Is there a mathematical way to work out uncertainty of something when that uncertainty has uncertainty and so on?

Statistics and the bell curve measuring standard deviation.

There are algorithms out there for almost every imaginable scenario. Weather computer modeling is one of the best examples and when the pizza delivery guy will get there is another.

Chaos Theory

The way statistics handles this is to provide a mean and a variance. The mean is the value that it predicts. The variance can be used to predict what percent of the time it varies by a particular +x to -x. Ordinarily, the +x to -x that they give you should have a 90% to 95% confidence range, meaning that the value should be within the range that percent of the time. If you are out of range more often than that, then they either used a smaller confidence interval or there is a defect in the device.

This is a very interesting question. I think the answer is yes. I’m not a
statistician, so take this with a grain of salt.

Let us take an example which is exactly not what you describe: we’ll pretend
the uncertainty is completely known. Say you have a GPS, and it gives your position
in meters from some landmark. And say you know, by really clever analysis of the
device, that the error always takes a bell-curve shape around the correct value,
+/- 10 meters.

Now say you read the device and it gives you 40 meters. You can imagine your
“belief” in your actual position as consisting of a really plausible
possibility of 40 meters, and getting less plausible further out—10 is
pretty implausible.

That’s a simple situation and not what you were asking. So let’s change it.

Say the manufacturer has put on the box that the measurement is a bell curve
+/- 10 meters around the actual value. But you know, from careful analysis of
GPS manufacturers, that when they say +/- 10, this could be off by a factor
of around +/- 3 (uniformly distributed).

The interesting thing here is that there are now two unknowns: your position,
which is what you were interested in in the first place, and the actual
reliability of the device.

Now say you read the device, and it shows 40 meters. Now we want to ask: how
plausible is it that the real number less than 36? What you now need to do is imagine
all possible manufacturers, and the corresponding device reliabilities.
So we said the box number may be off by around +/- 3. So what we can do is set
up an integral: we will integrate from the lower bound, 7, to the upper bound,
13, and at each of those points, imagine that that is the true reliability,
and find the plausibility of <36.

In other words, we are averaging the plausibily of 36 over all possible values
for the device reliability.

So this integral could look something like this: ∫ from 7 to 13 of P((36 -
40)/x) dx, all divided by 6 (the width of the interval). P is the norrmal
probability function. (the answer you get is ~.34—could have made a mistake
though).

This is an example of bayesian prediction. It is used all over the place. You
can probably find all sorts of far better explanations. I thought
http://www.probabilistic-robotics.org/ was a great book.