Bruno, as I see it, my ontology, whose relational aspect is defined by the relation of similarity (and its special kinds - instantiation and composition), includes your ontology, because pure set theory includes arithmetic. You may be right that arithmetic is sufficient to define physics but reality may also contain more than arithmetic. On the other hand, if I understand Godel's second incompleteness theorem correctly, as far as the relational/mathematical aspect of reality is concerned, we will never be able to prove that there exists more than arithmetic (because we will never be able to prove that it is consistent). And if we are not able to interact with infinite objects, we will never be able to observe them either.
But I don't see a reason to exclude infinite objects from existence. Some say that an infinite collection can never be "completed", as if mathematical objects are created by some kind of *process *that must reach completion*. *They are not created by a process; they exist timelessly; there is nothing to complete. Only inconsistency would prevent their existence. You said you don't really believe in sets. But a set is just a combination of objects, where the combination is another object, isn't it? Everything you see around you is structurally a set. About category theory vs. set theory, this is how I understand it: more general (more abstract) mathematical objects are instantiated in more specific mathematical objects (e.g. "geometric object" is instantiated in "triangle") and ultimately in concrete mathematical objects (e.g. in concrete triangles), which are not instantiated in anything else. (Those objects that can be instantiated in other objects are also called properties.) All concrete objects are concrete collections, that is, collections of concrete objects, so all mathematical objects are ultimately instantiated in concrete collections. This fact is used in set theory, where every mathematical object is represented as a collection (set), and that's how set theory can be a foundation of mathematics. The collections referred to in set theory are not concrete collections though but abstract collections (generalized collections), because differences between concrete collections of the same kind are not relevant for mathematical purposes. So for example, set theory does not refer to concrete empty sets but to one abstract empty set (which is instantiated in all concrete empty sets). (Although I have also heard of the extension of set theory to so-called "multiset" theory, which admits copies (instances) of the same object as distinct members of a set.) The approach of category theory is not to represent mathematical objects as collections but to study similarities (morphisms) directly between mathematical objects themselves. Collections, there, are treated just as one of many kinds of mathematical objects. About qualia, some time ago I imagined that maybe Godel sentences could explain qualia, as Godel sentences depend on an axiomatic system and yet cannot be proved from that system, similarly like qualia seem to depend on a neural system and yet cannot be proved from it. But then I grew skeptical of this idea because it seemed to me that numbers will always be just numbers, even if they are infinitely big, and an infinitely big number may be beyond our grasp in a sense but it will not somehow turn into red color, for example. Gradually I started to lean to the idea that numbers and mathematics in general are about the relation of similarity; that mathematics basically says that something is similar to something else but never says what that "something" is. So now it seems more plausible to me that qualia are those "somethings" that stand in similarity relations. Russellian monism is a similar explanation of qualia. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
