Mathematical and Object Oriented Data Models
for the Open Simulated General Intelligence
(Open-SGI) Data Multiverse
Working Draft Version 0.02
Abstract
Overview
Introduction
In order to create something artificial requires an understanding of the ‘real thing.’ Since Science has yet to solve the mystery of general intelligence we should adopt a simulation based paradigm for developing autonomous and general intelligence systems. Hence the term simulated general intelligence (SGI or Siggy).
An example of a Siggy is a personal virtual assistant (PVA) such as Siri, Cortina and Alexa that can look, listen and simulate general intelligence. A Siggy could also be a plugin, for example for a home security system that could see an animal approach my front door, determine it is my cat Henry, and make sure Henry isn’t trying to bring one of his friends (a mouse, chipmunk or other small animal) into the house to play, before unlocking the cat door. Another example of a Siggy is a phone plugin (or standalone system) for walkers, joggers and bike riders that would hear an approaching vehicle, countdown the arrival time, and warn the user to get off he road if the approaching vehicle is going to be too close for comfort. Siggies would manage our tasks, personal and family inventory, security system, environment systems (temperature, humidity) and entertainment systems, our finances, our medical and insurance information. In our car, Siggies would interact with our navigation systems. Siggies would also interact with other Siggies. The list of things that Siggies will be able to do is probably infinite since society and associated data needs are constantly evolving. Siggies could pretty much manage all of our data. Soon Siri, Cortina, Alexa, and their lesser known brethren will be able to do some of these things. I’m sure their parents are working on it. But the internet of things is vast and growing, ever changing. How could any one PVA do everything? How could any one PVA interact with all of the possible product brands? Perhaps Apple will make a deal with Whirlpool to support all of their appliances. Google would do the same with LG and Amazon would support Bosch and Haier. But what if I like Haier kitchen appliances and LG washer/dryers? Clearly, in order to fully realize the potential of SGI common and open data/service models and supporting protocols are needed. Until this happens, plug in implementors will have to rely on the monolithic systems and their Big Data to be willing and able to interoperate, or do it themselves.
The Open-SGI Data Multiverse provides the foundation for construction of open, interoperable and ethical data/service models and supporting protocols that will foster innovation and competition and democratize data resources. Systems based on the Open-SGI Data Multiverse will be testable and will be able to demonstrate and certify compliance with emerging ethical recommendations and standards such as those specified in the IEEE Ethically Aligned Design for Autonomous and Intelligent systems.
Organization
• The topic os Section 2 is a brief overview of the Open-SGI Data Ontology, which is based on a form of philosophical realism I have discovered in the course of my elaborations to Holism
and Evolution . 1
• The topic of Section 3 is Axiomatic Semantics. I introduce, in brief layperson terms the 2 concepts that establish the mathematical basis of the Open-SGI Data Multiverse.
• The topic of Section 4 is the axiomatic semantics of a Data World, which combines the axiomatic semantics of data level and the axiomatic semantics of a unity of opposites data sphere.
• The topic of Section 5 is Holistic Correspondence, also discovered in the course of developing my elaborations to Holism and Evolution. Holistic Correspondence is a philosophical hypothesis that I have adopted to classify objects in the Open-SGI Data Multiverse.
• The topic of Section 6 is the Guidelines for Definition of Intelligence Data Objects (GDIDO) abstract syntax notation which is based on the concepts of open network management and its 3 associated abstract syntax notation, the Guidelines for Definition of Managed Objects (GDMO) and the Common Management Information Service Element and supporting Protocol (CMISE/CMIP).
• The topics of Section 7 are a summary of the key concepts, a reality check couched in terms of semantic holism, a reemphasis of the need for open standards and a trailer of sorts of my work in progress to use MOO Models to demonstrate conformance to the IEEE Recommendations for Ethically Aligned Design of Autonomous and Intelligent Systems.
• Section 8 is the Appendix and includes the topics noted above.
Intended Audience
This document is intended for:
• Autonomous and intelligent system architects, data scientists and developers.
• Venture capitalists. I believe that there is need for start-ups that could provide packaging, version control, internal/interoperability testing services and also pioneer functional area specific implementation.
• Academia, for submission with ,my applications for graduate study, ideally in philosophy or computer science but mathematics, language, psychology, political science and sociology are also candidates.
• Representatives of standards organizations initially IEEE and NIST. Currently I am a contributing member of the IEEE SA via their open public engagement initiative.
• Representatives of other AI non profit organizations e.g. the Allen Institute for AI, OpenAI, European Association for Artificial Intelligence, Future of Life Institute, Partnership on AI
This is the topic of my next paper Introduction to Holism and Mathematics
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The supporting mathematics for Axiomatic Semantics are provided in the Appendix.
Objective
Aside from informing my objective is to elicit critical commentary and collaboration. In short to help or join in my mission and vision at Open-SGI.org to define standards for simulated general intelligence and create the framework for an open, interoperable and ethical network of autonomous and intelligent systems, the Open-SGI Data Multiverse.
Scope
This paper covers many areas at a high level. Initially I had intended to write separate papers covering the Holistic Mathematics, Holistic Correspondence, the Open-SGI Data Multiverse and the Guidelines for Definition of Intelligence Data Objects. In combining them the paper is long and may need more descriptive detail and examples.
References
My primary references are as follows:
• Holism and Evolution 2nd edition copyright 1927 The McMillian Co. https://archive.org/ stream/holismandevoluti032439mbp/holismandevoluti032439mbp_djvu.txt
• Philosophy of the Yi Unity and Dialetics Supplement to Volume 36 2009 Journal of Chinese Philosophy edited by Chung-ying Cheng and On-cho Ng
• An Introduction to Axiomatic Systems by Burnett Meyer copyright 1974 Prindel, Weber 7 Schmidt Boston Massachusetts
• ISO/IEC 10165-4:1992, Information technology - Open Systems Interconnection - Structure of management information - Part 4: Guidelines for the definition of managed objects
Reason for Re-issue
Data Ontology
Simulation of general intelligence requires a data model that embodies a complete monistic ontology. We can visualize such a model as a unity of data about physical and metaphysical events that is analogous to a magnetic field, which has
a three dimensional spherical structure.
