The joy of error bars

We normally think of approximation and imprecision as an unfortunate fact of our epistemic lives. But if the arguments of my previous two posts are correct, then there is a really serious problem with updating our information on precise data, such as that a spinner landed exactly at 102.34^∘, or that my height is exactly 182.01 cm, etc. Basically, it seems, in a number of probabilistic cases that information is useless, because it has zero probability on all the hypotheses under considertion. (A technical way to see the problem is that in contemporary probability theory, when dealing with continuous probability distributions, conditional probabilities are only defined up to sets of measure zero.)

But there is no problem at all in dealing with ranged information, such as that I am 182.0 ± 0.3 cm tall, since that is apt to have non-zero probability on the hypotheses that I am likely to be evaluating (say, hypotheses about the distribution of heights among male members of different professions). That our observations have error bars is essential for us to make use of them, at least sometimes.

Curious, isn’t it? It suggests something deep about the connection between our epistemology and our sensory limitations. But I don’t know what exactly it suggests.