I was hoping someone would bring this up! This is why I was careful to specify that the temperature was taken outside my window, and the strain was measured in a tuning fork in some unspecified location. In that situation, time really is the only correspondence between the points.
But your example brings up a much more general (and much more interesting) problem of identifying points. I’ll illustrate with another example. Suppose we measure a bunch of physiological variables in mice. We get a bunch of tuples mapping mice to the relevant variables, and we find lots of correlations. But then we lose our mouse id’s! Suddenly we have no idea which mouse each measurement came from. As before, everything gets scrambled and correlations disappear. We conclude that the measurements cause the mouse, or more accurately, the measurements cause the id of the mouse.
In the mouse example, notice that giving the mice actual names or id numbers wasn’t really necessary. We could just identify each mouse by its tuple of measurements. The identity of the mouse is mathematically just a mapping to match up the data points from different sensors.
Going back to your aptly-named thermostrainometer, we see a similar situation. Time is no longer the variable used to identify data points with each other. Instead, T and e points are associated through both space and time, and the whole mapping is conveniently handled inside the sensor itself and given to us in a convenient tuple structure. But the sensor itself still needs to associate the T and e values somehow, which is where space and time come in.
I was hoping someone would bring this up! This is why I was careful to specify that the temperature was taken outside my window, and the strain was measured in a tuning fork in some unspecified location. In that situation, time really is the only correspondence between the points.
But your example brings up a much more general (and much more interesting) problem of identifying points. I’ll illustrate with another example. Suppose we measure a bunch of physiological variables in mice. We get a bunch of tuples mapping mice to the relevant variables, and we find lots of correlations. But then we lose our mouse id’s! Suddenly we have no idea which mouse each measurement came from. As before, everything gets scrambled and correlations disappear. We conclude that the measurements cause the mouse, or more accurately, the measurements cause the id of the mouse.
In the mouse example, notice that giving the mice actual names or id numbers wasn’t really necessary. We could just identify each mouse by its tuple of measurements. The identity of the mouse is mathematically just a mapping to match up the data points from different sensors.
Going back to your aptly-named thermostrainometer, we see a similar situation. Time is no longer the variable used to identify data points with each other. Instead, T and e points are associated through both space and time, and the whole mapping is conveniently handled inside the sensor itself and given to us in a convenient tuple structure. But the sensor itself still needs to associate the T and e values somehow, which is where space and time come in.