08 Apr Symmetrical Logic and Lineage
Symmetry may not immediately appear as a principle of logic or reason, but it should. In mathematics we learn the commutative and associative properties of addition and multiplication. These represent a mirror-like symmetry. Symmetry, or invariance against change, is a fundamental principle of physics and an underlying assumption driving some logical decisions. Causality, for example, is a symmetrical principle of logic, as evidenced by the physical axiom “for every action, there is an equal and opposite reaction.” Action and reaction are clearly symmetrical. We could restate this in psychological/behavioral terms: “for an observed effect, the inquiring mind seeks to find a cause.” This is part of the broader cognitive task of classification.
Causality can also be described as a chain such that the first link in the chain is the fundamental cause. Each successive effect until the ultimate result acts as a cause for the next link in the chain. Here is an example:
I walked far > I grew tired > I tripped > I hurt myself
Some causal chains are circular, such as the nitrogen chain or the water cycle:
evaporate > form clouds > rain > drain > accumulate > evaporate
Whenever you say to yourself, “that just doesn’t look right,” check to see if the asymmetry is your stumbling block.
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|A Good Excuse for Heuristic Logic
|Abstract Contexts and Fuzzy Reasoning
|pattern classification chaos
|Lanius on self-symmetry
We often think of symmetry in terms of mirror images, but identical twins are rather extreme examples of symmetry. Perhaps the definition “invariance against change” is a little broad for simple coverage. Once again we sit on the shore and touch the surface of a deep pond. For a simpler perspective, let us consider symmetry to be a principle of self-similarity. Similarity can include exact duplication, but it does not preclude slight differences between like objects or phenomena.
Hierarchies and Invariance
Now that we have considered the importance of symmetry in logic, let us consider its place in the structure of knowledge. If knowledge has a structure as indicated in Joe’s Theory of Everything (JTE), is it possible that there is a macroscopic structure governing the microscopic interconnection between all associated concepts? Hierarchies provide symmetrical structure to the overall network of connected information.
There are two distinct manifestations of symmetry throughout hierarchical chains: vertical and horizontal. Vertical symmetry, which occurs between different levels on the chain, can be expressed by the two axioms:
- For an element to become a Parent, it must have a Child and vice versa.
- A knowledge object is a valid Child if it inherits some significant attributes from the Parent.
The possibility of multiple parents for a single child or multiple children for a single parent obscures the symmetry on a macroscopic level, but the symmetry is still perfect at the object level. Horizontal symmetry is self-evident in that a sibling relationship is inherently self-similar by virtue of inheriting from the same parent, and sharing a generation, even though:
- descendent lineages may differ,
- any child object may have different parent objects from which it inherits different attributes,
- different children may operate in different contexts, often defining the differences in lineage.
Does the case of the unrelated element (one that is neither a parent, child or sibling) disrupt this theory? Perhaps it would if such a case existed! It would take a vast amount of convoluted logic and proof to persuasively argue that any abstract or concrete thing in this universe is totally unrelated to anything else. This discussion leads back to our basic assumption of the quantum interconnectedness of all things described in Joe’s Theory of Everything.
Invariance is one of the principles that is often invoked in pattern resolution in chaos theory. In the midst of apparent chaos, seeking self-symmetry, geometric or general, and invariance, can lead to finding order, especially in large sets. Human language is a large set, rich in variety, representing human knowledge; which is even larger and richer. Finding, capturing and classifying the patterns is the beginning of the solution.
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