The Sleeping Beauty Problem: Any halfers here?

[PeterDonis]

The only thing she know for a fact, is that she is Monday.

But only on your idiosyncratic definition of "it is Monday". That is, she knows for a fact that she can't remember past Sunday. But she does not know for a fact that it is actually Monday, the way everyone but you uses that term. For example, she does not know for a fact that if she were to ask the experimenter who has just awakened her "what day is it?", he would answer "Monday". Or that if she looked at a computer-driven clock/calendar that always showed the current day of the week according to local time, it would say the day of the week was Monday. Or, etc., etc.

 

[Boing3000]

Everyone else is defining "it is Monday" as it actually being Monday,

We are not discussing time. There is no calendar in the room. I am sticking to the exact and precise explicit data that is found in the definition of the problem.

regardless of anyone's memory or lack thereof;

That is quite wrong. Time is a continuum, and memories is the only human clock.

thus, according to the way everyone else is using language, "I can't remember anything past Sunday" is consistent with either "it is Monday" or "it is not Monday, it is Tuesday, but my memory of Monday has been erased by the drug".

Or we are a Friday in year 2859 and I have taken a wormhole.
The way people connect event is the same as clock connect second. Monday is what follow Sunday.

I suppose this is an extreme example of the vagueness of ordinary language. But in your idiosyncratic use of language, we would still need to somehow distinguish the cases "I can't remember anything past Sunday because it is actually Monday"

A would like to see a non-idiosyncratic calendar where Monday popup randomly instead of every single time after Sunday.

and "I can't remember anything past Sunday because it is actually Tuesday and my memory of the actual Monday has been erased". So how, in your use of language, would you distinguish those cases?

The one (that have been drugged) cannot distinguish those two, which is the purpose of this explicit criteria.

Once you answer, then just go back and substitute your answer everywhere that anyone except you says "it is Monday" or "it is Tuesday", and so forth. Then you will be talking about the same actual math as the rest of us.

I have no problem with the math of the rest of you. You have problem understanding the problem of the problem.
There is no math covering time travel backward in time with probability attached.

 

[stevendaryl]

We are not discussing time. There is no calendar in the room. I am sticking to the exact and precise explicit data that is found in the definition of the problem.

Then you are arguing about a different thought experiment than the rest of us are.

 

[Boing3000]

The procedure is designed so that Beauty cannot condition on Monday or Tuesday, but she can still marginalize over them. (Although even that is not necessary)

So the condition left is she is in a room (awake, because some here suppose is is even relevant), you are making my points.
If the procedure does not change anything to her actual way to give credence, what's the point again?

 

[Boing3000]

Then you are arguing about a different thought experiment than the rest of us are.

I 'll let you bicker another half century with you pairs, about such a simple problem which allow so many solution.
OK, I have been told enough those kind of nonsense. My idiosyncratic use of language does not allow me to understand the following.

Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before

 

[stevendaryl]

So the condition left is she is in a room (awake, because some here suppose is is even relevant), you are making my points.
If the procedure does not change anything to her actual way to give credence, what's the point again?

Sleeping Beauty wakes up, and the experimenter tells her that she is in one of three situations:

1. His coin flip (hidden from her) resulted in heads, and today is Monday.
2. His coin flip resulted in tails, and today is Monday.
3. His coin flip resulted in tails, and today is Tuesday, and her memory of Monday has been erased.

She knows that those are the possibilities. She doesn't know which one is her actual situation.

Everybody else has known from the start of this thread that those were the possibilities, and that Sleeping Beauty was aware of those possibilities. If you want to talk about a different situation, where she believes that today is definitely Monday (even if it's not), fine. But that should be a different thread.

 

[stevendaryl]

OK, I have been told enough those kind of nonsense. My idiosyncratic use of language does not allow me to understand the following.

Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before

That means that she doesn't know whether it is Monday or Tuesday. It doesn't mean that she believes that it is Monday.

 

[Boing3000]

Everybody else has known from the start of this thread that those were the possibilities, and that Sleeping Beauty was aware of those possibilities.

Actually you were not aware of them before I had to explains them to you.

If you want to talk about a different situation, where she believes that today is definitely Monday (even if it's not), fine. But that should be a different thread.

I talk about the explicit situation, while you are not.

 

[Dale]

So the condition left is she is in a room (awake, because some here suppose is is even relevant),

The condition is that she was awoken in a room as part of the study protocol. That condition is correlated with Heads.

