Some big thinkers have gotten their reputation for bigness by saying things are not the way people think they are. For example: Socrates said everything we think might be the definition of “courage” is wrong, but he proposed no alternative, better definition. Others (Kant, Nagarjuna, the Upanishads, the pseudo-Dionysus) have done their work by saying no as well.
A worry I’ve always had is — what’s the difference between negating a sentence and affirming a sentence? Why should it be deep? Take a simple case: I have written an integer from one to three on a paper and not shown it to you. If you ask “Is it 2” and I say “No” you have learned that it’s either 1 or 3. If you asked “Is it 1 or 3” and I said “Yes” you would have learned exactly the same thing. So in this case negation and affirmation supply precisely the same information.
Why then do thinkers like Socrates et. al. seem deep?
Imagine a different game. I have told you that I have hidden a pawn on a chessboard which you do not see. If you ask me “Is it on square A1” and I say “No” it might seem that I have told you the same as if I had affirmed the pawn is on one of the other 63 squares of the chessboard.
However that’s not true. In fact, in this example, it is a very, very tiny pawn and I have placed it on the black line that separates square A1 from square A2.
Negation may negate our accustomed categories in other words and point towards a solution that we have not imagined yet. You assumed in this case that the pawn hidden on the chessboard had to be on one of the 64 squares. But you were wrong to do so. The negation of your guess — A1 — was correct. But the affirmation that you thought was logically equivalent to it — it is on one of the other 63 squares — was also incorrect.
Of course if the negator offers us literally no help at all at imagining these other possibilities he is not so helpful. But he’s still right.