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Notational Systems and 

Abstractions  

Author: Jeffrey G. Long (jefflong@aol.com) 

Date: March 24, 2004 

Forum: Talk presented at the Capital Science 2005 Conference, sponsored by the 

Washington Academy of Sciences.

  

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Pages 7‐28: Slides (but no text) for presentation 

 

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Page 1

Notational Systems and Abstractions

Jeffrey G. Long

Independent Researcher

Submitted: 1/23/2004

Key Words: ontology, cognition, language, music, cartography, set theory, time, symbol system

Cell Phone: 202-277-7268 Home Address: 13432 Burnt Woods Place, Germantown, MD 20874

Email: jefflong@aol.com

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Abstract The notion of “abstractions” is used in many different ways. Before developing a taxonomy of abstractions it will be necessary to clarify the various kinds of entities that are often subsumed under the rubric of “abstractions.” This paper makes an attempt at defining the notion of abstraction, and distinguishing it from the many other kinds of entities that are often called abstractions, by looking at several notational systems that seem to reify or tokenize abstractions.

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Notational Systems and Abstractions

Introduction There is little consensus among thinkers about the nature and reality of abstractions. The most common view is that in abstracting we are taking away all properties of an entity except one; e.g. abstract value takes away all properties of an object except its value; abstract number takes away all properties of a class of objects except its number of members. Another, “Platonic” viewpoint postulates that an “ideal” version of everything truly exists as an abstraction somewhere, and that the entities we see physically are but poorly realized implementations of that ideal. In this paper I would like to bring to bear an alternative perspective which states that:

abstractions exist prior to and independently of minds, in the form of “abstraction spaces” (defined below)

each major notational system (defined below) tokenizes a different abstraction

space, and essentially creates a map of that space in terms of the rules of that space

the effectiveness of any notational system says something important about both

the nature of reality (metaphysics) and the nature and limitations of knowledge (epistemology).

Definitions

By the term “notational system” I mean any system of tokens having a defined syntax and semantics, and a community of users which is larger than one person. When using “abstraction“ as a noun I do not mean to limit the term to classes or the names of classes, which are the sense most people assume when considering this subject. The class of “dogs,” for example, is indeed an abstraction, but it is merely an instance of one category of abstractions, namely sets. Set theory is the notational system that addresses this area. As important and fundamental as this area is, it is merely one of several dozen major abstractions. Set theory can itself be reduced to the notion of a “stroke” that divides the world in two (Spencer-Brown, 1972), and proceeds from there to create many other distinctions within distinctions. By the term “abstraction space” I mean an n-dimensional noumenal space that has many kinds of entities in it, which are all of the same class. An abstraction space might exist for shape, for example, and be populated by shapes we are familiar with, such as circles and triangles; it may also be populated by stranger entities such as fractal shapes, non-Euclidean spaces, etc. The entities in a single abstraction spaces follow local rules. Human understanding makes a great step forward when it discovers a new abstraction space, although it usually requires centuries or more for the abstraction space to be “settled” by explorers, i.e. fully mapped by a notational system. There are perhaps two dozen different abstraction spaces, of which we have settled maybe ten in the past 50,000 years.

