ok; I think we might be going somewhere. When we measure the acceleration of the rock towards the earth, aren't we not measuring, at the same time, the acceleration of the earth towards the rock? How could you know the difference? — Gampa Dee

No, because you have to measure the acceleration from an inertial frame of reference, which is neither the earth nor the rock in our example if we are including the earth's acceleration.

"Acceleration of the rock towards the earth" implies measuring the acceleration of the rock from earth's frame of reference. If the earth is accelerating towards the rock, this is not a valid frame of reference.

The same goes for " the acceleration of the earth towards the rock" as the rock is not a valid frame of reference.

Instead, what we should be doing is measuring the acceleration of the earth in 3d coordinates from a point in space that is not accelerating (any non accelerating point), and measuring the acceleration of the rock from this same point. And from the point, the rock will be seen to accelerate quickly towards the earth, while the earth will barely accelerate towards the rock.

This way of analysis is not just correct for gravity, but for anytime you are using classical mechanics to analysis motion. Think of a stationary passerby watching a car hitting a tennis ball. The ball will have a high acceleration as a result of the collision, while the car will hardly accelerate at all. From an inertial frame of reference, the car and tennis ball do NOT both have the same acceleration from the collision.

ok; so, what would be this equation? — Gampa Dee

a1 = Gm2r/r3

Where a1 is the acceleration of body 1 from an inertial frame of reference. Note this is a vector in 3d.

For example see: https://orbital-mechanics.space/the-n-body-problem/two-body-inertial-motion.html

First of all, for Newton's equations as you and I have written, they only apply when used from an inertial frame of reference. The derived acceleration equation is from the frame of reference of the larger object and assumes this object (the Earth or Moon, etc) is not accelerating, which simplifies the math.

If you want to model a system where both bodies are accelerating you will have to use the two-body equations of motion, and set up a inertial frame of reference as the origin (eg: the barycenter). In the simplified calculations, the larger body can be taken as the inertial frame of reference as it is not accelerating, this is not true if it is accelerating.

Secondly why would we include the acceleration of the earth caused by the second mass? That is not what we were calculating - we were calculating the acceleration of the rock. If you wanted to calculate the acceleration of the Earth caused by the rock, that is a separate calculation

I'm not sure where your formula comes from, but let me see if I can understand and answer your question.

Take a rock of mass m, on earth of mass M, a distance r from the center of mass of the earth, and let us ignore air resistance.

The force on the rock:
F=G*Mm/r^2 (1)
Also
F=ma (2)

substituting 2 into 1 gives

a=G*M/r^2

So the acceleration of the rock in earth's gravitational field is not dependent on the mass of the rock. It is dependent on the distance of the rock from the center of mass of the earth, and on the mass of the earth.

You seem to be asking, what if M were different? If M were different, then the acceleration due to gravity will also be different. This can obviously be seen in that the same rock will have a lower acceleration dropped on the moon than dropped on earth. Galileo was talking about objects of different masses being dropped in the same gravitational field and at the same distance.

-1kg and 100kg dropped from the same height on earth in a vacuum, will have the same acceleration. -1kg and 100kg dropped from the same height on the moon in a vacuum, will have the same acceleration.
-1kg dropped from the same height on earth and on the moon will NOT have the same acceleration.

Is this what you are getting at?

It’s not the same because she isn’t given a randomly selected waking after 52 weeks. She’s given either one waking or two, determined by a coin toss.

The manner in which the experiment is conducted matters. — Michael

Also, take this further evolution of Problem B that I outlined earlier. The SB experiment is done every week for a year. Each week she is woken once if the coin lands heads, twice if it lands tails.

But she is only asked a question once in the whole year. One of the wakings is randomly selected to be the one where she is asked the question. On this randomly selected waking, she is asked the question "what is the probability that this randomly selected waking shows a heads." The answer is 1/3, as per Problem A in my previous post.

She is given one waking or two, determined by a coin toss, and this is repeated 52 times. When she wakes up she has no idea which of the wake up events this is - so from her point of view it is a randomly selected wake up event.

