I was reading about a famous article by physicist Eugene Wigner titled *The unreasonable effectiveness of mathematics*, where, citing Wikipedia

In the paper, Wigner observed that the mathematical structure of a physics theory often points the way to further advances in that theory and even to empirical predictions, and argued that this is not just a coincidence and therefore must reflect some larger and deeper truth about both mathematics and physics.

I’ll write about this in a later post, but for now this brings me to consider what we mean by **coincidence** and how we think about them.

In the above, a coincidence is remarked between two apparently independent domains, that of mathematics, and that of the structure of the world. In general, when finding striking coincidences our instinct is to reach for an explanation. Why? Because by definition a striking coincidence is basically something of a-priori very low probability, something implausible that merits investigation to “make sense of things”.

An **explanation** of a coincidence is a restatement of its content that raises its probability to a level such that it is no longer a striking state of affairs, *the coincidence is dissolved*. Example:

**Bob**: *Have you noticed that every time the sun rises the rooster crows? What an extraordinary coincidence!*

**Alice**: *Don’t be silly Bob, that’s not a coincidence at all, the rooster crows when it sees the sun rise. Nothing special*

**Bob**:* Erm… true. And why did David choose me to play the part of fool in this dialogue?*

Alice’s everyday response to coincidence is at heart nothing other than statistical inference, be it bayesian or classical hypothesis testing[1]. The coincidence at face value plays the role of a hypothesis (null hypothesis) that assigns a low probability to the event, ie the hypothesis of a chance occurrence between two seemingly independent things. The explanation in turn plays the role of the accepted hypothesis by virtue of assigning a high probability to what is observed.

So one could say that the way we respond and deal with coincidence is really a mundane form of how science works, where theories are presented in response to facts, and those that better fit those facts are accepted as explanations of the world.

But how do explanations work internally? The content of an explanation is the establishment of a relationship between the two a-priori independent facts, typically through **causal mechanisms**. The causal link is what raises the probability of one given the other, and therefore of the joint event. In the example, the causal link is ‘*the rooster crows when it sees the sun rise*‘.

But the links are not always direct. An interesting example comes from what in statistics is called a spurious relationship. Again, Wikipedia says:

An example of a spurious relationship can be illuminated examining a city’s ice cream sales. These sales are highest when the rate of drownings in city swimming pools is highest. To allege that ice cream sales cause drowning, or vice-versa, would be to imply a spurious relationship between the two. In reality, a heat wave may have caused both

although the emphasis here is about the lack of *direct* causal relationship, the point regarding coincidence is the same. Prior to realizing that both facts have a common cause (the explanation is the heat wave), one would have regarded the relationship between ice cream sales and drownings as a strange coincidence.

In the extreme case the explanation reveals that the two facts are really two facets of the same thing. The coincidence is dissolved: any given fact must necessarily coincide with itself. Before the universal law of gravitation, it would have been regarded as extraordinary that both the apples falling from a tree, and the movement of planets in the heavenly skies had the same behavior. But we know now that they are really different aspects of the same phenomenon.

Notes

[1] The act of explanation is, in classical statistics language, the act of rejecting the null hypothesis. In the Bayesian picture, the explanation is what is probabilistically inferred due to the higher likelihood it assigns to the facts (and its sufficient prior probability)