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I don't know if this is more a question for mathematicians or physicists (or even philosophers), but what would be the difference between time having a beginning infinitely long ago and time having no beginning (apart from the obvious, one had a beginning and the other didn't)? How could we tell whether it has always existed or began infinitely long ago?

In other words, did time have a beginning or did it begin an infinitely long time ago? How could we tell?

It appears, from Ryder Rude's answer and the embedded link, that the problem is addressed in questions about the shape of the universe.

Michael Greaney
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According to general relativity, spacetime is a four dimensional manifold. The most trivial topology of spacetime would be that of an infinite $R^4$ space. This sort of universe has no beginning. The geodesics can be extended in the past infinitely far. There are other interesting topologies too. You can read about them here in the Global Universe Structure section.

"Having a beginning infinitely long ago" is an oxymoron. There can be no beginning if something has existed for infinitely long.

About the space dimensions, it is hypothesized that they might be compact, like some 3D equivalent of the 2D surface of a cylinder or a torus. I don't know if a similar claim can be taken seriously for the time dimension. Nevertheless, if you hypothetically consider that time is like that, then time doesn't have a beginning but it didn't exist infinitely far back.

Ryder Rude
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