Fair points, though there is in fact a lot of disagreement about what are the basic relata of the causal relation: see the SEP entry for example. When we apply causation to entities (which we can sometimes do, as in your example) then "A causes B" probably means something like "at least one event in which A is involved is a cause of every event in which B is involved".

On counterexamples to "what causes the whole, causes the part" : possibly an even stronger counterexample considers just one of the atoms in the cake. However, we must be careful here: it is only some temporal part of the egg (or of the atom) which is part of the cake; the eggs/atoms in their full temporal entirety are NOT parts of the cake in its full temporal entirety. We could perhaps treat the relevant temporal part ("egg mixed into cake" or "atom within cake") as an "entity" in its own right, but then it does seem that by making the cake, I am a cause of all the events which involve that particular "entity" (since I put the egg/atom into the cake in the first place).

In any case, note that the most recent version of the argument doesn't actually need to assume this "cause-whole => cause-part" applies to C, since it only ever uses the constructed relation C* instead. The conclusion is still interesting, since if nothing C*s the entity g, then nothing Cs it either, and if g causes some whole of which each entity is a part, that is still an interesting property of g. The argument makes no assumptions on whether C is reflexive or not.

On A3, I'm not totally sure of the circumstances in which we can aggregate entities together and treat them as parts of a single entity, but if the entities are causally related (and particularly if they are causally-related in an odd way, like an endless chain), then it does make some sort of sense to do this aggregation. After all, we immediately want to ask the question "How could there be an endless chain?" a question which does treat the "chain" as some sort of an entity to be explained. If entities are not causally related (they are in different universes), lumping them together seems much less natural.

Finally, on the "maximal entity" approach, CCC I believe discussed this in the original thread before I lifted here, and he seems to find it theologically interesting.

x C* y, and y P z. Therefore, x C* z.

No, you need x C* y and z P y to get x C* z (be careful about which way the P relation is going).

The intuition is "Anything which is a cause of the whole is a cause of the part", not "Anything which is a cause of the part is a cause of the whole". Again, there are intuitive examples here. (Compare me baking a cake for a child's birthday party vs just buying the cake from a shop, and putting a few sprinkles and candles on the top. In the second case, I am a cause of some part of the cake as presented to the child, but not the whole cake, and if someone says "Wow that cake tasted delicious!" I'd have to admit I didn't make it, only decorated it).

Hmmm... one example of a randomised universe might be one wherein any matter can accelerate in any direction at any time for absolutely no reason, and most matter does so on a fairly regular basis

Well let's take that example, since the amount of "random acceleration" can be parameterised. If the parameter is very low, then we're never going to observe it (so perhaps our universe actually is like this, but we haven't detected it yet!) If the parameter is very large, then planets (or even stars and galaxies) will get ripped apart long before observers can evolve.

So it seems such a parameter needs to be "tuned" into a relatively narrow range (looking at orders of magnitude here) to get a universe which is still habitable but interestingly-different from the one we see. But then if there were such an interesting parameter, presumably the careful "tuning" would be noticed, and used by theists as the basis of a design argument! But it can't be the case that both the presence of this random-acceleration phenomenon and its absence are evidence of design, so something has gone wrong here.

If you want a real-word example, think about radioactivity: atoms randomly falling apart for no apparent reason looks awfully like objects suddenly accelerating in random directions for no reason: it's just the scale that's very different. Further, if you imagine increasing the strength of the weak nuclear force, you'll discover that life as we know it becomes impossible... whereas, as far as I know, if there were no weak force at all, life would still be perfectly possible (stars would still shine, because that 's the strong force, chemical reactions would still work, gravity would still exist and so on). Maybe the Earth would cool down faster, or something along those lines, but it doesn't seem a major barrier to life. However, the fact that the weak force is "just in the right range" has indeed been used as a "fine-tuning" argument!

Dark energy (or a "cosmological constant") is another great example, perhaps even closer to what you describe. There is this mysterious unknown force making all galaxies accelerate away from each other, when gravity should be slowing them down. If the dark energy were many orders of magnitude bigger, then stars and galaxies couldn't form in the first place (no life), but if it were orders of magnitude smaller (or zero), life and observers would get along fine. By plotting on the right scale (e.g. compared to a Planck scale), the dark energy can be made to look suspiciously small and "fine-tuned", and this is the basis of a design argument.

Do you see the pattern here?

Consider any endless chain consisting of at minimum two elements. Consider two elements in that chain, x and y, such that x C* y. x and y are both proper parts of z. Therefore, x C* z, and z C* y.