This is an ambiguous claim. It is true that if you randomly select a seeing from the set of all possible seeings then it is twice as likely to be a tails-seeing, but the experiment doesn't work by randomly selecting a seeing from the set of all possible seeings and then "giving" it to Sleepy Beauty. It works by tossing a coin, and then either she sees it once or she sees it twice. — Michael

That is the difference in interpretation I am talking about. For you the question is asking about how a fair coin toss will land. For me it is about the seeings of the coin, not the coin toss.

Let's say The SB experiment is carried out every week for a year - 52 time in total. On a given time that SB wakes up, what should her credence be that the coin at that given wake up event is heads? I'll come back to this - let's call it problem A.

Well let me do this experiment again with one small change - call it problem B. The SB experiment is done every week for a year - 52 times. But instead of asking SB her credence at each wake up event, we are going to take a picture at each wake up event, then select one for her to answer about at the end. At the end of the year we will have on average 26 pictures of heads and 52 pictures of tails, each representing a unique wake up event. They are all put in a bag and one is picked out, then SB is asked "at this particular wake up event, what is the probability that the coin was showing heads, as seen in the picture?" Of course the answer is 1/3.

The answer to problem B is clearly 1/3 and I think we both will agree here. The problem A is the same question that is asked to SB - on a given wake up event, she is asked in the moment about the probability of the coin showing heads. So the answer in problem A is also 1/3.

No. That I get to see something twice doesn't mean that I'm twice as likely to see it. It just means I get to see it twice. — Michael

You are not twice as likely to see it. A given seeing of it is twice as likely to be tails. Those two are very different things.

You are interested in "it." That is your event for which you are calculating your probabilities

I am interested in "seeing of it." That is my event for which I am calculating probabilities.

Those are two very different events - "it" and "seeing of it" are different events.

Ok, let me try a different method. Will your stance change if the question asked to sleeping beauty is "What is the probability that you see the coin with it's heads up when you look at it now."

Would this change your stance? Maybe it is clearer that the frequency with she looks at each outcome affect the probability that she will see that outcome. If she looks more often on tails (twice as often), then she is more likely to see tails on a given look.

Indeed, not only would their expected value (EV) be positive, but it would be positive because the majority of their individual bets would be winning bets. Michael, it seems, disagrees with the idea of individuating bets in this way. However, this resistance appears to stem from an unwillingness to assign probabilities to the possible involvement of epistemic agents in specific kinds of events. Instead, like sime, Michael prefers to attribute probabilities to the propensities of objects being realized as seen from a detached, God's-eye-view perspective. — Pierre-Normand

Exactly, the disagreement stems from the perceptive from with the probability is being calculated.

I think using frequencies over multiple games to argue for the probability in a single game is a non sequitur. — Michael

I simply can't agree with this. Using frequencies over multiple games to argue for the probabilities in a single game is a fundamental way probabilities are calculated.

If you ask me what the probability of this dice in my hand will roll a 6, I can roll the dice a million times and that will give me credence for the probability that a 6 will roll the next time I roll the dice. And so on.

If 6 rolls 900,000 out of the million times, I am completely justified having credence that 6 is more likely to roll on the next single, one off roll I will do.

You seem to be suggesting that 6 having rolled 900,000 times out of a million should not affect my thought on the probability of 6 being rolled in the next one off event at all. That makes no sense to me.

It would be rational also in the sense that you are more likely to win on a particular guess (which would not be the case in a normal large prize 2,000,000 and a probability of winning of 1/1,000,000 at £1 each, as you point out).

If you repeated the experiment a trillion times, and kept a note of whether you guess was correct or not each time, and I did the same. We would find that I got it correct more than you. By the law of large numbers that would mean the outcome I guessed for was more probable than yours.

Will you bet that the coin landed heads 100 times in a row? I wouldn't. My credence is that it almost certainly didn't land heads 100 times in a row, and that this is almost certainly my first and only interview. — Michael

Fair enough, but then a person betting that it did land on heads 100 times in a row will have a greater expected value for their winning (as long as the winnings for heads are greater than 2^100 than for tails). And their position would be the rational one - maximizing your expected value when betting is the rational position.

