Hmm, you are right. Thanks for the correction!

I think this example brings out how Pearlian causality differs from other causal theories. For instance, in a counterfactual theory of causation, since the negation of a mathematical truth is impossible, we can't meaningfully think of them as causes.

But in the Pearlian causality it seems that mathematical statements can have causal relations, since we can factor our uncertainty about them, just as we can other statements. I think endoself's comment argues this well. I would add that this is a good example of how causation can be subjective. Before 1984, the Taniyama-Shimura-Weil conjecture and Fermat's last theorem existed as conjectures, and some mathematicians presumably knew about both, but as far as I know they had no clue that they were related. Then Frey conjectured and Ribet proved that the TSW conjecture implies FLT. Then mathematician's uncertainty was such that they would have causal graphs with TSW causing FLT. Now we have a proof of TSW (mostly by Wiles) but any residual uncertainty is still correlated. In the future, maybe there will be many independent proofs of each, and whatever uncertainty is left about them will be (nearly) uncorrelated.

I also think there can be causal relations between mathematical statements and statements about the world. For instance, maybe there is some conjecture of fluid dynamics, which if true would cause us to believe a certain type of wave can occur in certain circumstances. We can make inferences both ways, for instance, if we observe the wave we might increase our credence in the conjecture, and if we prove the conjecture, we might believe the wave can be observed somehow. But it seems that the causal graph would have the conjecture causing the wave. Part of the graph would be:

[̶P̶r̶o̶o̶f̶ ̶o̶f̶ ̶c̶o̶n̶j̶e̶c̶t̶u̶r̶e̶ ̶-̶>̶ ̶c̶o̶n̶j̶e̶c̶t̶u̶r̶e̶ ̶-̶>̶ ̶w̶a̶v̶e̶ ̶<̶-̶ ̶(̶f̶l̶u̶i̶d̶ ̶d̶y̶n̶a̶m̶i̶c̶s̶ ̶a̶p̶p̶l̶i̶e̶s̶ ̶t̶o̶ ̶w̶a̶t̶e̶r̶)̶ ̶]̶

[Proof of conjecture <- conjecture -> wave <- (fluid dynamics applies to water) ]

"Actually" isn't intended in any sense except emphasis and to express that Eliezer's view is contrary to my expectations (for instance, "I thought it was a worm, but it was actually a small snake").

Eliezer does seem to be endorsing the statement that "everything is made of causes and effects", but I am unsure of his exact position. The maximalist interpretation of this would be, "in the correct complete theory of everything, I expect that causation will be basic, one of the things to which other laws are reduced. It will not be the case that causation is explained in terms of laws that make no mention of causation". This view I strongly disagree with, not least because I generally think something has gone wrong with one's philosophy if it predicts something about fundamental physics (like Kant's a priori deduction that the universe is Euclidean).

I suspect this is not Eliezer's position, though I am unsure because of his "Timeless Physics" post, which I disagree with (as I lean towards four-dimensionalism) but which seems consonant with the above position in that both are consistent with time being non-fundamental. If he means something weaker, though, I don't know what it is.

Imagine a universe that is made only of ideal billiard balls eternally bouncing around on a frictionless, pocketless billiard table. Essentially the same thing as selylindi's idea of a gas in thermodynamic equilibrium. Imagine yourself observing this universe as a timeless observer, or to aid the imagination, that it's "time" dimension is correlated to our space dimension, so we see the system as an infinite frozen solid, 11 by 6 by infinity, with the balls represented by solid streaks inside that go in straight lines except where they bounce off each other or the boundary of the solid.

Now, this system internally has a timelike dimension, except without increasing or decreasing entropy. And the physics of the system are completely reversible, so we have no basis of saying which way is the "future" and which way is the "past" in this system. We can equally well say a collision at one time is "caused" by the positions of the balls one second in the "past" or one second in the "future". There is no basis for choosing a direction of causality between two events.

In our universe, our time is microscopically reversible but macroscopically irreversible, because of the fact that the universe is proceeding from a low entropy state (we call that direction the "past") to a high entropy state (the "future"). I am curious, can anyone coherently describe a universe with nothing similar to irreversible time, but with a useful notion of causation? Or with something like irreversible time, but no causation whatsoever? I have tried (for much more than five minutes!) and not succeeded , but I am still far from sure that it's impossible to do. It might be too much to ask to imagine actually being in a universe without causation or time, but perhaps we can think of how such a universe could look from the outside.

Consider "Elaine is a post-utopian and the Earth is round" This statement is meaningless, at least in the case where the Earth is round, where it is equivalent to "Elaine is a post-utopian." Yet it does constrain my experience, because observing that the Earth is flat falsifies it. If something like this came to seem like a natural proposition to consider, I think it would be hard to notice it was (partly) meaningless, since I could still notice it being updated.

This seems to defeat many suggestions people have made so far. I guess we could say it's not a real counterexample, because the statement is still "partly meaningful". But in that case it would be still be nice if we could say what "partly meaningful" means. I think that the situation often arises that a concept or belief people throw around has a lot of useless conceptual baggage that doesn't track anything in the real world, yet doesn't completely fail to constrain reality (I'd put phlogiston and possibly some literary criticism concepts in this category).

My first attempt is to say that a belief A of X is meaningful to the extent that it (is contained in / has an analog in / is resolved by) the most parsimonious model of the universe which makes all predictions about direct observations that X would make.

Your view reminds me of Quine's "web of belief" view as expressed in "Two Dogmas of Empiricism" section 6:

The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges. Or, to change the figure, total science is like a field of force whose boundary conditions are experience. A conflict with experience at the periphery occasions readjustments in the interior of the field. Truth values have to be redistributed over some of our statements. Reevaluation of some statements entails reevaluation of others, because of their logical interconnections--the logical laws being in turn simply certain further statements of the system, certain further elements of the field.

Quine doesn't use Bayesian epistemology, unfortunately because I think it would have helped him clarify and refine his view.

One way to try to flesh this intuition out is to say that some beliefs are meaningful by virtue of being subject to revision by experience (i.e. they directly pay rent), while others are meaningful by virtue of being epistemically entangled with beliefs that pay rent (in the sense of not being independent beliefs in the probabilistic sense). But that seems to fail because any belief connected to a belief that directly pays rent must itself be subject to revision by experience, at least to some extent, since if A is entangled with B, an observation which revises P(A) typically revises P(B), however slightly.

No 2^alpeh0=aleph1 is the continuum hypothesis, which is independent of the standard axioms of math, and can't be proven. I think maybe you mean he was close to showing 2^aleph0 is the cardinality of the reals, but I think he knew this already and was trying to use it as the basis of the proof.

Making mistakes like Eliezer's is a big part of learning math though, if we are looking for a silver lining. When you prove something you know is wrong, usually it's because of some misunderstanding or incomplete understanding, and not because of some trivial error. I think the diagonal argument seems like some stupid syntactical trick the first time you hear it, but the concept is far-reaching. Surely Eliezer came away with a bit better understanding of its implications after he straightened himself out.

The specific problem with calling the last game a "prisoner's dilemma" is that someone learning about game theory from this article may well remember from it, "there is a cool way to coordinate on the prisoner's dilemma using coin flips based on correlated equilibria" then be seriously confused at some later point.