On Hell and Forgiveness

[Angrboda]

Okay, a simple thought experiment.  Let's say that you and I each have a lump of gold, me at my location, and you at yours.  Now we know that these two lumps of gold are not part of the same lump of gold, the atoms from each are separated by considerable distance.  Any two atoms in the respective lumps of gold are not part of the same lump of gold.  Now take a single lump of gold.  There will be two atoms in it that are adjacent to each other.  Both atoms are a part of the same lump of gold.  Between the atoms is empty space.  Now suppose we start gradually expanding the space between the atoms, first by 5%, then 10%, eventually by 100%, 200%, 3000% -- eventually the individual atoms will be as far apart as the lumps of gold in our respective pockets.  Keep expanding and eventually the two atoms are farther apart than the entire width of the universe.  The immediate question that comes to mind is, at what point do the two atoms stop being a part of the same lump of gold, but more importantly, why?  What is it about the distance between them that makes one pair of atoms, separated by an arbitrary amount of empty space, different from two other atoms, equally separated by an arbitrary amount of empty space?  This is similar in many ways to what is known as the Problem Of The Many, and as a glance at that page from the Stanford Encyclopedia of Philosophy will tell you, there are multiple incompatible answers to the paradox.  Some suggest that this is just an example of the sorites paradox or the paradox of the heap, that it is a problem in the vagueness of the boundary, that something changes, even though there is no clear line or point at which it changes.  Others disagree with this, and see the problem of the many differently.  One solution which occurs to me immediately is that when the atoms are close together, they interact with each other by way of forces such as gravity and the other four forces.  But this doesn't solve the problem, as no matter how far apart the atoms are, they still interact, albeit weakly, it is only a change in the strength of the interaction.  To argue that it has to do with interaction would then be to assert that there is an arbitrary strength of interaction which defines whether two atoms belong to the same object or not.

Now, I don't really expect to convince you that parts and wholes do not exist.  I suspect your intuitions about the matter, as well as perhaps your prior philosophical commitments make that unlikely.  But if you'll recall, the original complaint was that the position that number, which requires parts and whole distinctions, being subjective was not so far fetched that, as Neo put it, no rational person would believe that number, like parts and wholes, is subjective.  At minimum, I think this example, as well as the problem of the many, is not as far out as your intuitions, Steve's hyperbole, and Neo's contemptuous remarks made it out to be.  If not, then what is your answer to the lump of gold problem, and your answer to the problem of the many?  (And ultimately all objects are like the cloud in the problem of the many, a cloud of particles, interacting in various strengths through various forces.  What makes the cloud of atoms that is my desk a whole with parts, and a cloud composed of water droplets problematic?  There is the problem of the boundary in the case of the desk, as the surface of the desk trails off and it becomes difficult to say which atoms are a part of the desk, and which are not, and that indeed may be an example of the sorities paradox, yet the question remains, what makes this particular "desk-like ensemble of particles" a whole in the first place?)

I don't think that I understand how this has anything to do with whether the nature of numbers is objective or not.

Can you count without making a figure / ground distinction? If you have four, does not that require you to conceive of a "set" of objects which are parts of a whole? Number by its very nature is a relationship built on parts & wholes. If you only have one, undiffferentiated whole with no parts, how can you have number? Number only exists in so far as we can posit identities (another concept dependent on parts & wholes), sets (parts belonging to a whole), and relations (reflexive consideration of the set of objects both as parts [multiple members of the set] and as a whole [as composing the set as a whole]). Without part and whole relationships, we don't have sets, and without sets, we don't have number. So you have to ground what makes something a part and something a whole in order to talk about sets, and you have to be able to have sets to have number. The concepts are dependent on each other. To put it in concrete terms, I can't talk about four apples if I can't identify individual apples as being wholes (an apple) and as being parts belonging to a set (the collection of four apples), which are part of a whole (a universe that has parts that are not apples). So you can't have number without part / whole relationships. Until you ground the concept and application of part / whole distinctions as objective, you can't ground number as objective. So I'll ask you to reconsider my prior post in that light and answer the questions.