Abstractions in general and mathematical abstractions in particular are much better described as ‘items’ than as ‘objects’. The advantage is that it is on one hand much nearer to how abstractions are treated by mathematicians and on the other hand no issue of existence arises. The whole issue of Platonism arises from the treatment of abstractions as objects without having any reason to do so. This leads immediately to the Tausendfüßler-Paradox: mathematicians create their own world exactly in the same way in which a child creates an own world when listening to a fairy tale. For the child there is no difference between fiction and reality. The fictional story is perceived as ‘real’ and there is nothing wrong with this. Every request to speak about the story, to explain the story will pull the child (the mathematician) out of the fiction. When the child matures, it is less and less able to ‘dive’ into the fictitious world of a fairy tale. Of mathematicians (and, by the way, also of musicians) another development is expected – they should retain as much as possible their (childish) ability to dive into a fictitious world. A multitude of such ‘worlds’ can be created from (mathematical) abstractions by interpretations.
The ‘characters’ of such a story of fiction, which might be called a ‘piece of mathematics’, are ‘identifiable items’ and not ‘existing objects’. By the way, it is very difficult to avoid becoming immediately linked to ‘social constructivism’ with such a line of thoughts although it is not so clear how this ‘diving’ into the story by a child or by a group of children should be classified.
How does this relate to the physical world? What about Wigner’s ‘unreasonable effectiveness’ of mathematics in physics? Maybe it is in the process of abstraction. Mathematical items are derived from properties of a physical object or of a conglomerate of physical objects. Why should such a fairy tale, located in the fictitious world of such abstractions not lead to something that can be translated back as a hypothetical property of a physical object? Maybe this is the way the whole enterprise of science works? But then the effectiveness of mathematics is reasonable as much as abstraction is reasonable and as a consequence – as much as every kind of representation including language is reasonable.