^Interesting. I'd like to hear more.
it can't be elliptical or a circle
it would have to be some weird shaped blob to fill out the appropriate 3D space to represent % of a characteristic because the shape may have to pass through the center of the cube in a 2D plane, and expand 3D to take up the appropriate space for all the 8 possible sub cubes if they are 0/100% for one, and 50/50 for another measurement.
definitely not a smooth ellipse. think of like a cingular logo for extreme situations.
Feeling and Thinking can come to the same conclusion, they just get there different ways.
When deciding between two things you can do what "feels right" (feeling) or what is "logical" or "makes most sense" (thinking). Sometimes the most logical thing is also what feels right. For example, I eat healthy because it feels like the right thing to do and makes me feel better (feelings) and also because there are scientific studies saying I am decreasing my chance of having medical problems by eating healthy. I could have reached the same decision with either thinking or feeling.
Ilah
The fact that not every function is 100% is due to nature. All personalities are made of nature (your mbti) + nurture (how you were raised, your environment, etc). There are 8 functions and they vary in strength due to orientation and the nature nurture aspect.
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Perception does a lot of the work you're talking about. Feeling literally can either say "good" or "bad". Thinking can either say "true" or "false". If you guys have heard of Turing machines, Perception is like the tape, and Judgment is like the head. Perception is what you're thinking about -- it does every single step that is not deductive. (In MBTI, a Thinker focuses more on "true" or "false", a Feeler focuses more on "good" or "bad".) If you string together a bunch of Feelings (coupled with a bunch of Perceptions and Thinkings, Perceptions to put the thoughts in your consciousness, Thinkings to answer Perceptions' yes or no questions), that's when you get the sort of thoughts you're talking about -- "This makes me feel good". That thought, though, is completely impossible without Thinking to answer questions like "is it true that 'this makes me feel good'?" Or, "is there a causal relation between feeling good and eating well?".
You get the picture, I hope. Every complete thought is really an interaction of all four functions. Sensing takes in data, Intuition makes the connections (metaphor is the basis of all language), Thinking checks logical consistency, Feeling checks motivation level.
Well, in theory a 3-D ellipsoid has 8 degrees of freedom (3 degrees for the origin, two degrees of freedom for the three axis, and 3 for the length, width, height along those axis). 8 degrees of freedom,8 functions--seems plausible.
I was actually trying to think through the equations of projections of the ellipsoid on the various "plane vectors" representing function scores.
Any 3-D ellipsoid can be specified by a 3x3 positive definite matrix, B, (defines axis and "lengths" along those axes), and a 3-D vector, r, (defines the origin).
The bounds of the ellipsoid are defined by the solutions to the equation:
[(x-r)^T][B^-1](x-r)=1
Each cognitive function vector would have a 3x3 projection matrix, P, of rank 1. These matrices are symmetric and have the property that P^2=P.
Now the projection of the solutions to the equation above using P becomes the solutions to:
[(y-Pr)^T]P[B^-1](y-Pr)=1
I was thinking that I could make the function score of the ellipsoid to be given by the maximum valued solution to (8 versions of) the above equation.
For convenience, lets denote the Projection Matrix by the actual function name.
Si score = max y such that, [(y-Sir)^T]Si[B^-1](y-Sir)=1
Ni score = max y such that, [(y-Nir)^T]Ni[B^-1](y-Nir)=1
Se score = max y such that, [(y-Ser)^T]Si[B^-1](y-Ser)=1
Ne score = max y such that, [(y-Ner)^T]Ni[B^-1](y-Ner)=1
Fi score = max y such that, [(y-Fir)^T]Si[B^-1](y-Fir)=1
Ti score = max y such that, [(y-Tir)^T]Ni[B^-1](y-Tir)=1
Fe score = max y such that, [(y-Fer)^T]Si[B^-1](y-Fer)=1
Te score = max y such that, [(y-Ter)^T]Ni[B^-1](y-Ter)=1
Note: in all the above equations the "y" is independently bound, that is, each y is a different y (I just didn't want to do y_Si, y_Ni, etc.).
Other things to note:
-The directions of the axes of the ellipsoid are given by the eigenvectors of B, and the "half-axis length" along those axes is given by the square-root of the corresponding eigenvalues.
-B must be positive definite, and because of that, it must be symmetric (we're dealing completely with real-numbers here).
With the things noted above, we've now created a framework of 9 scalar variables.
--b1 b2 b3
B=b2 b4 b5
--b3 b5 b6
--r1
r=r2
--r3
Given the eight constraints above and one more for positive definiteness, it seemed like it was doable.
The issue is that I haven't yet thought thought what vectors should represent the functions, and I need to make sure that equations given from the projections are independent (or at least not contradictory).
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