Section Headings
Metaphoric Logic, along with Redundancy Logic and Explanatory Power, are part of the validation network for the Attention Matrix. Metaphoric Logic is the process of employing the logic of a known system to understand the logic of an unknown system. This type of logic provides the foundation of verbal abstractions; exerts a significant influence upon human behavior; and is a powerful technique of science. The web of logical symmetries between Data Stream Dynamics (DSD) and Attention-related phenomena provides strong validation for the Math-Fact Matrix that defines the Realm of Attention. The isometric relationship also indicates that we can employ the mathematical logic of DSD to better understand the logic of Attention.
Redundancy Logic, Metaphoric Logic and Explanatory Power are the three sources in the validation network for the Math-Fact Matrix that defines the Realm of Attention and places it on an equal footing with the Material Realms. This article focuses upon how Metaphoric Logic provides validation for the Attention Matrix that addresses the monitor-adjust feature of living systems. This crucial form of logic holds the Matrix together. It is the glue that binds evidence and mathematics.
Two systems are isomorphic when they share a symmetrical logical structure. In this case, we can frequently employ the logic of known system to better understand the unknown system. Metaphoric logic is the name we have given to this process.
Metaphoric logic is significant in many ways. It provides the basis of abstract thought, influences human behavior, and is a powerful tool for science. As Nature via evolution chose this powerful form of logic to form verbal abstractions, isometric logic must be is a powerful form of reasoning.
There are many logical symmetries between the LA’s mathematical system and a system of scientific evidence relating to Attention. Rather than isolated threads, an interlocking logical web unites the two components of the Attention Matrix. Because of this isomorphic relationship, we can employ the LA’s mathematical logic to better understand Attention.
The isomorphic relationship between Attention and the Living Algorithm (LA) provides significant validation for our model. The remainder of this article examines these topics in greater detail.
Metaphoric logic plays a crucial role in the validation network for the Math/Fact Matrix that defines the Realm of Attention. This common yet unsung form of logic is an incredibly important feature of everyone’s life from infanthood onwards. In addition to influencing human behavior and providing the logical foundation for abstract thought, isomorphic logic is also an essential tool for the scientific community. Our express intent is to familiarize the reader with the concept and its implications. Let’s start with the basics.
The prefix ‘iso-‘ derives from the Greek word ‘isos’ meaning equal. It is a combining term meaning alike, the same, or equal; denoting equality, similarity, or identity. The suffix ‘-morph’ is also a combining term meaning characterized by a specific form. Together the term ‘isomorphic’ means of like or similar form.
The word is used in many disciplines. We are using it to compare the logical structure of two systems. Every system has an implicit logic that links its components. Sometimes the logic of one system matches the logic of another system. When two systems share a similar or equal logical structure, they are said to have an isomorphic relationship.
Logic A ≈ Logic B
Identifying logic equivalency between two systems can have great utility. When two systems seem to have an isomorphic relationship, it is possible to employ the logic of a known system to better understand the logic of an unknown system. This is what we call isomorphic logic. Correctly identifying isomorphic systems enables predictions regarding the behavior of the unknown system.
Metaphoric logic is incredibly important for every human being, not just scientists or logicians. Although it might sound like an esoteric form of reasoning that is reserved for specialists, all humans have used it regularly and inadvertently since early childhood. Indeed isomorphic logic is the foundation of the verbal abstractions that informs our choices and thereby influences our behavior. Playing this crucial role, this common, although previously unnamed, form of reasoning is a hardwired cognitive process.
Rather than just a clever conjecture, this process is instead what would be considered a scientific fact. A virtually universal consensus among cognitive scientist is that humans learn to make useful abstractions about their world via conceptual metaphor. We apply the logic of what we have experienced – the known – to better understand the unknown. Humans develop conceptual metaphors from both sensory experience and hard wiring. These conceptual metaphors are then combined into metaphorical blends. The underlying logical schema of these known systems provides the foundation for abstraction.
