‘All right. Think of it in terms of a set of laws. No matter how wise your law-making, there will always be cases that can’t be decided within that framework. What you have to do, of course, is add another ruling. But that shows that the law can never be made complete. That’s not surprising to anybody who’s come up against a county magistrate, such as myself.
‘But what Godel did was to find a worm of doubt in mathematics itself, at the heart of what seems the most abstract and logical of all our intellectual pursuits. Even simple arithmetic can never be made complete. I mean, this doesn’t stop maths working, from day to day; but we know there will always be limits to what we can say about it. There is a gap between our intuition and the truth about mathematics, you see. And having dropped that bomb, Gödel turned his attention to Einstein ...’
As he studied relativity, Gödel became concerned about another gap between intuition and theory, concerning the nature of time.
‘We know Gödel read thinkers like Plato and Heidegger. Following them, he thought carefully about our inner experience of time.’ He studied her. ‘Our intuition is that only the present is real. Yes? Here we are in this farmhouse, as moment after moment ticks by. The future is only potential, undecided; it will emerge from the moment, but does not yet exist. The past too is gone for ever, leaving only traces, just as the long-gone Romans left their footprints here at Birdoswald. So time is like a fabric, built up strand by strand, each new strand coming into existence to be laid over the last.’
‘Time’s tapestry,’ she said, thinking of Geoffrey Cotesford’s phrasing.
‘Well, quite. That’s how time feels. Unfortunately that isn’t how relativity describes time at all ...’
In relativity there was no global time, no single instant that spanned the universe to unite all observers. Instead there were only ‘events’, points swimming disconnected around a graph of space-time. ‘It’s all to do with the speed of light. It would be, wouldn’t it? And without a global time there can’t be a common present, coming into existence across the universe for all of us. Relativity is a very well established theory. And yet the time it describes seems to have nothing to do with inner time, time as we experience it.’
‘I think I see,’ she said.
‘You’re being very brave, Mary, but you’re not fooling me! Now Gödel isn’t the only one troubled by this stuff. Some physicists, such as James Jeans, have argued that perhaps there is a universal time after all, a cosmic time based on the expansion of the universe, or perhaps the mean motion of matter within it. You see? Some feature of the universe on which all observers could agree, whatever their state of motion.
‘But that seemed a bit arbitrary to Gödel; he wasn’t reassured. So he - and we think in collaboration with Kamen - dug into Einstein’s mathematics. I admit I’m winging it a bit here; Gödel hasn’t actually written up his results formally, and until he does we’re all a bit in the dark. But what he seems to be trying to do is to find a description of a possible universe, consistent with Einstein, in which there could be no cosmic time.’
‘And he’s found one.’
‘I’m afraid so. He says that if the universe rotated, in a grand and global way, it would be possible for me to fly around it in some super Spitfire - heading off into the future, but at last arriving in the past.’
‘So time travel would be possible in such a universe.’
‘Well, that’s the implication you and I would naturally draw. I think Gödel, however, doesn’t care about time travel. He’s a logician; he’s simply trying to establish a contradiction. For you see if there is no global time in his model universe, you can’t assume a global time exists in ours, for there is no difference in principle between the universes. And so our sense of inner time, our most basic apprehension of the universe, is actually an illusion.’ His eyes were unfocused. ‘Must say that when I lie awake at night, this sort of stuff scares me more than all the Panzers in Germany. Poor old Gödel.’
‘But what we have to be concerned about,’ Mary said, ‘is the practical application.’
‘Quite.’ He tinkered with his pipe, tucking the last strands of his precious tobacco into place, and lit it. She’d observed this sort of finicky fiddling in smokers ever since tobacco had gone on the ration. He blew smoke up towards the ceiling. ‘Now we must come back to Ben Kamen again - and having led you off into the depths of time and space, I’m going to ask you to take another leap in the dark. For at Princeton Ben developed yet another new set of ideas, with a student called Rory O’Malley ...
‘Look, Gödel’s theory suggests that a path might exist between present and past - between this room, say, and the Roman fort of AD 200 from whose ruins it was built. The question is, how do you find your way along that path? Have you ever heard of a writer called John William Dunne?’
‘I think so,’ Mary said. ‘Something to do with J.B. Priestley?’
Dunne was British, an aeronautical designer. He had served as a soldier, and was invalided out of the Boer war. He had become interested in the perception of time, having had, he believed, a glimpse of the future through a dream.
