Friday, July 24, 2015

The Barber who did not shave himself

In a somewhat weird town there was a group of barbers who followed a simple rule - they shaved only those who did not shave themselves. Now let us make an assertion (which can either be true or false) - "There was a barber in the group who did not shave himself". Well possibly. What is the big deal about it? If the above statement is true, then by the definition of collection (that barbers only shaved men who did not shave themselves) the barber must shave himself. But also, no barber in the collection can shave himself. The statement can neither be true nor false. Something is seriously wrong.

In early 20th Century, Bertrand Russell's paradox, outlined above, lent a severe blow to the philosophy of mathematics that was trying to find its footing among established disciplines (or find it again after Archimedes, Plato etc) .

Looking at the problem more closely a bit of help from mathematics might come handy. The “group”” as used in the paradox mentioned above is often referred to as a SET. A Set of things is different from the thing itself. A SET of humans is not a human. Similarly a SET of humans in love is not the same as a human in love. There is simple way of explaining this idea in mathematics (precisely Set Theory).

If x is free variable (Humans for example) and there is a property λ(Pronounced Lambda)  limiting it (Humans in Love, African Elephants, North Indians in Bangalore etc). Then we can define a set

A = {x:x is in love}. This would be read in English as A is a set of humans who are in love. Or more precisely.

A = {x:λ(x)} where λ is a property which means - " Is in Love".

I used hate those braces "{}" in set theory, but I guess she is like an elegant and exotic woman who wears braces to ward of insincere suitors.

Back to What Russel pointed out

Let us define a set R where R is a set of things that not members of themselves. In mathematics it would loosely translate as

R = {x: ~λ(x)}

Where λ(x) stands for  x x . The ~ at the beginning of the property denotes the inverse. If λ(x) means a property where x belongs to x then  ~λ(x) would refer to a property where x does not belong to x.

If R defined above is a set of things that are not members of themselves and we start with the assertion, is R a member of itself? If R is a member of itself, then it must satisfy the condition of not being a member of itself so it is not. If it is not then it must not satisfy the condition of not being a member of itself, and so it must be a member of itself.

Would it be fair to say that such set like R cannot exist? Why not? Well it obviously leads to a contradiction, but that is none of R's problem. What is a set that is not a member of itself. If we look at a set of all things. All animals , plants, bananas, apples, Macs, iPods, men, women, mountains, Andromeda galaxy, Space, time and everything else that senses can perceive or cannot perceive. When such a set is created it should have as its members all that has been, is and will be. Even that which could not be. That would be everything, but  everything is also something so this set should contain itself.

For no mathematical or philosophical reason, I would like to name this set as ∞.

= {x:x is everything}

But a set cannot be a member of itself. If there is something outside the set then
∞ is not a set of everything.

How can infinity be contained within itself. If it cannot then there is something outside
infinity so obviously its not infinity. Indian philosophy terms it as Infinity's constraint.
Just being infinite, you have to be infinite in infinite number of ways, constantly evolving andchanging in dynamic infinity.

In Indian philosophy there is an apparent reconciliation of the problem posed by Russel. What if there are two infinities that can mutually exist without making the other impossible? What if there is an infinite that is also a null set, that is zero that is not manifest. That infinity can coexist with the manifest infinity. You could take out of infinity from the infinite that is not manifest but it will still remain zero. As the sage Kapil goes on to explain to his mother the theory of Samkhya, he says that the manifest infinite is nothing but knowledge (Vedas) defining tangible things out of infinite that is not manifest.

As Tulsidas puts it,

नेति नेति कर वेद पुकारा !

When Vedas tried discovering the non manifest infinite, they looked at everything they discovered in the process and cried out - ït is not this, it is not this. The universe (The manifest infinite was created in the process).