For you the Monday heads interview is an A event. The Monday tails interview is a B event, the Tuesday tails interview is a B event as well (it is the same tails event as far as you are considered.)

For me the Monday heads interview is an A event. The Monday tails interview is a B event, the Tuesday tails interview is a C event (they are different interviews as far as I am concerned).

You care about the coin only, I care about the interview moment combined with the coin.

For me the coin showing tails on Monday is a completely different showing to it showing tails on a Tuesday. For you it is the same flip so it is the same showing.

What is the probability of the Mona Lisa showing? For you it will be 100% as we know it has been painted. For me it would depend on the time of day and the date. The probability of it showing during the day is higher than the probability of it showing at night, as the Louvre is only open on some special nights.

Different interpretations of what it means for the Mona Lisa to be showing.

Yes, an individual tails interview event is twice as probable. A tails interview where Monday and Tuesday interviews are grouped together and not seen as different events is equally likely as a heads interview. It comes back to the language of the question and interpretation.

That a correct prediction of tails is twice as frequent isn't that a correct prediction of tails is twice as probable – at least according to Bayesian probability. — Michael

This is one part of halfers thinking I don;t understand. There is something in statistics called law of large numbers that allows just this inference.

We have a disagreement over a dice. I say that it is more likely to land on 5, you say all numbers are equally likely to happen. What can we do?

The answer is simple, do an experiment where we roll it many times. If it lands on heads 5 million times out of 10 million rolls, then I will have high credence to say that 5 is more likely to roll on the next one off roll we do.

Halfers seem to be saying that we can;t use the results of the 10 million rolls to give credence to what we think is more probable for the next one off roll. I disagree, we certainly can.

Sorry for the repeated posts here is another thought experiment to see the difference between the two camps. Both camps are given a camera and told to take pictures of the coin at what they consider to be the important moments.

The experiment is done twice and the coin lands on heads once and on tails once from the perspective of an independent observer.

Halfer camp

The coin is tossed and lands on heads on Sunday. They take a picture of it showing heads.
The coin is tossed and lands on tails on Sunday. They take a picture of it showing tails.

They come away with two pictures - one heads and one tails. This is what the question is asking for as far as they are concerned.

Thirder camp

The coin is tossed and lands on heads. They don't take a picture - they are not interested in the toss itself as sleeping beauty is asleep. Sleeping beauty wakes up on Monday, now they take a picture of the coin at that moment showing Heads.

The coin is tossed and lands on tails. They don't take a picture - they are not interested in the toss itself as sleeping beauty is asleep. Sleeping beauty wakes up on Monday, now they take a picture of the coin at that moment showing Tails. Sleeping beauty wakes up again on Tuesday, now they take a picture again of the coin at that moment showing Tails

They come away with three pictures - one heads and two tails. This is what the question is asking for as far as they are concerned.

You can see the difference in what the halfers and thirders think is important based on when they took their pictures.

Let's look once again at two wordings of the SB problem, I have bolded what I see as the key part of them. Both are questions to SB:

From Scientific American: "What is the probability that the coin shows heads"
From wikipedia: "What is your credence now for the proposition that the coin landed heads?"

From the thirder position, both of these questions are asking about something that is intrinsically linked to her mental state at a particular moment in time, what she can see, etc. From the halfer position they are only asking about the coin. The disagreement is about what exactly the question is asking.

Let me rewrite these statements in more obvious ways, sorted into how the two camps interpret them.

Thirder camp:

"The coin is behind you. When you turn around and look at it, what is the probability that you will see heads at this very moment in time"
"What is the probability that photons leaving the coin and entering your eyes at this very moment in time will encode an image of heads?"

Halfer position

"What is the probability that the coin landed on heads when it was flipped regardless of this moment in time when i am asking you the question?"
"What is the probability that the coin landed on heads, as seen by an independent observer watching the whole experiment without sleeping."

You are missing out that she will also wake up on the Tuesday if the coin landed on heads - you are only looking at Monday.