Mackie waved a hand, a bit dismissively. ‘Sort of half-baked stuff that becomes fashionable from time to time in certain arty circles, playwrights and novelists and that sort of crew. That’s why you connect him with Priestley, I imagine. But what gives Dunne’s work credibility is a patina of science. He was an engineer. So he is methodical even about his dreams; he derives statistics about them; he couches his ideas in language that sounds almost Einsteinian. Dunne says essentially that when we sleep we come untethered in time. He imagines time as an extra dimension, a landscape you can go exploring. Some remember what they see during their dream journeys, some don’t. And some may be able to direct where their dreaming selves travel.’
It felt very strange to Mary to hear these ideas expressed by a serious middle-aged man in a sober Navy uniform.
Mackie ploughed on, ‘It was Rory O’Malley who introduced these ideas of Dunne’s to Ben Kamen. Now, Ben may have such a facility, this “dream precognition. Or he may not - interestingly he seems to deny it himself.’ He looked at her. ‘Perhaps you can see where I’m going with this.’
She nodded. ‘If you put it together - Godel shows that paths from the present to the past may exist. And Dunne argues that it might be possible to explore such paths.’
‘To dream yourself from present to past - and perhaps to do a bit of mucking about when you get there. We don’t think it would be possible with this method to travel to the past, but you could perhaps send back information - perhaps in the form of a dream or a vision implanted in another wandering soul.’
Which, she reflected with growing excitement and dread, was exactly how many of the historical ‘deflections’ in the testimony of Geoffrey Cotesford were supposed to have originated.
‘But it would take a Ben Kamen to do it, perhaps,’ Mackie said. ‘A man who has, or may have, both this peculiar precognitive facility, and the brains to understand Gödel’s mathematical solutions.’ Mackie smiled. ‘It’s a wonderful idea, isn’t it, to be able to run around in future and past, as freely as one runs as a child loose in a meadow of grass?’
‘Wonderful, yes,’ Mary said. ‘But is it true?’
‘We have reason to believe the Nazis take it seriously. Indeed they killed for it.’
Mary was shocked. ‘Who was responsible?’
‘Actually not a German. A British woman called Julia Fiveash. Holds a rank in the SS. Took part in the invasion - on the German side.’
‘I know,’ Mary said. ‘I met her.’
‘Did you, by God?’ Mackie listened as she told him the story of how she had run into Fiveash at Battle, with her accomplice Josef Trojan. ‘Well, that could be useful. Very nasty piece of work, that young lady. And now,’ he said, ‘we believe they are at a Nazi research centre at Richborough. And that’s where they’ve taken Ben. He managed to hide away in the POW camp for the best part of a year, i
t seems. But at last they flushed him out.’
‘And that’s why you say the situation is becoming urgent - why you contacted me now.’
‘Yes. For, you see, if they have Ben, they may have everything they need to make their wretched scheme work - if there’s anything in it at all.’ He stood up, holding his pipe. ‘I feel a bit stale, do you? I could do with a walk, I think. And there’s something else I should show you of what we’re doing here ...’
He led her to what appeared to be a converted barn; it was stone-built, and she wondered if this was the building with the Roman god built into its wall, but Mackie didn’t mention it. Inside, the barn seemed to have been converted into a workshop, the walls panelled with whitewashed wood, and a bright light glowed from bulbs suspended from the ceiling; Mary surmised the fort must have its own generator, for no mains electricity bulb burned so bright these days. ‘We do try to keep this place clean,’ murmured Mackie. ‘All the small parts, you know ...’
The centrepiece of the room was a table bearing an elaborate mechanical device, a rectangular array screwed together from fine strips of green-painted metal, with tiny pulleys and gears and motors and threads of string - and, in one corner, two discs of what looked like ground glass. Elaborate graphs had been prepared on drafting tables, set up under the lights for visibility. It was all very complicated, but toy-like, like a model of something else rather than anything significant in itself. But it was being taken very seriously, Mary realised. Around the walls were shelves bearing spare parts, and racks of tiny screwdrivers and spanners.
Mackie asked, ‘Any idea what you’re looking at?’
Mary shrugged. ‘Some kind of game?’
‘Not exactly, but you’re close. Mary, we live in a mathematical age - indeed, this is a mathematical war. And we need new mathematical techniques to cope with it all. There is a class of analyses based on differential equations, which—’
‘Please, Captain. Gödel and his undecidability are enough for me for one day.’
‘Quite so. Look - let’s suppose you want to compute the trajectory of a shell from a new breed of gun. Very necessary for firing tables, as you can imagine. Now you can list the impulse of the propellant, the angle of the barrel, gravity, air resistance and so forth. But to work out how the shell will fly you must put all that together, step by step, mapping the trajectory as a whole.’
‘And that’s what this thing does, right?’
‘We call it a differential analyser. It’s a sort of mechanical brain, if you will. You can input your requirements by using this stylus - you see, you manually push it along the curves, here. The motion is transmitted through these levers and gears and so forth to the glass discs; roughly speaking the spinning of those discs is a model of the variables of interest - I mean, the numbers that describe the shell trajectory, or whatever.’
