I take your point about mathematics, its pedigree is long and good, but John Barrow, "The Infinite Book" makes an interesting point about mathematical "existence": “Cantor’s most dramatic discovery was that infinities are not only uncountable, they are insuperable. He discovered that a never-ending ascending hierarchy of infinities must exist." ..... "Before we see how he did that, it is important to say a little about the meaning of the word ‘exists’ in this context. We are used to using the word on an everyday basis without any ambiguity. ‘Cambridge exists’, ‘inflation exists’, seem to be assertions that are clear enough. They are about physical existence. Up until the early nineteenth century, mathematical existence was rather similar. Euclid’s geometry existed because it was manifested in the physical world. Indeed, it was believed for thousands of years that there could not be another logically consistent and complete geometrical system. The discovery of non-Euclidean geometries which described the topography of curved surface changed that view. Gradually mathematicians lighted upon a new concept of existence. Mathematical ‘existence’ meant only logical self-consistency and this neither required nor needed physical existence to complete it. If a mathematician could write down a set of non-contradictory axioms and rules for deducing true statements from them, then those statements would be said to ‘exist’.”
This seems to suggest that mathematical existence, or reality, might not necessarily be the same thing as physical existence or reality.
