Would you say that there exists something in common between the 21 sheep and the 21 dry peas in the shepherd's pocket which he uses every evening to see if any of the sheep have been lost? That common which exists, in a very clear and practical sense, is called a Number.
No, I would not say that there exists something in common between those. Cardinality is an emergent property of a set, but that doesn't mean that cardinality exists on its own.
Did Plato ever make the argument that because his Shapes have corners, edges and angles, that therefore corners, edges and angles exist in the same way as the Shapes? I don't recall if he ever described Shapes in those terms.
What does it mean for something to exist then? Does an apple exist in the same sense that a number exists?
There is something in common between two chairs; there is something in common between a red ball and a red book; there is something in common between 21 sheep and 21 dry peas. It is _emphatically_ unclear that each of these three statements of 'existence' have anything at all to do with each other except in the most intuitive sense - that they refer to things which are as opposed to are not. At some point the bar for existence becomes so low that Harry Potter exists - really exists - because we can conceive of the idea of him. The question of whether numbers 'exist' is more than that.
Language, having evolved over tens of thousands of years of the human practical use, is often smarter than us, and where we are forced to say that something exists, it usually does. It means that more often than not something that we describe in words is not merely an idea but has an objective meaning, i.e. has to do with reality and therefore exists, independently from from our minds. Thus not only “objects” exist, but also their properties as well as relations between them. A tip of the needle enjoys the same objective existence as does the needle itself, and this fact is easy to verify by sticking it in you finger. What other existence do you want?