I view data ontology as a sub branch of philosophical ontology that deals with identifying the types and nature of data that can exist. All software systems are predicated on a narrow, rudimentary data ontology and exhibit a corresponding narrow, rudimentary simulated intelligence. They ‘know’ all of the possible valid equivalence classes of input data that exist and they ‘know’ how to perform input driven processing that is creative, as evidenced by the output equivalence classes of data that are a result of ‘thinking’ and ‘doing.’ The ontology for the Open-SGI Data Multiverse requires more than this. It must be complete in the philosophical sense. A Siggy must be able to differentiate between data that is descriptive of physical subjects, and events, and data that is descriptive of metaphysical subjects and events. “What is happening? And “What does it mean?” History has produced many theories but there is no form of realism that is generally accepted by all who understand and practice Philosophy.
In addition to being complete the ontology for the Open-SGI Data Multiverse must also be monistic. Aside from the philosophical preference for optimal simplicity (Occam’s razor) from a programming point of view it would add unmanageable complexity to have different models for physical and metaphysical data (a dualistic ontology) let alone different models for equivalence classes within these two primary classes (a pluralistic ontology). Therefore I have established it as a requirement that a Siggy needs to be able to deal with metaphysical information-data as if it is real, and further it needs a rational and consistent way to represent the unity of metaphysical information-data and the associated physical physical information-data.
In developing the mathematical and object oriented data model (MOO Model) for the Open-SGI Data Multiverse, I have gained significant insights from my mathematics education, Holism, as defined by JC Smuts in Holism and Evolution and the Yi unity of opposites concept. I have also read extensively on a variety of topics including but not limited to recent work on semantic holism, the many forms of philosophical realism, theories of mind, personality and psychology. The formal development of Holism and Mathematics is a work in progress and will probably take me as long to complete as it would take to get a doctorate degree.
In this paper I will only touch on the philosophical concepts when I believe it is necessary to explain the MOO Modeling methodology, and such is the case for the Yi philosophy assertion that the unity of opposites is primarily ontological and ontologically primary which I interpret 4
related noetic phenomena; and that reality is a unity of physical and not-physical (metaphysical) aspects. We have already seen speech and image recognition become a commodity. The logical next step will be the recognition of video and audio events that occur in physical space and time. As our ability to examine and compare physical event data increases, it will become apparent that the Siggy data must reflect a unity of opposites: physical and metaphysical event data. 5 We can establish the basis for such a model when we we view the unity of physical and metaphysical event data as being analogous to a magnetic field. The reader may recall doing the magnetism experiment; you started by placing a piece of paper over a bar magnet and then sprinkled iron filings on the paper around the magnet. You observed that each iron filing became magnetized and that they aligned themselves within the magnetic field. This experiment demonstrates that at each point in a magnetic field the charge is not entirely positive or negative, it is both positive and negative; the degrees of each being a function of the location of the filing in the magnetic field.
An analogous observation can be made with respect to the conceptual field of a unity of opposites. Each point in the field would represent both the physical and the metaphysical data the relative degrees a function of the location in the field. It bears noting that the sphere has historical acceptance as a symbol of wholeness for both Western and Eastern philosophy. Further, the mathematical concepts of diametric opposition, polar and degrees of differentiation appear to be well suited for conveying meaning, and the simple concepts of prime numbers appear to be well suited for representing the progressive acquisition of data via initialization, learning and upgrade.
Like a magnetic field, the field of a unity of opposites has a spherical structure. It is this spherical structure that provides the initial framework for axiomatic semantics.
This ontological view of reality data as a unity of physical and metaphysical event data is only one example of the
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Axiomatic Semantics
By applying the concepts of axiomatic systems, meaning can be represented and conveyed mathematically by the data world coordinates.
I view axiomatic semantics as becoming a separate branch of linguistics that is concerned with the mathematical representation of meaning, including:
• Formal axiomatic semantics, which would deal with the the logical aspects of data meaning, such as sense, reference and implication. 6
• Elemental axiomatic semantics, which is the way numbers are used to convey data meaning. The concept of elemental axiomatic semantics is similar to the concept of lexical semantics. • Dimensional axiomatic semantics, which the use of one, two and three dimensional coordinate
system systems to convey data meaning. The concept of dimension semantics is similar to the conceptual semantics.
• Discipline specific semantics, which is the way the mathematical disciplines, such as Arithmetic, Geometry and Algebra are used to convey data meaning.
Axiomatic Semantics is based on the concepts of axiomatic systems which is typically learned as an advanced topic of mathematics. Most people learn the basic mathematical disciplines (arithmetic, algebra, geometry and perhaps trigonometry and calculus) without first learning the underlying theory of numbers and how we are able to operate on numbers based on the principles of axiomatic system. I first learned these concepts in the course of obtaining my BA in Mathematics.
I recognize that for most non-mathematics majors, the mere mention of number theory and axiomatic systems has throw them into brain-lock. The good news is that acquiring an understanding of, and using, the Open-SGI Data Multiverse does not require an understanding of of number theory and axiomatic systems. Some of the details related to number theory and axiomatic systems is provided in the Appendix.
Data World
A data world is a spherical three dimensional model that represents data meaning in terms of three data dimensions: Data Level and the two dimensions of a unity of opposites sphere: Data Latitude and Data Longitude
Data Level
The data level represents the ordered, progressive, named and numbered emergence of meaningful data elements and are represented by the numbers starting with 0 and theoretically going to infinity.. There are two types of meaning associated with each data number, one that is semantic and one that is quantitative. The semantic aspect of meaning is conveyed by the assignment of the name. The quantitative aspect reflects the traditional numbered ordering system.
Multiple data level line segments can be used, it is not required that one ordered progression of numbers represent all possible meaning. Multiple data level lines can exist within the same or different data system, galaxy or universe.
Prime Data
The concept of prime data level identical to the familiar concept of prime numbers which means it is dependent on the concept of data multiplication, which I discuss in the section that follows. Prime elements of data only have themself and the data multiplication identity 1 as factors. Defining, in any ultimate sense a standard, common progression of prime data levels would be an intensely philosophical and linguistic exercise, having interdependencies, and requiring collaboration, with many academic disciplines. For Open-SGI and the Data Multiverse, while it is still an enormously complex undertaking it is still much easier to contemplate the standardization of of prime data levels because the objective is not to represent ultimate reality, the objective is to create good data models that can support ethical interoperability within well defined functional areas.
individual standards or recommendations. Taking this one step further, the implementation specific data primes would begin with M + 1. The prime data that follows would reflect incremental learning and learning from upgrade, which is included separately because it entails different types of processing.
The implementation of traditional audit logging will be able to provide a look-back capability that tells us why an SGI made the decision it did in terms of ethical and functional data factors. This will enable us to identify data that requires refinement, and ensure that the refined data is automatically applied in each case where it is a factor.
The following is an example of how the core data primes could be defined. The focus is on the polar concepts of the core unity of opposites and can support the Open-SGI data epistemology aka the Data Multiverse.
1. Data element 1 represents the concept of being, the quality or state of having existence. 1 is 7 the data identity element for the set of all Open-SGI Data Multiverse elements. All data elements have the data of existence as a factor. Even data elements that are associated with future events or possibilities, their mere contemplation is existence.
2. Data element 2 represents the metaphysical quality of being. The metaphysical quality of being is a factor of all even number data numbers. Data is fundamentally of a metaphysical nature and would represent universals; such as Platonic forms, properties, attributes and other metaphysical existents.
3. Data element 3 represents the physical quality of being. The physical quality of being is a factor of all odd number data numbers, but all odd data does not have 3 as a factor. Note that when we think about the data numbers in their alternate sense of magnitude 2 data is closer to the concept of fundamental existence (1) than 3 data.
4. Data element 4 represents two orders of reinforcement of the metaphysical quality of being. 8 5. Data element 5 represents universality.
6. Data number 6 would represent the conceptual unity of physical and metaphysical data. All data that is directly tied to experience will have 2, 3 and 6 as factors.
7. Data element 7 represents individuality.
8. Data element 8 represent three orders of reinforcement of the metaphysical quality of being 9. Data element 9 represents two orders of reinforcement of the physical quality of being 10. Data number 10 represents metaphysical universality.
11. Data number 11 represents benevolence
12. Data number 12 represents two orders of reinforcement of metaphysical combined with physical
13. Data number 14 represents malevolence
I may want to provide another example of assigning semantic content to the initial primes that follows the Yi
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One of the problems with language is that there are multiple meanings for the same word and multiple combinations of different words can convey the same meaning. This redundancy combined with the reality of semantic holism manifests itself in the misinformation, misunderstanding, confusion and conflict that arises and is inherent in our traditional languages and interpersonal/international dialogue. All of this would be largely addressed by a rational set of core and standardized ontological, epistemological and ethical prime.
Binary Data Operation
When we are dealing with data numbers semantically there is only one binary semantic operation; multiplication. Semantic multiplication will produce the usual numerical result that uniquely identifies an element of data as the product of semantic factors. For example, if the numbers 1, 2 and 3 represent existence, university and individuality respective than the number 6 represents the semantic content semantic content of the first data element 2 x data element 3 equals data element 6 that can not be represented by a single word. Loosely translated, 6 means “the semantic product of one universality and one individuality semantic factor” The number could also be assigned the meaning “the unity of universality and individuality.”
Implicit to the concept of data multiplication is the concept data factors. Whenever two or more semantic elements of data need to be or are found to be related, that relation is represented by their product. The semantic data elements are multiplied to create a new semantic data elements. Semantic element multiplication combines data element meaning in both a conceptual and a quantitative manner. When we are dealing with the elements of data within the quantitative system traditional addition and multiplication are defined. The difference between data elements can provide an indication of complexity or conceptual separation. For example, the conceptual separation between existence and the unity-of-universality-and-individuality = 6 whereas the conceptual separation between universality and individuality = 1; the concepts of universality and individuality are more closely related by the numerical (not semantic) factor 6.
Data Identity Element
When we are dealing with data numbers semantically there is only data multiplication and the assignment of the meaning identity element, which is 1, will be implementation specific and a candidate for standardization by functional area. This is especially important if the objective is to define a semantic group. If we are able to do then all theorems and proofs that have been developed for groups would be at our disposal as semantic tools. The multiplication identify element has to be the number 1 in order for us to define a consistent axiomatic group.
To could represent the entirety of initialization, it could represent a single concept, such as existence or it could represent an SGI’s core values or ethical standards; ideally as specified by formal standards such as the IEEE Global Initiative on Ethics of Autonomous and Intelligent Systems and standards that are to be developed and published by NIST. 1 could also represent the concept of God because God is the only thing that is a factor of all that exists.
For the Open-SGI Data multiverse my current thinking is that 1 will represent the concept of existence.
Data Inverse Elements
As noted earlier, inverses are required if the objective is to define an axiomatic group. I invested a significant amount of time attempting to make this work consistently that was demonstrable by examples and was unable to do so for reasons as follows:
Each time a new element of data comes into existence it would be required to instantiate the inverse simultaneously. From the earlier example, the initial instantiation of 6 would require the instantiation of 1/6 which would be an unnecessary complication, and not straight forward due to the fact that the inverse data element in all likelihood would not be initially associated with a physical event.
When defining data element inverses we are essentially introducing data division; just as is the case with ordinary arithmetic, data division would be defined as data multiplication of the data inverse. For example, data element 210 would be created by three separate data operations involving the elements 2, 3, 5 and 7. If we wanted to remove the 2 data fact we would do so by multiplying 210 by 1/2. Data inverse elements would have to be initialized and maintained to be different in every way possible, aside from sharing the same data identity element. I have a strong doubt that this can be done in a way that is demonstrably consistent.
This distinction between a data element and it’s inverse does represent duality, but it does not represent the unity of opposites. Visualization of the progression of data from 1 to infinity and at the same time looking at the infinite progression of inverse data elements between 0 and 1 does not look like a whole, it looks like two different and disconnected sets of infinities.
Commutativity, Associative and Distributive Properties
butter and celery” means the same thing as “I know how to prepare a peanut butter and celery snack and I have both of them” followed by “I’m hungry.”
I am close to certain that the semantics of axiomatic rings and fields must be limited to numerical semantics such as the order, magnitude or distances associated with data numbers. There are cases where this can be useful, for example two global properties, currentDataLevel and lastDataInitializationLevel:
IF currentInformationLevel > lastDataInitializationLevel THEN CALL update_learned_information_function
ELSE CALL update_initialized_information_function ENDIF
Unity of Opposites Data Sphere
Dimensional Semantics
The unity of opposites data sphere is a two dimensional model (a unit sphere). The latitude dimension of a unity of opposites sphere represents degrees of meaning relative to the diametrically opposed points designated as the conceptual poles of the unit of opposites sphere. The latitude at the equator of a unity of opposites sphere is 0 degrees, the latitude at the North Pole is 90 degrees and the latitude at the South Pole is -90 degrees.