It follows that z C* y but it does not follow that x C* z or that y C* z.. The "whole" z may be a cause of its parts, without in turn being caused by its parts. Note that by construction of C* it is true that if x is a cause of y and x is a part of z, then z C* y. However, it is not generally true that if x is a cause of y and z is a part of x then z C* y.

As an example of the intuition behind this: suppose I have a thermostat box containing two circuit boards. Board 1 is connected into my home heating system; Board 2 is a spare not connected into anything. It is true that Board 1 causes my heating to come on. It is true that the thermostat (of which Board 1 is part) causes my heating to come on. It is false that Board 2 (which is part of the thermostat) causes my heating to come on.

But then we have a longer chain; using x C* z C* y in place of x C* y.

You are right that when adding z, we now get a longer chain {x, y, z}, but this won't in general be an "endless chain" (the new z may well be an end).

As discussed, I have a new version which preserves the proof structure, but weakens the premises about as much as possible.

A1. The collection of all entities is a set E, with a causal relation C and a partial order P, such that x P y if and only if x is a part of y.

Note: This merges the assumption that P is a partial order into the overall set-up; that feature of P now gets used earlier in the argument.

A2. The set E can be well-ordered.

This ensures we can apply Zorn's Lemma when considering chains in E, but is not as strong as the full Axiom of Choice. If the set E is finite or countable, for instance, then A2 applies automatically.

Definitions: We define a relation C* such that x C* y iff there are entities v, w such that v P x, y P w and v C w.

Note: This gives a broader causal relation which automatically satisfies "if x C* y and x P z then z C* y" as well as "if x C* z and y P z then x C* y", loosely "anything which is caused by a part is caused by the whole" and "anything which causes the whole, causes the part". So we don't need to state those as extra premises.

We then define a further relation <= such that x <= y iff x = y, or there are finitely many entities x1, ..., xn such that x1 = x, xn = y and xi C* xi+1 for i=1.. n-1.

Note: This construction ensures that <= is a pre-order on E.

Say that a subset S of E is a "chain" iff for any x, y in S we have x <= y or y <= x. Say that S is an "endless chain" iff for any x in S there is some y in S distinct from x with y <= x. We shall say that y is "uncaused" if and only if there is no z in E distinct from y with z C* y (this of course implies there is no z distinct from y with z C y, but it also implies that y isn't part of anything which is caused by something distinct from y). Say that x is a proper part of y iff x is distinct from y but x P y.

A3. Let S be any endless chain in E; then there is some z in E such that every x in S is a proper part of z.

Lemma 1: For any chain S in E, there is an entity x in E such that x <= y for every y in S.

Proof: Suppose S is not endless. Then there is some x in S such that for no other y in S is y <= x. By the chain property we must have x <= y for every member y of S. Alternatively, suppose that S is endless, then by A3, there is some z in E of which every x in S is a part. Now consider any y in S. There is some x not equal to y in S with x <= y, so there are entities x = x1... xn = y with each xi C* xi+1 for i=1..n-1. Further, as x P z we have z C* x2 and hence z <= y.

Lemma 2: For any x in E, there is some y in E such that y <= x, and for every z <= y, we must have y <= z.

Proof: This follows from Zorn's Lemma applied to pre-orders.

Theorem 3: For any x in E, there is some uncaused y in E such that y <= x.

Proof: Take a y as given by Lemma 2 and consider the set S = {s: s <= y}. By Lemma 2, y <= s for every member of S, and if S has more than one element, then S is an endless chain. So by A3 there is some z of which every s in S is a proper part, which implies that z is not in S. But by the proof of Lemma 1, z <= y, which implies z is in S: a contradiction. So it follows that S = {y}, which completes the proof.

We now partition E into three subsets. I are the "inert" entities, which do not cause anything and have no causes themselves. (Note that the new version allows there to be some of these, unlike the previous version; you can think of them as abstract entities like numbers, sets, propositions and so on, if you want to). Formally I = {x in E: there is no y distinct from x with x C* y or y C* x}. U are the "uncaused causes" - formally U = {x in E: there is no y distinct from x with y C* x, but there is z distinct from x with x C* z}. O are all the "other", caused entities, so that formally O = {x in E: there is some y distinct from x with y C* x}.

B1. If S is any subset of U such that for any x, y in S we have x P y or y P x, (call such an S a "chain of parts"), then there is some entity z of which all members of S are parts.

B2. Suppose that y C* x and z C* x. Then there is some entity w such that: w <= x; w <= y or y P w; w <= z or z P w.

Informally, the idea is that y and z can't independently cause x without any further causal explanation. So there must be some common cause, however each of them may be part of that common cause.

