“The abstract structure that all progressions have in common”
Hello! I’ve been terribly caught up with research lately, but the next several posts on categorical logic are in the pipeline, and will surface sooner or later. Until then, here’s a nice quote about representation independence in mathematics.
Therefore, numbers are not objects at all, because in giving the properties (that is, necessary and sufficient) of numbers you merely characterize an abstract structure—and the distinction lies in the fact that the “elements” of the structure have no properties other than those relating them to other “elements” of the same structure. [...] To be the number 3 is no more and no less than to be preceded by 2, 1, and possibly 0, and to be followed by 4, 5, and so forth. And to be the number 4 is no more and no less than to be preceded by 3, 2, 1, and possibly 0, and to be followed by…Any object can play the role of 3; that is, any object can be the third element in some progression. [...]
Arithmetic is therefore the science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions. It is not a science concerned with particular objects—the numbers. The search for which independently identifiable particular objects the numbers really are (sets? Julius Caesars?) is a misguided one.
— Paul Benacerraf, What numbers could not be (1965)