The semantic longitude dimension of a unity of opposites sphere represents the degrees of meaning differentiation around the great circle equator of the unity of opposites sphere from an arbitrarily assigned starting point. You can draw any half of a great circle starting from the North Pole and ending at the South Pole and designate it as 0 degrees longitude. Although it’s assignment is arbitrary 0 degrees longitude plays a crucial role in establishing a standard basis for representing locations on maps of the unity of opposites sphere. On Earth zero degrees longitude is defined as a point in Greenwich, England and it is called the Prime Meridian. When modeling, the use of 0 degrees longitude would be implementation specific and again, a candidate for standardization. One possible use would be establishing a convention that 0 degrees longitude contains the location of the dictionary and directory of maps for the semantic sphere.
Elemental Semantics
The reason why a circle is measured in degrees, and why 360, has not known. The reason that is commonly given refers to the Egyptian calendar which had 12 months of 30 days each for 360 days; there were five additional days that the Greeks considered to be outside of the months bringing the count to 365 days per calendar year. The radian is a more accurate unit of measure that is used extensively in caucus and is expressed in terms as follows: 2 x pi x the radian = the circumference of the circle; one full revolution. Since Pi is a transcendental number (more than merely irrational) and difficult to calculate with, the radian method is not used. Each degree can be divided into 60 minutes and each minute can be divided into 60 seconds and further subdivision is possible if needed.
Recall that with the data dimension we were not able to use the numbers to represent conceptual differentiation due to the discontinuity between a number and it’s multiplicative inverse. Within the unity of opposites of sphere the elemental semantics is primarily related to conceptual differentiation, e.g. degrees of difference.
If we define leftward movement as evolution, there are progressively 180 degrees of evolutionary difference between any point on the surface of a unity of opposites sphere and it diametrically opposed point. There are likewise 180 degrees of devolution difference between this point and the original starting point.
A coordinate system is used uniquely represents points on the unity of opposites sphere. All of the information that is conveyed by the unity of opposites sphere is done by or around the concept of data location which is a named coordinate expressed as an ordered pair of the degrees longitude and the degrees latitude.
Have shown that the data dimension supports the establishment of hard coded data during initialization and subsequent learning proceeded as a result of the never-ending emergence of prime data and by the binary operation of data multiplication. There are no elemental semantics on the unity of opposites sphere. There are no restrictions as to the number or types of locations that can be instantiated on a unity of opposites sphere other than that they be located in terms of Longitude and Latitude.
diametrically opposed information that have importance within the named unity of opposes coordinate system.
I recommend that there be a convention and that the first point that is defined on a unity of opposites sphere is the Directory location which would contain the information or pointers to functions that describe the structure and content of information locations within the unity of opposites; the Directory is the storehouse of maps, and other important information associated with the unity of opposites sphere. I have decided that either by standard or by convention; for the Data Multiverse of Open-SGI, the directory location will always have as it’s coordinates 0 degrees longitude and 0 degrees Latitude.
There is an infinity of points on the unity of opposites sphere so it should be obvious that not all points will have information associated with them. All points have the potential of being on a circle of meaning, but it is not always the case. Aside from coordinate locations, all points of a unity of opposites sphere will be contained on an arc because there will always be some conceptual degrees of differentiation between meaningful points.
Meaning is not fully conveyed by a location. Recall that coordinate systems are based on making arbitrary assignments and using them as points of reference. We would like our use of mathematics; our axiomatic semantics, to make information more precise and deterministic because that is how many if not most view mathematics. As a very ridged and precise discipline. In fact, at the very heart of it, the more we understand mathematics we realize that it speaks equally and in fact at time more so, to the holistic aspect of meaning. Case in point, a named location. Does it contain or point to something that is instantiated, that can be seen as being disabled or enabled? If it is enabled is the meaning associated with the location in motion or is it stationary? All of this information and basically any information that is important to SGI will be represented using the an object information model that has been specified using using the Guidelines for Definition of Intelligence Objects (GDIO) notation or some other form of abstract syntax.
Cursory research indicates that the term ‘right angle’ comes from the Latin angulus rectus; rectus can be interpreted as meaning “upright" and hence the term right angle might simply refer to the geometric property of the vertical perpendicular to a horizontal base line as being upright. Rectus also Latin for right as proper, or in accordance to justice or equality. When two straight lines intersect they form two angles. When these angles are equal then this is a fair or just situation and the angles are “right.” From this meaning the application to the unity of opposites becomes apparent; for any pair of diametrically opposed meaning locations the right meaning is 90 degrees, exactly halfway between the two. Eastern philosophy refers to this as the Middle Way, but even with the Middle Way there must be some latitudinal variation.
point of view would need to equally reflect both points of view; essentially saying that there are cases where it is proper and appropriate and cases where it is not.
Meaning is not discrete or static in the sense that it is for the data dimension. Meaning is also fluid, relative and contextual. The terms left and right, up and down, north or south and east and west can be used to describe movement along the surface of the unity of opposites sphere, but the more accurate way to think about movement as either being towards or away from some point of reference. All movement will be along the circumference of some information circle or arc that includes the point of reference. As we move away from this point of reference the distance we travel is expressed in terms of degrees of differentiation. When we get conceptually as far away as possible from the point of reference the distance we will have traveled is exactly half of the circumference of the circle. The line drawn between our current position and the point of reference is a diameter. This is where the term diametric opposition come from. Having travelled 180 degrees of difference and reaching the point of diametric opposition, if we continue moving on the same information circle we are now moving towards our original point of reference. Since our coordinates are express in terms of longitude and latitude, movement is typically expressed in this way as well. When our location and movement needs to be represented in terms of a common point of reference, the conceptual poles (latitude) and our arbitrarily assigned 0 degrees longitude. In fact, the unity of opposite sphere is comprised of potentially an infinite number of diametrically opposed meaning locations; each potentially representing a unique unity of opposites.
Further Axiomatic Semantics
Since the elements represent degrees of differentiation within fixed ranges e.g plus or minus 90 degrees representing the degrees of differentiation between the polar unity of opposites meanings, and 360 degrees of conceptual differentiation associated with any arbitrary meaning circle. There are additional semantics but these are associated with the individual branches of mathematics that can used to convey meaning, for example Algebra, Geometry, Trigonometry, Calculus, Differential Equations, Statistics, etc. When once understand the breath and depth of applicability it becomes clear that fully developing these ideas wold best be achieved collaboratively. The following high level overviews of, Geometry, Field and Algebra give some indication.