When you add in the probabilities of the Tuesday as well, then you find that amnesia is no better or worse as per my previous post.

The latter is obvious but not what is asked about when asked her credence that the coin landed tails. — Michael

This right here is the key disagreement.

Taking the wording from the Scientific American article, it says "What is the probability that the coin shows heads?"

It does not say "what is the probability that the coin landed heads."

The linguistic difference between the coin showing heads and landing on heads when flipped is the disagreement.

Thirders take the position that "shows" is referring to the event of light reflecting from the coin and then entering SB retina, which is dependent on when and how many times she opens her eyes. The showing happens multiple times in the experiment if the coin lands on heads, it is not a one off event.

Halfers take the position "shows" is only referring to how the coin landed when it was flipped. That is independent to when and how many times SB opens her eyes. The coin lands only once per experiment - it is a one off event as you say.

The disagreement is linguistic rather than statistical.

In your example using the thirder position,

If she has amnesia she should guess heads and will will 2/3 of the time.

If she doesn't have amnesia she should guess either on the first wake up (1/2 probability so doesn't matter which she guesses) and she should guess tail with absolute certainty if she remembers having woken up before (ie on her second wake up). Again she will win 2/3 of the time.

So having amnesia or not does not change the probability that she will win, but the tactics she should use are different.

But honestly, all this talk of successes is irrelevant anyway. As I said before, these are two different things:

1. Sleeping Beauty's credence that the coin tossed on Sunday for the current, one-off, experiment landed heads
2. Sleeping Beauty's most profitable strategy for guessing if being asked to guess on heads or tails over multiple games

It's simply a non sequitur to argue that if "a guess of 'tails' wins 2/3 times" is the answer to the second then "1/3" is the answer to the first. — Michael

I took some time to read through the many good arguments on this thread, I agree with the above 2 different things you mentioned above.

The crux of the matter as I see it is whether the question about what the coin "shows" (taking the Scientific American article) is relevant to 1 or 2.

And the way I interpreted what the coin "shows" is that at the moment SB opens her eyes photon leave the coin and hit her retina. What is the probability that those photons contain information for heads? This related to 2 and is 2/3

While as far as I can tell you interpreted "shows" to mean what the coin landed on when it was flipped. This is related to 1 and is 1/2, absolutely.

And I put forward that this kind of difference in interpretation is what is happening in the two different position in the SB problem.

Can this not be experimentally validated using a simulation? Write a computer program simulating SB and the experiment. Run the simulation 1 million times. Each time SB wakes up make a note of whether the coin was seen on heads or tails.

Will the number of heads and tails seen on wake up be 1/2 and 1/2 or 1/3 and 2/3?

Surely someone must have thought of doing this? I might have a search.

That is how the conditional probability works in this instance. If I am SB and I wake up, I know it could be (Heads and Monday), (Tails and Monday), (Tails and Tuesday) all with equal probability. The probability that it is heads is therefore 1/3 and tails 2/3.

There are 3 possible wake up event with equal probability, and only 1 of them is heads.

The conditional probability is dependent on the frequency in this case. Because SB wakes up more on tails, a given wake up event is more likely to be caused by a tail flip that a head flip.

1/3 of wake up event are caused by a head flip. 2/3 by a tail flip. So the conditional probability is influenced by the frequency.

I am saying you are wrong. And in my example, where you wake up once for heads and never for tails shows that the probability of you seeing heads when you wake up is conditional on how often you wake up for heads and how often for tails.

There is an above post by fdrake that sets this out more clearly in a table for the SB example.

I was talking about frequency not probability.

1. — Michael

And yet the probability of a heads being flipped is 0.5. So you see that the probability of you seeing a heads is conditional on the head being flipped and the criteria for you waking up. In my example the criteria of waking up is 1 for heads and 0 for tails. In the SB problem it is 1 for heads and 2 for tails. The probability of you or SB seeing a heads is not 0.5 in either case

I think the reasoning that leads you to this conclusion is clearly wrong, given that it’s an absurd conclusion. — Michael

I do not think it is an absurd conclusion at all. The set up in your example is absurd in the practical sense of waking someone 2^101 times, but in order to explore your though experiment we suspend that absurdness.

