Infinite universe

 If the universe is infinite, we will never know it.  It will always be a case of the end might be just over the next rock.

That's essentially correct. An infinite universe is not something we could establish by direct observation, because our observations are always limited to a finite region.

The situation is a bit like standing on a vast plain in a fog. No matter how far you can see, there is always the possibility that the fog clears a little farther on and reveals an edge. Seeing more and more territory never proves there is no edge anywhere.

Modern cosmology faces exactly this problem. We can observe only the observable universe, the region from which light has had time to reach us since the early universe became transparent. Beyond that horizon, we have no direct access. The entire universe may be much larger than the observable part, perhaps finite, perhaps infinite.

What cosmologists can do is look for clues. Measurements suggest that space is very nearly flat on the largest scales we can observe. A perfectly flat universe can be infinite, though it can also be finite if it has a more complicated global topology. So even if every observation continues to support flatness, infinity itself remains an inference rather than a directly observed fact.

There is an interesting philosophical asymmetry here. We can sometimes prove a thing is finite by finding a boundary or by measuring a closed geometry. But proving infinity is much harder. No finite amount of evidence can completely rule out the possibility that "the end is just beyond the next rock," as you put it.

This resembles your earlier point about maps and dates. A map can keep expanding as new territory is surveyed, but no matter how large the map becomes, its size alone does not demonstrate that the territory has no end. The claim "infinite" always reaches beyond what has actually been mapped.

We know that the universe as we find it is amazingly flat in Euclidean terms, within the limits of our measurement.  As we refine measurement, would finding curvature demonstrate anything?

Yes, potentially quite a lot, depending on what kind of curvature we found.

If measurements eventually showed that space has positive curvature—the three-dimensional analogue of the surface of a sphere—then that would be strong evidence that the universe is finite in volume. In such a universe, if you could travel far enough in a straight line (ignoring expansion and practical difficulties), you might eventually return to your starting point.

If measurements showed negative curvature—the analogue of a saddle shape—then the universe would almost certainly be spatially infinite, assuming the simplest topology. Negative curvature naturally produces an unbounded space.

The difficulty is that the observed curvature is astonishingly close to zero. Imagine trying to determine whether the Earth is curved while standing on a billiard table-sized patch of ground. The Earth is curved, but locally it looks flat. Likewise, the observable universe is only a portion of whatever the whole universe may be, and that portion appears extremely flat.

There is another complication. Curvature and finiteness are not exactly the same thing. A flat universe can still be finite if it has a nontrivial topology. The classic two-dimensional example is a video-game screen where exiting the right edge brings you back on the left. Locally everything is flat, yet the total area is finite. Mathematicians describe analogous possibilities for three-dimensional space.

So if future measurements found significant positive curvature, that would tell us something profound: the universe is probably finite. If they found significant negative curvature, that would strongly suggest infinity. But if measurements continue to converge toward exactly zero curvature, we may never know. A perfectly flat universe could be either finite or infinite depending on its global structure.

This is one of the places where cosmology runs into a deep epistemological limit. We can measure local geometry ever more precisely, but the global question—"Does space go on forever?"—may remain forever beyond direct observation. The end may indeed always be just beyond the next rock, except that in this case the next rock lies beyond the cosmological horizon.

Even finding positive curvature would not settle the issue as it could be some localized curvature.  The idea of non-trivial topology seems a huge stretch.

Your first point is exactly the sort of caution cosmologists themselves have to exercise.

If we measured positive curvature within the observable universe, we would not have directly proved that the entire universe is a positively curved closed space. We would have shown that the region we can observe has that geometry. The inference to a globally closed universe would depend on assuming that the observable region is representative of the whole. That assumption is reasonable under the standard cosmological principle, but it is still an assumption.

There is, however, a difference between ordinary localized curvature and the curvature cosmologists discuss. General relativity allows local distortions caused by stars, galaxies, and clusters, but when cosmologists speak of the universe having positive or negative curvature, they mean the large-scale average geometry after smoothing out those local lumps. If a persistent positive curvature signal appeared on the largest observable scales, it would be difficult to explain as merely a local bump in an otherwise flat universe. Not impossible, but increasingly contrived.

As for non-trivial topology, many physicists share your instinct. The idea is mathematically respectable, but there is currently no observational evidence for it. It arises partly because once you allow the mathematics of general relativity, such spaces are perfectly legal solutions. A flat three-dimensional space wrapped around on itself is not inherently stranger mathematically than the surface of a cylinder or torus.

The question is whether nature actually does this. So far, no compelling evidence says it does.