Geometry of Meaning in a Unity of Opposites Sphere
It is easier to explain the concepts of geometry within a Cartesian plane. On the surface of a sphere, all lines that can be drawn are curves, there are no straight lines. Mathematics is somewhat different when performed on a spherical surfaces compared to the Cartesian plane, and it is not common for people to have the degree of mathematical literacy that is required to understand and implement. That may not be a problem for SGI ultimately because that’s what programing is all about, encapsulating complex information processing into functions, methods or apps so that they can be utilized by higher level code. But for the sake of explanation in my parallel effort Holism and Mathematics I will limit myself to Cartesian systems.
Field of Meaning in a Unity of Opposites Sphere 9
The meaning of named locations expressed by explicit coordinates provide the basis of meaning associated with metaphysical events. From this initial location a shape that represents a well defined area of meaning differentiation is chosen that further delineates the concept of the metaphorical field. The concept of cooperating meaning is also a part of the meaning field. Cooperating meaning is meaning that is operating at the same physical and/or metaphysic time within the meaning field, and is the most basic and common form of meaning interaction types. The other meaning event types that may be included in the meaning field are coordination and regulation.
The concepts of present, past and future, established by Holistic Correspondence, are also reflected once the concept of the surface of meaning is established. Meaning that has occurred in the past and includes memory and knowledge would be lower than the surface of meaning. Meaning that is associated with the present would occur on the surface of meaning and meaning that is associated with the future would occur above the surface of meaning. These concepts will be more fully developed in which will be discussed in section XX after the next section that deals with the combination of data and the unity of opposites sphere and creates the modeling concept of a unity of opposites world of meaning.
Meaning Algebra in a Unity of Opposites Sphere
Like the concept of field, Algebra can not be fully developed until we get to the 3 dimensional unity of opposites world of data. When introducing the concepts it is again much easier to work in a Cartesian plane rather than a spherical surface. One way to use Algebra is to perform rate, time and distance calculations. For example, when the completion of tasks towards a well defined purpose are seen as the transportation of data meaning, and if there are different programmatic forms of dat meaning transportation, decisions can be made based on the criticality of time and availability of . There are many ways Algebra can be used when associated with the areas and distances between meaning locations rather that the meanings that are associated with the locations by naming.
The concepts of field directly reflect the concept as initially defined in Holism and Evolution and elaborated in my
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Unity of Opposites World of Data
Dimensional Semantics
A unity of opposites world of data combines the axiomatic semantics of the data level dimension and the unity of opposites sphere. Each instantiation of a data number carries the referential semantics associated with the longitude and Latitude coordinates. There are now three dimensions of meaning associated with each named location on the unity of opposites world of data.
While there are no new dimensional semantics added, I believe it is noteworthy to highlight what it means, to both the data counting number dimension and the unity if opposites dimensions, when the two are combined. The data level are no longer restrictive; there is now an entire 2 dimensional unity of opposites sphere for each element of data. Instead of each data element being represented by a zero dimensional point there is now an infinity of points that can be used to represent each element of data.
It is not correct to view the unity of opposites world of data as a series of co-centric surfaces. There is not an entire unity of opposites sphere for each progressive element of data. There is at least one point on the unity of opposites data sphere surface that is a named location and instantiated with meaning.
Consider as an analogy, mountain ranges. They are areas where the surface has become extended and higher in elevation as a result of the seismic factors below. The highest elevation on Earth; the very concept of highest, can only be assigned to one physical or metaphysical location. If the concept of most complex data element is represented as the instantiation of many if not all of the most recent prime data elements the resulting data product would be significantly more elevated than local named locations that have not been upgraded with the new data prime factor. In using this example I am again foreshadowing the Holistic Correspondence method for classifying named locations (which will also be referred to as intelligence objects). The unity of opposites sphere had a constraint associated with its dimensional semantics. There was potentially an infinity of information that could be associated with a named location, but it was not possible for 2 or more named location to share the same location, exactly the same longitude and latitude coordinate. This constraint is removed in the unity of opposites word of data because the dimensional semantics the concept of meaning areas and volumes, layering; the equivalent of semantic skyscrapers and semantic knowledge ranges are a natural outcome.
Elemental Semantics
location that their potential usefulness became apparent when, for example when we assign names to locations that are descriptive of meaning; for example The Directory of Intelligence Objects, The Abstraction Archives, The Purpose Manager. It is the coordinate that provides the real elemental semantics and in combining the data dimension with the spherical dimensions it is the three dimensional named location that is the locatable element of meaning.adding the information level line segment as the 3rd dimension in the unity of opposites world the range of semantic content that can be represented in unlimited.
Further Axiomatic Assessment
As was the case with the unity of opposites sphere, there is no further axiomatic assessment for the unity of opposites world of data because we can’t operate on locations arithmetically. As was the case with the unity of opposites sphere, there is mathematics discipline specific semantics that becomes even more useful with the three dimensional model.
Geometry of Meaning in a Unity of Opposites World of Data
With the two dimensions of the unity of opposites sphere I introduced the concept of data areas in terms of shapes and areas. Now in the three dimensions unity f opposites wold of data we extend the concept of data area to data volume. As was the case with the unity of opposites sphere one would expect that experientially the boundaries and shapes of most, if not all, volumes of data will be irregular, unless we make a conscious effort to define them as regular which we can do, and for the same reasons noted for the unity of opposites sphere and/or to make the boundaries associated with initialization to be more regular which will lend itself to ethical recommendations dealing with transparency. As was the case with the unity of opposites sphere it is easier to explain and use the concepts of geometry within a Cartesian solid.
One example is to establish a correlation between data volumes and processing time. An area of 1000 cubic meters will take 10 times longer to process than an area of 100 cubic meters. The area that is 1000 cubic meters contains 900 more data locations than the area that is 100 cubic meters.
Another example of the use of data volumes is to support the concept of containment. If a compare process is optimized relative to N number of data sets that designate a match, and if the number N is a configureable system parameters the concepts of adding, deleting and modifying data sets can be represented in terms of containment within the volume of data.
