A point in Omnipotence vs Omniscience

In Dawkins' The God Delusion there is an argument showing why omniscience and omnipotence are mutually incompatible. Roughly it goes like this :

If God is omniscient he must be knowing what he is going to do at some point of time (say one year later) in the future. Then he is powerless against changing it. Hence He cannot be omnipotent

As we understand, defining omnipotence and omniscience is a troublesome task at best. And to give it a concrete unambiguous meaning seems impossible. We have tried various tacks of trying to define both the things, none of which works. Ultimately these concepts are ill defined and logically contradictory. Hence they cannot be used to form unambiguously meaningful sentences. So the above proof stands invalid. We are not saying that omnipotence and omniscience are mutually compatible. Just that it doesn't make sense to ask this question. In this respect it is like many english questions which are grammatically correct but meaningless to answer. like - what is the shape of red color ? flaw : shape is not defined for color.

In fact as we have seen here, if at all omniscience and omnipotence should be expected to go hand in hand. But again it is meaningless to explore this thing further.

What is God : Omniscience - C

This article is a follow-up to part B.

Before going further, we show that a K-entity and a entity is one and the same.

Since a K-entity is already an entity by definition, it suffices to show that an entity is also a K-entity. Any entity is supposed to be able to finish a task. i.e, after a finite number of steps the task is over and the result is obtained. So we define a formal system whose axioms are the steps of the task and rules of derivation are the transition from one step to next. Since the task finishes (in technical jargon, terminates) this must constitute a formal system. We also need to include the elementary logical rules too. This system may not be minimal, so we make in into a minimal system. Under this formal system, the performing of the task is same as a proof of the initial input leading to the result of execution of the task. Consequently the above mentioned entity has completed a proof in a system and hence is a legitimate K-entity.

The above argument shows that there is no difference between an entity and a K-entity. This is the reason we defined types of God as G1, G1* and so on. The star versions are the same concepts but related to knowledge. Therefore an omnipotent God is same as an omniscient God and subject to the same contradictions. But we will spell out everything in detail and leave no room for doubt. We have already seen that G1 and G1* Gods are contradictory. Later we will see that G2, G2*, G3, G3* are also contradictory but that needs more work and we will revisit this issue after learning some more things.

For now let us try a different tack for defining God. Instead of defining God to be an entity, we define him to be knowledge itself. It is not too far off course as this is one of the glib interpretations of omniscience. This is tantamount to defining God to be a Formal system rather than somebody who works with the system. Let us call such a God as G4. The question is which system is G4? Since we have many (far too many) systems, all of them cannot be God. Think about the elementary examples we considered in the earlier part. This is not what we intuitively understand by God. So we consider some stronger systems. Any God worthy of his name must be at least capable of doing arithmetic and be able to prove arithmetical statements (or at least understand the proofs if presented). This simple requirement actually poses the most formidable difficulty.

As proved by Gödel, any system which includes ordinary arithmetic cannot be both consistent and complete. This is referred to as the Incompleteness theorem. Meaning, however many axioms we include in our system, there will necessarily be left out statements which will neither be false nor true in the system. So the system can always be enlarged by adding one more axiom and there is no canonical choice of any largest system. That makes the choice of G4 very difficult. On the one hand we want to include all the arithmetical theorems in our system and on the other hand we want G4 to be as large as possible (so that God comes closer and closer to 'omniscience'). But whatever system we choose as G4, there are competing systems which are as large and also many (far too many) which are strictly larger. This dilemma is akin to choosing the largest natural number. Whatever number you choose, there exist a (and infinitely many of them) larger number. Hence, G4 also becomes non-existent. In hindsight we can see that G1* was a defined to be a k-entity in possession of the formal system G4. Since G4 itself is problematic, G1* too cannot exist. So we see that all the grandoise talks of God being 'Pure Knowledge' is just vacuous.

Actually even if G4 existed, it makes little sense to worship a list of axioms. And in any case, this is not the popular understanding of God. As we have seen till now, the ontological status of God seems problematic and is getting mired into more and more contradictions. It leads us to suspect whether God is just a psychological construct. Or worse : an illusion constructed by the way our brain works. Not unlike the illusion we have of sun moving round the earth or the desert mirage. See this for a good example of an optical illusion :
If you move your head towards the screen while looking at the center black dot, the two outer circles seem to rotate.

If the above is true, then to be an atheist is to be free of precisely such illusions. In the future we will also discuss the emotional / psychological apologies for God / religion and examine whether religiosity is really needed for our life. Whatever we conclude, our motto is not to take any thing on faith, not even atheism. Whatever stands up to rigorous scrutiny will be our stand.

In the next article we will take a break from this grind of God and take a look at the real world.

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.