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Background Borrowing from the foundations of mathematics, some people might say that notational systems are a mere formalism whose syntactical and semantic rules are social conventions observed by the group of users; this may be thought of as a “formalist” perspective on the nature of notational systems. Others may say that notational systems are essentially intuitive and they therefore reflect the nature of the human mind; this may be thought of as an “intuitionist” perspective. Others may have something equivalent to a “logicist” view, saying that the wide variety of species of notational systems can ultimately be reduced to but one, such as natural language perhaps, or logic, from which the others may be derived. In contrast to all of these positions, I’ve come to believe a “(notational) realist” perspective that states that notational systems map portions of a noumenal world that I call abstraction spaces. To a notational realist, notational systems reify the noumenal world for us, assigning tokens to the (known) objects of the space and syntactical rules to match the (known) local rules that define each object and its interactions. We become “literate” in a new notational system via a process that fundamentally involves learning about how to perceive a new set of entities and relations in the world. By thus expanding our vision intellectually the way a telescope expands it physically, each new notational system can help, and historically has helped, to solve a whole class of intellectual and practical problems at once. Each different notational system is a different cognitive tool that must continually prove its worth in order to continue being used. It must either evolve in the face of new challenges or be superseded by new and better notational systems. To a notational realist, notational systems are not merely useful formalisms; the fact that any notational system is successful at all implies something very important the nature of reality. The notational realist differs from the platonic realist by not believing that there is, for example, a “perfect chair” or a “perfect red” that truly exists anywhere. To explore how notational systems may shed light on the nature of reality, we must look at various notational systems.

Examples from Different Notational Systems Musical notation addresses the question of how a composer can communicate musical ideas to a performer. It typically provides a graphical set of instructions to the performer. If the performer follows those instructions in even a crude manner, knowledgeable listeners will be able to identify the name of the composition and may render an opinion on the performer’s success in following the composer’s instructions. The notational system of music wisely also gives performers some latitude in interpreting a composer’s instructions; I may decide that Rachmaninoff’s instructions allow me to play one of his pieces slowly, while another performer may wish to execute them quickly. Both performers are following the instructions literally, but adding in their own judgments too. A music critic may not agree with any particular musical judgment made by a performer, but so long as the instructions that do exist are followed the critic can not offer any technical criticism. And if a composer wishes to define performance speed she may include a word such as “allegro,” or even specify a metronome setting for defining the duration of (say) each quarter-note in the piece.

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Cartography is a notational systems that has had enormous practical effect on the development of civilization, for it defines relationships. Thus a globe may show the spatial relationships among cities, or terrains, or both. The same information, removed from a spherical surface, may be presented on a flat two-dimensional map or chart, but in moving from the three-dimensional medium to the two-dimensional medium some information is inevitably lost. How to make this transformation, using different map projections, has been as central to the evolution of cartography as have the facts that maps convey such as the borders of countries the locations of cities, and other matters of fact. Even the presentation of so-called matters of fact by map-makers can be warped, like any technology can be, by the powers that be, which may show one’s own country in disproportionate size to another “enemy” country, or show one’s own country as the center of the map, and by subconscious implication, the center of the world. Not all maps or charts show geographic structures or relationships; mathematical functions relate a given input to a given output, and mathematical graphs show interconnections among nodes; they too thereby reify abstract relationships. If someone were to perform a comparative study of the evolution of these three representations (maps, functions, and graphs) they might arrive at a fundamentally deeper understanding of the nature of relationships, and this could in theory be parlayed into a new notational systems that included and yet superseded maps, functions and graphs. Natural language is a particularly difficult notational system to discuss because we are so steeped in its everyday use that we almost cannot imagine how a mind might work before it was invented. It is the cognitive water that we fish swim in, and its existence is so basic that we cannot get out of it and see it with any true perspective. Nevertheless, I’ll proceed with a preliminary analysis. Natural language is the notational system that assigns names to various configurations of sense data. While each notational system takes the continuous flux of sense data and parses it according to the filters that it uses, language then takes these distinctions and associates them with vocal patterns (i.e. one word or a set of words, e.g. “dogs” and “brown dogs” and “friendly brown dogs”), giving them a name and thus a higher epistemological status. Language cannot assign a sound pattern to anything that has not previously been parsed out of the flux of language by the rules of some notational system; such things have no words to describe them and remain ineffable until a new distinction is made, possibly by a new notational system. While notational systems give us a framework in which to parse reality, language makes the resulting entities more visible and communicable. Another difficult and fundamental notational system is money. Money is designed to notate value, but it does a poor job of doing so, for it can assign value only to those things that can be traded in a marketplace. It thus does not work for anything that cannot by offered in a marketplace, such as clean air or friendship, resulting in the absurdity that such things are formally valueless. It is dangerously flawed since it leads us to make corporate and public policy decisions on issues in which non-market things are literally not accounted for. We may try to bring these non-market items into a market, as economists are trying to do with the markets being set up for pollution rights. Or, hopefully, we may eventually create a new and more powerful notational systems for value, based on abstractions yet to be discovered.

