Changes

As agents of systems, whether that system is our own physiology or our own society, we move through what we commonly imagine to be a continuous manifold of states, but with distinctions being drawn in that space that are every bit as compelling to us, and often quite literally, as the difference between life and death.  So the relation of discretion to continuity is not one of those issues that we can take lightly, or simply dissolve by choosing a side and ignoring the other, as we may imagine in abstraction.  I'll try to get back to this point later, one in a long list of cautionary notes that experience tells me has to be attached to every tale of our pilgrimage, but for now we must get under way.

As agents of systems, whether that system is our own physiology or our own society, we move through what we commonly imagine to be a continuous manifold of states, but with distinctions being drawn in that space that are every bit as compelling to us, and often quite literally, as the difference between life and death.  So the relation of discretion to continuity is not one of those issues that we can take lightly, or simply dissolve by choosing a side and ignoring the other, as we may imagine in abstraction.  I'll try to get back to this point later, one in a long list of cautionary notes that experience tells me has to be attached to every tale of our pilgrimage, but for now we must get under way.

Returning to ''En'' and ''Ex'', the two most popular interpretations of logical graphs, that happen to be dual to each other in a certain sense, let's see how they fly as "hermeneutic arrows" from the syntactical domain '''S''' to the objective domain '''O''', at any rate, as their trajectories can be spied in the radar of what George Spencer Brown called the "primary arithmetic".

Returning to <math>\operatorname{En}</math> and <math>\operatorname{Ex},</math> the two most popular interpretations of logical graphs, ones that happen to be dual to each other in a certain sense, let's see how they fly as ''hermeneutic arrows'' from the syntactic domain <math>S\!</math> to the object domain <math>O,\!</math> at any rate, as their trajectories can be spied in the radar of what George Spencer Brown called the ''primary arithmetic''.

Taking ''En'' and ''Ex'' as arrows of the form ''En'', ''Ex'' : '''S''' &rarr; '''O''', at the level of arithmetic taking '''S''' = {rooted trees} and '''O''' = {Falsity, Truth}, let's factor each arrow across the domain of formal constants '''S'''₀ = {O, |}, the domain that consists of a single rooted node plus a single rooted edge.  As a strategic tactic, this allows each arrow to be broken into a purely syntactic part ''En''_syn, ''Ex''_syn : '''S''' &rarr; '''S'''₀ and its purely semantic part ''En''_sem, ''Ex''_sem : '''S'''₀ &rarr; '''O'''.

Taking <math>\operatorname{En}</math> and <math>\operatorname{Ex}</math> as arrows of the form <math>\operatorname{En}, \operatorname{Ex} : S \to O,</math> at the level of arithmetic taking <math>S = \{ \text{rooted trees} \}\!</math> and <math>O = \{ \text{Falsity}, \text{Truth} \},\!</math> let's factor each arrow across the domain of formal constants <math>S_0 = \{ \ominus, \vert \} = \{</math>[[Image:Cactus Node Big Fat.jpg|16px]],&nbsp;[[Image:Cactus Spike Big Fat.jpg|12px]]<math>\},</math> the domain that consists of a single rooted node plus a single rooted edge.  As a strategic tactic, this allows each arrow to be broken into a purely syntactic part <math>\operatorname{En}_\text{syn}, \operatorname{Ex}_\text{syn} : S \to S_0</math> and its purely semantic part <math>\operatorname{En}_\text{sem}, \operatorname{Ex}_\text{sem} : S_0 \to O.</math>

As things work out, the syntactic factors are formally the same, leaving our dualing interpretations to differ in their semantic components alone.  Specifically, we have the following mappings:

As things work out, the syntactic factors are formally the same, leaving our dualing interpretations to differ in their semantic components alone.  Specifically, we have the following mappings: