None of these bijections have nice properties, though. There are bijections between R³ and R², but no continuous ones. (I’m not sure if they can even be measurable.) One might criticise Eliezer’s mention of the Pigeonhole principle, but the point that he is making stands: a three-dimensional space cannot be mapped out by two real-valued parameters in any useful way. A minimal notion of “useful” here might be a local homeomorphism between manifolds, and this is clearly impossible when the dimensions are different.
There is a big leap between there are no X, so Y and there are no useful X (useful meaning local homeomorphisms), so Y, though. Also, local homeomorphism seem too strong a standard to set. But sure, I kind of agree on this. So let’s forget about injection. Orthogonal projections seem to be very useful under many standards, albeit lossy. I’m not confident that there are no akin, useful equivalence classes in A (joint probability distributions) that can be nicely map to B (causal diagrams). Either way, the conclusion
This means the first causal structure is falsifiable; there’s survey data we can get which would lead us to reject it as a hypothesis
can’t be entailed from the above alone.
Note: my model of this is just balls in Rn, so elements might not hold the same accidental properties as the ones in A and B, (if so, please explain :) ) but my underlying issue is with the actual structure of the argument.
None of these bijections have nice properties, though. There are bijections between R³ and R², but no continuous ones. (I’m not sure if they can even be measurable.) One might criticise Eliezer’s mention of the Pigeonhole principle, but the point that he is making stands: a three-dimensional space cannot be mapped out by two real-valued parameters in any useful way. A minimal notion of “useful” here might be a local homeomorphism between manifolds, and this is clearly impossible when the dimensions are different.
There is a big leap between there are no X, so Y and there are no useful X (useful meaning local homeomorphisms), so Y, though. Also, local homeomorphism seem too strong a standard to set. But sure, I kind of agree on this. So let’s forget about injection. Orthogonal projections seem to be very useful under many standards, albeit lossy. I’m not confident that there are no akin, useful equivalence classes in A (joint probability distributions) that can be nicely map to B (causal diagrams). Either way, the conclusion
can’t be entailed from the above alone.
Note: my model of this is just balls in Rn, so elements might not hold the same accidental properties as the ones in A and B, (if so, please explain :) ) but my underlying issue is with the actual structure of the argument.