Your skepticism reflects a broader pattern in science. When confronted with two explanations,

    The universe is approximately what it appears to be on the largest scales.

    The universe has an elaborate global topology that makes appearances deceptive.

most scientists would prefer the first unless observations force the second.

What fascinates me is that cosmology may never escape this dilemma. Even if measurements become a hundred times more precise, we may only learn that the observable universe is flatter than we thought. That narrows the possibilities, but it does not eliminate the fundamental underdetermination. There may be truths about the global structure of reality that are simply inaccessible from within one finite observational patch.

In that sense, the question "Is the universe infinite?" resembles the medieval question "What lies beyond the edge of the map?" Except now the map itself may have a permanent boundary imposed not by ignorance but by the finite age of the universe and the finite speed of light. No matter how good our instruments become, some territory may never be available for inspection.

You appear to think I have a log in this fire; I am only responding to the possibilities you raise, and fully think this question represents a limit on possible human knowledge.  The difference between the sort of curvature you describe closing the universe and local curvature is only one of degree.  

Fair point. I was reading more commitment into your remarks than was actually there.

You seem to be approaching this less as a cosmological question than as an epistemological one: what, in principle, can be known from inside the system? On that question, I think your position is quite defensible. Even if the universe has a definite global structure, there may be no observational path that allows beings confined to one causal region to determine it with certainty.

On curvature, I would make a small distinction. In general relativity, local and global curvature are described by the same mathematics, so in that sense you are right: the difference is one of scale and degree, not kind. A star bends spacetime in its vicinity; a positively curved universe would bend spacetime on the largest scales.

The reason cosmologists treat them somewhat differently is that local curvature tends to fall off with distance from the source, whereas a globally curved universe has curvature as a persistent property of the large-scale geometry itself. But from a geometric standpoint there is no sharp conceptual boundary between the two. The equations do not suddenly switch to a different category when one moves from a galaxy to the cosmos.

What I find interesting is that infinity itself may be a red herring. Suppose future measurements established beyond reasonable doubt that the observable universe is extraordinarily flat. We would still be left with several possibilities:

    Infinite and flat.

    Finite and flat with some nontrivial topology.

    Vast but finite and so close to flat that we cannot distinguish it.

    Something more exotic outside our horizon.

The observational consequences within our accessible region might be nearly identical. At that point the dispute becomes less about physics and more about what level of inference one is willing to accept.

Your "next rock" analogy gets at this nicely. If every time we look farther, the universe continues in much the same way, we may become increasingly confident that it extends indefinitely. But confidence is not the same thing as demonstration. The edge, if there is one, can always retreat beyond the horizon of observation.

There is a recurring theme in science that some questions are not merely unanswered but may be structurally unanswerable. The global extent of the universe is one of the strongest candidates. Not because we lack sufficiently advanced instruments, but because the information needed to answer the question may never be available to observers situated as we are. That is a different kind of ignorance from not yet having measured something accurately enough. It is closer to a built-in limit of perspective.

I don't see where that makes infinity a red herring.  It is there as a possibility or even, given present observations, a probability.  There may be an ontological thing here in the sense that "non-existence cannot exist." 

I think you're right to push back on my use of "red herring." That phrase suggests infinity is somehow irrelevant, and that's stronger than I intended.

What I was trying to get at is that the question of infinity may be more accessible to ontology than to observation. It is certainly not irrelevant. If the universe is infinite, that is a profound fact about reality. The issue is whether it is a fact that can ever be established empirically.

Your remark about "non-existence cannot exist" points toward a very old line of thought. Going back to Parmenides, some philosophers have argued that "nothing" is not a coherent state at all. If one takes that intuition seriously, then one naturally becomes suspicious of boundaries. A boundary seems to invite the question: what is beyond it? If the answer is "nothing," one is tempted to reply that "nothing" cannot be a place or a thing.

Historically, this intuition has often supported belief in an unbounded cosmos. Not because observations proved it, but because a finite universe with an edge seemed conceptually unstable. If you can point to a limit, then what lies beyond the limit? If the answer is "absolutely nothing," many people find that difficult to make sense of.

On the other hand, modern geometry complicates the issue. A finite universe need not have an edge. The standard analogy is the surface of the Earth. It is finite in area but has no boundary. One can travel indefinitely without encountering an edge or a region of "non-existence." In three dimensions, analogous geometries are mathematically possible.

Of course, that does not settle the ontological question. One might still ask why reality itself should be finite rather than infinite. But it does mean that finiteness and boundaries are no longer synonymous.

