In a Kantian or even platonic context, there should always be natural explanations found (even if having to appeal to statistics / probability) since any noumenal "provenances" for happenings / situations would still be converted into either a concrete or an abstract physics member of the former's arrangements. [As represented in extrospective perceptions / reflective thought; or the sensible world.] But let's explore the possibility of synchronicity qualifying for "natural" instead of a speculative denizen of the transcendent side.
Is there room for conceiving an extra system of
events connected by "meaning" or whatever alternative, that can co-habit with or supervene over the traditional conception of the world being a succession of events / effects and an interdependent maze of causal relationships? Difficult to imagine such ever being accepted by science unless principles could be found for it that yield accurate forecasts or whatever. Nevertheless, what would have opened-up if the era of positivism had fully succeeded in banishing causality along with the rest of what it declared "metaphysical" mumbo-jumbo?
Karl Pearson's historic rejection of causation (an applicable quote at bottom) almost seems to imply the consequence of making the pattern of events / effects that has been transpiring for billions of years into nothing more than an extremely long-lived sequence of orderly coincidences falling out of probability. [Akin to immortal monkeys eventually typing out
Hamlet over the course of infinity, without the aid of creative or rule-following intellect.] Accordingly, it would also require the vast majority of existence to lack this "mere appearance" of causal-organization enabled by principled forces. So as to allow this otherwise extremely unlikely possibility of a lengthy, unbroken pattern of a "developing universe". In essence, "chance" or "accidental illusion of orderly agencies at work" [like paredolia affairs] would not be confined anymore to just "content of the moment" or happening over short spans, but could also rarely involve huge tracts of time and space (like the duration of a cosmos).
IF the above was the case, then our conceiving of this "apparently"-regulated structure of unfolding change as the "handiwork of causation" would be erroneous. Yet still a practical conception, which likewise might open the doors to multiple ways of interpreting / understanding its patterns [equally untrue yet providing useful purposes]. One of those contenders might be synchronicity. But again, that bugbear of "what use is it, is there a formulation that could yield predictive results, etc" returns...
Judea Pearl: These investigations, drove Galton to consider various ways of measuring how properties of one class of individuals or objects are related to those of another class. [...] Here we have, for the first time, an objective measure of how two variables are "related" to each other, based strictly on the data, clear of human judgment or opinion.
Galton's discovery dazzled one of his students, Karl Pearson, now considered the founder of modern statistics. Pearson was 30 years old at the time, an accomplished physicist and philosopher about to turn lawyer, and this is how he describes, 45 years later, his initial reaction to Galton's discovery: "I felt like a buccaneer of Drake's days -... I interpreted that sentence of Galton to mean that there was a category broader than causation, namely correlation, of which causation was only the limit, and that this new conception of correlation brought psychology, anthropology, medicine, and sociology in large parts into the field of mathematical treatment."
Pearson categorically denies the need for an independent concept of causal relation beyond correlation. He held this view throughout his life and, accordingly, did not mention causation in ANY of his technical papers. His crusade against animistic concepts such as "will" and "force" was so fierce and his rejection of determinism so absolute that he EXTERMINATED causation from statistics before it had a chance to take root.
It took another 25 years and another strong-willed person, Sir Ronald Fisher, for statisticians to formulate the randomized experiment - the only scientifically proven method of testing causal relations from data, and which is, to this day, the one and only causal concept permitted in mainstream statistics.