In his famous book Laws of Form, George Spencer-Brown tries to construct the world from the most simple. He begins with a simple distinction, a circle on a structureless plane for example. He does not say that this is the only way to begin, but makes the reader simply do it and see what is developing. In addition, he gives more handling instructions, which reduces the most possible to the least possible.
I focus on his beginnings – the distinction – and the way how this is interpreted by him (indeed): ”… every duality implies triplicity: what the thing is, what it isn’t, and the boundary between them.” By this the world is digitalized. Subsequently, he examines extensions (“… to recross [the boundary] is not to cross [the boundary]”) and reductions (“What a thing is and what it is not are identical in form”). In any case, the boundary remains only boundary, i.e. distinction.
However, the boundary has been created just by crossing itself: It only exists by changing the sides. If I stand still on one side, all vanishes.
But when I change the side, I must change to another. Therefore, both sides are never identical in distinction, because then the crossing, the alternation, that creates them, would vanish.
If we reduce everything to distinction (“form”) as such, as Spencer-Brown sometimes does, the distinction still must be perceived as such. This means, the alternation condenses. And this it does (without changing everything) only in the middle. So the boundary is unity in extreme, moment of identity of both sides. The alternation then takes place between the center of identity and periphery, which leads to infinitesimality structure.
What Spencer-Brown wants to rationalize out of existence, therefore, is alternation itself – the prerequisite of his whole operation! By that he simplifies (identifies) more than he says. And he does not say all that is important.
If one follows his instructions, no contradiction results. But who says, that one has to restrict oneself in this manner?
As for the extension “to recross is not to cross”: It shows that identity always points beyond itself, since strictly speaking it came into being by recrossing – there is no closed operation, the unlimited world is connected. With this even the extension of the Brownian principles is justified.