What strikes me about your observation is that it is not really an argument from physics at all. It is closer to a metaphysical intuition: reality seems more intelligible if existence is in some sense unbounded. That intuition has appeared repeatedly throughout intellectual history, from the ancient Greeks through thinkers such as Giordano Bruno and beyond.

Whether that intuition is correct is another matter. Physics can tell us that a finite but unbounded universe is mathematically consistent. It is much less clear that physics can tell us whether existence itself must be finite or infinite. That may be one of those questions where ontology gets farther than observation, though not necessarily far enough to produce certainty.

The interesting thing is that your argument does not rely on the universe looking infinite. It would still arise even if tomorrow we discovered unmistakable evidence that space closes on itself. The question would simply shift from "Does space go on forever?" to "Why is reality structured this way rather than some other way?" In that sense the ontological issue survives whatever cosmology eventually finds.

As you point out, not my argument, so don't treat it as it were.  I used the word "may," not "is." To me that seems wordplay.

That's a fair correction.

You did not advance an ontological argument; you merely noted that there may be one lurking in the background. I responded as though you were endorsing a particular metaphysical position, which goes beyond what you actually said.

As for the "finite but unbounded" idea seeming like wordplay, I can understand that reaction. Mathematicians and physicists would insist it is not merely verbal, because they can construct precise geometrical models with those properties. The surface of a sphere is the standard example: finite area, no edge.

But there is a sense in which the phrase can feel unsatisfying when it is imported into broader philosophical discussions. If someone asks, "Does reality end?" and receives the answer, "No, it is finite but unbounded," one may reasonably feel that the language has shifted from the original concern rather than resolving it. The mathematical concept is clear enough; whether it addresses the deeper intuition behind the question is another matter.

More generally, I think your earlier point about the limits of knowledge remains the stronger one. Even if we set aside ontology entirely, there is a gap between what we can observe and the global structure of the universe. An infinite universe, a sufficiently large finite universe, and certain topologically exotic universes could all look essentially the same from our vantage point.

That does not mean the question is meaningless. It means there may be no observational procedure capable of settling it conclusively.

One thing I have noticed in our discussion is that you tend to separate three issues that are often run together:

    What the observations currently suggest.

    What can, even in principle, be known.

    What may be true for philosophical reasons independent of observation.

Many discussions of cosmology blur those together. You have been fairly careful not to. When you say infinity may be a possibility or even a probability given current observations, that is a different claim from saying it is demonstrable, and different again from saying it is required by ontology.

Keeping those distinctions separate probably avoids a lot of confusion. The fact that infinity may be plausible does not mean it is knowable, and the fact that it may be unknowable does not mean it is implausible. Those are different questions.

The wordplay is not the idea of an unbounded but finite space.  I can picture that well enough, especially when time is part of the structure.  The wordplay is the sentence that nonexistence cannot exist.

Ah, then I misunderstood you again.

Yes, "nonexistence cannot exist" is perilously close to a linguistic trick. The sentence uses the verb exist to deny existence, which makes it sound profound even before we've established that it means anything coherent.

One can unpack it in several ways:

    If "nonexistence" means "the absence of existence," then saying it cannot exist is almost tautological.

    If "nonexistence" is being treated as a thing or state, then the sentence may simply be objecting to reifying an absence.

    If it is meant as a substantive metaphysical claim, then much more work is required than the sentence itself provides.

This is why philosophers spend so much time trying to distinguish between nothing as a word and "Nothing" as if it were a peculiar kind of something. Ordinary language encourages us to treat absences as objects. We speak of darkness, silence, emptiness, or nothingness as though they were things that occupy a place.

In that respect, "What is beyond the edge of the universe?" and "Can nonexistence exist?" may suffer from a similar problem. The grammar suggests a question, but the grammar itself may be smuggling in assumptions.

One reason I tend to be cautious around these discussions is that many apparent metaphysical conclusions evaporate when translated into more precise language. Sometimes a deep mystery remains. Sometimes one discovers that the mystery was generated by the structure of the sentence.

Your use of "may" was appropriately tentative. There may indeed be an ontological issue hiding behind the cosmological one. But the slogan "nonexistence cannot exist" does not, by itself, get us very far. It may be a profound truth, a trivial tautology, or a grammatical illusion. Determining which of those it is requires considerably more analysis than the slogan provides.

In fact, one test I often apply is: what observation or logical consequence follows from accepting the statement? If the answer is unclear, that is usually a sign that the sentence is doing more rhetorical work than explanatory work.

So on reflection, I think your characterization of it as "wordplay" is a healthy suspicion rather than a dismissal. Many famous philosophical arguments have turned out either to depend on such verbal pivots or to require much stronger foundations than the memorable phrase that first expressed them.