Definition: Say that entities x and y are causally-connected if and only if x=y or there are finitely-many entities x=x1,..,xn=y with each xi C* xi+1 or xi+1 C* xi for i=1..n-1.

B3. Any two entities x, y in O are causally-connected.

Informally, O doesn't "come apart" into disconnected components, such as a bunch of isolated universes. Premises B1-B3 turn out to be necessary for Theorem 6 to hold, as well as sufficient (see below). So they can't be made any weaker!

Theorem 4: For any x in O, there is a unique entity f(x) in U such that: f(x) <= x, and any other y in U with y <= x satisfies y P f(x).

Proof: For any x in O, define a subset U(x) = {y in U: y <= x}; this is non-empty by Theorem 3. Consider any chain of parts S that is a subset of U(x). If it has at least two members, then by B1 there is some z in E of which all members of S are parts, and such a z must be in U. (If not, then note any w C* z would also satisfy w C* s for each member s of S, which would require them all to be equal to w). Also since y <= x for any member of S and y P z we have z <= x. So z is also a member of U(x). Or if S is a singleton - say {z} - then clearly all members of S are parts of z, and z is also in U(x). By application of Zorn's Lemma to U(x), there is a P-maximal element f(x) in U(x) such that there is no other y in U(x) with f(x) P y. By B2, for any other y in U(x) there must be some z in U(x) with f(x) P z and y P z; given f(x) is maximal we have z = f(x) and so y P f(x). This makes f(x) the unique maximal element of U(x).

Theorem 5: For any x, y in O, f(x) = f(y) if and only if x and y are causally-connected.

Proof: It is clear that if f(x) = f(y) then x is causally-connected to y (just build a path backwards from x to f(x) and then forward again to y). Conversely, consider any two x, y in O:

a) If x C* y, then f(x) is in U and satisfies f(x) <= y so we have f(x) P f(y). Since x is not f(x), we have f(x) C* x2 <= x for some x2, and hence f(y) C* x2 <= x i.e. f(y) <= x which means f(y) P f(x), and so f(x) = f(y).

b) If z is in U with z C* x and z C* y, then z P f(x) so f(x) <= y and f(x) P f(y); similarly, f(y) P f(x) so that f(x) = f(y).

The result now follows by induction on the length of the causal path connecting x to y.

Theorem 6: If O is non-empty, then there is a single entity g in U such that: f(x) = g for every x in O, and y P g for every y in U.

Proof: Assuming O is non-empty, take any element y in O, and set g = f(y); then the result that f(x) = g for any x in O follows from Theorem 5 and B3; further, for any y in U, there is some x in O with y C* x, so by Theorem 4, y P f(x). If there are no elements of O (meaning there are none in U either) then the Theorem is trivial.

Finally, note that B1, B2 and B3 are entailed by the statement of Theorem 6. For B1, we can just take g as the relevant z. For B2, we can take g as the relevant w. B3 follows using using the first part of Theorem 5 (just track from x back to g, then forward to y again).

I'm just about done now, so unless there are errors in the above proof will leave it. What are the residual weak points? Well, B2 and B3 have been weakened a bit, but are still basically unjustifiable (we can imagine them being false without absurdity) and the above re-work shows they are needed for the uniqueness conclusion (Theorem 6). Also, we have the weakness of not deriving anything else useful about g.

No, I omitted that step for reasons discussed in the earlier thread: this gives too weak a "God" to be any interest to anyone, and is downright confusing.

The only way I can think to get back to some form of traditional theism is to add a premise saying that "every entity not of type G has a cause" (insert your favourite G) and then perhaps to pull the modal trick of claiming all the premises are possible...

I have spotted an error in the statement (and proof) of Theorem 5, and then Corollary 6. The issue is that for any uncaused y we must have f(y) = y, so if there are several uncaused entities then they can't all have f(y) = g. The revised statements should go like this:

Theorem 5: Let x and y both have causes. Then f(x) = f(y) if and only if x and y are causally connected.

Proof: It is clear that if f(x) = f(y) then x is causally-connected to y (just build a path backwards from x to f(x) and then forward again to y). Conversely, suppose that x C y, then f(x) is uncaused and satisfies f(x) <= y so we have f(x) P f(y); since x is caused, there are f(x)=x1,...,xn=x such that each xi C xi+1 for i=1..n-1, then by A3 we have f(y) C x2 and hence f(y) <= x, which implies f(y) P f(x) and so by B1 f(x) = f(y). Next, suppose that for some uncaused z we have z C x and z C y; then z P f(x) which implies by A3 that f(x) C y and hence f(x) P f(y); similarly, f(y) P f(x) so by B1 f(x) = f(y). By an induction on the length of any other path connecting x to y, we have that f(x) = f(y).

