there are two questions basically:
1. Is human's thought algorithmic or not?
2. Is human being able to reach the "final truth"?

The first question is arguable. Some people think human's brain is just a powerful Turing machine, the others think it is more than a machine--it has some abilities that a machine cannot have. Roger Penrose had an extensive dicussion about this issue in his famous writing "The Emperor's New Mind". I donot want to repeat it here. We just need to admit that the question is still open.

The second question seems related to the well-known Godel incomplete theorem, which states that any mathematical system which is powerful enough to embrace the arithmetic contains unprovable statements. In light of this theorem, it looks like "no" will be the only answer to the second question. But it is a misleading argument. The second question has nothing to do with the Godel incomplete theorem, because people do not need a "fixed" theory to explain everything. People always change a theory when they have new findings, so that their theories are becoming more and more powerful and form a sequence of theories. Now the second question becomes: what is the limit of this sequence? Is it the final truth or final falseness or not convergent at all?
My opinion is : it is convergent and is convergent to the final truth. It is so even when the human's brain is algorithmic, even when any individual theory of the sequence is recursive, and even when the sequence of theories itself is recursive enumarable. Yes, the "final truth" (which is the theory containing all true statements about the universe) is not recursive, but it is still approachable by a sequence of recursive theory, just like any irrational number can be appoached by a sequence of rational numbers.
The process of people's knowlege evolvement is just like a boy who is counting the number 1, 2, 3 ....
Given unlimited time, can he cover all numbers at any time? No, obviously. Can any number be hidden from his counting for ever? No, either.