Field of Meaning in a Unity of Opposites World of Data
The meaning of named locations as points, which started to lose its significance in the unity of opposites data sphere, becomes even less significant when the field of unity of opposites data can be based, at least in part, on areas and volumes within the unity of opposites data world. The following illustrates one way that the concept of data areas and volumes might be useful. 10 A data building, is an area that is protected from the external data environment and contains intelligence data objects that support data input, processing and output. The area of a data building is determined by its shape, eg. rectangle, square, circle, triangle, etc and the sizes and number of floors. A data building may contain the data of an individual, a family or of an organized data group, institution or organization. A data building may contain one or more floors each of which is occupied by data intelligence objects that may or may not be related e.g. the first floor may be designated as being foundational
The concept of cooperating data is also a part of the meaning field. Cooperating meaning is meaning that is operating at the same physical and/or metaphysic time within the meaning field, and is the most basic and common form of meaning interaction types. The other meaning event types that may be included in the meaning field are coordination and regulation.
The concepts of present and past can be reflected once the concept of the surface of meaning is established. Meaning that has occurred in the past and includes memory and knowledge would be lower than the surface of meaning. Meaning that is associated with the present would occur on the surface of meaning. Meaning that is associated with the climate of data would occur above the surface of meaning. These concepts will be more fully developed in section XX. Clearly the significance of the data location is minor relative to the significance of the data field.
Meaning Algebra in a Unity of Opposites World of Data
Holistic Correspondence
Analogies, or other methods, can be used to classify intelligence related data
The Holistic Correspondence hypothesis maintains that there is a holistic correspondence between the natures of the physical and metaphysical aspects of reality. I discovered the hypothesis in the course of my work to provide elaborations on Holism that (1) addressed the 11
metaphysical difficulties and omissions that Smuts acknowledged in his preface, (2) revisited the inductive argument for holism in light of scientific advancements over the past century and (3) demonstrated the general concept and functions of holism for the metaphysical classes of wholes. Holism, and Holistic Correspondence are not required concepts for the development of mathematical and object oriented models of the unity of opposites to serve as the basis for a data ontology for SGI. The elaborated general concept of holism is important because it is reflected in the Guidelines for Definition of Intelligence Objects notation, and it is also reflected in the specification of the inheritance class for the Open-SGI data multiverse. Holistic Correspondence is important because without it, I do not believe it is possible to demonstrate the concepts of structure and field for metaphysical existents. In short, Holism is broken beyond repair. Holistic Correspondence is also important because it is the method I have chosen to classify the intelligence objects that comprise the Open-SGI data universe. Other Open-SGI data universes can be defined based on different methods of classification that have a nature that does not (attempt to) correspond to the nature of physical reality.
When I say that the correspondence is holistic, I mean that it reflects the general concept and functions of holism. As will be shown in the following section the combination of the GDIO notation and the specification of the inheritance class for the Open-SGI Data Multiverse also reflect the general concept and functions of holism.
I first read Holism and Evolution in the late ‘70s as an undergraduate Mathematics student. At the time I was filling free electives with a Theories of Personality course and one of the topics was Jung’s collective conscious. The combination of Jung, Smuts theory and mathematics led to the rough beginnings of a blueprint for what I then referred to as the Holistic Mathematics of Personality. Later when I became more aware of the ontological and epistemological implications (Smuts wanted his work to be considered scientific as opposed to philosophic) I read Russell’s History of Western Philosophy to find out: have any published philosophers suggested spherical models, or correspondence; to what extent am I seeing something for the first time? More importantly, if holism represents all wholes, and I am saying it can be represented as a three dimensional sphere, can all of these differing views be represented along with Holism in a single model?
Jan Christiaan Smuts Holism and Evolution, 1926 Macmillan, London
In the 40 years since I began my journey I have explored a very small fraction of the surface of many areas of knowledge that I considered to be tangental: neuroscience, relativity, quantum physics, complexity, chaos, string theory and the concept of a theory of everything, psychology, language, western and eastern philosophy, religion to name a few. It was not until relatively recently that I began to piece together the puzzle that became axiomatic semantics.
The following section identifies the concepts and functions of Holism that I have incorporated into the design of the GDIDO and the inheritance class for the Open-SGI Data Multiverse.
Synopsis of Holism and Evolution 12
After identifying the need for reform in the fundamental concepts of matter, life and mind (chapter 1) Smuts examines the reformed concepts (as of 1926) of space and time (chapter 2), matter (chapter 3) and biology (chapter 4) and concludes that the close approach to each other of the concepts of matter, life and mind, and the partial overflow of each other's domain, imply that there is a fundamental principle (Holism) of which they are the progressive outcome. [8]:86 Chapters 5 and 6 provide the general concept, functions and categories of Holism; chapters 7 and 8 address Holism with respect to Mechanism and Darwinism, chapters 9-11 make a start towards demonstrating the concepts and functions of Holism for the metaphysical categories (mind, personality, ideals) and the book concludes with a chapter that argues for the universal ubiquity of Holism and its place as a monistic ontology.
The following is an overview of Smuts' opinions regarding the general concept, functions, and categories of Holism; like the definition of Holism, other than the idea that the whole is greater than the sum of its parts, the editor is unaware of any authoritative secondary sources corroborating Smuts' opinions.
Structure
Wholes are composites which have an internal structure, function or character which clearly differentiate them from mechanical additions, aggregates, and constructions, such as science assumes on the mechanical hypothesis. The concept of structure is not confined to the physical domain (e.g. chemical, biological and artifacts); it also applies to the metaphysical domain (e.g. mental structures, properties, attributes, values, ideals, etc.)
12/22/19 I just noticed that one of the Wikipedia editors created a Holism and Evolution entry that contains the
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Field
The field of a whole is not something different and additional to it, it is the continuation of the whole beyond its sensible contours of experience. The field characterizes a whole as a unified and synthesized event in the system of Relativity, that includes not only its present but also its past—and also its future potentialities. As such, the concept of field entails both activity and structure.
Variation
Darwin's theory of organic descent placed primary emphasis on the role of natural selection, but there would be nothing to select if not for variation. Variations that are the result of mutations in the biological sense and variations that are the result of individually acquired modifications in the personal sense are attributed by Smuts to Holism; further it was his opinion that because variations appear in complexes and not singly, evolution is more than the outcome of individual selections; it is holistic.
Regulation
The whole exhibits a discernible regulatory function as it relates to cooperation and coordination of the structure and activity of parts, and to the selection and deselection of variations. The result is a balanced correlation of organs and functions. The activities of the parts are directed to central ends; co-operation and unified action instead of the separate mechanical activities of the parts.
Creativity
It is the intermingling of fields which is creative or causal in nature. This is seen in matter, where if not for its dynamic structural creative character matter could not have been the mother of the universe. This function, or factor of creativity is even more marked in biology where the protoplasm of the cell is vitally active in an ongoing process of creative change where parts are continually being destroyed and replaced by new protoplasm. With minds the regulatory function of Holism acquires consciousness and freedom, demonstrating a creative power of the most far-reaching character. Holism is not only creative but self-creative, and its final structures are far more holistic than its initial structures.
Causality
added in the second edition, and of course will not be found in reprints of the first edition; nor is it included in the most recent Holst edition. It is the second edition of Holism and Evolution (1927) that provides the most recent and definitive treatment by Smuts.
The whole is greater than the sum of its parts
The fundamental holistic characters as a unity of parts which is so close and intense as to be more than the sum of its parts; which not only gives a particular conformation or structure to the parts, but so relates and determines them in their synthesis that their functions are altered; the synthesis affects and determines the parts, so that they function towards the whole; and the whole and the parts, therefore reciprocally influence and determine each other, and appear more or less to merge their individual characters: the whole is in the parts and the parts are in the whole, and this synthesis of whole and parts is reflected in the holistic character of the functions of the parts as well as of the whole.
Progressive grading of wholes
Smuts provided the following "rough and provisional" progressive grading of wholes that comprise holism:
1. Material structure e.g. a chemical compound 2. Functional structure in living bodies
3. Animals, which exhibit a degree of central control that is primarily implicit and unconscious 4. Personality, characterized as conscious central control
5. States and similar group organizations characterized by central control that involve many people
6. Holistic Ideals, or absolute Values, distinct from human personality that are creative factors in the creation of a spiritual world, for example Truth, Beauty and Goodness.
Examples of Holistic Correspondence in the Open-SGI Data Multiverse
The following are examples of how holistic correspondence will be used in the Open-SGI. Data Multiverse.
• Within a data world the points of experience might initially be contained within one continents of experience, eventually there is continental separation and drift.
• The concept of a multiverse of data each data universe is comprised of clusters of galaxies. • Each data galaxy is comprised of large numbers of unity of opposites data stars and data
systems. Each data star provides it’s own conceptual light that holds all planetary worlds of data in orbit through the force of its gravity.
• Data worlds can have multiple data satellites.
• The concept of societies of data and different social structures. We already know that data can belong to an individual or a family; an organization or a business, in shot data exists at each social level that we encounter in our human experience.
• Data can either be animate or inanimate; animal, mineral or vegetable.
The foundational metaphysical sciences that are implied by Holistic Correspondence include the fundamental metaphysical forces, metaphysical chemistry, metaphysical biology and the data world sciences. Construction of an epistemology and language using MOO models classified by Holistic Correspondence would have to begin with the fundamental metaphysical forces followed by the subatomic data types and then progressing to the identification of the natural elements of data.
• Metaphysical gravity is the force that attracts a metaphysical existent to the center of a metaphysical (data) world, or towards any other metaphysical existents that have significantly more mass.
• Metaphysical electromagnetism is an interaction that occurs between metaphysically electrically charged particles. Note that I use the term ‘data world’ when I am referring to the Open-SGI data ontology and I use the term ‘metaphysical world’ when I am referring to the metaphysical aspect of reality that we are attempting to simulate. Presumably the concepts of life, mind, consciousness, knowledge, memory and related phenomena are somehow reflected by the metaphysical sciences which would exist if the Holistic Correspondence hypothesis is true.
• The strong metaphysical force is responsible for the binding together of metaphysical protons a n d m e t a p h y s i c a l n e u t r o n s i n t h e m e t a p h y s i c a l a t o m i c n u c l e u s a n d i s the strongest metaphysical force, but acts only over distances comparable to those between nucleons in an atomic nucleus.
• There would also be be a weak metaphysical force that accounted for the metaphysical phenomena of radioactive beta decay.
Data World Sciences
The Earth sciences use the term “sphere” to designate the constituent planetary areas e.g. atmosphere, biosphere, hydrosphere, etc/. For example, the data biosphere is the regions of the surface, atmosphere, and hydrosphere of a unity of opposites data world that is occupied by living data organisms. Earth’s mantle, comprised of upper and lower portions would have a 13 corresponding concept in a world of data as the regions that contain the fundamental data prime and composite levels.
Fundamental Data Forces
Due to the breath of information in this paper, I have decided not to provide details on most of the concepts that I
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Data Chemistry
Data Biology
Guidelines for Definition of Intelligence Data Objects (GDIDO)
To enrich the information conveyed by classified intelligence related phenomenaHolism, object oriented analysis and the concepts of open network management are combined in specifying the
Guidelines for Definition of Intelligence Objects (GDIO) abstract syntax notation.
Overview
Using the Guidelines for Definition on Management Objects (GDMO) as a reference, I created the GDIDO abstract syntax notation. GDMO is an open industry standard that is used to define the managed objects of interests that comprise communication networks e.g voice, data, wireless, etc. I envision the GDIDO as becoming an open industry standard that can be used to define the intelligence data objects of interest that comprise the Open-SGI data multiverse.
Intelligence data objects are computer programs that interact with data devices. Examples of data objects include:
• Perception objects; the programatic systems or functions that drive the data perception devices and are responsible for managing and effectuating the input, processing, storage and output of data e.g. RAM, ROM, Cloud, audio, video, keyboard, file transfer, external messaging, etc. • Abstraction objects; the programmatic systems or functions that perform pattern matching
intended to delineate or parse the data input into meaningful parts. An example of a data abstraction function is one that would scan an image looking for a specific class of image data e.g a specific kind of animal, a face, or the various objects that make up our personal (home) environment.
• Recognition objects; the programmatic systems or functions that perform pattern matching between classified and unclassified abstracted data objects
• Data conditions objects which are multi-termed logical data comparisons that are performed within a data object in the course of purpose-driven intelligence data interactions.
• Data planning functions; the programmatic systems or functions that follow specific purpose and information field driven steps in order to determine the appropriate next action and (expected) subsequent actions.
• Data communication functions; those that apply the SGI to the perceived, abstracted and recognized data objects. Generally this is expected to involve many data conditions and planned data transfer steps.
interactive and interoperable SGI. Interactions are ASN.2 events expressed in terms of 14 attributes, properties, conditions, purposes, actions and notifications.
