Hey Scott,
Nice to see you here on Toequest. I am interested in the subject matter of three; I have been writing about it myself also. I have some information on three from a mathematical perspective.
As we all know, Aristotle created the very familiar standard structure based on duality, that there is no third way. Rotterdam-born mathematician L.E.J. Brouwer showed, however, in the twenties of the last century that Aristotle wasn't completely right with his vision of the absence of the third way: a highly remarkable feat. This intuitionistic mathematician was able to calculate that a third way was possible on the condition that the third way had nothing in common with the set-up of the investigation. So, in ordinary words, that would mean Aristotle made the right comment, but that his third absent position actually is a real third position as well. If you have read any of my other contributions, you'd know I use the term "phenomenon of nothing" to describe the importance of this position.
According to me, a simple example of the absence of a third position is found in an investigation of opposites of male and female. If we would investigate the gender of a gender-neutral bacteria, for instance, we would end up empty handed. The contradictory first step of this investigation, expressed in binary numbers, must conclude that this gender neutral bacteria delivers two sets of opposition. (0,1) stands for the facts that it is not male, but it is alive, while (1,0) stands for the fact that it is alive, but it is not female. The structure delivers both sets of 1s and 0s, and therefore becomes unfit to use, because the required match of (1,1) is not available. As shown, a third way does exist, but only on the condition that it has nothing in common with the first two ways. It points to the limitations of the set-up of the used structure.
Again, concepts are at the heart of the matter, because the used structure determines the perceived reality, but is itself limited. If you divide the world into male and female you may come a long way — and explain a lot — but you'd come shy to explaining everything. The conclusion that everything can be seen in pairs is still valid, but not as the structural absolute truth. The ultimate structure must contain duality, but should not be limited by it.
However, Scott, I claim that the ultimate structure is not based on three either, rather that it is a pyramid with either four positions (the grounded positions) or five positions (with the abstract overall position in top included as well). My mathematical evidence — and I leave it available on this site for you to read in the next two weeks:
http://www.pentapublishing.com/Math.html — actually has a basis of six, though one of the six positions in this matrix must be given to the phenomenon of nothing (in numbers: zero). Three only has its place under very special circumstances, and mainly based on the incompleteness of duality to explain the whole (2 X 3 = 6).
What is very interesting according to me is that — when we say that the fundaments are four-fold — that the level up is then still not 4 X 4 = 16. Rather, we find twelve as a standard. I believe this 4 X 3 = 12 format is explained by one of the four positions being the phenomenon of nothing. Allow me to use the holy trinity here to explain these words.
Where the holy trinity is father - son - holy spirit, it is a gender limited answer. Next to this holy trinity we must (sorry, but I consider this indeed a must) place the other holy trinity of mother - daughter - holy spirit. As you can see when combining both holy trinities, we can find four grounded positions of father, mother, son, daughter, and either state there are two holy spirits (six positions) or just one and the same holy spirit in both trinities (five positions). If one wishes, we can even fold the holy spirit of one set into being the unspoken expression of the members in the other trinity, and vice versa; we would then have just four positions, all included. I hope this explains the six-position as found in the matrix, and the five-position of the pyramid.
The four-position (and with it the twelve-position) are positions that all contain a fundamental duality: the self and the other. On the one hand, there is the experience of oneself (which is subjective), and on the other hand there is the experiencing of the three others (which are objective experiences). It is impossible to experience oneself from the position of the other (though we are all smart and try to use our imagination to circumvent this obstacle somewhat). Hence, there are only 4 X 3 objective positions available. The subjective self is locked in with the four positions, of course, and should be regarded as the internal unity. Yet as an external objective experience it comes up as nothing — it ain't there.