Weather forecasts is mentioned everywhere I look for information on chaos theory. I can't help but wonder how many variables are included in forecasting the weather. As you say, weather forecasts diverge from real outcomes after just a few days. But what affects air? Or more generally, what affects substances that are in a state that allows them to be affected by events that concern weather? With all the knowledge we currently command, and the computing power we are capable of, are we able to identify and calculate all the variables that apply in weather forecasting? I mean, even the amount and temperature of co2 released from Giotto's shoe factory in italy makes a difference... Even if that was the only unknown variable, given enough time, predictions would inevitably diverge from actual observation.
And it seems to me that if "random" is a property of our system of measurment rather than the world we measure, it is merely an indication that our methods, however nearly acurate, are not completely applicable to the observable world.
You are correct that forecasting the weather potentially involves more details and small forcing factors than are usually included even in the largest and most comples analyses. However, chaos, unpredictability and randomness also occur in much simpler processes. Consider the flow of water in a smooth pipe, without any heat transfer or like effects. In some classical experiments, done well over a century ago, Osborne Reynolds demonstrated the sudden transformation of a smooth ("laminar") flow to chaotic turbulence that occurs when the average flow velocity exceeds a certain value (that also depends on the density & viscosity of the fluid in motion). Indeed this, on a larger scale, is what makes the weather unpredictable. The equations of motion are well known and exact. However, because they are highly non-linear and coupled, they cannot be solved, except in certain simplified (and rather trivial) cases. Piping systems and aircraft wings are designed with empirically derived fudge factors that account for some of the average effects of this turbulence, but the details remain unpredictable and describable with random models. This is not a matter of perception, it is objective fact.
Consider also another well-known number, "e" , the base of natural logarithms. This number is defined as the sum of an infinite, convergent series; e = sum 1/n! for values of n from 0 to infinity. (n! simply means the product of all integers up to and including n, with 0! defined to equal 1). Like all irrational numbers, e involves an infinitely long string of decimal digits which involve no cyclic repition at any period - they satisfy all the statistical tests for randomness, despite the fact that the rule for calculating them is known and definite.
So you have these examples, plus the fact of quantum uncertainty that Thomas just noted, all illustrating objective randomness independent of the perceptions of any observer, and all involving processes that are ultimately deterministic.