What makes you so sure that having x^2==2 can be instantiated? What about, say, "an even number greater than 4 that cannot be expressed as the sum of 2 primes"? It's ridiculous to say that that exists and we're arguing over whether it has some property or not; far more natural and practical to say that it's an open question whether an integer like that exists.

> Sure, but that property is possessed by some x.

No, you can't say that until you've proven it. If you just assume that such an x exists, that might turn out to be just as incoherent as assuming that an integer that's both even and odd exists.

There's a mathematical urban legend about the Journal of Foo Manifolds; these were manifolds defined in some particular way, and proved the object of much interesting study, so eventually got their own journal. In particular, mathematicians wondered about the Bar property, and published various incrementally improving papers about what kind of Foo Manifolds did and didn't have the Bar Property.

Eventually, a paper was received proving that all Foo Manifolds had the Bar Property, and another proving that no Foo Manifolds had the Bar Property. Both were carefully checked and found to be valid. The final issue of Ruhr Journal of Foo Manifolds consisted of both papers and an announcement that it was closing down.