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Some say I'm impossible,
So try me if you dare.

Some things, before you tackle this beast:
Before me even old men have ceased.

Liars I have found defeated,
Countless animals I have also cheated.

Don't point arrows at me,
(You won't get anywhere anyway)
Don't forget I can be constructive,
As much as I am destructive.

Try as hard as you might,
You won't find the answer to me,
For then I'd surely be no more.

What am I?

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Perhaps you are a

paradox.

Some say I'm impossible / So try me if you dare

A paradox is something that appears impossible.

Some things, before you tackle this beast: / Before me even old men have ceased.

Perhaps a reference to the "grandfather paradox" that apparently forbids time travel? Alternatively, "old" may be "a long time ago" rather than "aged"; some of these paradoxes go back a long long way.

Liars I have found defeated,

A reference to the Epimenides paradox or liar paradox ("this statement is false").

Countless animals I have also cheated.

Not sure what this one is about. Perhaps just the fact that several paradoxes feature animals? (Buridan's ass, Hempel's ravens, the proof that all horses are the same colour, etc.)

Don't point arrows at me, / (You won't get anywhere anyway)

A reference to Zeno's paradox (an arrow shot from A to B has to get half-way from A to B, then half-way from there to B, etc., so how can it ever reach its target? Indeed, before getting half-way from A to B it has to get half-way from A to there, etc., so how can it even get started?)

Don't forget I can be constructive, / As much as I am destructive.

Not sure about this; perhaps the idea is that sometimes a paradox can lead to enlightenment (by forcing you to find a clever way out or -- see e.g. koans in Zen Buddhism -- by bypassing rational thought altogether).

Try as hard as you might, / You won't find the answer to me, / For then I'd surely be no more.

A paradox ceases to be a paradox if a convincing resolution is found.

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  • $\begingroup$ Yep, you've got it! (I'll leave the rest for you to explain) $\endgroup$
    – Inazuma
    Jun 5, 2016 at 13:57
  • $\begingroup$ Is the line "Countless animals I have also cheated" refering to Achilles and the tortoise maybe? And I'd say that the constructive/destructive line might refer to Russell's paradox, as it was constructive since set theory was built upon it (Zermelo Fraenkel set theory), and destructive (it blew up Frege's previous work). Those are only guesses anyway. $\endgroup$
    – user314159
    Jun 5, 2016 at 18:35
  • $\begingroup$ Yup, could be Achilles and the tortoise -- though it's Achilles rather than the tortoise who gets cheated there. And i agree that the history of set theory provides a nice example where a ------- is constructive because it leads to a clearer and more rigourous understanding. $\endgroup$
    – Gareth McCaughan ♦
    Jun 5, 2016 at 21:22
  • $\begingroup$ Yep, the 'animals' was referring to them all. Originally I had Shrodinger's cat/ Achilles's tortoise, but I thought explicitly mentioning individuals might give it away too easily. Set theory was definitely one of the constructive ideas I had in mind, and old men did refer to the Grandfather Paradox. Well done again! $\endgroup$
    – Inazuma
    Jun 6, 2016 at 1:29
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Are you a

Lie

Before me even old men have ceased.

Liars I have found defeated,

Countless animals I have also cheated.

Everybody lies.

Don't point arrows at me,

You're the one who started this.

Yet I can be constructive,

As much as I am destructive.

A well constructed lie can easily become a truth. It could be used constructively and destructively.

Try as hard as you might,

You won't find the answer to me,

For then I'd surely be no more.

If we try the answer to a lie, you are looking for the truth. If you have the truth, you have no lie.

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  • $\begingroup$ This is somewhat along the right thinking, but not the answer. $\endgroup$
    – Inazuma
    Jun 5, 2016 at 13:47
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I don't much like this answer -- it seems too prosaic and there's one bit of the riddle that doesn't seem to match -- but you could possibly be

death.

What is (perhaps) impossible is

defeating death (though I also have a feeling some eminent philosopher -- maybe Socrates as portrayed by Plato -- argued that death was clearly impossible because souls are naturally immortal, and therefore life must somehow continue after bodily death).

Old men, liars and animals

all die (as for that matter do young women, truth-tellers, and plants).

I don't have a good explanation for "Don't point arrows at me, you're the one who started this". Constructive/destructive:

the process of evolution depends on a whole lot of death, and if death were somehow abolished without a lot of other radical changes in the world we'd all rapidly be in a lot of trouble.

"You won't find the answer to me, for then I'd surely be no more":

If we found an "answer" to death, we would stop it happening.

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  • $\begingroup$ I like your reasoning, but this isn't the answer. $\endgroup$
    – Inazuma
    Jun 5, 2016 at 13:49
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    $\begingroup$ Good! I bet the correct answer is more interesting than mine :-). $\endgroup$
    – Gareth McCaughan ♦
    Jun 5, 2016 at 13:51

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