In that case it is more likely that given an instance I wake up I will see the coin has been flipped heads 100 times in a row.

Could you address my counter extreme scenario that I proposed?

I flip a coin and if it lands heads I wake you up tomorrow, if it lands tails you never wake you up. If you wake up and are asked the probability the coin landed heads, what would you say?

The coin being flipped and you waking up are not independent in any of the scenarios. We are looking at conditional probabilities here.

It is more likely that you wake up and not have got heads 100 times in a row. The probability that heads lands 100 times in a row is in 8^-31, while the probability that you wake up to see heads 100 time in a row is 8^-25.

So the condition of you waking up to see it still changes the probability, but since you are starting off with a very unlikely outcome, you still end up with an unlikely outcome.

The key point is that SB waking up is liked conditionally to the coin toss. They are not independent of each other.

Imagine a different extreme scenario. I flip a coin and if it lands heads I wake you up tomorrow, if it lands tails I never wake you up. If you wake up and are asked the probability the coin landed heads, what would you say?

In both your and SB case the waking up and coin toss are linked, just yours is in a more extreme way.

it isn't a non sequitur. The probability that see sees tails at the point she wakes up is partly dependent on how often she wakes up for each outcome.

Isn't the confusion here in the ambiguity of the question put forward to sleeping beauty? Are not the two camps interpreting the question put to SB differently?

Camp 1 - The probability that the coin landed on heads. This is 1/2 for a fair coin regardless of anything else.

Camp 2 - The probability that the coin is showing heads on the day she awakes. This is 1/3 as it is a combination of the probability of the coin and the probability of her waking up. She is more likely to wake up and see a coin showing tails, as she will wake up more often if the coin lands on tails.

So perhaps SB should ask the experimenters to clarify what probability they are after, exactly.

I think the problem with a tough non-compassionate approach to these problems is that there are others with worse world views than us ready to extend a hand to these groups. So if you or I or mainstream society don't reach out with compassion, these groups will turn to the Andrew Tate's of the world.

My observation is that in a debate, if the strong claim—the claim that (A) wants to prevail—fails, then retreating to a more defensible position is a tactic still to make the strong claim prevail. I think it’s fair to call this a fallacy. — Jamal

I agree with you on your analysis of what you call the motte and bailey fallacy. I would like to extend your analogy of the motte and bailey to describe something I have noticed in recent years when it comes to debate.

Recently I have noticed a lot of people debating and arguing about technical points, that they could not care less about, as a proxy for what they would actually like to argue.

If I may use this thread's analogy, Frank lives in a land that he has grown up in, let's call it Frankia. He believes this is his land and should be run by his rules and values. He builds a motte and bailey to defend this land run by his values.

Someone comes along and proclaims that Frank's values are immoral and this needs changing, let's call this attacker John. He attacks the bailey, and after a brief battle the bailey falls. However the bailey was simply a means to an end for Frank - the end being that Frankia be run my Frank's values. Frank retreats to the motte and still proclaims Frankia to be run by him.

You see even if the motte falls one day, Frank will fall back to a ditch, or fight an asymmetric war from the underbrush, because the motte and the bailey and even the ditch are not the point. The point is that Frank thinks Frankia should be run by his values, not John's.

This is what I have experienced in discussion and debated in the last few years from many sides with many different viewpoint - there are many Franks out there. Frank argues about the bailey and the motte and maybe even the ditch and underbrush. But even if they all fall, he will keep believing Frankia belongs to him and should be run by his values.

In effect people take what they consider to be brute facts about certain value positions, and then argue various technicalities. But no matter of technical arguments are going to make them change their mind over what they see as brute facts (this is from their perspective).

We find MACS’ ultimate source by answering “Why do cultural moral norms exist?”

As I have been saying, cultural moral norms exist because they were selected for by their ability to solve cooperation problems. Domination norms which exploit outgroups are creating cooperation problems for the outgroup – the opposite of MACS’ function and therefore automatically excluded (pruned) from the start. — Mark S

But this is not correct. Cultural moral norms exist because they were selected for by their ability to solve cooperation problems in the in group. The out group is often disposable for cultural moral norms, historically. This is what is observed as "is."