What is God : Omniscience - A

Apart from Omnipotence, the other putative property of God is Omniscience. Let us try to see how valid this is. To be omniscient is to know everything. We would require any knowledgeble being (entity ?) to know of *true* things and only true things. It doesn't make much sense to know of false things and certainly not from God. That brings us to meaning of *truth*. But here is a problem.

Strictly speaking there is no such thing as absolute logical truth. Truth only has a conditional status. Every statement can be analyzed for its truth value, which rests on *accepted* truth of some earlier statements, which has similar status themselves. It boils down to *truth* of some irreducible statements, which are accepted as true. These statements are called Axioms. Axioms are defined to be true beyond question, they need not be self-evident, realistic, intuitive or even simple but most of them are. Based on these axioms, certain statements are analyzed for truth value and thereby declared (true) Theorems if found to be derivable from the axioms. A Proof (or Derivation or Deduction) is a sequence of logical steps (which themselves have axiomatic status but are self evident) by which a set of axioms lead to the given statement whose truth is being analyzed, and then that statement is considered to be Proved. The axioms themselves cannot be arbitrary, but a strong condition of Consistency is being imposed, i.e a self contradictory statement should not be a proved theorem. We also require the set of axioms to be mutually Independent, i.e not derivable from each other. A classic and typical self contradictory statement is "The umbrella is black and not black". A consistent set of axioms and all of its (or some) theorems is said to constitute a Formal System. The word 'formal' should warn readers that *truth* is decreed formally and has no other justification. As I clarified in my earlier post, I am not being pedantic in definitions, just careful enough without compromising readability. Let me illustrate Formal systems with some examples. The examples are from Physics so that things seem familiar to many readers.

1. Classical mechanics (with the Newtonian Gravitation) as propounded by Newton, and formalized by Lagrange is a typical Formal System. Force, potential energy, kinetic energy, mass, velocity, momentum etc are the elements of the System either defined or axiomatically decreed. The laws of motions (aptly called laws, as they must be accepted without question) and Gravitation are the axioms and rules of derivation are elementary logical rules plus the mathematical formalism of calculus. The theorems of this system are the physical predictions viz, keplers planetary laws of motion, orbits of asteroids, etc. We distinguish this formal system from the physical world it is supposed to model. The formal system is no more true or false if it fails to describe the physical world, it is just a model after all. If some real phenomena is not described by this model, we try to construct a new Formal System (or model) which would be appropriate for our job.
2. If we impose a limit on velocity (in this case that of speed of light) that any body may achieve, it gives us the Special Theory of Relativity. Among many of its astonishing predictions are time dilation, length contraction, etc. Again this is just a model and is not required to correspond to physical reality. But if not, we would abandon studying this model and find a better a model instead. This is the usual paradigm of Science. Experimentation / observation lead to more data, which helps in refining (or renewing) the model which suggests experiments in a particular direction to refine the model even more. This is how science progresses.
   
3. Special relativity along with the principle of equivalence and some more laws (essentially imposing speed limit in accelerated frames as well as inertial frames) gives us the General theory of Relativity.
4. All the above examples are conventionally considered classical physics. Quantum mechanics is a system which is obtained by changing much of classical physics. The postulates are counter-intuitive as well as its predictions (theorems). Nevertheless till date no experiment has been found to contradict these predictions. It is beyond the scope of this article to describe the axioms. I can do no better than to suggest you an expert, who is also a highly skilled expositor. Please refer Mathematical foundations of Quantum Mechanics, K.R. Parthasarathy (Hindustan publishing, TRIM-35) to study the formal system.
   

Actually in the above examples, to clearly identify the axioms and rules of derivation is a tough task and requires extreme originality. But then who can say Newton was not a genius of the highest rank ?

As such to decide consistency / independence of any Formal System is also a highly non trivial task and Category theory is the usual playground of such considerations. In mathematics, one usually plays in a universe called ZFC, the usual formalism of set theory. Many professional mathematicians leave such task to professional logicians and are content that consistency is guaranteed. As we will soon see God can have no such luxury. Historically this type of understanding came much later. We already had a System (in mathematics) and nobody seriously doubted it. Only after Cantor's seminal work on infinite sets, the need to explicitly spell out the axioms arose. Let me give you a quick example of a toy non-system.

The undefinables are umbrella, black, cloth, cover of an unbrella, color of a cloth. First axiom : cover of an umbrella is made of cloth. We define black to be color. By color of an umbrella we mean color of the cloth of its cover. Second axiom : the color of all umbrellas is black. Third axiom : the color of all cloth is not black. Clearly it implies that all umbrellas are black as well as not black. This contradiction violates consistency requirement and so cannot be a formal system.

The above discussion makes it very clear that there is no unique Formal system. We shall see many more examples too. Changing an axiom leads us to a new system and and any omniscient being is supposed to *know* all the Formal systems. Just what could this preposterous idea might mean ? We also need to undertand to the concept of *knowing*. We will see this in the next article.