Wednesday, 29 July 2020

Heads SRH is true, Tails it is false

"This is a lie" is the classic contradiction. Sometimes Godel's theorems are styled as logical examples of that.

If we use random coins flips to make decisions as I have been for a while we can generate similar problems.

One such problematic decision is whether we should use coins to make decisions at all. We can ask the coin:

Heads: coins are fine for decisions.
Tails: never use coins to make decisions.

Fine if the result is Heads but what to do with the Tails decision. It says never use coins, so presumably we dismiss the result as it was made with a coin. But to dismiss the result is to make a decision based on a coin. So we keep the result which tells us to dismiss it and we are trapped. Its isomorphic with "This is a lie" and "This statement is not provable i.e. is not a member of the set of our logical statements"

SRH is an intuition I have that there was a problem with self-reference which I called the "Self-Reference Hypothesis" although I don't yet know what this is.

It seems that you don't need very strong self-reference to get into difficulties. It is sufficient for the player of "coins" to just ask questions about Coins.

Indeed any decision mechanism which can make decisions about itself has problems. And "itself" just needs to be its name, not "itself" in the complex "self-conscious" ways I was initially envisaging.

To recap. There are many types of self-reference. The loosest is using names. "Statement 5 is false" when put at position 5 in a list becomes a problem. The context of that "world" understands the "5" to mean position in the list. It requires no work "within" the statement. It's all up to the reader and inferred context. But "This statement is false" seems to contain its own context and world. It is always problematic regardless of context. Well not entirely true, the context must be English reading population. Such a statement in a world where English did not exist is just marks on a page and no problem exists. We might certainly decide there was information present if we found it on a beach as marks like that are rarely natural. But with no other context it remains a mystery.

Returning to coins. That decision seems to have us trapped. Every time I think about this I get to the same position something unsatisfying like Russel's Types or Cantor's hierarchy of Infinities. Systems that enable decisions about themselves seem to form a hierarchy. That self-reference like the cross-product generating a statement that points outside.

The point is that to ask the paradoxical question: "Can Coins be used to make valid decisions?" presupposes a context *outside* of Coins. The player is clearly more sophisticated than a machine that just makes decisions by flicking a coin. Such a machine couldn't ask the question. To ask such a question already requires a world in which coins are just a part.

As one sceptical paper on Godel argued: contradictions do not exist in reality. The "real" version of this paradoxical coins question is a machine that blows itself up if it gets a tails result. But obviously "time" is a constant here that is *outside* the question. If we allow the machine to go back in time and stop itself being created on a tails result then the paradox returns. 

It appears that to ask a "self-referential" question of a decision device requires the question to exist in a wider context than the device. If we allow the device to make decisions about the context of the question we run into problems.

There does seem to be some necessary hierarchy with the context being a larger order than (and *outside*) the device.

Returning to the Coin Paradox. The coins cannot make meaningful decisions about the whole of the Coin game. To play the game puts the player in a different place to the coins. The coins know nothing of the world, even of decisions far less they can make decisions. It is up to the user to interpret the result and make the decision based upon the result. And the user is fundamentally and necessarily not a coin. This the paradox is being misunderstood. The coins are not the point. Really the user is asking "should I play coins?" this is something they must decide. Now they could decide to play coins to answer the answer, but the point is they already made the decision to play coins. The contradiction is no more paradoxical than someone flicking a coin and then flicking it again and getting a different answer. The first flick was the decision to play coins actually made by the user, and the second flick said don't play coins. It is not a paradox, but requires understanding the contexts outside the game.

Th paradox occurs if we forget the context and try and understand the game as "self contained". And this was the original insight into SRH (or God Principle as it was also called) that any system exists embedded in a context and so cannot become self-contained and so self-knowing or entirely self-referential within itself.

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