Purpose of the GDIO notation
Although it was designed for and is only used by the Open-SGI Data Multiverse GDIDO is designed to be implementation agnostic, so if the structure of packages as currently portrayed is not generic enough modifications will be required. As I am the only known user; a data scientist that has a need to model the data objects for Open-SGI, the notation will be used to formally describe the common Open-SGI data Multiverse so other data scientists who are engaged in related AI/AGI/SGI autonomous and intelligent theoretical and algorithm development can assess the usefulness the notation, and Open-SGI for that matter, relative to their efforts. Can they envision a usefulness of using GDIDO to define their intelligence products in a way that will allow they to interact; interoperate within the Open-SGI data multiverse?
The first data intelligence class that I will define is the single inheritance class for Open-SGI; The Whole. As I have alluded earlier, I have modeled the GDIO to be able to support the specification of the general concept and functions of Holism. This is reflected by the combination of the GDIO itself and the specification for the Whole.
GDIO also provides a means to support accountability, transparency and testability and it is my intention to demonstrate how can support or contribute to supporting many of the IEEE Ethically Aligned Design recommendations and any standards that follow: 15
Notation Conventions
The GDIDO notation conventions are exactly the same as the conventions adopted in the specification of the GDIO notion and are provided in the Appendix.
Templates
Object Class
The Object Class is the key template in GDIO forming the basis for the formal definition of an intelligence data object class. The template allows for the definition of the intelligence data
I envision ASN.2 to be the existing open standard ASN.1 plus a standard pseudo code notation that will be used to
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specify data interactions. Pseudocode is a structured set of keywords that describe data processing algorithms. It allows the program designer to focus on the logic of the algorithm without being distracted by details of language syntax. The pseudocode will need to be complete so as to be able to describe the entire logic of the data algorithm so that implementation becomes a rote mechanical task of translating line by line into source code. In general the vocabulary is the vocabulary of the problem domain, not of the implementation domain.
The specification for ASN.2 is outside the scope of this paper
In the US, the National Institute for Standards and Technology (NIST) has been designated as being responsible
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object class in terms of its parent class (or classes) and mandatory and conditional packages of attributes, properties, conditions, actions, notifications and interactions.
Condition and Condition Group
In order to be instantiated, an intelligence object class must contain at least one condition. The template allows the definition of the condition type in terms of either its parent condition it is derived from, or its own syntax.
Condition Groups are named collections of related conditions.
Attribute and Attribute Group
Attributes are characteristics of an intelligence object that do not have an information location. The template allows the definition of the attribute type in terms of either its parent attribute it is derived from, or its own syntax.
Action
The ACTION template enables the definition of an action type in terms of information syntax and reply syntax. Actions can be defined in either a confirmed or unconfirmed mode and the specification of action parameters is supported.
Notification
The NOTIFICATION template enables the definition of a notification type in terms of information syntax and reply syntax, can be defined in either a confirmed or unconfirmed mode and supports the specification of notification parameters.
Interaction
The Interaction Template enables the definition an interaction in terms of information syntax, parameters, attributes, properties, conditions and purposes.
Name Binding
Conclusion
Standards will mitigate the challenges posed by semantic holism and foster interoperability, innovation and adoption of the IEEE recommendations for
Ethically Aligned Design of Autonomous and Intelligent Systems. https://www.frontiersin.org/articles/10.3389/fpsyg.2013.00296/full
Appendix Axiomatic Semantics
Since our objective is to model intelligence data in a manner that is mathematically consistent it is mandatory that we adopt an axiomatic systems methodology. The following definition of mathematics in terms of axiomatic systems establishes the theoretical basis for axiomatic semantics:
We consider a set of objects, called elements, and a set of relations on the elements. The elements and relations are known as primitive terms or undefined terms. The primitive terms obey certain rules known as axioms or postulates. These axioms must be consistent. All technical terms introduced later must be defined by means of the primitive terms. Any statement deduced logically from the axioms is called a theorem. A collection of primitive terms, axioms, and all theorems deducible from them is known as an axiomatic system or mathematical discipline. Mathematics is the study of all such axiomatic systems.
Rather than attempting to create new axiomatic systems from scratch with SGI specific elements, relations and axioms I have focused my efforts on exploring the possibility of using the familiar axiomatic systems that are associated with algebra, geometry, trigonometry, calculus and advanced topics.
Dimensional Semantics
Dimensional Semantics refers to the manner in which information is represented by a model’s dimensions. Mathematical models can have any number of dimensions; however one, two and three dimensional models are easier to work with because they are somewhat familiar and can be visualized.
Elemental Semantics
Elemental Semantics refers to the manner in which information is represented by the individual elements of a model, which for my purpose here are numbers. As shown below, there are different types of numbers and each conveys a different types of information within an axiomatic information system. Real numbers underlie algebra and advanced topics. When defining the elemental semantics it is essential that the type of number that can be used is identified
Binary Semantic Operation
A Binary Semantic Operation is a semantic function F on a set S such that, from an ordered pair of elements of S (a, b) we get a third element of S: a F b = c. Addition and multiplication are the binary operations for ordinary arithmetic.
Semantic Identify
A Semantic Identity is a semantic element e such that for all semantic elements a of S: a F e = a. Each binary semantic operation may or may not have an identity element. The identity elements for addition and multiplication are the numbers 0 and 1 respectively.
Semantic Inverse
For each binary semantic operation on a set S, the semantic inverse is a semantic element a’ (a prime) such that for all semantic elements a of S there exists an a’ such that a F a’ = e. For addition the negative and positive numbers are inverses of one another and for traditional multiplication the inverse of each number N is the fraction 1/N.
Commutative Semantic Property
A binary semantic operation on a set of semantic elements exhibits the commutative semantic property if for all semantic elements a and b of S and the binary semantic operation F, a F b = b F a. Traditional addition and multiplication are commutative.
Associative Semantic Property
A binary semantic operation on a set of semantic elements exhibits the associative semantic property if for all semantic elements a, b and c of S and the binary semantic operation F, a F (b F c) = (a F b) F c. Traditional multiplication and addition are both associative.
Distributive Semantic Property
Two binary semantic operations on a set of semantic elements exhibits the distributive semantic property if for all semantic elements a, b and c of S and the binary semantic operations F and F’, a F (b F’ c) = (a F b) F’ (a F c). The traditional operations of addition and multiplication exhibit the distributive property.
Semantic Group
