GODËL'S INCOMPLETENESS THEOREM AND AL-GHAZALI'S DOUBT AT CAMERON HIGHLAND

I have always been fascinated with Kurt Godël's Incompleteness Theorem since it was first introduced to me by Arash (the same Arash who taught my maths; my senior at Imperial College).

I am fascinated to it because I believe, somehow, the theorem is related to Ghazali's doubt. I believe somehow the former originated from and was the answer to the latter.

What is so great about this theorem?

The theorem is so great, Stephen Hawking himself used this theorem to denounce his attempt to derive the Theory of Everything; a theory that describes completely the universe. In other words, Hawking himself has confessed that a complete knowledge about the universe is impossible because Godël's theorem said so.

What is this theorem actually?

The true understanding towards Godël's Incompleteness theorem can only obtained through rigourious and difficult mathematical logic.

But it is sufficient for laymen like us to understand it as follows.

The theorem says that, any statements, facts, formulae hence knowledge always requires more fundamentals statements, facts, formulae to explain it.

For example, we understand 4x4 equals 16 because we understand in prior what 4 is, what multiplication is, what equal is and what 16 is.

And in turn, we understand what equal is because we understand what not equal is and so on.

But there will be a point where "we dont know anymore", "we are not certain anymore", "we cannot distinguish between true and false anymore". This point will be the "end" or the "collapse" of our knowledge.

The best way to describe this point (where our knowledge is made limited) is by considering the following paradox.

Lets say we are given a statement and we are required to write a condition code (if, else) to decide whether it is true or not. It will be sbown that, for the given paradox, our algorithm (if,else) will fail to decide.

Consider the following statement:

"I am lying"

As mentioned, the question is whether it is possible to determine (or decide) whether I am lying or not?

If I am really lying, which implies that the above statement is true, then it is false because I am lying.

Can you see the paradox?  Can you see that, if we write such an algorithm, our computer will end up with infinite loops or infinite iterations hence can not decide. The running will not end up actually.

So this is what Godël's incompleteness is, in laymen terms.

But this attempt about trying to identify what is true and what is false hence what is reality actually really sounds like Ghazali's doubt and Godël's theorem seems to be the answer to it. It confirms the proposition of Al-Ghazali that we can never know the truth through our five senses since we can always argue (doubt) everything.

This is why I am so fascinated to the theorem. But the mathematics is too deep and involved, I dont think I will able it to grasp it in the way it is supposed to be understood.

We are still limited by 24 hours a day and 7 days a week, remember? I still got my "engineering maths" to finish :)

AIRIL
- 3 June 2016, 5.53pm.
- Written at the balcony of Copthorne Hotel, Cameron Highland.