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.

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.

- 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.
- 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.
- 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.
- 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.

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.

## No comments:

Post a Comment