After successfully mapping the logic of an understood system onto an unknown system, we then use this new system logic as a fresh foundation. On the simplest level, we apply this known logic to other unknown systems. On more complex levels, we also blend it with the logic of other systems to create massive structures of logic to understand our complex world. This process of building logical systems from the logic of known systems, especially sensory, is the basis of abstract thought.
As the basis of the abstract thought that informs our behavior, isomorphic logic pervades human cognition, hence our existence. We employ the logic of known systems to both better understand and make predictions regarding unknown systems. In addition to our cognition, this reasoning process also inadvertently influences our behavior in a non-verbal, imitative fashion.
Simply speaking, we humans inadvertently learn the logic of abstraction from the logic of familiar experiences with our environment. Further this logic informs our behavior. While beginning with our sensory experiences, this process includes our experiences with nature, city, other humans, culture and even animals. There are innumerable examples.
We tend to absorb and utilize the logic of our physical environment. For instance, Concrete Logic is static, rigid and permanent, while Nature Logic is dynamic, flexible and transitory. Those who spend a lot of time surrounded by Concrete tend to project a more static logic upon reality than those who are surrounded by dynamic Nature. Could this be one reason that city folks tend to benefit from a nature fix? Could it be that city planners have incorporated parks into their vision to give urban dwellers a fresh logical perspective? Could this be why Nature can be so refreshing?
We also absorb political logic from our social environment. As an example, those who live in the sparsely populated countryside with low ethnic diversity tend to have a far less tolerant political agenda than those who live in high-density cities with high ethnic diversity. The political battles between country folk and city folk have been ongoing ever since the rise of large international population centers. The differences in the logic they have absorbed from their environment could certainly be a contributing factor to this perennial conflict.
We also absorb and emulate the behavioral logic of our fellow humans – for better or worse. After watching professional athletes, we inadvertently attempt to emulate their behavioral logic, whether in sports, martial arts or dance. Disciples emulate the emotional intelligence (logic) of their Master. The masses emulate the emotional logic of their leaders. (Let’s go to war over national pride or let’s negotiate to avoid unnecessary bloodshed.)
This tendency to absorb the logic of other humans and apply it to our behavior could be a significant factor in the collective behavior of any group, whether united by gender, tribe, race, economic class, culture, country or empire. The collective logic of both tolerance and racism/sexism can spread from individual to individual, from community to community, and even from generation to generation. This collective mindset that derives from the logic of our human environment can result in entire classes or even countries of people going to war or participating in the construction of gorgeous religious architecture, e.g. cathedrals.
The frequency of group or mob action indicates that this tendency to absorb human logic must be a very powerful binding social force. An immersion in the logic of Nature via regular trips to the countryside is an effective antidote to our propensity for collective insanity. The isolation of meditation also serves to neutralize this follow-the-leader mentality.
Hopefully the preceding explication and examples have rendered the Reader at least a little more comfortable with the importance of ‘isomorphic logic’. Reviewing: isomorphic logic is the process of employing the implicit logic of a known system to understand the implicit logic of an unknown system. As we employ this technique inadvertently to form abstractions and inform our behavior, this reasoning process is hardwired.
As an important part of our genetic code, Nature, via evolutionary forces, must have chosen this form of logic for a reason. Why? Metaphoric logic must be a very powerful and effective form of reasoning for negotiating our dynamic and turbulent world. Employing isomorphic logic must have enhanced our chances of surviving to pass on our gene pool. With Nature’s endorsement, it seems fair to claim that isomorphic logic is a strong means of validating the Math-Fact Matrix that defines the Realm of Attention.
In addition to being the logical process behind abstract thought and playing a significant role in human behavior, isomorphic logic is a fundamental technique of science. Scientists regularly employ the known logic of mathematics to better understand the unknown logic of empirical systems. After establishing the validity of a Math-Fact Matrix, scientist regularly ask: what does the mathematics imply? This is the essence of the isomorphic logic discussed. Let’s examine these ideas in more detail.
We build our initial abstractions from the underlying logic of our senses, e.g. sight. However, sensory logic lacks precision in that it is inadvertently applied to real life phenomena in a general and approximate fashion. In contrast, mathematical logic is specific and precise.
Logic of Sensory Systems = General and Approximate
Logic of Mathematical Systems = Specific and Precise
Due to this precision, the scientific community prefers to employ mathematical systems, when possible, as a metaphorical map to better understand unknown systems. They love the absolute precision and transparency of mathematical logic. Me too!
There are many famous examples of this congruence between mathematical logic and empirical reality.
Examples of Math/Fact Congruence:
Newton: Force = Mass x Acceleration
Gas Law: Volume = Pressure x Temperature
Ohm’s Law: Voltage = Induction x Resistance
Simple experiments can verify these equations. Change the variables, and observe the predictable results.
After identifying logical symmetries between a mathematical system and a particular physical phenomenon, scientists are able to make precise predictions. For instance, Kepler found that the mathematical ellipse matched the planetary orbits about the sun. This discovery enabled astronomers and astrologers to better predict planetary position. This strategy has proved successful beyond imagining in the material world.
Due to this almost miraculous success, many scientists have attempted to map the mathematical logic of the material world onto the living world of Attention. However, Attention has proved resistant to this type of explanation. As this approach has provided very little explanatory power, if any, many consider it to be an unsuccessful abstraction. Could Material Mathematics be the wrong mathematical map?
In contrast, applying the logic of DS Dynamics has proved to be a useful way of understanding the phenomenal network of the Realm of Attention. Because this approach provides an abundance of explanatory power, it is a useful abstraction. This congruence/isomorphism between the mathematical logic of Data Stream Dynamics and the underlying logic of Attention enables us to better understand and make predictions regarding the behavior associated with Attention.
This is a crucial point. It is important to find the right mathematics for the job. Just because a mathematics is able to accurately characterize one aspect of existence, does not imply that it is the right map for another phenomenal web. We do not hire a lawyer to fix our plumbing, nor a plumber to tend our legal affairs.
Indeed, mathematican-scientists have regularly developed new mathematical systems to deal with phenomena that old mathematical systems couldn’t address. For example, Maxwell developed field theory to encompass the behavior of electricity and magnetism. Newton (calculus), La Place (differential equations), and Dirac (quantum mechanics) all originated new mathematical systems to provide a better map of a unique phenomenal network. In similar fashion, I developed Data Stream Dynamics (DSD) to address the attention-related behavior of living systems.
In each case, the logical symmetries between the new mathematical system and the scientific data far surpass those of the older systems in both precision and explanatory power. In the terminology of the current discussion: when scientists are unable to establish an isomorphic relationship between an older system and a phenomenal web, they attempt to develop a new system that does. This is indeed the case with the older material mathematics, Attention’s phenomenal web, and DSD, as we shall see.
Why identify a new type of logic? What’s the matter with the old types of logic? Aren’t they enough?
Deduction and Induction are single chain logic. Deduction works on assumptions to draw necessary conclusions. Induction makes general inferences from specific examples.
Based upon a single chain, these fundamental types of logic are fragile. A faulty assumption voids any deductive conclusions. A single contrary example voids inductive inferences.
Metaphoric and Redundancy Logic (addressed in a subsequent chapter) are based in networks – chain-mail logic. Because of logical reinforcement from many angles, chain-mail logic is very stable. A finely woven basket can still hold water even if a single thread breaks. The integrity is due to the stability of the unbroken threads.
This is also true of chain-mail logic. Metaphoric logic is based in a network of logical symmetries between 2 systems. If one of these symmetries is flawed, refinements are made. The isomorphic relationship between systems is only invalidated if multiple symmetries are flawed.
Redundancy logic is based upon multiple sources of cross-validation. Metaphoric systems could be one of these sources of validation. If a single thread is challenged, this doesn’t necessarily threaten the web. Rather than abandoning the matrix, a new strand could be created or old ones strengthened.
It could be argued that Metaphoric Logic is a type of Induction, which indeed it is. After all, this type of logic is based upon a simple assumption; If there are multiple logical symmetries between two systems, then there could be more. Corollary: By understanding the logical structure of one, we can better understand the other. We inductively infer a general proposition from multiple individual cases.
However, it is a very special type of Induction. Rather than logical propositions, the elements of isomorphic logic are logical symmetries. Logical symmetries are not solitary, but instead are between two systems. Further our given is that there are multiple symmetries between the two systems not just one. In other words, the two systems are logically connected in multiple ways.
Because of interactive complexities, it could the argued that Metaphoric Logic is orders of magnitudes more complex than the simple propositions that are generally associated with Induction. To say that Metaphoric Logic is a form of Inductive Logic would be akin to claiming that a B-52 Bomber is a type of balloon. Both are misleading statements that obscure some real differences.
The same analysis holds true of Redundancy Logic. The basic proposition is that the greater the number of sources in a validation network, the higher the likelihood of truth. For instance, if all five of our senses tell us the same thing, we believe them. If we only hear something, we look for other sources of validation before jumping to conclusions.
Like Induction, a generalization is made from individual examples. Again rather than logical propositions, the elements of redundancy logic are validation sources. Instead of only one, there are multiple sources that point to the same conclusion. The given, while based in multiple examples, is a complex network. Redundancy logic can’t even by applied until there are at least two sources that point the same direction.
Due to this interactive complexity, Redundancy Logic is, like Metaphoric logic, orders of magnitude more complex than ordinary Induction. Again, calling it a form of Induction is a simplification that obscures rather than clarifying the importance of this special form of logic. In similar fashion, calling a cell a type of non-living matter obscures rather than clarifying the importance of this special form of matter.
As types of chain-mail logic with multiple layers of complexity, the conclusions derived from Redundancy and Metaphoric Logic can be very stable, especially as compared to the single-chain logic of ordinary Deduction or Induction. We employ the stability of these unique forms of logic combined with Explanatory Power to establish the validity of the Attention Matrix. Recalling the greater context, this Math-Fact Matrix with its unique ‘physics’ and network of phenomena defines the Realm of Attention and places the realm that is unique to living systems on an equal footing with the Material Realms.
We are now going to develop an algebraic language for the isomorphic logic. We will substitute symbols for concepts and equivalencies for sentences. This symbolic language will enable us to express the concepts we have developed with more brevity. In such a way, we can avoid the lengthy wordiness of sentences and paragraphs. By adopting this form of precise expression, we hope to crystallize understanding of our unorthodox way of thinking.
We use the term ‘algebraic’ in the verbal sense that symbols are substituted for words and equivalencies are substituted for sentences. This mathematical language is for explication only, with virtually no relationship to numbers. Unlike ‘real’ algebra, these equivalencies cannot be employed to construct mathematical derivations. Our ‘explanatory’ algebra is simply a method for making our positions as clear as possible by eliminating all superfluous verbiage.
Let’s commence this unusual discussion with some notational clarifications. In the following series of algebraic statements, we compare the logic of one system, S, with that of another system, T.
What is the definition of ‘system logic’?
A system has components that interact with each other in a relatively regular way to produce relatively regular results. The system’s logic consists of the underlying logical network of the system’s interactive processes. When we compare systems we are comparing their underlying logical structures, not their components.
Let S = any kind of system, e.g. mathematical, emotional, physical
Let S Logic = the underlying logic of the system
The ‘approximately’ symbol ‘≈’ indicates that the logic of the two systems is fairly equivalent/isomorphic. The ‘does not equal’ symbol, ≠ , indicates that the logic of two systems is not equivalent/isomorphic.
