I didn’t—but I did do a back-of-the-envelope calculation, which predicts that there are something like a googolplex times more graphs with one cycle than there are acyclic paths, assuming 10^60 nodes (the number of Planck times since the beginning of the universe.)
And I don’t have a prior that says that an acausal universe should have a probability penalty of one over googolplex.
a googolplex times more graphs with one cycle than there are acyclic paths
(I assume you meant “acyclic graphs”)
And I don’t have a prior that says that an acausal universe should have a probability penalty of one over googolplex.
If this sort of reasoning worked, you could find strong arguments for all sorts of (contradictory) hypotheses. For instance:
Surely, the universe has an underlying n-dimensional (topological) manifold. Since all of the infinitely many n-dimensional manifolds are not homotopy equivalent to the n-sphere, except for the sphere itself, the universe must not be an n-sphere. Therefore, there are holes in the universe.
or
Surely, the universe has an underlying set. Let A be the cardinality of that set. Then, since all cardinalities (except for countably many) are larger that Aleph_0, the universe is not countable. Therefore, the universe is not Turing computable.
By the Poincare recurrence theorem, all of this has happened before and all of this will happen again.
I mean, your observation is interesting, but I don’t think it constitutes a “large argument”. You can’t just slap reasonable-ish priors onto spaces of mathematical objects, and in general using math for long chains of inference often only works if it’s exactly the right sort of math.
No, I meant acyclic paths—I am not at all sure that that is the correct term, but I meant something like “no branches”—there is only one possible path through the graph and it covers all the nodes.
And, well, point granted. Honestly I was expecting something like that, but I couldn’t see where the problem was*, so I went ahead and asked the question.
… yeah, in retrospect this allows for silly things like “surely rocks must fall up somewhere in the universe.”
I didn’t—but I did do a back-of-the-envelope calculation, which predicts that there are something like a googolplex times more graphs with one cycle than there are acyclic paths, assuming 10^60 nodes (the number of Planck times since the beginning of the universe.)
And I don’t have a prior that says that an acausal universe should have a probability penalty of one over googolplex.
(I assume you meant “acyclic graphs”)
If this sort of reasoning worked, you could find strong arguments for all sorts of (contradictory) hypotheses. For instance:
or
or
or
I mean, your observation is interesting, but I don’t think it constitutes a “large argument”. You can’t just slap reasonable-ish priors onto spaces of mathematical objects, and in general using math for long chains of inference often only works if it’s exactly the right sort of math.
No, I meant acyclic paths—I am not at all sure that that is the correct term, but I meant something like “no branches”—there is only one possible path through the graph and it covers all the nodes.
And, well, point granted. Honestly I was expecting something like that, but I couldn’t see where the problem was*, so I went ahead and asked the question.
… yeah, in retrospect this allows for silly things like “surely rocks must fall up somewhere in the universe.”