No clearly defined boundary doesn't mean no distinction. Two (actually from real work) examples: one may have several (for example 3) sets of measurements with the result that:
The mean of set A is not significantly larger than that of B.
The mean of set B is not significantly larger than that of C.
The mean of set A is significantly larger than that of C
A second example is a table of "random" numbers (for example, bits). (It's not clear that such a thing exists, but the "A Million Random Digits with 100,000 Normal Deviates" from Rand was used successfully for years.) Again, I'm assuming the Abnormal Deviates were taken by Eyegor and given to Doktor Frankensteeeen. The question is, if one number is changed in a table of random numbers, is the table still random? The answer would seem to be yes. Of course still "yes" for two, or three changes; but what about changeing half or even all the numbers? At the exremes, the table appears either random or completely structured. It's not clear that there is a specific number of changes that make the difference.
I would like to (randomly?) amplify the stuff in post 665.
More effort is spent in reducting randomness or uncertainty than in amplifying it. (In general only for simulation or cryptography or sampling, etc. does one want more uncertainty.)
One obvious method to reduce uncertainty is to measure with more precise instruments.
A less obvious method (which can sometimes be used when the previous idea fails) is to average values. Under some technical conditions (finite variance or satisfyint the Lindeberg-Feller conditions, for example), an average is a better estimate of the "center" of a distribution of values than is any individual value. Bigger samples also yield smaller uncertainties which is why political pollsters try for big samples.
Good example: in a sample from a normal distribution (a "bell curve" for those with degrees in Education); the sample average is a better estimate for the center than any individual.
Borderline counterexample: in a sample from a Cauchy distribution (the distribution of brightness from a flashlight that shines on an wall), the sample average is no better than any single sample as an estimate of the center. This is a strange distribution with undefined mean and infinite variance.
Really ugly counterexample: take a sample from a normal distribution and for each item take its reciprocal. The sample average is much worse than any single sample value as an estimate of the center.
The point is, the "rules and procedures" for working with uncertainty are not always obvious or simple. Intuition must be developed. Trick question: what is the probability of gettin HH in a series of coin throws before getting TH?
I do appreciate what you are saying, but in this discussion of splitting rocks, it must be noted that there is no such thing as a table of random numbers – since all the numbers are the effect of a cause, whatever that might be – algorithm, tossing of die, your ingenuity, even Brownian motion. (based on Wolfram’s response to Chaitin’s Omega)
From the frog’s perspective (Tegmark) that observation makes no difference – the table of evidently random numbers could just appear in his timeline. But from the bird’s perspective, there is nothing random at all.
Yours is a tipping point example, much like Schrödinger’s cat with greater obscurity. Schrödinger gives us two choices, dead cat/live cat – but you suggest a tipping point in the eye of the beholder.
I counter that at the root, what is obscure or distinctive is a mathematical structure to the observer, i.e. the answer depends on whether the observer is a frog or a bird. The bird would see a distinctive tipping point which the frog could not see.
Nevertheless, frog or bird, I would suggest that the question the frog could ask is: what is the Kolmogorov complexity of the unaltered table? If it is a low complexity, then what he would observe to be a range of tipping possibilities is knowably false from his frogness view. If it is a high complexity, he’ll have to ask the bird.