Do human beings search for questions that cannot be answered on purpose, or does it happen by chance?

Hmmm...so do you think this question is also unanswerable? How do you know the questions you refer to are actually unanswerable (as opposed, say, to be not yet answered)?

At the beginning of Aristotle's Metaphysics, he says that philosophy begins in wonder. It seems pretty clear that the human race is strongly inclined to this activity--wondering. If sometimes we wonder about questions that actually can't be answered, I doubt if we got there on purpose. When we wonder, it seems that we seek answers!

Anyway, I would humbly propose to you that neither you nor I are in a position to say that some (wonderful) question is actually unanswerable--so far unanswered, perhaps, but not necessarily unanswerable.

I suppose there are some questions that are unanswerable (or at least some that would need further specification before being answered)--for example, if I were to ask now, "Is it?" (Is it what? What "it"?) Similarly, there are questions that cannot be answered in the form they seem to require (presumptuous questions), for example, "Have you stopped kicking your dog?" asked to someone who either doesn't have a dog or never kicks the one he has. But even these are answerable ("I don't have a dog"), just not in the terms they seem to require.

I'm inclined to think that if a question really were unanswerable, it wouldn't be a very interesting question...

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