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We may get an idea for some possibilities for the future of money by studying the evolution of the concept of number, in which the real number line was eventually supplemented with an imaginary number line. Both number lines are equally real and important, and both are used in many practical areas such as electrical engineering. Perhaps true value could be better represented by a complex number, where the “real” component was established by a market (as is done currently), and the “imaginary” component is defined using another source of knowledge and authority such as governmental or industrial standards. The net result might be a balance sheet that could be read to say (e.g.) “Company X has a lot of (real-axis) financial assets but has a huge (imaginary-axis) liability in terms of customer and employee attitudes towards the company.”

Conclusion In this brief paper we can’t review and discuss all major notational systems, although time, chemistry, logic, software, architectural and engineering diagrams, and other areas are each fascinating and informative; so I will close here with a few last points:

to a notational realist, the set of all abstractions that are reified by notational systems is a small subset of the class of all possible abstractions; others can and must be discovered and reified by practical notational systems if we are to address the challenges facing us today

The systematic and comparative study of notational systems is not currently an

academic discipline, but it should be, and it should be supported by public funds. I call this proposed field “notational engineering,” as it must not only study the historical structure of notational revolutions, but must also create and test new notational systems that solve hard practical problems in science, government, business, and even the arts. It is only by building practical new notational systems that we will truly appreciate the nature and power of notational systems.

Notational systems and cognition, under notational realism, co-evolve; the evolution of one requires, facilitates and in some real sense causes the evolution of the other. Notational systems and civilization also thus co-evolve. As Alfred North Whitehead (1948) said, "By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race." Understanding this, we must be prepared to greatly change and broaden our concept of the nature and reality of abstractions.

References Spencer-Brown, George (1972). Laws of Form. New York, Julian Press Whitehead, Alfred North (1948): An Introduction to Mathematics. New York: Oxford University Press

Notational Systems and yAbstractions

Jeffrey G LongJeffrey G. Long

jefflong@aol.com

March 20, 2004

There Are Many Definitions of Abstraction

• Anything not concrete or physically perceivable

• Ideal forms in the noumenal world

• Ideas or classifications formed by mental separation from• Ideas or classifications formed by mental separation from particulars

• Entities lacking causal powers

Referents of words that are not proper nouns• Referents of words that are not proper nouns

These have not been very useful distinctions

– they conflate things that should be distinguished

Most So-Called AbstractionsMost So-Called Abstractions are Merely Instances

• “red” and “green” are possible values for a color variable

• “human” is a possible value for a species variable

“ ti ” i ibl l f liti l t t i bl• “nation” is a possible value for a political status variable

• “125” and “” are possible values for a quantity variable

These are not really fundamental

What Does an Analytical Tool that WorksWhat Does an Analytical Tool that Works Say, if anything, About Ontology?

OntologyNotational

OntologySystems

Any connection?

We Have Many Mistaken AssumptionsWe Have Many Mistaken Assumptions About Notational Systems

• Notational Systems are sets of written marks, e.g. , , , , , a, b, c, 1, 2, 3...

• Notation is merely abbreviation, a minor communicationNotation is merely abbreviation, a minor communication convenience

• Notation is incidental to perception

• Notation is incidental to cognition• Notation is incidental to cognition

• Notational evolution and revolution is incidental to civilization

What is a Notational System?