 

[PeterDonis]

The one (that have been drugged) cannot distinguish those two, which is the purpose of this explicit criteria.

Once again, you are using language in a highly idiosyncratic way. To you, "cannot distinguish those two" means "the first of the two must be true". To the rest of us, "cannot distinguish those two" means "must allow for the possibility of both". And the latter is what is specified in the problem. So, once again, you need to rewrite the problem statement in your idiosyncratic language so that you are talking about the same thing as the rest of us.

 

[Boing3000]

That means that she doesn't know whether it is Monday or Tuesday. It doesn't mean that she believes that it is Monday.

What you believe she believe is irrelevant to the situation. She is not asked about here believe to be Monday, nor is it relevant, because she is, as far as she can tell.
What new information she discover after awakening which is different from Sunday evening is relevant.
50 posts down the drain, and I am still waiting an answer...

 

[PeterDonis]

And the latter is what is specified in the problem.

Unless, I guess, we want to look at this as an even more extreme example of the vagueness of ordinary language.

 

[PeterDonis]

50 posts down the drain, and I am still waiting an answer...

Read my post #290 (and read #281 again). The answer is that you are insisting on interpreting language in a way that is very, very different from everyone else. I can't say that your interpretation is "wrong", since I have made the point multiple times in this thread that ordinary language is vague; but you also can't insist that your interpretation is "right" and everyone else's is "wrong". So the answer that you say you are waiting for has already been given to you, multiple times now: you are answering a different question than the rest of us, and you don't seem to even comprehend the possibility that the interpretations the rest of us are putting on the ordinary language in the problem statement are valid interpretations.

 

[stevendaryl]

What you believe she believe is irrelevant to the situation. She is not asked about here believe to be Monday, nor is it relevant, because she is, as far as she can tell.

You don't seem to understand how conditional probability works. If you don't know whether today is Monday or Tuesday, and someone asks you what the probability of X is, then you use conditional probability formula:

P(X) = P(X | Monday) P(Monday) + P(X | Tuesday) P(Tuesday)

So you don't need to know whether it is Monday or Tuesday, but you need to know what the probabilities of it being Monday versus Tuesday are.

Your approach seems to be: I don't know whether it is Monday or Tuesday. So I'll just assume it is Monday.

 

[stevendaryl]

What new information she discover after awakening which is different from Sunday evening is relevant.
50 posts down the drain, and I am still waiting an answer...

The fact that she's awake is itself information! Consider a different problem where if the coin is heads, she isn't wakened at all. Then do you agree that upon being awakened, she will know that the coin flip was tails?

 

[Boing3000]

To you, "cannot distinguish those two" means "the first of the two must be true".

I certainly don't do that. She certainly can distinguish Monday from Tuesday with the explicit apparatus called a brain, with her explicit memory erase (which is explicitly identical to no having those memory yet)

To the rest of us, "cannot distinguish those two" means "must allow for the possibility of both".

That's lab frame information, no relevant to putting her to sleep.
Why not trying to explain what this "putting into a sleep" means in mathematics language, and how it will change here credence computable the Sunday evening ..
That would certainly be more useful than insulting people.

And the latter is what is specified in the problem. So, once again, you need to rewrite the problem statement in your idiosyncratic language so that you are talking about the same thing as the rest of us.

Nobody can because the problem statement is irrational, and that "the rest of us" is split in 3 "camps" at least.
You should be aware that you being wrong does not make everyone else right.

 

[Dale]

What new information she discover after awakening which is different from Sunday evening is relevant

She discovers that she is currently in the trial, which is also when she is asked to assess her credence.

 

[stevendaryl]

I certainly don't do that.

You said it many times: For Sleeping Beauty, it's always Monday.

Maybe you didn't mean it, but what do people have to go on, other than what you say?

Nobody can because the problem statement is irrational, and that "the rest of us" is split in 3 "camps" at least.
You should be aware that you being wrong does not make everyone else right.

The problem statement is perfectly clear: Sleeping Beauty wakes up knowing that she is in one of three situations:
(1) Monday and Heads, (2) Monday and Tails, (3) Tuesday and Tails. She knows that if she were in situation #3, that means that her memory of what happened on Monday was erased. She doesn't know which of the three is the case, but she's being asked to quantify her uncertainty by giving a subjective likelihood that she's in situation #1.

There might be multiple plausible answers, but the problem statement is clear enough.

