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It's true that there's an (usually) unspoken assumption in the setup, that Monday and Tuesday are both guaranteed to occur and there's no subjective difference between them. I think that's what you're calling out? So, what Wikipedia calls a "principle of indifference" occurs: if there were an argument for weighting Monday/tails higher than Tuesday/tails, then the same argument could be flipped to show the reverse too.
You could alter the experiment to violate this indifference. For instance, if there's a 1/3 chance that the experiment will be halted Monday night because of weather (so Tuesday might not happen). Or if Sleeping Beauty knew there was a 0% chance of rain on Monday and a 10% chance of rain on Tuesday, and she can see outside (so she has more subjective information about what day it is). You can still list the four states as {Monday,Tuesday} x {heads,tails}, but in the former case, they don't have equal weight (Bayesians would say there are different priors), and in the latter case, she has to apply two pieces of information (waking up, and whether it's raining outside).
I know the principle of indifference, but youve talked about mathematicians who know what probability measures, and the indifference principle isnt a mathematical result, or obligatory to use them. Its something we use to come up with some probabilistic model when we dont have any better idea. It doesnt really make sense to use it to refute someone elses probability claims. Either they have a reason that applies, in which case indifference doesnt apply, or they dont have one, in which case that is what you need to argue.
I already told you the actual proof: if somebody had "a reason that applies", you can swap Monday/Tuesday in it and it would give the opposite result, which is a contradiction unless the probabilities are the same. Whether you think that's called the "principle of indifference" or not doesn't matter. Like several other people in the thread, it just sounds like you're here to argue for your own variant of philosophy. But the measured result is 2/3 regardless of whether you think your version of probability is better than a mathematician's. "Reality is that which, when you stop believing in it, doesn’t go away."
That would be true if all your knowledge where symmetric about them. But you know that heads/Tuesday is impossible, that Tuesday comes after Monday, and much more. You only have that its subjectively indistiguishable which one youre in in the moment.
Im also a mathematician, and not arguing towards either result. The halfers dont even object here. I just thought this argument is weird.
Ok, as long as you're not challenging the actual correct result, I can relax and accept that, sure, there's some philosophical weirdness about the whole thing. Sleeping Beauty's consciousness ends up "splitting" into both Monday and Tuesday, which is not something we normally encounter. So you could imagine some philosophical argument that her "soul" is more likely to experience the "qualia" of Monday than of Tuesday (if, say, "souls" jump into random time slices of a fixed universe, and earlier ones are more likely than later ones), so when it "picks one" to "be", it's not evenly apportioned. To an outside observer (i.e. for all practical purposes), across repeated experiments her body still experiences twice as many tails as heads, but her "soul" might not.
Is that a fair representation of what you think is "weird"?
This has some application to various anthropic arguments (and if we ever start simulating human brains or worrying about AI welfare, this is going to be a HOT topic of debate). Indeed, "souls" floating around independently and "picking someone" to "be" in a fixed universe is also a requirement for the Doomsday Argument to work. But personally I just think there's no disconnect between observers and physical bodies/brains (and everything I put in quotes above is nonsense). It's not something that can be settled with evidence, though.
I meant that the argument quoted in my first comment is weird, for the reasons Ive outlined. This doesnt really say anything about the sleeping beauty disagreements, except by pointing to the difficulty of establishing probabilities for "its Monday" and "its Tuesday" without indifference.
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