🔵 **π的极高位数与人口创造的关联：** 文章设定了一个基于π的第10↑↑100位数字奇偶性的条件，来决定创造蓝色眼睛和棕色眼睛的人口数量。如果该数字是偶数，则创造10↑↑10个蓝眼人和1个棕眼人；如果是奇数，则创造1个蓝眼人和1个棕眼人。这引出了在数学上确定性结果如何映射到概率推断的问题。

🟠 **量子测量与主观概率：** 通过一个粒子自旋测量的例子，文章阐述了在观察到特定结果（如棕色眼睛）后，如何推断之前量子测量状态（如自旋向上）的概率。作者引用了埃弗雷特解释（Everett Interpretation），认为在观察到棕色眼睛的情况下，粒子被测量为自旋向上的概率是50%，因为在所有可能的世界中，一半的棕眼人对应于粒子自旋向上的测量结果。

🌌 **多重宇宙、泛心论与自我指示假设：** 文章的核心论点在于，如果“自指示假设”的确定性版本成立，这将成为支持Tegmark多重宇宙理论和泛心论的有力证据。这意味着，在概率上，我们观察到的现象（如作为大脑而非原子存在）并不能作为反驳某些哲学命题（如原子具有意识）的证据，从而迫使我们对这些命题赋予更高的可能性。

Published on October 21, 2025 5:45 AM GMT

I am stuck on the following problem:

If the 10$\uparrow \uparrow$100th digit of pi is even then 10$\uparrow \uparrow$

10 blue eyed people and one brown eyed person gets created. Otherwise one blue eyed person and one brown eyed person gets created.

What is my subjective probability of the 10$\uparrow \uparrow$100th digit of pi being even? 

If you don't like me talking about probabilities in a deterministic context then pretend that I am asked to bet on whether the 10 $\uparrow \uparrow$100th digit of pi is even and I don't have access to a galaxy-sized supercomputer. I want to know what odds to bet at.

Let us first go over an easier problem:

A particle is initialised spin up in the z direction. Then its spin is measured in the x direction. If the spin is measured to be up then 10$\uparrow \uparrow$10 blue eyed people and one brown eyed person gets created. Otherwise one blue eyed person and one brown eyed person gets created.

If I observe myself to be brown eyed then what is the probability that the particle was measured to be spin up in the x direction?

Here the answer is 50%. To see this, first assume that the Everett Interpretation of quantum mechanics is correct because none of the other interpretations make sense. Then 50% of all the brown eyed people in all the universes in which the measurement took place were in universes in which the particle was measured to be spin up.

Aside: I have cheated a bit here. If there are infinitely many universes then there might be a bit of work to do to make this argument rigorous, which I shall leave as an exercise for the reader.

Anyhow, the reason I am asking this question is because if the deterministic version of the Self Indication Assumption were to turn out to be true then that would be a very strong reason to believe both in the entire Tegmark multiverse hierarchy and in panpsychism. Now there's nothing remotely weird about believing in multiverses. 

On the other hand, consider the proposition P: 'atoms are just as conscious as brains'. If P is true then I should be very surprised indeed if I were to observe myself to be a brain rather than an atom. 

Since I know so little about the phenomenology of consciousness, I don't have any strong evidence against proposition P except for the fact that I don't observe myself to be an atom. If the deterministic version of the Self Indication Assumption is true then the fact that I don't observe myself to be an atom does not count as evidence against P. So I am forced to assign a fairly high probability to statement P, meanwhile wondering how breathtakingly improbable it is that I am not observing myself to be an atom right now.