 

[Boing3000]

You don't seem to understand how conditional probability works. If you don't know whether today is Monday or Tuesday, and someone asks you what the probability of X is, then you use conditional probability formula:

P(X) = P(X | Monday) P(Monday) + P(X | Tuesday) P(Tuesday)

That the 10 times now that I explicitly say that this the lab knowledge compute Sunday evening. Do you copy ?

Your approach seems to be: I don't know whether it is Monday or Tuesday. So I'll just assume it is Monday.

That's not my approach. That's the one explicitly described in the article as the only reason to change her credence between Sunday evening and awakening.
I am still waiting your other alternative explanation.

The fact that she's awake is itself information

No it's not. She cannot update any likelihood/credence with this information. I am just sorry you cannot understand that.

 

[stevendaryl]

No it's not. She cannot update any likelihood/credence with this information. I am just sorry you cannot understand that.

Because it's false! If heads versus tails changes the number of times she is awakened, then the fact that she is awake changes the subjective likelihood that it's heads or tails. That's the way conditional probability works.

 

[stevendaryl]

That's not my approach. That's the one explicitly described in the article as the only reason to change her credence between Sunday evening and awakening. I am still waiting your other alternative explanation.

Once again, could you give me the answer to the following modified Sleeping Beauty experiment:

Suppose that in the case of tails, she is awakened twice, but in the case of heads, she is not awakened at all. Then do you agree that when she is awakened, she will know for certain that the coin is heads? Yes or no?

 

[PeterDonis]

She certainly can distinguish Monday from Tuesday with the explicit apparatus called a brain, with her explicit memory erase (which is explicitly identical to no having those memory yet)

In other words, she can distinguish Monday from Tuesday, except when she can't? So which of the two applies to this experiment, in your view?

 

[Dale]

However, I'm wondering how you would compute the expected payoff for bets in this way in a scenario like the second one in post #67 (where only one bet is paid off even if Beauty is awakened twice).

Going back to this, I don't think that Beauty can formulate a winning strategy if the wager is malicious, meaning she is unaware of the possibility of a bet not being honored. But if she is made aware of the possibility then she can probably marginalize over the possibility of a malicious wager.

However, since she is being asked for her credence "now", I think that the relevant wager would be an immediate wager offered, resolved, and honored "now". Since she knows in advance that she will be asked the question on each awakening the wager would need to be offered at each awakening in order to be an indication of the probability asked.

 

[PeterDonis]

I don't think that Beauty can formulate a winning strategy if the wager is malicious, meaning she is unaware of the possibility of a bet not being honored.

As I understand that scenario in post #67, it isn't that a bet isn't honored. It's that Beauty is told in advance that, if it should happen that she is awakened multiple times and asked to bet, only the last bet will count. Of course, while the experiment is in progress, she won't know whether the bet she is being asked to make is one of multiple bets she will make or not. But she will know that it is possible that she will be asked to bet multiple times, and if so, only the last one counts, so she can take that into account in deciding how to bet.

if she is made aware of the possibility then she can probably marginalize over the possibility of a malicious wager

That is my understanding of what she is expected to do in the scenario in question. And given that she knows in advance how the bets will be processed, I don't think the wager is "malicious"; unusual, perhaps, but not malicious.

Since she knows in advance that she will be asked the question on each awakening the wager would need to be offered at each awakening

In this scenario, as I understand it, Beauty is indeed asked the question on each awakening. But you might be putting a tighter meaning on the word "wager" than was intended in that scenario, since the scenario explicitly says that the "wagers" are resolved at the end of the experiment, and that if Beauty is awakened multiple times, only the last "wager" will actually be paid off (any others will be "resolved" by being discarded). This might be an unusual use of the word "wager", but that can be fixed by changing the word if it's thought necessary. The actual conditions of the experiment are, as far as I can see, the way I have described them.

 

[Marana]

Edit: I tried to come up with an equivalent situation but I don't think it worked.

 

[Stephen Tashi]

That's the challenge: to figure that out.

I agree.

It's interesting that students in introductory probability courses are admonished that the probability of an event is undefined until the probability space containing the event is stated - and they are assigned exercises where they must state the probability space for situations that are verbally described; yet when experts tackle verbal problems they often charge in and offer solutions without saying what probability space they are talking about.

1. (tails, monday, awake)
2. (tails, tuesday, awake)
3. (heads, monday, awake)
4. (heads, tuesday, asleep)

Those are possible states for Sleeping Beauty and the coin. To define a probability space also requires saying what probability is assigned to each event and that is the subject of debate.

