A new Perspective on Dichotomies in Socionics - Pyramid Diagrams, Draft

[Exodus]

This is my delima - I can either write a popular paper that is easy to understand but not very technical, or I can use mathimatical proofs to prove this not only works but it is a mathematical fact. I've been trying to write this damn thing for over a year, but I finally got closure when I talked to Reinin and Gulenko and they both said that they don't know the math, so I decided to write a popular paper and trust they will see the truth in it. I was thinking of including annotated proofs in the appendix, but it is a low priority now.

The basic flow of the proofs would be:
Prove the reinin dichotomies are abelian groups
Use the "fundamental theorem of abelian groups" to turn the abelian group into the internal direct product of Z2 cyclic groups
Show the order of this group is Z2^4
Show that is Isomorphic to the pyramid diagram
Use this property to project each trait onto the diagram to create a type of block design called a steiner system
Show the meaning of each subgroup

Just in case you're still trying to do this:

Dichotomies form an abelian group under the XOR/interleaving operation (which has (A XOR B) and (A == B) as its two halves). It's pretty easy to show that this operation is commutative/abelian, and that X + X = 0 for all X. This immediately gives you that it's a product of Z2's. If you also define Reinin dichotomies as being generated by four independent dichotomies, then you get Z2^4.

That's just the math part though. I'm highly skeptical of typing people by tallying up Reinin dichotomies, to say the least.

[Lao Tzunami]

Just in case you're still trying to do this:

Dichotomies form an abelian group under the XOR/interleaving operation (which has (A XOR B) and (A == B) as its two halves). It's pretty easy to show that this operation is commutative/abelian, and that X + X = 0 for all X. This immediately gives you that it's a product of Z2's. If you also define Reinin dichotomies as being generated by four independent dichotomies, then you get Z2^4

Yes, I am still interested. However, I would like to define each Z2 element as a sign group (the multiplicative group of +1 and -1) not with Boolean logic. It's been a while since I've written a proof. I think we know each other from the world socionics society. Are you interested in helping me or do you know of any good resources?

[Exodus]

Yes, I am still interested. However, I would like to define each Z2 element as a sign group (the multiplicative group of +1 and -1) not with Boolean logic. It's been a while since I've written a proof. I think we know each other from the world socionics society. Are you interested in helping me or do you know of any good resources?

If you define it as a product of multiplication groups then it's basically abelian by definition and there is nothing to prove. The interesting part is that you can "multiply" the natural groupings of types (dichotomies) in such a way that they form a vector space.

[Lao Tzunami]

If you define it as a product of multiplication groups then it's basically abelian by definition and there is nothing to prove. The interesting part is that you can "multiply" the natural groupings of types (dichotomies) in such a way that they form a vector space.

If possible, I would like to write a long proof that does not take short cuts or apply laws only known to mathematicians to arrive at the answer. I'm assuming that almost everyone who reads this has not even heard of a graded vector spaces. Ideally, I would use only the most basic opperations and have a column next to each step explaining with words what I am doing and giving the appropiate background. If there is not a reasonable chance the layman could follow my proof, then it is not worth writing.

In school, I've been asked to recreate importaint proofs, like in physics and calculus. These proofs are facts, we know they work 100%, but the exercise shows "why" they work. I would like to include that "why" in all of my proofs.

Also, my final proof will be quite a bit more complex than just proving a single vectorspace or geometric projection. Eventually, I am going to take the direct product of multiple vector spaces. A great example is how the direct producted of the information elements and functions creates the 16 types or how the direct product of two types creates the intertype relations.

There is some nuance to this, as I think you have commented in your own article, because you can make the model appear unbalanced with out a full explanation. I think you define the supervisions rings as Dih4 if I remember right. This is unnecessary if you account for the rationality dichotomy, but i cannot just define the type dichotomies as z2 sign group and have that point understood.