Today Uncertainty Wednesday continues with limits to observations. Limits to observations are — along with limits to explanations — the key sources of uncertainty that we face. Following the discussion of foundational limits to observations from last week, next up is observational resolution (this is a category I forgot in my initial list).
Last year we went on an amazing trip to India. One of the many highlights was a visit to the Jantar Mantar in Jaipur. Right in the middle of the city are huge sundials and other astronomical instruments. Here is a picture I took showing a mid size sundial in the foreground and the largest one in the background
A sundial is used to measure time based on observing the position of the sun in the sky. Why would you want to build ever larger sundials? Because by increasing the size you can improve the “resolution” of your observation. I am using resolution in exactly the sense one might for a digital camera. A better camera has a higher resolution which gives you a more detailed picture (meaning you can zoom in and see details that would otherwise be lost). Similarly a larger sundial gives you a higher resolution on the angle of the sun. You can think of this as to whether you are just getting the degree of the angle or also the minutes (and possibly even seconds). Incidentally here is the historical connection for why we use the terms minutes and seconds in describing fractional units of degrees.
As it turns out the largest sundial in the Jantar Mantar of Jaipur has a resolution of minutes of time. So what does this mean? It means no matter how well you use it, you cannot observe the angle of the sun more precisely than one minute of time. This limit on your observation results in uncertainty. How much uncertainty and whether that matters is something we will get to later, for now let’s just note that all observations have a limit on their resolution and that this limit introduces uncertainty. This will be true as long as the underlying reality is more finely subdivided than the resolution of your observation.
The only way that resolution of observation will not introduce uncertainty is if the underlying reality is quantized and the observation can happen at that resolution. If the reality in question for instance is a Minecraft World, then block size resolution is lossless and introduces no uncertainty. If I tell you there are 1,128,960 blocks in a particular Minecraft world that number is precise. But keep in mind that — and this was the topic of last Wednesday — while there is now no uncertainty about the number of blocks that number itself is still very much a summary of the underlying reality as it says nothing about the location of the blocks, type of block, etc.
In summary then, the resolution of an observation will almost always be lower than the granularity of the underlying reality which limits our knowledge and thus introduces uncertainty.
