Perspective-based reasoning suggests there is a possibility for a question to have different answers from different perspectives. In daily life, this only seems to occur in obvious problems involving the perspective center. (e.g. Am “I” a man? Its answer changes depending on whose perspective we are taking.) Yet, in anthropic reasoning, perspective disagreement can occur in a less trivial way. To illustrate this consider the following thought experiment.
1. Toss and Fission
Imagine you are taking part in this experiment. Tonight during your sleep, an advanced alien Omega would toss a fair coin. If the coin landed Tails, Omega would make a clone of you down to the details at the atomic level. The process is highly accurate even memories are preserved. So the created clone would not be able to tell if he is physically new. If the coin landed Heads, Omega would not perform the cloning. The copy/copies would be randomly put into the two rooms. On the next day, one of your friends would randomly choose a room to visit. Both you and the friend know the complete experiment setup. Suppose on the next day you wake up, moments later your friend comes in. The two of you can communicate however desired. Both of you need to answer the question: “what is the probability that yesterday’s coin toss landed on Heads?”. What answer should you and your friend give?
2. The Disagreement
According to the experiment setup, cloning does not apply to the friend. To him, this is not an anthropic problem. If the coin landed on Heads one of the two rooms would be empty. If the coin landed on Tails both rooms would be occupied. The friend’s room selection is independent of the coin toss. The fact that a copy exists in the randomly chosen room is evidence favoring both rooms being occupied. Therefore the probability of Heads shall decrease from the initial half. By performing a simple update using Baye’s theorem the probability of Heads can be calculated to be 1/3.
From the subject’s first-person perspective there is no new information at the time of wake-up. Some arguments suggest there is new information that the subject learned “I” exist. However, from the first-person perspective, recognizing the perspective center is the starting point of further reasoning, it is a logical truth that “I” exist. Or simply, “It is guaranteed to find myself exist.” Therefore “my” own existence cannot be treated as new evidence. Furthermore, the friend’s selection is independent of the coin toss. So for the subject, meeting the friend does not give any new information about the coin toss either. As a result, the probability of Heads shall remain at 1/2.
Here the subject and the friend are giving different answers to the same question. They are free to communicate however they want. There is nothing one can say to the other that could change their answer. At the same time, both would have to accept the other’s reasoning is rational. That is, the disagreement is entirely due to their difference in perspectives.
3. Cause of the Disagreement
Just like my wife and I would give different answers to the question “Am I a man?”, the disagreement above is also because the question involves the perspective center. Only this time it is more subtle. Even though “What is the probability of Heads?” does not directly mention any indexicals, the question has an implied condition. The complete question should be “given the current information, what is the probability of Heads?”. One important information here is that the subject and the friend have met each other. Yet the subject and the friend have different interpretations of who exactly is in this meeting.
From the friend’s perspective, he has met an unspecific copy. If the coin landed Tails so there exist two copies, he has no way to specify which exact one is in the chosen room. However, from the subject’s first-person perspective he knows exactly who is in the chosen room. “I” am. This specification is based on his perspective. It is indexical and incommunicable. The subject could tell the friend “It’s me you are meeting with” yet it would mean nothing to the friend.
Another way to tell their difference in interpretations is to look at the complementary events. From the friend’s perspective, the alternative outcome of the meeting would be “the chosen room is empty”. From the subject’s perspective, the alternative outcome of the meeting would be “My room is not chosen by the friend”. It is not the same as “the chosen room is empty” due to the possibility of the friend meeting “the other copy”. This is because as the first person, the subject inherently focuses on “I” while the friend is indifferent towards the two copies (if there are two). Thus the friend shall treat the meeting as evidence suggesting there are more copies while the subject should not.
4. Different Relative Frequencies
If a large number of experiments are performed, among cases where the meeting occurs, the relative fraction of Heads would be different for the subject and the friend. The friend is not a participant in the fission process. So he is an outsider to the anthropic problem. To repeat the experiment from his perspective, another Toss&Fission shall be performed on some subject. He then shall choose a random room to enter again. Imagine a large number of experiments are performed, among repetitions where the chosen room is not empty about a third would be after Heads. For example, if 1000 experiments are performed, the friend would meet a copy in the chosen room about 750 times. Among those 750 experiments about 250 would be after Heads. This is consistent with the probability of 1/3 from his perspective.
Repeating the experiment from the subject’s perspective is similar to Question 2 in Part 3. After waking up and potentially meeting the friend, “I” can take part in another Toss&Fission. After waking up from the second experiment “I” can take part in yet another one and so forth. “I” can experience a large number of repetitions as such. About half of the experiments “I” experienced would be Heads. Among experiments where “I” have met the friend, the fraction of Heads would still be 1/2. For example if “I” experienced 1000 experiments, “I” would have met with a friend about 500 times. Among those 500 experiments about 250 would be after Heads. This is consistent with the probability of 1/2 from the subject’s perspective. The different relative frequencies are due to experiments where the friend met “the other copy” instead of meeting “me” specifically.
5. Understanding the Disagreement
The above analysis shows there is nothing mysterious about the disagreement. It is merely because the question’s content depends on which perspective we are taking. E.g. the friend and the subject would include different sets of experiments when calculating the relative frequency. This change is due to the involvement of the perspective center in the question formulation. Effectively, the question is asking about two different probabilities for the friend and the subject.
In this sense, I feel “disagreement” might not be the right word for this difference in answers. Because both the friend and the subject in the meeting would agree the counter-party is correct as well. Just like to the question “Am I a man?”, where I would answer yes and my wife no. We won’t argue over the merit of one’s rationale over the other. It is obvious we are both right. However, in the literature, this difference in answers has been referred to as disagreements in the past. E.g. John Pittard’s When Beauties Disagree pdf link, Katja Grace’s well-written piece here, I called them disagreements before too. At this point coming up with a new name might bring more confusion than any benefit it could bring.
However, even by using the same name, we must be aware it is different from the type of disagreements we usually think of. The famous Aumann’s agreement theorem is not applicable here. Furthermore, it shows once again we shall not mix reasonings from different perspectives. Most notably, the first-person center cannot be regarded as a sampling outcome from another perspective. For it could alter the result of probability calculations.
So far I have discussed why self-locating probabilities are invalid and perspective disagreement can happen. That is everything I need to explain all anthropic paradoxes. Part 5 discusses the causes of anthropic paradoxes except for the Sleeping Beauty Problem. Part 6 discusses the sleeping beauty problem specifically. And Part 7 explains the idea of the fine-tuned universe and why it is another misconception of perspective inconsistency