Let ‘≈’ = Logic of 2 systems is equivalent/isomorphic
Let ‘≠’ = Logic of 2 System is not equivalent/isomorphic
The subsequent statements compare the logic of 2 systems (S and T). The first equation states that System S is logically isomorphic with System T. The second equation states that System S is not logically isomorphic with System T. Following are algebraic representations of these statements with our notation.
S Logic ≈ T Logic
S Logic ≠ T Logic
Recall that these are verbal, not mathematical, statements. As such, these equivalencies (equations) are relative rather than absolute. The logic of one system is never an exact replica of another. Rather than either-or, the logical equivalence (isomorphism) of systems is always ‘more or less’. Rather than ‘right or wrong’, ‘true or false’, the isomorphism is on a sliding scale from ‘approximately equal’ to ‘not applicable’. This is due to the fact that the logical relationship between systems is metaphorical.
Because the relationships are metaphorical, the equivalencies are never exact, except in the case of 2 mathematical systems. This is due to a simple fact – individual system logic is not precise, with one exception – mathematical systems. The logic of exclusively material systems is almost exact, but not quite. While consistent, the logic of Attention is approximate.
Further, these equivalencies might only apply to subsets of the system. Accordingly, the generalization of each symbol requires refinement. For instance, Attention Logic is a broad concept, which has many parts. Despite this fragmentation, there are enough unifying features to justify some generalizations, as we shall see. With these clarifications, descriptions and disclaimers out of the way, let’s get into the specifics.
Preceding chapters argue that Attention and Matter require different types of mathematics. Let us clarify our position with some explanatory algebra. Scientists have found that the logic of Calculus frequently matches the logic of Matter. My investigations indicate that LA Logic is a fairly good match for Attention Logic.
Calculus Logic ≈ Material Logic
LA Logic ≈ Attention Logic
It is amazing that a mathematical system’s logic can be almost exactly the same as the logic of a material system. The great mathematician scientists, (Ptolemy, Newton, Maxwell, Einstein, Dirac et al) have conclusively demonstrated this miraculous logical congruence by generating equations that match the patterns of empirical data. It is equally amazing that a mathematical system’s logic can be approximately the same as the logic of Attention. This mathematical approximation of living behavior, although not exact, still evokes awe. Indeed this wondrous match between Abstract Math and Empirical Reality remains one of the great mysteries of existence. Everyone knows What; No one knows Why.
Despite the mathematical miracles of the Regular Equations of Calculus, it is unable to match the logic of the interactive feedback loops associated with Attention. Similarly, the logic of a Reflexive Equation such as the LA is unable to match the logic of stimulus-response associated with Matter.
Calculus Logic ≠ Attention Logic
LA Logic ≠ Material Logic
These are general abstractions. Let us examine some specific examples. Earlier chapters developed the following two propositions with regard to choice. Open and iterative, the Logic of LA’s system provides a good match for our Common Sense Logic regarding choice. Closed and continuous (no input and no opportunity to choose), the Logic of Calculus is unable to encompass Choice Logic.
LA Logic ≈ Choice Logic (Common Sense)
Calculus Logic ≠ Choice Logic (Common Sense)
What about some evidence? Let’s examine some of the logical equivalencies that we will encounter between scientific facts and LA Logic.
The LA generates an entire mathematical system DSD, which consists of many subsystems. The Pulse is the most significant of these logical subsystems. A subsequent chapter illustrates that the logic of the Pulse is similar to the logic of the SAE, more popularly known as the Flow Experience. Both the Pulse and the Flow Experience require duration to achieve their ideal dimensions/potentials and interruptions have a negative/adverse impact upon these same ideals that is disproportionate to their size.
LA (Pulse Logic) ≈ Attention (Flow Logic)
The Triple Pulse is another significant mathematical subsystem of the LA. The Triple Pulse Logic is exceedingly similar to the internal logic of our human sleep-cognition system. Shortening or interrupting the Triple Pulse’s Rest Pulse exhibits a negative impact upon the ideal dimensions of the following Active Pulse, just as shortening or interrupting our sleep cycle has an adverse impact upon our cognitive abilities the following day.