The probability space you gave implies that in order to realize an event, we must pick the day to be Monday or Tuesday. The statement of the problem does not explicitly say anything about an experiment where we pick a day. So to compute the probability of an event involving picking a day, we need to use a different probability space where an event like "It is Monday" can be defined in terms of the events described in the problem.

A straightforward translation of the events described in the problem doesn't give a probability space where an event like "It is Monday" is defined.

For example, we could omit the state of the coin. The definition of a probability space requires specifying both the events and the probability assigned to those events. The coin is what implements the probability measure.

I'll use the word "treated" to mean "awakened and interviewed".

Event---------- Probability
A: SB treated on Monday and SB treated on Tuesday, 1/2
B: SB treated on Monday and SB not treated on Tuesday, 1/2
C: SB not treated on Monday and SB treated on Tuesday, 0
D: SB not treated on Monday and SB not treated on Tuesday, 0

In that probability space, how shall we interpret the verbal information "Sleeping beauty is awakened and interviewed"? when no day for the treatment is specified? I would interpret that event to be the event $A \cup B \cup C$.

In asking for the conditional probability that "It is Monday given that SB is awakened and interviewed", how do we define the event "It is Monday"? I don't see how to define such an event in the sample space above.

 

[stevendaryl]

It's interesting that students in introductory probability courses are admonished that the probability of an event is undefined until the probability space containing the event is stated - and they are assigned exercises where they must state the probability space for situations that are verbally described; yet when experts tackle verbal problems they often charge in and offer solutions without saying what probability space they are talking about.

I would say that in this case, it's pretty clear that there are exactly 4 events, or situations, or whatever. The question is how to assign probabilities to them.

The probability space you gave implies that in order to realize an event, we must pick the day to be Monday or Tuesday.

No. We're talking subjective Bayesian probability. The probability is the measure of uncertainty about the truth of a statement. There is no implication that we're picking anything at random. Sleeping Beauty is uncertain about whether today is Monday or Tuesday. That doesn't mean that she is randomly picking between those two possibilities, it just means that there are two possibilities consistent with her knowledge.

She's also uncertain about whether the coin was heads or tails. She quantifies her uncertainty by assigning likelihoods to them. The rules of the experiment imply that if she is awake, then she knows for certain that the combination (Tuesday, Heads) is ruled out. So the question is: how to sensibly assign the other three probabilities.

You can certainly reject the idea of subjective probability, but that's the whole basis of the Sleeping Beauty problem is to ask what is a sensible assignment of subjective probability.

 

[JeffJo]

One of the problems in this thread is the wide variety of alternative scenarios proposed, which seem to be having the opposite effect as intended regarding clarifying the original scenario.

Obviously, I disagree. And I think the length of this stalemate, which is more about what Beauty's point of view should be than probability, proves that trying to decide the proper point of view first is a futile exercise.

If the correct answer can be determined without having to decide point of view first - as I feel I have done - then you can use it to help settle your debate. That won't happen without an independently justified answer. Existing probability theories just don't support including the number of trails as a random variable.

So, counter to your intent, here is as another version of my variation. Use four volunteers, and the four cards I described before (with (H,Mon), (H,Tue), (T,Mon), and (T,Tue) written on them). Deal the cards to the four, and put them in separate rooms. Using one coin flip, and waken three of them on Monday, and Tuesday. Leave the one whose dealt card matches both the day, and the coin flip, asleep. Ask each for her confidence that the coin matches her card.

Obviously, if you show each Beauty her card, her answer has to be the same as the original Beauty's. Since it is the same regardless of what card is dealt, you don't have to show it to any of them. If you don't show it to any of them, you can put all three awake Beauties in a room together to discuss their answers. All have the same information, so all answers have to be the same. Since exactly one of the three has a card that matches the coin flip, that answer must be 1/3.

If you can find a flaw in my logic, I'd be happy to discuss why it is to correct to disregard the day and why Beauty has no new information to provide a probability update. But I feel my approach shows, quite trivially, that the answer is 1/3. So the proper discussion is why there is new information.

 

[Dale]

That is my understanding of what she is expected to do in the scenario in question. And given that she knows in advance how the bets will be processed, I don't think the wager is "malicious"; unusual, perhaps, but not malicious.

I agree. She would have to marginalize over the different options correctly, but it could be done. It would be complicated but, as you say, not malicious.

