Non-existent mathematical objects
An amusing passage from a recent paper by Erik and Martin Demaine on the hypar, a pleated hyperbolic paraboloid origami structure:
Recently we discovered two surprising facts about the hypar origami model. First, the first appearance of the model is much older than we thought, appearing at the Bauhaus in the late 1920s. Second, together with Vi Hart, Greg Price, and Tomohiro Tachi, we proved that the hypar does not actually exist: it is impossible to fold a piece of paper using exactly the crease pattern of concentric squares plus diagonals (without stretching the paper). This discovery was particularly surprising given our extensive experience actually folding hypars. We had noticed that the paper tends to wrinkle slightly, but we assumed that was from imprecise folding, not a fundamental limitation of mathematical paper. It had also been unresolved mathematically whether a hypar really approximates a hyperbolic paraboloid (as its name suggests). Our result shows one reason why the shape was difficult to analyze for so long: it does not even exist!
So the hypar joins the ranks of phlogiston, the planet Vulcan, the largest prime, or the quintic formula: objects of inquiry that turned out not to exist.
Ordinary language allows naming and quantifying over such objects, which suggests that they must have some kind of existence or being after all. Similarly, familiar systems of quantified modal logic entail that whatever might have existed (such as Vulcan or the hypar) actually exists. Since in fact none of these objects exist, we should reject the relevant systems of modal logic, and take care not to draw metaphysical conclusions from intuitive statements of ordinary language.
Metaphysicians often go the other way, claiming that phlogiston, Vulcan, along with Sherlock Holmes and Emma Woodhouse, all exist. On this view, when we allegedly discovered that Vulcan doesn't exist, we really only discovered that Vulcan is not a real planet but rather some kind of abstract object.
What about the hypar, the largest prime, or the quintic formula? These things were never meant to be concrete, so it is hardly reassuring to be told that they are real but abstract objects. Moreover, can't we prove that they don't exist? What's wrong with these proofs?
One might try to draw a distinction between Vulcan and the hypar and say that non-existent mathematical objects are genuinely non-existent. But that looks like a dangerous position, since most of the arguments that support the existence of Vulcan also support the existence of the hypar.