Coincidence, as I define it, is the association of two or more disparate events, and the added value attributed to that association.
If you travel around the world and happen to run into your next-door neighbor, that would be an obvious coincidence. Anybody with an understanding of geography would recognize the unusual nature of that meeting, but the deeper meaning of the event would be limited primarily to the two people involved. It may startle you to run into your neighbor on the other side of the globe, but chances are it won’t really affect your life, or the world around you, in any significant way.
Nor should it. Given what we understand about probability, it’s entirely reasonable to think that nearly impossible events are going to happen some day. It may be odd to run into your neighbor in a foreign country, but it could be that you’ve been traveling in the same circles for years, influenced in unseen ways by the common factors in your lives, and only once happened to recognize each other on the other side of the room. To the outside observer, there’s no coincidence at all. Imagine the random, uninformed villager who witnesses the random meeting of 2 random westerners on some random day. Would he/she recognize the significance of that event? Would he care?
Coincidence is an invention of the human mind, its ability to recognize patterns, and its desire to assign meaning where none exists. Coincidence doesn’t exist in the natural world. The predator attempting to kill his prey is not concerned with anything but survival. He isn’t concerned with his prey’s life story, and won’t take any significance from a random falling tree helping the prey to escape. He’ll simply go on with his life, looking for his next meal. Only a human would create a story of those events, or even recognize that this day was different from any other.
But even if coincidence is a human construct, that doesn’t mean it doesn’t exist. Nor does it mean that it shouldn’t be appreciated. That a human would assigning meaning where none exists doesn’t mean that the human is wrong or against nature. We, as humans, are part of the world, and our desire to recognize coincidence is part of us. We take pleasure in finding the story, finding the beauty in how events unfold. It’s part of what makes us what we are.
Whether we use coincidence to appreciate the random nature of things, or to find some magical pattern where one doesn’t actually exist, coincidence is there, for you to do with as you please.