However, I don't think that this wager would be the one that corresponds to the credence of heads. It would be the credence of something more convoluted, like the credence of heads and amnesia drug positive.

since the scenario explicitly says that the "wagers" are resolved at the end of the experiment,

I didn't see that. Are you talking about the Wikipedia article? If so, where does it say that?

 

[Dale]

In asking for the conditional probability that "It is Monday given that SB is awakened and interviewed", how do we define the event "It is Monday"?

You don't need to do that in order to calculate the requested probability. There is no need to bring Monday or Tuesday into the calculations at all.

 

[stevendaryl]

So, counter to your intent, here is as another version of my variation. Use four volunteers, and the four cards I described before (with (H,Mon), (H,Tue), (T,Mon), and (T,Tue) written on them). Deal the cards to the four, and put them in separate rooms. Using one coin flip, and waken three of them on Monday, and Tuesday. Leave the one whose dealt card matches both the day, and the coin flip, asleep. Ask each for her confidence that the coin matches her card.

Obviously, if you show each Beauty her card, her answer has to be the same as the original Beauty's. Since it is the same regardless of what card is dealt, you don't have to show it to any of them. If you don't show it to any of them, you can put all three awake Beauties in a room together to discuss their answers. All have the same information, so all answers have to be the same. Since exactly one of the three has a card that matches the coin flip, that answer must be 1/3.

If you can find a flaw in my logic, I'd be happy to discuss why it is to correct to disregard the day and why Beauty has no new information to provide a probability update. But I fell my approach shows, quite trivially, that the answer is 1/3. So the proper discussion is why there is new information.

I think it's a little difficult to see the connection with the original problem, so the fact that the results are the same are a little less than compelling...

Oh, now I get it! (I think). The original Sleeping Beauty problem singled out a single event, (H, Tues) as the day that Sleeping Beauty sleeps through the interview. You're restoring the symmetry by having 4 different beauties, each with a different event singled out. So no matter what the day is, and no matter what the coin flip result, 1 of the beauties is sleeping, 1 has a matching coin flip result, and 2 have a non-matching coin-flip result. So the odds of a match among those who are awake is 1/3.

Very clever.

 

[Dale]

Existing probability theories just don't support including the number of trails as a random variable.

That simply isn't true. In fact, it is one of the recognized advantages of Bayesian statistics.

That said, I have no objection to your alternative scenario other than just the fact that it differs from the scenario in the OP.

 

[PeterDonis]

Are you talking about the Wikipedia article?

No, I'm talking about post #67 by @Demystifier . Link here:

https://www.physicsforums.com/threa...m-any-halfers-here.916459/page-4#post-5777796

 

[Stephen Tashi]

No. We're talking subjective Bayesian probability.

The fact that a probability is called a "Bayesian" probability does not exempt it from being defined on a probability space. In the probability space that you gave, you may choose to subjectively assign probabilities to each of the events, but, naturally, that subjective assignment is not mathematical demonstration that your assignments are correct.

As you know, the usual scenario in applying Bayesian probability is to have a "prior" distribution on some probability space and then to compute a "posterior distribution" by conditioning on some event that can be defined in that space. So if we wish to apply the Bayesian method to compute "The probability that the coin landed Heads" given that "Sleeping Beauty is awakened and interviewed", we need to have a probability space where those events can be defined. And to have a convincing argument that our calculation is correct, we need to show that the information given in the problem implies the probabilities we assign in our probability space.

The probability is the measure of uncertainty about the truth of a statement. There is no implication that we're picking anything at random. Sleeping Beauty is uncertain about whether today is Monday or Tuesday. That doesn't mean that she is randomly picking between those two possibilities, it just means that there are two possibilities consistent with her knowledge.

As far as I can see, those statements have no mathematical interpretation in the theory of probability. The usual measure of "uncertainty" is the standard deviation of a random variable, which is not a probability. People do say informally that a probability is a "measure of uncertainty", but what is the mathematical definition for a "measure of uncertainty"?

I agree that the definition of a probability distribution (of Bayesian origin or otherwise) has no notion of picking something at random. It's the application of probability theory to specific problems that makes the connection between a probability distribution and an experiment where something is picked at random. If we are trying to apply probability theory to a problem where the object is to quantify a persons credence and we say credence has nothing to do with picking something at random then we'd better define "credence" and state some axioms that give its properties.

