Not sure if sarcastic. Causation definitely implies correlation.

Not always. There can be other more significant or equivalent causes.

Sure you can, "causation without correlation" happens all the time in systems with feedback from unmodelled variables. A canonical example is correlating braking force with vehicle speed while driving downhill; if you're trying to maintain speed, your data's going to look like your speed remains relatively constant no matter how hard you brake.

(Or, I guess relevant here, you're not going to find much correlation between, say, getting guillotined and being dead 150 years in the future, even though getting guillotined absolutely does count as a cause of death in the conventional sense.)

or, for a mathematical example x and sin(x) have zero correlation.

correlation implies a linear relationship.

Is that a definition of correlation inside the field of mathematics? Because the normal English definition of correlation just means a connection, and absolutely doesn't mean linear.

"a connection or relationship between two or more facts, numbers, etc." https://dictionary.cambridge.org/dictionary/english/correlat...

edit: Ah, Wikipedia intro suggests that yes it's specific within statistics (though still seems to not be a hard requirement to call something a correlation?): "In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it normally refers to the degree to which a pair of variables are linearly related." https://en.wikipedia.org/wiki/Correlation

The slightly more fun version of that example is that cos(x) and sin(x) have zero correlation. But if you plot a bunch of coordinates that are generated that way, you'll quickly notice a pattern...

You can easily view cos(x) as causing the value of sin(x) to change - if cos(x) starts going down, sin(x) will too soon afterward. If cos(x) starts going up, so will sin(x), soon afterward.

In fact, if you allow for a time lag between your data series, you will find that sin(x) and cos(x) are perfectly correlated.

I am in a sphere in space, and I throw a dart at a wall, and I measure the coordinates. I caused those coordinates to occur, yet there is no correlation between cause and coordinates. Or a coin toss.

Causation without correlation is the norm.