"There can be no integers which are both even and odd", sure. Such a number cannot exist because the properties can not be co-instantiated.
> Well the property of interest is having x^2 == 2.
Sure, but that property is possessed by some x. The interlocutor in the article seems to resist not that this equation could be satisfied, but whatever satisfied it wouldnt "count" because it didnt possess some touchy-feely "existential" property.
cf., at the end where concession is only given to defining existence within some system as opposed to some prior intuition of "Existence" which was being denied to sqrt(2).
My point is that whatever this intuitive notion of existence is it's basically unhelpful. It is embedding properties into the very criterion of there-being-a-quantity of something.
As in, it is not merely good enough that an x can be defined such that x^2==2 but that it must also possess the Property of Reality (whichever that might be, eg. spatial location, abbacusy, ..).
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.
His scepticism seems deeper. It isnt merely that he thinks that nothing satisfies that equation, he seems to in addition think, that even if something does it wouldnt count as "existing".
If the objection were merely that "x^2 == 2" isnt satisfied, then I agree, that is equivalent to saying "x does not exist" and it would be appropriate to say that.
I think that's just the only way to express scepticism about the existence of a solution to x^2 == 2 in a world where any schoolboy can tell you x = 1.414...
Well the property of interest is having x^2 == 2. If x^2 isn't 2 then x isn't in any meaningful sense "the square root of 2", and it's much clearer to say "the square root of 2 doesn't exist" than "the square root of 2 squared doesn't equal 2".
> I guess the hypothetical objector in this article wants numbers to possess the property of "being easily exemplified by stones" (or whatever), in which case sqrt(2) doesn't have that property.
> Why it should be that "2" exists because it can be exemplified by fingers, and sqrt(2) doesnt because it cant be -- is a bit strange.
By that logic you wouldn't be able to make any statements about general properties of numbers. E.g. we like to say that all integers are either even or odd. But what if I define a new integer, let's call it Banana, that is both even and odd. It's a lot more productive and clearer to say "Banana doesn't exist" than to talk about Banana not having particular properties.