to be true in a formal language. Now, this ‘higher’ language may have ambiguity and errors, whereas the formal language can’t have any ambiguity or errors. It seems to me that all you’re saying is this ‘higher’ language proof must have some ambiguity or error."

He watched Patricia in quiet satisfaction as she considered this. For a long time she was silent.

Eventually she replied, I’ve got to admit that what you’re saying is very persuasive, but as I said before, Gödel published his proof in 1931, and since that time hundreds of logicians and mathematicians have studied it – and that includes the best logicians and mathematicians in the world. Not one of them has been able to find a flaw in it. I’m not going to pretend that I’d be able to look at the proof and understand it – I’m basing what I’m saying on the work that other people have done.

Gödel’s proof, she said, "basically produces a sentence which says ‘This sentence can’t be proved to be true.’ That sentence can’t be false, because if it was false, then that would be saying that the sentence itself can be proved to be true. But if it can be proved to be true, then the sentence itself says that it can’t be proved to be true. And that would mean that the sentence would at the same time be false and could be proved to be true. That’s impossible. That means that the sentence can’t be false. So it must be true. At the same time, because it’s true, it must be unprovable."

Ralph snorted derisively. But that’s simply a play on words. It doesn’t mean anything.

No, you’re wrong. The whole point of Gödel’s proof is that he was able to put that sentence in precise mathematical terms, without any of the vagueness of English.

Huh. I find that hard to believe.

I can see that, said Patricia, raising her eyebrows. But it isn’t just Gödel’s proof. There are other similar theorems as well which lead to the same conclusion…

Ralph raised his hand, interrupting her.

Hey, hey, wait a minute. We haven’t finished dealing with this theorem, and now you’re bringing in other theorems. If the first theorem can’t stand on its own merits, it can’t be very plausible. Why do you need to pull in another theorem to prop up the first one? What does that prove? What if they’re all wrong?

That’s simply impossible, she said. All the evidence is that these theorems are right.

What do you mean, all the evidence? It just means that that a lot of people think it’s unlikely that they’re all wrong. That doesn’t mean that it’s impossible that they’re wrong. There’s a big difference.

Ralph noticed that she was looking at him with an amused look on her face.

You said that you weren’t a philosopher, she said, but now you’re starting to act like one.

Then she looked down at her watch. Look, it’s been nice talking to you but I’ve got to go now. I really appreciate what you did for me.

She thought for a moment and then said, I’d like to do something for you. I’ve got a couple of books on Gödel’s theorem. I’d like to lend them to you – you can have a look at them and then you can tell me if you still think it’s all rubbish.

Ralph sipped the dregs of his coffee slowly, and thought that he just couldn’t be bothered. On the other hand, there was something about Patricia that made him hesitate to turn down her offer. She was attractive, all right, but it wasn’t just her looks. Even in the short time they had been speaking, he had realised that there was something different about her, something teasing and intangible that he had not seen in any other girl. And while she did have some strange ideas, she was easy to talk to. Agreeing to borrow her books meant that he would have to return them. And to return them he would have to meet her again…

Okay, he said. I’ll have a look and see what I think. But don’t expect me to be won over. I can be very stubborn and obstinate. And unless I’ve picked you up wrong somewhere on this, my stubborn and obstinate instincts tell me that it has to be a load of twaddle.

She smiled. I can be very stubborn and obstinate too. Then her eyes took on a hard intensity as she continued, gesturing with her forefinger. And determined. When I set my mind on something I’m very hard to dissuade.

There was something faintly chilling in the way that she spoke that he was momentarily taken aback. For an instant he saw a glimmer of steel beneath the soft exterior. But as soon as she had said it, she grinned at him.

I need to go now, she said. I’ve already missed one lecture. I’ll bring the books in tomorrow – I could meet you in the café in the Arts Building, if that suits.

It didn’t really, but he found himself saying, Yeah, that’ll be fine – would around two o’clock suit you?

Okay, she said as she stood up. And thanks again for saving me from being another accident statistic.

Ralph stood up awkwardly, not wanting to appear overly polite. He was surprised to see that she was actually slightly taller than him – he hadn’t noticed that before.

No problem, he said. I’ll see you tomorrow.

* * *

The next morning Ralph was looking forward to meeting Patricia and rehearsed several lines that he thought would intrigue her. He made sure that he would be there by himself, not wanting any of his friends looking on when he spoke to her. However, after waiting twenty minutes there was no sign of her, so he picked up a tray and got into the queue for the counter. As he stood in the queue, Patricia tapped him on the shoulder.

Hi, Ralph, she said. Sorry I’m late. Here’s the books. Look, I’m with some friends here, so I’m afraid I can’t stay and talk.

Ralph’s face dropped. All the pre-prepared lines were immediately forgotten as he mumbled, Thanks.

She smiled as she turned away, I hope they’re of some use to you.

He watched wistfully as she walked away, then suddenly he ran after her and called her name. She looked around as he came up to her.

How am I going to get them back to you? he said. Can you give me your mobile?

She smiled at him. I hadn’t thought of that.

After she had given him her mobile number and had gone to rejoin her friends, Ralph realised he had lost his place in the queue. He suddenly decided he wasn’t that hungry after all. He would just go back to his flat and have a look at the books that Patricia had given him. After all, he could stop and get something to eat on the way back to the flat.

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Intimations Of Immortality

Brno, Moravia, April 1914

In a three-storey detached house in the fashionable Spilberk area of the town of Brno, a young boy lies in bed. Although it is midday and the sun is shining full in the sky, the curtains remain drawn. The boy, just eight years old,