Quote:
Originally Posted by Aaron W.
Do ideas have influence in the universe? That is, if I decide to kick the dog and then I kick the dog, did the idea to kick the dog have an influence on my kicking the dog?
Ideas and mathematics are not objects since all objects have non-structural properties, but ideas and mathematics do not.
There is nothing more to being the number 3 than having certain
intrastructurally defined relational properties, such as succeeding 2, being half of 6, and being prime. No matter how hard we study arithmetic and set theory, we will never know whether 3 is identical with the fourth von Neumann ordinal, or with the corresponding Zermelo ordinal, or perhaps, as Frege suggested, with the class of all three-membered classes (in some system that allows such classes to exist).
Therefore, numbers are not objects at all, because in giving the properties …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.
In other words, there can be no objects which have nothing but structural properties.
All objects must have some non-structural properties as well. (See Benacerraf 1996 for some later reflections on this argument.)