Hi Tomas,
> 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. OK. But it seemed to me you said that is better not to make unnecessary assumption. Indexical Digital Mechanism (the idea that we would personally survive in the usual clinical sense with a digitalis-able body) can be shown to make one simple inductive set enough, and actually needing the simplest model of arithmetic. Infinite computations do play an important role, but in the ontology, we can use only their finite portions. If we allow inductions and/or axiom of the infinite, things get awry with the first person indeterminacy, as you will need to take into account transfinite histories. I have not yet prove this, but I suspect the axiom of infinity making unnecessary mess. > You may be right that arithmetic is sufficient to define physics The arithmetical reality is sufficient, and eventually, even just the partial computable part (the Sigma_1 reality). But that is only true for the ontology. The phenomenology will not be bounded in any way. Even ZF + KAPPA (the existence of a inaccessible cardinal) can still only scratch on the arithmetical reality. > but reality may also contain more than arithmetic. If you are a digitalis able machine, arithmetic seen from inside *is* more than arithmetic. That is the reason why we would be foolish to commit oneself in any ontological commitment bigger than one universal machine/number/word/combinator/game-of-life-pattern, etc. > 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 No, but it is consistent to assume more. And the ontological arithmetical reality explains already, by incompleteness indeed, that the numbers will have to assume much more than the numbers to be able to understand themselves. Like complex analysis is useful in Number Theory, incompleteness justify the helpfulness of some strong axioms. Löbian machine, like PA or ZF, or ZF+KAPPA can prove their own incompleteness and contemplate the geometry of their ignorance. The fact that all that is phenomenological does not make it less real. There is a Skolem phenomenon. From outside the ontology is recursively enumerable, from inside it is above anything expressible.Somehow, the whole of possible math can only scratch it a little bit. > (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. Yes, indeed. Same with the non-computable. Ae^iH(omega)t, with omega = Caitin or Post number is a solution of the schroedinger equation, but is not computable, but we would never recognise it as such, and confuse it with (pure) noise. > > But I don't see a reason to exclude infinite objects from existence. They do exist, the machines met them all the time, and are keeping in touch with them. Phenomenological existence is not non existence. But we don’t need them in the theory: they are explained by the entity living in that theory, without f-havng them to commit an ontological act. So why would we do that? Plus the fact that if we do that, the semantic, which is what make the “consciousness flux” differentiate on the computations/histories. The problem of the machine is that it has the foot on the finite ground, living in the neighbourhood of 0. Bt its soul, where it truly live belongs to the neighbourhood of infinity. Infinity is a key notion, it is just that we don’t need to assume it, it is a necessary “meme” of all universal machine introspecting itself, and it is a key help to them in most situation. > 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. Yes. OK. > > 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. Yes, but with mechanism, that is the result of the work of a finite number, in front of (infinitely many variants) of a finite number. > > 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. Oh, I could argue that they can be instantiated in the (greek) triangular number 1, 3, 6, 10, 15, 21, … like many other shapes .. Of course, with mechanism, triangle are “construct of the mind”, like space, time, … but number relations can play both roles of 3p objects and collection of 3p objects analysing 3p object from the 1p views. > (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. Yes, but mathematical theories, when you instantiated them concretely like that get extended in the transfinite. A proof, when make concrete, is a transfinite object, same for computations if you add the axiom of the infinite in the ontology. Maybe we can do that if we have good reason to do it,(with our without mechanism) but that seems useless without extraordinary evidence, I would say. > 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. It is higher order logic in disguise, and it flattened a bit the structure, and then why ZF-like set, why not NF-type of sets. A set, for me, is typically phenomenological, like suggest the name of the axiom of comprehension and the name of the reflexion principle. > 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.) > It is the theory of bag. It is even worst. > 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. You can found the whole of math with pure category theory. Instead of set, you have only arrow (you don’t even need the points). But category theory is difficult, and the categorical approach to untyped partial sigma_1 complete reality cumulates the difficulties of recursion theory and category theory, so it has never really grow up. Now, in the constructive realm (which plays a role in the first person notion), categories provide good technic to build many models of applicative algebra. In my setting where self-reference plays an important role, Kripke semantics works for a good part, but other semantics are needed for the “proper” theological corona (G* minus G). > > 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, Nice. I exploit this methodically by distinguish G1 and G1*, and distinguishing Z1 and Z1* (the logic of bewesibar-and-consistent). Amazingly the “pure first person” is the same for G and G*: S4Grz1 = S4Grz1*. The “1” is related to the modal axiom p -> []p, which limit p to the sigma_1 sentences (the computable). > 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. With mechanism, “seeing red” result from small, but subtle relation between numbers, but the qualia is related to the infinite histories going through our state (it is somehow like an infinite of sigma_1 sentences, equivalent from a lint of view). > 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. Yes. We don’t need to say that something is, nor that it exists, except that we have to assume some relation to start the conversation. The beauty is that we don’t need to assume more than K and S and their laws to get the arithmetical “Indra net” of all universal being, including the infinite one (as seen from inside). > 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. I think that this is coherent with mechanism, once we get that incompleteness makes the machine looking at itself in many different modes (truth, proof, experience, observation, sensations). Bruno > > > > -- > 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 > <mailto:everything-list+unsubscr...@googlegroups.com>. > To post to this group, send email to everything-list@googlegroups.com > <mailto:everything-list@googlegroups.com>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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