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Lumifer comments on The best 15 words - Less Wrong
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I am confused, that doesn't seem to be true.
Consider a sine wave. It can be observed in a great number of phenomena, from the sound produced by a tuning fork to the plot of temperature in mid-latitudes throughout the year. All measurements which produce something resembling a sine wave are correlated. Remember that correlation (well, at least Pearson's correlation -- I assume that's what is meant here) is invariant to linear transformations so different scale is not a problem.
Correlation isn't a property of a pair of mathematical functions or a pair of physical systems, it's a property of a pair of random variables.
"A and B are correlated" means "Observing A can change your probabilistic beliefs about B".
If you already know that A and B are both sine waves, then neither has any belief-updating power over the others, there's no randomness in the random variables.
(I know that's not 100% precise... someone else please improve.)
In the vast majority of cases involving sine waves, the correlation between A and B is due to the common cause of time. Space is also a common cause of such correlations.
However, if you imagine a sine wave in time and another sine wave in space, they have no correlation until you impose a correlation between space and time (e.g., by using a mapping from x to t). In that case, Armok's comment about a logical rather than physical cause might apply.
I don't understand what does that mean. In which sense can time be thought of as a cause?
I started writing a reply to this comment, but as I was thinking through it I realized that the situation is actually WAY more interesting than I thought and requires a whole post. I've posted it in discussion:
http://lesswrong.com/r/discussion/lw/is7/the_cause_of_time/
Sorry if it's a bit unclear right now, hopefully I'll have time to add some diagrams this weekend.
This is a case of a common cause, in the form of a logical fact rather than a physical one.
I don't understand this. Which logical fact is the common cause? The fact that the measurements are correlated? Doesn't the whole thing collapse into a circle, then?
The fact of the shape of a sine curve.
Only if the frequencies are identical. In that case, follow the improbability and ask how they come to be identical.