(from Wikipedia)

A recent study within the Princeton Engineering Anomalies Research Lab has suggested that there is a small though statistically measurable link between human thought and patterns that occur in random data sets. There is no evidence as to if this is caused by individuals unintentionally recognizing complex patterns and then moulding their thoughts towards an unconsciously known result or if the thoughts of the individual are themselves affecting the random patterns in a manner of individuation. This study's results have not been replicated, and its methodologies are disputed. [1]

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Criticism

Since the theory of synchronicity is not testable according to the classical scientific method, it is not widely regarded as scientific at all, but rather as pseudoscientific or an example of magical thinking.

Probability theory can attempt to explain events such as the plum pudding incident in our normal world, without any interference by any universal alignment forces. However, the correct variables required for actually computing the probability cannot be found. This is not to say that synchronicity is not a good model for describing a certain kind of human experience - but it is a reason for refusal of the idea that synchronicity should be considered a "hard fact", i.e. an actually existing principle of our universe.

Supporters of the theory claim that since the scientific method is applicable only to those phenomena that are reproducible, independent of observer and quantifiable, the argument that synchronicity is not scientifically 'provable' should be considered a red herring, as, by definition, synchronistic events are not independent of the observer, since the observer's unique history is precisely what gives the synchronistic event meaning for the observer.

A synchronistic event appears like just another meaningless 'random' event to anyone else without the unique prior history which correlates to the event. This reasoning claims that the principle of synchronicity raises the question of the subjectivity of significance and meaning in the sequence of natural events.