We often read claims that such and such an observational anomaly cannot be plausibly explained by our current models of the universe so that the explanatory models needs tweaking. Here is an example, ras org :
"The voids we have detected cannot explain the Cold Spot under standard cosmology. There is the possibility that some non-standard model could be proposed to link the two in the future but our data place powerful constraints on any attempt to do that. If there really is no supervoid that can explain the Cold Spot, simulations of the standard model of the universe give odds of 1 in 50 that the Cold Spot arose by chance"
Should we care any more about an "unlikely" 2% than a 50-50 when it comes to a sample of one (universe)? I feel that we are being persuaded by an irrelevant "probability measure" argument which I am unable to debunk cogently but will try in the following. One first turns Anthropically to the following kind of argument. That it did happen (even under the guiding hand of Standard Model of Cosmology) seems as unremarkable as reflecting that there has been at least one instance of a single winner of two lotteries. carolina lottery winner. From the perspective of the winner, indeed a remarkable coincidence, one which they wouldn't have appreciated given their participation in the mad odds in the first place. To us, given all the irrational hopefuls in the world it was just a matter of time.
Moving away from such Bayesian prior arguments we can rather ruminate about what might constitute outrageous odds anyhow. That is how to ascribe some plausibility measure to a probability claim about the state of our universe given our model for it.
I will mix metaphors and mince the maths. We start with the paradox of the heap that reads: "you have a heap of sand, and take a single grain away, you still have a heap of sand. But if you keep removing one grain at a time, eventually you will only have a single grain remaining, and that’s clearly not a heap. So when did it stop being a heap?" Sorites_paradox
Compare now probability theory, in which we have a function called a "probability measure" if it assigns to each event in the collection of all events a non-negative real number. The "measure" is a scale telling you the weight of any subset (of the collection) where the total weight of everything is one. Whether you put a couple items on the scale separately one by one or you weigh them together at once, the sum of their weights will be the same.
Consider again the grains of sand that individually weigh zero, but collectively (as an uncountably large aggregate) in a jar have a weight bigger than zero.
Consider again the grains of sand that individually weigh zero, but collectively (as an uncountably large aggregate) in a jar have a weight bigger than zero.
Similarly there are uncountably many rationals within a number line interval.
If we cover all rational numbers between 0 and 1 by rational sized intervals the total length of the covering is an infinite series that sums to only a half. Despite the rational numbers being everywhere in between 0 and 1, and all being covered with its own interval we fill only half the number interval [0 ,1]. There are uncountably many real (rational and irrational pi, e and phi and the likes) numbers within a number line interval, so the probability of choosing an irrational number in the interval has a probability measure of zero even though the whole interval has some positive measure.
What we discern as distinguishable light brightness and differentiation in colours adheres to (log) power laws.
Colours are indiscernibly the same over a small enough wavelength interval.
What we can distinguish across the COBE pictures of the CMB as more than noise we would not deem as anything but noise across small enough neighbourhoods.
So to a self-selecting universe adhering to its own laws from an uncountably infinite set of possibilities, what is the appropriate "measure" to ascribe a probability "odds" to an observed behaviour?
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