# Extension of Inside Theory and Outside Theory

Jan 26 2021

Over at Applied Category Theory discord server Sven Nilsen and I with others have been having a conversation about his idea of Inside Theory and Outside theory that is relevant here. See https://advancedresearch.github.io/blog/2021-01-18-platos-cave-found-in-mathematics

Invite: https://discord.gg/hTEpgYv

I will reproduce the text my conclusion about our discussion here.

Hate to interrupt your discussion but lets connect this to something I can understand. Formal Systems have three properties which are Completeness, Consistency and Clarity (Well formedness). These properties are the relations between three of the aspects of Being which are Identity, Presence and Truth. But when we add Reality as the fourth Aspect then we get three other properties which are Verification, Validation and Coherence. That means there are six properties that are the relations between the Four aspects of Being. Now due to Godel we know we have to choose either Consistency or Completeness. You cannot have both. Similarly you cannot have both Verification and Validation at the same time. Nor can you have both Clarity and Coherence at the same time. This last part is my extension of Godel’s insight. If you take one as Absolute then the other has to be Relative from each pair. There is something that I call the Axiomatic Platform. That is the set of axioms taken together that is a platform for building convex theorems within an axiomatic framework. This kind of framework serves as the basis for the formal system. We want formal systems to be Complete, Consistent and Clear all at the same time but this is impossible. You get to choose either Completeness or Consistency but not both. And further outside the convex closure of the axiomatic system of the formalism in its extension there is always the question of the relation of the system to its environment, what is beyond its convexity. That of course is the meta-system which is what is beyond the system boundary to the horizon. We take what is beyond the formal system as reality and so then we have the other three properties which are Verification, Validation and Coherence. We cannot make validations absolute without forcing verifications to be relative and vice versa. Similarly with clarity and coherence. If one is absolute then the other is forced to be relative.

This is a speculative extension to the idea of Godel with respect to formal systems, taking into account that formal systems are defined by the different aspects whose relations produce the various properties of formal systems in relation to their meaning that comes from their relation to reality. Now I am not sure what inside and outside theories are. But lets pretend we do know and say that inside theories are within the concave closure of the axiomatic system while outside theories are not. In other words outside theories refer to reality beyond the closure made possible by the axiomatic platform. What Deconstruction tells us is that there are meta-levels of inside and outside the formal system. We know this is true because Tarski says that truth/falsehood is only at a meta-level. What Derrida has showed in relation to Husserl’s phenomenology in Speech and Phenomena is that if you push the inside to a further level deeper inside you end up outside and Zizek suggests that the same is true of the outside. So you notice that if truth/falsehood is at a meta-level from the first order language of the formal system of logic, then we need to push one level beyond truth/falsehood distinction and suddenly you are outside, i.e. the inside/outside distinction vanishes. And this can occur if we push into the fourth dimension. You can get inside or get out of a sphere in the fourth dimension without piercing the convex shell of closure. I call these openly/closed systems and believe Victor Frankl was the first to suggest that these exist. But we can see Leibnizian monads as an example of these openly closed systems.

It is basically this idea that Zizek is taking from Derrida and developing with those equations. It posits that there is a way to get beyond inner and outer through the inside of the convex formal system and this inconsistency is what Godel posits for every even modestly formal system via the liar’s paradox. There is always contamination that can get through the barriers we attempt to erect against paradox. This has a huge effect on set theory for instance. But does not effect Category Theory. That is why we advocate switching over to the weaker Category Theory representation of systems, but that takes us from entities to relations and functional bases for relations. Entities can drop out and identity arrows stand in for the missing entities.

Anyway perhaps you can explain what you are talking about in this framework, if you get what I am talking about here, i.e. apply it to formal systems of logic or set theory etc. I talk about these things in my tutorial on Schemas Theory at http://schematheory.net/

You know this also reminds me of what we are reading about with Zizek’s Sex and the Failed Absolute with regard to for instance Set theory where the bottom of the lattice is the null set and the top of the lattice is Universe instead of the Set of All sets. Top of the lattice for sets gives paradox that needs to be avoided. Bottom of lattice is the null set which is an exception that is not a member. Each one is an extreme that sets the limit on the Set. Null set 0 goes beyond empty set (). Zizek says that this is related to the difference between the antimonies which is related to what Lacan calls sexation. It seems like these formulas of Derrida might be a version of this strange structure of exception on one side and non-all on the other side. ALL is Outside3 and Null set 0 is Inside3. Non-All of Set Universe is Outside2 and Empty Set () is Inside2. Set with something in it is Outside1 and particulars in Set is Inside1. If this is the case it is equivalent to Lacanian Sexation which is also equivalent to the Antimonies in Kant.

However, if we instead use Category Theory which is weaker then we avoid paradox because element can be dropped for identity arrows, and Categories of Categories are possible because there is nothing inside that can be not-All. In other words everything are structural relations mappings and no entities. Category Theory does not have a membership function which is a Having. With the membership function Having is squared. Sets have to have things in them as members. Categories are pure relationships based on functions so they do not have to have anything in them. Having along with Being are always fragmented roots in Indo-European languages. Having produces paradox. Now we can see why Having and Being are both fragmented. Having is the equivalent of the membership function that places a particular within a set. The set can exist without anything inside it, just as category theory can drop its elements. That is why Badiou needs the multiple to produce particulars. Being on the other hand is the spectre of the Non-all in relation to the Set of all sets and that gives us the paradox of the transcendent in relation to the immanent.

Summary of discussion: I am thinking about trying to write a paper on our discussion here, perhaps you @Sven Nilsen could write a paper about it as well so I can refer to it. This is a breakthrough in the sense that several different models that were separate in my mind suddenly became related via your suggestion that there may be inside and outside theories. But when I think about writing about it, it still seems nebulous so I think more work needs to be done to reconcile the different models. But I want to try to capture it in a working paper while I still think I understand what has been said here. The major convergence is between Zizek’s use of Derrida’s deconstruction equations, what Zizek says about Kant’s Antimonies in Sex and the Failed Absolute and Sexation, and finally applying this to the Axiomatic Platform based on your distinction between Inside and Outside Theories. But for me the real breakthrough is the articulation in this context the relations between Aspects of Being and Properties of the Formal System.. This has been a long time speculative hypothesis of mine, extending the properties based on the relation of reality to identity, presence and truth. And further extending Godel’s indecision between Consistency/Completeness to Verification/Validation and Clarity/Coherence. But when one places these Godelian pairs of undecidable elements in the context of the overall structure of the Set that reflects Sexation differences then you get to see how these pairs operate in that context and that is very interesting. But it is a little bit complex. However, it seems to serve to validate the speculative hypothesis that these two things: differences between Godellian pairs and structure of the Set according to the layers of Inside and Outside model work together. And the fact that you @Sven Nilsen seem to have discovered this Inside/Outside relationship which I call the openly closed system on your own is further evidence that the hypothesis might be correct.