Fictionalism, and other formalist theories, will have to confront that problem that when we speak of mathematics as abstract rules governing strings of symbols, these rules themselves are mathematical.  So it only replaces &quot;numbers are real&quot; with &quot;abstract symbols are real&quot;.  There are axiomatic systems that are strong enough to express manipulations of abstract symbols, but weaker than the usual systems that mathematicians deal with (e.g. see the work of Edward Nelson on so called predicative arithmetic).  But to my knowledge these have not been explored much.