‘All right. So what’s it doing here?’
‘Well, Einstein’s equations of general relativity are just another example of a set of differential equations. It’s fiendishly difficult to extract any kind of analytical solution from them. And if you do need to extract solutions of Gödel’s kind, describing trajectories from present to past—’
‘Oh. You’d need a machine like this.’
‘We know that Kamen and O’Malley had access to an analyser in Princeton. And we believe, though we aren’t sure, that Fiveash and her Nazi companions are building such a device at Richborough. We, or the mathematical boffins I recruited to work on this, thought we should study Gödel’s solutions ourselves, if we were to try to make sense of it all. Hence the beast you see before you.’
‘Kind of Rube Goldberg, isn’t it?’ Mary longed to touch the gadget, to pull the little levers and turn the pulleys. ‘Did you have to get these teeny tiny parts specially made?’
‘Actually no. They come from a kit called Meccano.’
‘A kit?’
‘A construction kit for boys.’
‘A toy? You made your calculating machine from a toy?’
He coughed. ‘Rather embarrassing to have to admit that to an American - but, yes, afraid so. Rather British, don’t you think? Of course it will make it all the more satisfying if we were to beat the bad guys with it.’ He rubbed his hands together. ‘Let’s go back to the office, and you can tell me all about the history you’ve dug up.’
XII
Back in Mackie’s kitchen-study she opened her briefcase and spread the contents over the table.
‘It begins again with Ben Kamen. When he arrived in England he did a bit of research himself - he is a bright boy - and came up with a medieval study of historical anomalies.’
‘You’re kidding.’
‘Nope. He got to know Gary, and found out that his mother was a specialist in the period, and as soon as he met me he got me started on it.’
Kamen had found a memoir by a fifteenth-century monk called Geoffrey Cotesford. She raised a scrap of paper and read out: “‘Time’s Tapestry: As mapped by myself, that is Cotesford. “In which the long warp threads are the history of the whole world; and the wefts which run from selvedge to selvedge are distortions of that history, deflected by a Weaver unknown; be he human, divine or satanic ...”’
‘Deflections of history,’ murmured Mackie.
‘Yes. Suggestive, isn’t it? Cotesford believed he had lived through one such attempted deflection himself, and had been made aware of others. He did some research - he was a Franciscan monk, a scholar, and he knew what he was doing. Plus he had access to sources, such as from the Muslim libraries in Spain, which have now been lost to us. This is a sound piece of work, considering.
‘In all he found evidence of six deflections. He went right back to a prophecy supposedly intoned right here at Birdoswald by a Briton some decades before the Roman invasion of the country. It’s called the Prophecy of Nectovelin. That’s the one I’ve concentrated on first. Nectovelin itself is lost, at least the original. But Geoffrey was able to find extracts from it, in an old Moorish library in Toledo. Just a few lines - here.’ She passed him a paper.
Mackie read:
Ah child! Bound in time’s tapestry, and yet you are born free Come, let me sing to you of what there is and what will be, Of all men and all gods, and of the mighty emperors three ...
‘We don’t know how much was lost, and how much has been garbled in the repeated transcriptions. We don’t even know the purpose of the deflection, if it was a deflection. Geoffrey speculated it had something to do with the Emperor Constantine.’
‘Constantine? He was centuries later. What’s he got to do with the price of fish?’ He glanced down the lines again.
Remember this: We hold these truths self-evident to be -
I say to you that all men are created equal, free
Rights inalienable assuréd by the Maker’s attribute
Endowed with Life and Liberty and Happiness’ pursuit.
O child! thou tapestried in time, strike home! Strike at the root!
Hmm. I’m blowed if that doesn’t sound familiar.’
She smiled. ‘An American would recognise the reference straight away.’
‘Ah. Let me see if I can remember. “We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. The Declaration of Independence.’
‘Right—’
‘Good Lord.’ The implications seemed to hit him then, and he sat silently for a heartbeat. ‘I think I’m going to have something eke to keep me awake at night, apart from Gödel’s banishment of time ... So a modern figure sent this back to pre-Roman Britain? Why? To establish democratic values back in the Iron Age?’
‘Maybe. But, Tom, the important question is not why but who.’
‘Ben’s friend Rory O’Malley, perhaps.’
‘Yes. O’Malley was an idealistic Irishman who was a great admirer of the US. And he was a lapsed Catholic. He had grown u
p amid a lot of religious sectarian tension. In the US, before he went to Spain, where he met Ben, he wrote a whole series of articles attacking the Church’s oppressive nature, and the evils done in the name of organised religion.’
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