Bram, that's a slippery topic. If we talk about Model Theory we have a formal framework where interpretation or model is defined in terms of mathematical objects. But existence and interpretation in the sense you seem to use are another issue, which I am sure you know, only we have different opinions on this.

There are first-rate specialists in Foundations of Mathematics that doubt we humans have any clue of what the natural numbers actually are. We take some things for granted intuitively, but the model in one's head is likely to be different from the model in any other person's head, for they are inherently vague.

Some years ago there were some interesting threads in the FOM mailing list regarding these issues, I remember Professor Sazonov was one thinking in that line. This message from 1998 addresses it somewhat but there were more.