Thursday, August 16, 2007

What is God : Omniscience - B

This article is a follow-up to part A.

To 'know' something is to know it to be 'true'. Being 'true' means true in some Formal system. That means there is a formal system to which the statement refers to and is derivable from its axioms by a sequence of steps. Which means, the given statement is a proved theorem in some formal system. Which formal system is immaterial (it may be implicitly assumed, but once at least we should do the exercise of concretely identifying the axioms and rules) and it also doesn't matter whether the supposed statement is elementary or involved. We only want it to be a theorem to say that it contributes to our knowledge. Also a related point is the likelihood of 'truth'. For this we incorporate the usual rules of probability in our formal system and demand the probability of the said statement being true given all the data.

Motivated by the above discussion, let us define an K-Entity to be an entity who can prove a theorem of some formal system (K stands for knowledge). Recall (from the omnipotence article) that an entity is supposed to be capable of carrying out a task. Also recall that a task is a sequence of well defined steps explicitly spelt out to the entity. By our definition any computational program such as Matlab or Mathematica is a K-entity, so are all the humans and (supposedly) God too. Knowledge of a k-entity is the collection of all the statements he has proved till date. The true statements that one 'knows' must be a part of her/his knowledge indeed. A is said to More knowledgeable than B if A's knowledge subsumes that of B and is strictly larger. As is clear we are allowing one's knowledge to increase with time. Before going further we will see some examples of 'proof' required of somebody to be a K-entity. We will also see later that a K-entity is actually no different than a entity.
  1. Our formal system consists of 1, 0, + and =. The axioms are 1+1=0, 1+0=1, 0+1=1. So let us try to see some of the theorems of this system. 1+1+1+1=0 is a theorem as ca be clearly seen by repeatedly applying either axioms. Similarly 1+1+1=1 is also a trivial theorem. And anybody who can perform this task is a K-entity according to our understanding.
  2. We include some more symbols now. Let us have 0,1,2,+ and =. The axioms are 1+1=2, 1+2=2+1=0, 0+1=1+0=1, 0+2=2+0=2. It is easy to see that 2+2=1 is a theorem.

  3. In real world, we deal with physical evidences. Let us say a piece of skull is found at a archaeological site with cut marks on its surface. Also let us say that by radio carbon dating, its age is estimated to be contemporaneous to stone age. This is a probabilistic situation, and all we infer from this is the likelihood of that person being butchered by its fellow human beings and possibly cannibalized. But alternate interpretations are also possible and wherever possible the likelihood estimates can be done too. All this may be done by a computer or a human being, both of which are K-entities. Here it is cumbersome to pinpoint the exact formal system, but we assume the truth of physical laws, and also some earlier estimates of similar sites. This formal system is very special and we will need to explicitly elaborate it. I will come to it in a future article.
The above two examples illustrate the essential common-sensical nature of our definition. Also in the above examples the symbols 1,2,0,+,= are used just because of ease of typing and familiarity. You may use any symbols which you may as well have designed yourself, it is totally superfluous, and doesn't carry any meaning. The theorems are arrived at just mechanically. in practice for a complicated formal system such as Algebraic Topology, it may require considerable ingenuity to prove theorems of interest. But just to be a K-entity, it doesn't take much effort.

Having done all this, now we ask the following question : since God is supposed to be omniscient, exactly what is meant by omniscience in the above context ? Intuitively, God's knowledge must include all the true statements possible. This implies one of the three things (in cases I and II God is supposed to be infinitely knowledgeable) -

  1. God is in possession of a 'divine' system which tells him proofs of theorems of all the possible formal systems. We will define such a God as type G1*. If we demonstrate the existence of two formal systems which are mutually incompatible, then these two systems can't be part of a larger single formal system. Hence, God's ultimate weapon (the 'divine' system) cannot exist. It is a routine task to come up with such examples. But historically such a thing has seen much acrimonious debate, most notably between Hilbert vs Brower. Category theory provides the usual framework to settle such issues. This is a issue of completeness of a formal system, which we discuss more fully while considering G4 in the next article. Suffice it to say that the idea of a unique formal system giving all the true statements is logically absurd hence impossible hence G1* is non-existent.

  2. He picks one by one every possible formal system and diligently proves all its theorems and consequently 'knows' it, moreover he must already have done all this analysis (maybe much before the universe began, else what use as a God He is) as his knowledge includes all the true theorems. We define such a God as of type G2*. This gives God a quality of timelessness or eternalness. We will have occasion to deal with eternalness and a related concept omnipresence in future. For the moment we are leaving this issue and will come back to it.

  3. He is in the process of proving all the true statements and will eventually complete the task. We define such a God as of type G3*. This tells us that God is essentially no different from us. We are also in the process of finding all the true theorems and 'potentially' can find it. However we may not uncover all the truth in any conceivable time but God may do it. So we will also deal with God 'eventually' completing the task (of knowing all true theorems) when we deal with 'eternalness' of God and see how this leads to contradictions.

So is God really not omniscient or are we missing something ? None of the above three cases are really tenable. These cases actually rule out omniscience as a property of any entity, God or otherwise. However there is still one way for 'something' to be omniscient. But here we run into Godel's incompleteness and will see this in the next article.

Note : Observe that G1* is analogous to G1 (defined here). G2 and G3 types are defined here.

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