The real point is that neither of these options is satisfactory, and the origins of space and time fit neither model. The universe is not like a village, which ends at a fence or an imaginary line on the ground, neither is it like the distant desert which seems to vanish into eternity but actually just gets too far away for us to see it clearly. Time is not like a human lifespan, which starts at birth and ends at death, nor is it like the extended lifespan found in many religions, where the human soul continues to live indefinitely after death, and the much rarer belief (held, for example, by Mormons) that some aspect of each person was somehow already alive in the indefinite past.
So how did the universe begin? 'Begin' is the wrong word. Nonetheless, there is good evidence that the age of the universe is about 15 billion years, so nothing, not space, not time, existed before some instant of time roughly 15 billion years ago. See how our narrativium-powered semantics confuses us. This does not mean that if you went back 15 billion and one years, you would find nothing. It means that you cannot go back 15 billion and one years. That description makes no sense. It refers to a time before time began, which is logically incoherent, let alone physically so.
What cosmologists are pretty sure happened is this. The universe came into being as a tiny speck of space and time. The amount of space inside this tiny speck grew rapidly, and time began to elapse so that 'rapidly' actually had a meaning. Everything that there is, today, right out into the furthest depths of space, stems from that astonishing 'beginning'. Colloquially, the event is known as the Big Bang. The name reflects several features of the event -for example, that tiny speck of space/time was enormously hot, and grew in size exceedingly rapidly. It was like a huge explosion, but there was no stick of cosmic dynamite, sitting there in no-space with its non-material fuse burning away as some kind of pre-time pseudo-clock counted down the seconds to detonation. What exploded was, nothing. Space, time, and matter are the products of that explosion: they played no part in its cause. Indeed, in a very real sense, it had no cause.
The evidence in favour of the Big Bang is twofold. The first item is the discovery that the universe is expanding. The second is that 'echoes' from the Big Bang can still be detected today. The possibility that the universe might be getting bigger first appeared in mathematical solutions to equations formulated by Albert Einstein. Einstein viewed spacetime as being 'curved'. A body moving through curved spacetime deviates from its normal straight line path, much as a marble rolling on a curved surface does. This deviation can be interpreted as a 'force', something that pulls the body away from that ideal straight line. Actually there is no pull: just a bend in spacetime, causing a bend in the body's path. But it looks as if there's a pull. Indeed this apparent pull is what Newton called 'gravity', back in the days when people thought it really did pull bodies together. Anyway, Einstein wrote down some equations for how such a bendy universe ought to behave. They were very difficult equations to solve, but after making some extremely strong assumptions, basically that at any instant of time space is a sphere, mathematical physicists worked out few answers. And this tiny, very special list of solutions, the only ones their feeble methods could find, told them three things that the universe could do. It could stay the same size forever; it could collapse down to a single point; or it could start from a single point and grow in size without limit.
We now know that there are many other solutions to Einstein's equations, leading to all sorts of bizarre behaviour, but back in the days when today's paradigm was being set, these solutions were the only ones anybody knew. So they assumed that the universe must behave according to one or other of those three solutions. Science was subliminally prepared either for continuous creation (the universe is always the same) or for the Big Bang. The Big Crunch, in which the universe shrinks to an infinitely dense, infinitely hot point, lacked psychological appeal.
Enter Edwin Hubble, an American astronomer. Hubble was observing distant stars, and he made a curious discovery. The further away the stars were, the faster they were moving. He knew this for distinctly indirect, but scientifically impeccable, reasons. Stars emit light, and light has many different colours, including 'colours' that the human eye is unable to see, colours like infra-red, ultra-violet, radio, x-ray ... Light is an electromagnetic wave, and there is one 'colour' for each possible wavelength of light, the distance from one electromagnetic peak to the next. For red light, this distance is 2.8 hundred thousandths of an inch (0.7 millionths of a metre).
Hubble noticed that something funny was happening to the light emitted by stars: the colours were shifting in the red direction. The further away a star was, the bigger the shift. He interpreted this 'red shift' as a sign that the stars are moving away from us, because there is a similar shift for sound, known as the 'Doppler effect', and it's caused by the source of the sound moving. So the further away the stars are, the faster they're travelling. This means that the stars aren't just moving away from us, they're moving away from each other, like a flock of birds dispersing in all directions.
The universe, said Hubble, is expanding.
Not expanding into anything, of course. It's just that the space inside the universe is growing. That made the physicists' ears prick up, because it fitted exactly one of their three scenarios for changes in the size of the universe: stay the same, grow, collapse. They 'knew' it had to be one of the three, but which? Now they knew that, too. If we accept that the universe is growing we can work out where it came from by running time backwards, and this time-reversed universe collapses back to a single point. Putting time the right way round again, it must all have grown from a single point -the Big Bang. By estimating the rate of expansion of the universe we can work out that the Big Bang happened about 15 billion years ago.
There is further evidence in the Big Bang's favour: it left 'echoes'. The Big Bang produces vast amounts of radiation, which spreads through the universe. Because the universe is spherical, the radiation eventually comes back on itself like a round-the-world traveller. Over billions of years, the remnants of the Big Bang's radiation smeared out into the 'cosmic background', a kind of low-level simmering of radiant energy across the sky, the light analogue of a reverberating echo of sound. It is as if God shouted 'Hello!' at the instant of creation and we can still hear a faint 'elloelloelloelloeiio ...' from the distant mountains. On Discworld this is exactly the case, and the Listening Monks in their remote temples spend their whole lives straining to pick out from the sounds of the universe the faint echoes of the Words that set it in motion.
According to the details of the Big Bang, the cosmic background radiation should have a 'temperature' (the analogue of loudness) of about 3° Kelvin (0° Kelvin is the coldest anything can get, equivalent to about -273° Celsius). Astronomers can measure the temperature of the cosmic background radiation, and they do indeed get 3° Kelvin. The Big Bang isn't just a wild speculation. Not so long ago, most scientists didn't want to believe it, and they only changed their minds because of Hubble's evidence for the expansion of the universe, and that impressively accurate figure of 3° Kelvin for the temperature of the cosmic background radiation.
It was, indeed, a very loud, and hot, bang.