I think your mistake here is saying that observing how cultural moral norms are selected is in their ability to solve universal cooperation problems for everyone. That is simply not what is observed. Rather we see many instances of cultural moral norms that are selected to strengthen cooperation in the in group, while dominating the out group.

So what you claim as "is", is not really what is observed as "is."

What may not be obvious is that these principles innately exclude domination moral norms – no sneaky separate pruning required. Domination moral norms are excluded because their goals of exploiting outgroups are excluded. Exploiting outgroups creates cooperation problems for the outgroup and are therefore immoral (even while solving cooperation problems for the ingroup). — Mark S

But this is not what is observed in past societies

Let's take a step back and look at your original question in the other thread: "What if cultural moral norms track cooperation strategies?" I agree, in the "is" form, cultural moral norms often track cooperation strategies, often to the exclusion of the out group. That is what is observed.

So the foundation of your theory, is based on observing past societies. And in this observation we see that total cooperation including the outgroup is not what is the moral norm, rather the moral norm includes domination of the outgroup.

And so your pruning of the domination moral norm is not justified by the method you use. You claim that the "is" excludes domination moral norms. But the "is" that is observed includes domination moral norms.

If I were to base my morality on past societies, it would be to form an in-group and then dominate the out group - that is what many of the great past civilizations did. To get around the problem, I need a moral theory based on "oughts" - just because past successful civilizations dominated the out group doesn't mean I should too.

True, if the conditions for a snowflake to grow infinitely large were present, it could grow infinitely large. This is if there was infinite water, infinite space at the perfect temperature, etc.

But the same is not true going smaller. As you zoom in, and get to the molecular level and smaller, the snowflake will not look like a snowflake even with infinite water, infinite space at the perfect temperature, etc. Thus it can never be an actual fractal.

But I think that is beside the point as the universe is not a giant snowflake - i.e in our observable universe there are no true fractals, just approximations. In a perfect mathematical universe, there may be.

Fitness is something specific: "Survival of the form (phenotypic or genotypic) that will leave the most copies of itself in successive generations."

Natural selection is all about the survival of the genotypic line over successive generations. The genes that survive are fit, those that do not are not fit.

In that way it is applicable to all life on Earth, including humans. The genotype of some of us will have have more copies over more generations than that of others.

I did read your first post, it runs into the same problems. You are flitting between 2 similar but different theories, as per the description in my previous post.

For example:

"Domination moral norms (and sometimes Marker moral norms) violate MACS’s “Do not act to create cooperation problems”. These violations make them immoral in an absolute sense. "

That is not what is observed. Observation of past societies show that domination moral norms are just as effective at cooperation. However you are pruning away the domination moral norms by using some other "ought" based morality, but then presenting it as if it were an "is" observation.

So this is the version with pruning.

I still don't get it. Not the paradox of a set of all sets not members of themselves - that I know. I don;t understand how that applies to 3D configuration trying to contain a 4D configuration. Perhaps you can point me to where Frege conceptualized this particular paradox with 4D and 3D shapes?

Since a hypercube, being 4D, has 3D boundaries, it occupies four distant 3D locations, i.e., the same object in four places simultaneously. This type of spatial expansion, i.e., spatial dimension, deals a fatal blow to logical consistency at the level of 3D spatial expansion. At the level of 4D spatial expansion, logical consistency, i.e., one object being in two places at once is natural not fatal. — ucarr

I'm afraid I don't understand where the paradox is in 4D hypercubes. Let's simplify for a moment to better visualize the problem. A 2D square has 1D boundaries (lines) in 4 different locations, meeting at the edges. This is the same relationship that a 3D cube has with its 2D sides, and that a 4D hypercube has with its 3D sides.

What is the contradiction in a square having lines in different locations, meeting at the edges? What is the contradiction in a 4D hypercube having 3D sides in different locations, meeting at the edges? And what has that to do with Russel's paradox?