Examples of Notational Systems

If We Want to Understand AbstractionsIf We Want to Understand Abstractions, We Should Study Notational Systems

• It is hard to get a handle on the nature of abstractions

• We are familiar with the technology of notational systems

• Notational systems reify abstractions: they are essentially designed to provide systematic access to abstractions

Each Abstraction Space ContainsEach Abstraction Space Contains Many Abstraction Instances

Entityhood: things, actions, events

Grouphood: classes, sets

Relationhood: graphs charts mapsRelationhood: graphs,charts, maps

Formhood: maps, geometries

Quantityhood: numbers

Notational Systems MapNotational Systems Map “Abstraction Spaces”

• Each notational system maps a different abstraction space

• A revolutionary notational systems arises from the discovery or substantial extension of an abstraction spacesubstantial extension of an abstraction space

• A useful notational system says something about the nature of reality and the nature of cognition

• New media are critical to the degree they permit new or• New media are critical to the degree they permit new or improved tokenization

Five Levels to Any Notational System

Changing Our Minds

• The minds of individuals, and of the species, evolve via the discovery and use of new abstractions

• Each major new abstraction is reified by a new notational system, and permits the creation of a newnotational system, and permits the creation of a new kind of ontology

• There is a chasm between people having different ontologies

Develop Complete List of Current andDevelop Complete List of Current and Potential Abstraction Spaces

• Identify all current notational systems (20+)

• Determine uniqueness, i.e. inter-translatability (6+)

• Is there any pattern a la Mendeleev? (probably not!)• Is there any pattern, a la Mendeleev? (probably not!)

• Are there practical and/or logical limitations for each abstraction space?

Study of Revolutionary NotationalStudy of Revolutionary Notational Systems can be Useful

• Discovery of new abstraction spaces

– quantities, sets, infinitessimals, value, form, relation

• Progressive extension of abstraction spaces• Progressive extension of abstraction spaces

– imaginary numbers, fractal geometry, fuzzy sets

• Improved praxis with better tokens, media and teaching

– Leibniz’ versus Newton’s tokenization, printing versus hand-lettering, writing versus oral tradition

Settling an Abstraction Space Is Difficult

• Settling means exploring and mapping into tokens and syntax

• Each abstraction space by definition was never before imagined (discoverer seems nuts)(discoverer seems nuts)

• There is no predefined language available for the concepts involved

• The notational systems requires training and practice for new• The notational systems requires training and practice for new users to “see” the entities (literacy)

• The notational systems is fully accepted only when it is seen to provide significant practical benefit in the real worldprovide significant practical benefit in the real world

An Alternative PhenomenaAn Alternative Concept of Signs

Phenomena

T kMi d AbstractionTokenMind Abstraction Space

Cognitive Lens Reality

FidoAlternative View: Example

Ab t t S t“Dog”Mind Abstract SetsAbstract ID

Notational Systems as Cognitive Lenses

• The notational systems we are literate in affect how we see reality, a la Sapir-Whorf

• The limitations of our notational systems are the limitations onThe limitations of our notational systems are the limitations on our perception

• Revolutionary notational systems open up whole new worlds to usus

Notational Systems as Maps

• Each notational systems maps a different abstraction space

• A revolutionary notational systems arises from the discovery or substantial extension of an abstraction spacesubstantial extension of an abstraction space

• Abstraction spaces are the ontological dimensions of reality

• Abstractions are not “forms”, a la Plato

Literacy is the Process ofLiteracy is the Process of Learning to See an A-Space

• Prior to literacy, notational systems tokens are nonsense or magic

• Literacy is part rote memorization, part practiceLiteracy is part rote memorization, part practice

• Net result: user sees new abstraction space

Fundamental Hypothesis of NotationalFundamental Hypothesis of Notational Engineering

Many problems in government, science, business, the performing arts, and engineering exist solely because of the way we currently represent them Theseof the way we currently represent them. These problems present an apparent “complexity barrier” and cannot be resolved with more computing power or more money. Their resolution requires a new abstraction which becomes the basis of a notational revolution and solves a whole class of previously-p yintractable problems.

References

• Long, J. (Guest Editor), Semiotica Special Issue on Notational Engineering, Volume 125-1/3 (1999)