I agree with most of that, but I don't think the "science of cause and effect," as you call it, should be completely ignored. After all, it can shed light on some coincidences. If for example, the reason an event is seen as significant is closely tied to the reason that it happened, or if it turns out that one of the two coinciding events was actually quite probable or inevitable, that would be relevant to our perception of the coincidence.
But more important than cause and effect is probability. I'll take your example of the bus: Considering the vast number of buses in the world, and the vast number of people who take them, the scenario you describe is inevitable, given enough time. Of course, it will seem very monumentous to the individual concerned, and fairly enough, I admit, but on a global scale, it's rather mundane. It seems likely to me that this scenario has already happened to some people, out there, (whether I believe individual anecdotes or not), especially if we expand it to consider regularly scheduled trains, charter planes, and ferries (which we should, I think, since we would concider an event involving one of these just as significant.)
It is impossible to prove anything.
That's true (to an extent, at least.) I don't necessarily know what I would think if it happened to me. As a skeptic and an atheist, I like to think that--although I would be profoundly glad to have my life--I would accept it as merely fortuitous chance. But, since it has never happened to me, it is impossible for me to say how I would react if it did. I can admit that.
However, even if that event caused me to, let's say, completely devote my life to religion or cherish every item I wore that day as "lucky," my personal, emotional reaction would change nothing about the probabilities involved.
Does the coincidence mean that two events are meaningful? Or does the coincidence mean that two events are actually meaningless because it's just cause and effect? The term coincidence has been used to mean both.
This is a good question. Strictly speaking, the word "coincidence" doesn't mean either of those. The word is neutral on the question of meaning. It simply means that two events "coincided." In fact, two events can still constitute a "coincidence," even if they are not related in any apparently meaningful way. If you see two sixes, and then roll a two and a five, that's still technically a coincidence, as long as those two things happened closely enough in time.
So, when you hear someone using the term in a loaded way, regardless of whether they are attributing meaning or dismissing it, they are misusing the word.
It is acceptable, though, if they separate the word itself from the meaning or lack thereof, by using the word "just" or "merely," or something similar.
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