On Wednesday, October 24, 2018 at 3:47:39 PM UTC-5, Tomas Pales wrote:
>
> 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.
>
>
>
The future practice of mathematics is as an empirical science:
https://www.youtube.com/watch?v=BkKBgA902Uk
Category theory is type theory "shorn of syntax":
https://en.wikipedia.org/wiki/Type_theory#Relation_to_category_theory
(or: Category theory is a conspiracy to obfuscate type theory.)
Qualia from pure numbers (immaterial information) are like category theory
relative to type: They are "bodiless".
- pt
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