In statistics, apophenia is called a Type I error, seeing patterns where none, in fact, exist. It is highly probable that the apparent significance of many unusual experiences and phenomena are due to apophenia, e.g., EVP, numerology, the Bible code, anomalous cognition, ganzfeld "hits", most forms of divination, the prophecies of Nostradamus, remote viewing, and a host of other paranormal and supernatural experiences and phenomena. (Carroll)
The human brain is a powerful pattern-recognition machine, evolved over the millennia. Ray Kurzweil: "Humans are far more skilled at recognizing patterns than in thinking through logical combinations, so we rely on this aptitude for almost all of our mental processes. Indeed, pattern recognition comprises the bulk of our neural circuitry. These facilities make up for the extremely slow speed of human neurons." (quoted in Steven Johnson's Emergence). We evolved to see connections.
And there is some research that indicates that dopamine levels in the brain enhance pattern recognition. In a New Scientist article, "People with high levels of dopamine are more likely to find significance in coincidences, and pick out meaning and patterns where there are none." ("Paranormal beliefs linked to brain chemistry", New Scientist, July 27, 2002.)
To the skeptic, Carl Jung's "synchronicity" is an example of apophenia. Synchronicity suggests an acausal connection between phenomena, that two apparently unrelated events reveal something. (Again, see The Skeptic's Dictionary entry on Jung for an unsympathetic view.)
Saying the world can only be understood empirically, and that empirical tests are the basis of knowledge, is problematic without taking into account the role of the brain in organizing sense data into recognizable phenomena. Inasmuch as consciousness has evolved (and continues to evolve), that science has passed through various "paradigms" since Copernicus and Galileo, and that there are "qualitative" modes of scientific practice in addition to quantitative or deductive modes suggests at least that we keep an open mind about ways to knowledge.
In this vein, one would have to say exactly what "meaning" or "meaningfulness" is, in light of coincidence and metaphor, before dismissing Jung. (Per Agent Dale Cooper: "When two separate events occur simultaneously pertaining to the same object of inquiry, we must always pay strict attention." or "Coincidence and fate figure largely in our lives. Ah! Damn good coffee!" I could swear I remember him saying "Never laugh in the face of coincidence", but that could be false or true-but-clouded memory.)
The ability of the brain to see patterns, or connections, or to make connections or leaps between two seemingly unrelated concepts -- e.g., "all the world's a stage" -- is a basic, perhaps universal, human device for exploring and understanding the new, the complex, the difficult, and perhaps the unfathomable.
Along these lines, see the Journal for Patterns Recognised: "The Journal for Patterns Recognised is a journal dedicated to the study of pattern recognition. We can recognise sheep in clouds, faces in 4 well-placed rocks and a tree in a mathematically produced set of lines. This ability to recognise familiar objects in formlessness is said to be the engine behind imagination. Therefore we understand pattern recognition gone wrong as the well from which human culture, roughly defined as the framework of socially accepted interpretations of the real, flows." For more, see the background essay, "Why a Journal for Patterns Recognised?!".
The "perceived connection where none exists", a pattern recognized where no obvious or detected material interaction is taking place between the two nodes. But a connection erupts in the mind, rich in meaning maybe, revealing something. Something has happened.
This past spring, under the Kennedy Expressway on Fullerton Avenue here in Chicago, someone observed that the water dripping down the concrete underpass had created a form like Mary, the Mother of God. There is something inherently un-testable (and what does it say about someone who wants to test it?) about images of the Virgin Mary anywhere, but regardless, the pattern recognized becomes in itself a material force as people act on the idea. As word of the Virgin of the Kennedy spread, hundreds came to pray and leave flowers, candles and other offerings. "Faithful Call Image On Underpass Wall 'Beautiful' Others Call Image Salt Stain." A few weeks later, police charged Victor Gonzalez with defacing public property by writing "Big Lie" across the image with shoe polish. Police then asked the Illinois Department of Transportation to paint over the image to prevent further trouble.
There is a network of interactions, let's say, independent of human consciousness. Say salty melted snow running off of a highway, rusting steel, dripping down concrete. Molecules interacting, the workings of ice and salt, bonds breaking and re-forming, themselves influenced by Winter yielding to Spring, by the motion of Earth around the Sun. (Is there not something wonder-full even in that process?) And then there is the interconnection of consciousness with sense perceptions, the sparking of recognition of a pattern on the concrete, yielding meanings, shared by some, contested by others. Another network, this one of ideas, thoughts, understandings. Why not the Goddess revealing herself, exerting herself, in the most mundane, dismal, anti-Nature of places, the expressway underpass, the literal underside of another network, the infrastructure of Global Warming and Planet Destruction? Or maybe the pattern recognized is "just" illusion, or dis-illusion. Prompted by the hunger of the dispossessed who beg under that underpass, and the workers crawling along Fullerton in perpetual rush-hour Chicago traffic on the way home from another day of mind-dulling, soul-stealing work; the hunger for a Sign that they might achieve a future worth living in? Or does it say that that which you long for is nothing, void of meaning, a stain? Which is the Big Lie?
Something in the "perceived connection", even if it is not agreed upon by all parties.