Corollary 6: There is a single g in E such that: f(x) = g for every x in E with a cause, and every uncaused y P g.

Proof: Suppose there is at least one entity x with a cause, then set g = f(x). For any other caused entity y, f(y) = g by Theorem 5 and B5, and for an uncaused y, B5 implies y <= x, so that y P g. Lastly, if there are no caused entities, then B5 implies that E = {y} for some uncaused y, so we can just pick g = y.

I have also spotted a way of weakening or removing some of the premises (in particular A3, and B1 to B4). I will update with that later today.

Set Theory and Uncaused Causes

I'm relocating part of a thread that was originally on "Welcome to Less Wrong" but has wandered way off topic. It also seems that a remote ancestor comment was heavily downvoted, discouraging further contributions in the original place. So I'm moving into the Open thread.

(Huh. One of the ancestors to this comment - several levels up - has been downvoted enough to require a karma penalty. I wonder if there should be some statute of limitations on that; whether, say, ten levels of positive-karma posts can protect against a higher-level negative-karma post?)

Here are links to my latest version of the "recipe", and to CCC's response

anthropic principle is arguing from the existence of an intelligent observer; I'm arguing from the existence of an orderly universe. I don't think that the existence of an orderly universe is necessarily highly correlated with the existence of an intelligent observer.

This depends on the direction of correlation doesn't it? It could well be that P[Observer|Orderly universe] is low (plenty of types of order are uninhabitable) but that P[Orderly universe|Observer] is high since P[Observer|Disorderly universe] is very much lower than P[Observer|Orderly universe]. So, for example, if reality consists of a mixture of orderly and disorderly universes, then we (as observers) would expect to find ourselves in one of the "orderly" ones, and the fact that we do isn't much evidence for anything.

Another thought is whether there are any universes with no order at all? You are likely imagining a "random" universe with all sorts of unpredictable events, but then are the parts of the universe dependent or independent random variables? If they are dependent, then those dependencies are a form of order. If they are independent, then the universe will satisfy statistical laws (large number laws for instance), so this is also a form of order. Very difficult to imagine a universe with no order.

For uniqueness, we build on the idea of all uncaused causes being part of a whole. The following premises look interesting here:

B1. If x P y and y P z then x P z; x = y if and only if x P y and y P x.

This states that P is a partial order, which is reasonable for the "part of" relation.

B2. If S is any chain of parts, such that for any x, y in S we have x P y or y P x, then there is some z in E of which all members of S are parts.

This states that E is inductively ordered by the "part of" relation.

B3. If x C z and y P z then x C y.

Informally, "a cause of the whole is a cause of any part".

B4. Suppose that y <= x and z <= x and both y, z are uncaused. Then y P z or z P y, or there is some w of which both y and z are proper parts.

Informally, two uncaused y and z can't independently conspire to cause x unless they are parts of a common entity.

Definition: Say that entities x and y are causally-connected if and only if x = y, or there are entities x=x1,..,xn=y with either xi C xi+1 or xi+1 C xi for each i=1..n-1.

B5. Any two entities in E are causally-connected.

Informally, E doesn't "come apart" into completely disconnected components, such as a bunch of isolated universes.

Theorem 4: For any x in E, there is a unique entity f(x) in E such that: f(x) is uncaused, f(x) <= x, and any other uncaused y with y <= x satisfies y P f(x).

Proof: For any x, define a subset E' = {y in E: y <= x, y is uncaused}. Consider any chain of parts S in E' with at least two elements. By B2 there is some z in E of which all members of S are parts. By B3, z must be uncaused (or else some w C z would also be a cause of all the members of S, which would require them all to be equal to w, so S would be a singleton), and by A3, z <= x. So z is also a member of E'. By application of Zorn's Lemma to E', there is a P-maximal element f in E' such that there is no other y in E' with f P y. But then, by B4, for any y in E' we must have y P f; this makes f unique.

Theorem 5: For any x, y in E, f(x) = f(y) if and only if x and y are causally-connected.

Proof: It is clear that if f(x) = f(y) then x is causally-connected to y (just build a path backwards from x to f(x) and then forward again to y). Conversely, suppose that x C y, then f(x) is uncaused and satisfies f(x) <= y so we have f(x) P f(y). This implies f(x) = f(y). By a simple induction on n we have that if x is causally-connected to y, then f(x) = f(y).

Corollary 6: There is a single entity g in E such that f(x) = g for every entity x in E.

Proof: This follows from Theorem 5 and B5.

Done!