LA (Triple Pulse Logic) ≈ Human Behavior (Sleep/Cognition Logic)
Triple Pulse Logic is also similar to the Biological Logic of Sleep, as characterized by Dement’s Opponent Process Model.
LA (Triple Pulse Logic) ≈ Biology (Dement’s Sleep Logic)
Combining the statements, Triple Pulse Logic is fairly equivalent to both the biological and behavioral (cognition) logic of sleep. This commonalty extends the dimensionality of the LA system.
LA (Triple Pulse Logic) ≈ Biology + Behavior (Sleep Logic)
There is a scientific consensus that the interaction (x) between evolutionary mechanisms and living systems has produced every biological system along with its logical substrate. In our terminology, evolutionary logic applied to Life produced the biological system associated with sleep described by Dement.
Evolution x Life = Dement’s Sleep Biology
Rather than an entire living system, evolution’s natural selection generally operates upon a specific feature. For instance, canine evolution focused upon the sense of smell associated with the nose. What individual feature of living systems did evolutionary processes interact with to give birth to Dement’s biology of sleep?
Due to the logical congruence between them, we suggest that evolutionary forces must have worked upon the LA to produce the biological systems associated with Sleep. But in what capacity? How is the LA connected with Life?
Evolution x Living Algorithm = Sleep Biology
Could the LA be Life’s computational tool? It certainly fits the requirements. If so, the biological system associated with sleep could have evolved to take advantage of the LA’s capabilities. Why? Could it be to maximize our relationship with information?
Living Algorithm = Life’s Computational Tool
As a validation of this analysis, LA Logic is also a good match for the Logic of Posner’s Attention Model, another scientific fact. Like Dement’s model, Posner’s model is linked to biological systems. While it has broader ramifications, Posner’s model reveals the logical structure of the Biology of Attention. By extension, LA Logic is similar to this logic (shown below in algebraic format).
LA Logic ≈ Attention Biology Logic
Employing the same line of reasoning, the interaction (x) between evolutionary forces and living systems generated the biology of attention over an exceptionally long period of time. What feature of living systems did evolution operate upon? Again due to matching logic, we suggest natural selection operated upon the LA, acting as Life’s computational tool, to generate the biological systems associated with Attention.
Evolution x Living Algorithm = Attention Biology
Can the LA fulfill the requirements for this crucial role? Subsequent chapters illustrate that the LA’s capabilities are ideally suited to satisfy Life’s computational needs. In contrast, traditional systems (Physics, Probability, Calculus and Info Theory) have innate features that prevent them from fulfilling this role.
LA Logic ≈ Life’s Computational Logic
Physics/Probability Logic ≠ Life’s Computational Logic
Calculus/Info Theory ≠ Life Computational Logic
It seems that LA Logic fits many features of Attention: 1) our Common Sense Logic with regards to Choice; 2) behavioral, biological and evolutionary evidence regarding Attention; and 3) Life’s computational needs.
It is evident from this analysis that the LA’s connection with Attention is not based upon a single logical strand. Rather the logic of the LA’s mathematical system, Data Stream Dynamics, has a similar structure to the logic of Attention’s system, which consists of a broad array of scientific facts from multiple disciplines. This interlocking web of multi-dimensional logical strands creates a tightly woven tapestry that provides credibility for the Math/Fact Matrix that defines the Realm of Attention.
It seems at least plausible that LA Logic and Attention Logic are isomorphic. Could there be a better fit? Is it possible that theoretical physicists are on the verge of a momentous discovery that will link some branch of Material Mathematics (perhaps Quantum Mechanics) with Attention? Or could it be that the Logic of Matter and the Logic of Attention are inherently incompatible? Check out the next chapter for an in depth discussion of this topic.