You’ve misunderstood me. It’s really not at all conspicuous to allow a none-empty “set” into your ontology, but if you’d prefer we can talk about heaps; they serve for my purposes here (of course, by “heap”, I mean any random pile of stuff). Every heap has parts: you’re a heap of cells, decks are heaps of cards, masses are heaps of atoms, etc. Now if you apply a level filter to the parts of a heap, you can count them. For instance, I can count the organs in your body, count the organ cells in your body, and end up with two different values, though I counted the same object. The same object can constitute many heaps, as long as there are several ways of dividing the object into parts. So what we can do, is just talk about the laws of heap combination, rather than the laws of numbers. We don’t require any further generality in our mathematics to do all our counting, and yet, the only objects I’ve had to adopt into my ontology are heaps (rather inconspicuous material fellows in IMHO).
I should mention that this is not my real suggestion for a foundation of mathematics, but when it comes to the challenge of interpreting the theory of natural numbers without adopting any ghostly quantities, heaps work just fine.
(edit):
I should mention that while heaps, requiring only for you to accept a whole with parts, and a level test on any gven part, are much more ontologically inconspicuous than pure sets. Where exactly is the null set? Where is any pure set? I’ve never seen any of them. Of course, i see heaps all over the place.
You’ve misunderstood me. It’s really not at all conspicuous to allow a none-empty “set” into your ontology, but if you’d prefer we can talk about heaps; they serve for my purposes here (of course, by “heap”, I mean any random pile of stuff). Every heap has parts: you’re a heap of cells, decks are heaps of cards, masses are heaps of atoms, etc. Now if you apply a level filter to the parts of a heap, you can count them. For instance, I can count the organs in your body, count the organ cells in your body, and end up with two different values, though I counted the same object. The same object can constitute many heaps, as long as there are several ways of dividing the object into parts. So what we can do, is just talk about the laws of heap combination, rather than the laws of numbers. We don’t require any further generality in our mathematics to do all our counting, and yet, the only objects I’ve had to adopt into my ontology are heaps (rather inconspicuous material fellows in IMHO).
I should mention that this is not my real suggestion for a foundation of mathematics, but when it comes to the challenge of interpreting the theory of natural numbers without adopting any ghostly quantities, heaps work just fine.
(edit): I should mention that while heaps, requiring only for you to accept a whole with parts, and a level test on any gven part, are much more ontologically inconspicuous than pure sets. Where exactly is the null set? Where is any pure set? I’ve never seen any of them. Of course, i see heaps all over the place.