You can certainly reject the idea of subjective probability, but that's the whole basis of the Sleeping Beauty problem is to ask what is a sensible assignment of subjective probability.

Then what the problem asks is undefined until we define the meaning of "a sensible assignment". I don't reject the idea of assigning prior distributions and I don't reject assigning them according to certain criteria such as maximum entropy or symmetry. I do object to the verbal description of an event without specifying what probability space contains that event - for example "P( Monday)" or "Given that Sleeping beauty is awakened".

The following problem is similar to the Sleeping Beauty problem except that it poses some well defined mathematical questions.

Experiment A outputs two possible strings. String "h, M" has probability 1/2 of being the output. String "t,M,T" has probability 1/2 of being the output. Experiment B consists of running Experiment A N times (where N is some given number) and concatenating the outputs of these repeated experiments. For example, a possible result of Experiment B when N = 4 is "h,M,t,M,T,t,M,T,h,M". Experiment C consists of performing experiment B and then selecting a lower case letter from the output of experiment B according the following procedure. First pick an occurrence of a capital letter from the output of experiment B at random, given each occurrence of a capital letter an equal probability of being selected. Then pick the lower case letter to be the first lower case letter preceding the occurrence of the capital letter that was selected.

Question 1: What is probability the lower case letter selected in experiment C is "h" when N = 1?
Question 2: What is the limiting value of the the probability that the lower case letter selected in experiment C is "h" as N approaches infinity?

It seems to me the 1/3 vs 1/2 debate amounts to asking which of those two questions defines Sleeping Beauty's "creedence".

 

[stevendaryl]

The fact that a probability is called a "Bayesian" probability does not exempt it from being defined on a probability space.

But it means that any question that your uncertain about, such as "Is today Monday?" can have an associated probability. The issue is then just to assign probabilities to the exclusive and exhaustive cases: Monday & Heads, Monday & Tails, Tuesday & Heads, Tuesday & Tails. So your claim that Monday isn't an event is not true for Bayesian probability. Any statement with uncertainty can be an event.

As you know, the usual scenario in applying Bayesian probability is to have a "prior" distribution on some probability space and then to compute a "posterior distribution" by conditioning on some event that can be defined in that space.

Yes, the obvious prior for the coin toss is $P(H) = P(T) = 1/2$. The difficulty is to calculate priors for P(Monday) and P(Tuesday). However, there is really only one choice that makes sense: P(Monday) = P(Tuesday) = 1/2. (Argument at the end)

Given these priors, we compute:

P(Awake) = P(Monday) + P(Tuesday) P(T)

(she's always awake on Monday, and she's awake on Tuesday only if it's tails).

So we conclude: P(Awake) = 3/4

Now finally we compute:

P(H | Awake) = P(H & Awake)/P(Awake)

Since P(H & Awake) only happens if it's Monday, then we can write: P(H & Awake) = P(H & Monday) = P(H) P(Monday) = 1/4

So we conclude:

P(H | Awake) = (1/4)/(3/4) = 1/3

Here's an argument for why P(Monday = 1/2):

First, note that if you told Sleeping Beauty that today is Monday, then she would have no reason to think heads more likely than tails, since the difference between them only shows up on Tuesday. So P(H | Monday) = 1/2.

Second, note that if you told Sleeping Beauty that the coin result was tails, then she would have no reason to think Monday more likely than Tuesday, since they are only different in the case of heads. So P(Monday | T) = 1/2.

Now, compute the conditional probability: P(Monday | H)

We can use Bayes' theorem to write that as:

P(Monday | H) = P(H | Monday) P(Monday)/P(H)

But we already have P(H | Monday) = P(H) = 1/2. So we get:

P(Monday | H) = P(Monday)

Finally, we compute:

P(Monday) = P(Monday | H) P(H) + P(Monday | T) P(T) = 1/2 P(Monday) + 1/4

which implies that P(Monday) = 1/2

As far as I can see, those statements have no mathematical interpretation in the theory of probability. The usual measure of "uncertainty" is the standard deviation of a random variable, which is not a probability. People do say informally that a probability is a "measure of uncertainty", but what is the mathematical definition for a "measure of uncertainty"?

It's subjective. There isn't a right or wrong answer, except that it has to obey the laws of conditional probability. However, in many cases of interest (such as this one), we can appeal to symmetry: If there are two possibilities, and there is no reason to think one more likely than the other, then the probability of each should be 1/2. That principle alone is enough to get unique probabilities in this case.