She studied him. 'My son is in a POW camp, and I haven't seen him for a year. I lost my daughter-in-law to an SS killer squad. I've followed up this crazy stuff to keep myself busy, this and my WVS work and my bits of journalism. Better than sitting around on my ass while the bombs fall. But a game, it isn't.'

'Sorry. No. Quite right. It's just, you know…' He sighed. 'Shut up, Tom. Now – Ben Kamen. Nine or ten years ago he worked in Vienna, as a student of a man called Kurt Godel. I don't suppose you've heard of him.'

'Only in connection with Ben,' Mary said.

'Eventually Kamen made his way to Princeton, the Institute of Advanced Studies, where he met up again with his old mentor Godel. Who by this time was working with Einstein himself.' Mackie pulled a pipe out of his breast pocket, dug a tobacco pouch from a drawer in the table, and began to fill the bowl meticulously, strand by strand. 'Remarkable chap, actually. Godel, I mean. One of the top mathematicians of his generation. And Kamen must have been pretty sharp to keep up with him. He was very young.

'Now, Einstein's theories are physical. They concern the nature of space and time, the structure of the universe, that sort of show. But they are couched in mathematics, and when it comes to maths Godel has a peculiar forte. He is a master of doubt, you might say. He studies formal systems, that is mathematical theories, and niggles away at them until he finds inconsistencies. His most famous result is a proof that even simple arithmetic – you know, just adding and subtracting – can contain a statement you can neither say is true nor false. And so arithmetic, and by implication all of mathematics, can never be made complete and decidable.'

'You're already losing me,' Mary admitted.

'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, Godel turned his attention to Einstein…'

As he studied relativity, Godel became concerned about another gap between intuition and theory, concerning the nature of time.

'We know Godel 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 Godel 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 Godel; 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; Godel 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 Godel, 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 Godel.'

'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, Godel'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.