Ah, I rewrote my comment a few times and lost what I was referencing. I originally was referencing the geometric meaning (as an alternate to your statistical definition), two vectors at a right angle from each other.
But the statistical understanding works from what I can tell? You have your initial space with extreme uncertainty, and the orthogonality thesis simply states that (intelligence, goals) are not related — you can pair some intelligence with any goal. They are independent of each other at this most basic level. This is the orthogonality thesis.
Then, in practice, you condition your probability distribution over that space with your more specific knowledge about what minds will be created, and how they’ll be created. You can consider this as giving you a new space, moving probability around.
As an absurd example: if height/weight of creatures were uncorrelated in principal, but then we update on “this is an athletic human”, then in that new distribution they are correlated! This is what I was trying to get at with my R^2 example, but apologies that I was unclear since I was still coming at it from a frame of normal geometry. (Think, each axis is an independent normal distribution but then you condition on some knowledge that restricts them such that they become correlated)
I agree that it is an informal argument and that pinning it down to very detailed specifics isn’t necessary or helpful at this low-level, I’m merely attempting to explain why orthogonality works. It is a statement about the basic state of minds before we consider details, and they are orthogonal there; because it is an argumentative response to assumptions about “smart → not dumb goals”.
Ah, I rewrote my comment a few times and lost what I was referencing. I originally was referencing the geometric meaning (as an alternate to your statistical definition), two vectors at a right angle from each other.
But the statistical understanding works from what I can tell? You have your initial space with extreme uncertainty, and the orthogonality thesis simply states that (intelligence, goals) are not related — you can pair some intelligence with any goal. They are independent of each other at this most basic level. This is the orthogonality thesis. Then, in practice, you condition your probability distribution over that space with your more specific knowledge about what minds will be created, and how they’ll be created. You can consider this as giving you a new space, moving probability around. As an absurd example: if height/weight of creatures were uncorrelated in principal, but then we update on “this is an athletic human”, then in that new distribution they are correlated! This is what I was trying to get at with my R^2 example, but apologies that I was unclear since I was still coming at it from a frame of normal geometry. (Think, each axis is an independent normal distribution but then you condition on some knowledge that restricts them such that they become correlated)
I agree that it is an informal argument and that pinning it down to very detailed specifics isn’t necessary or helpful at this low-level, I’m merely attempting to explain why orthogonality works. It is a statement about the basic state of minds before we consider details, and they are orthogonal there; because it is an argumentative response to assumptions about “smart → not dumb goals”.