Notational systems and abstractions


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March 20, 2004: “Notational Systems and Abstractions”. Presented at the Capital Science 2005 Conference, sponsored by the Washington Academy of Sciences.

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Notational systems and abstractions

  1. 1. Cover Page   Notational Systems and  Abstractions  Author: Jeffrey G. Long ( Date: March 24, 2004 Forum: Talk presented at the Capital Science 2005 Conference, sponsored by the Washington Academy of Sciences.   Contents Pages 1‐6: Preprint of paper Pages 7‐28: Slides (but no text) for presentation   License This work is licensed under the Creative Commons Attribution‐NonCommercial 3.0 Unported License. To view a copy of this license, visit‐nc/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA.  Uploaded June 26, 2011 
  2. 2. Page 1 Notational Systems and Abstractions Jeffrey G. Long Independent Researcher Submitted: 1/23/2004Key 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:
  3. 3. Page 2 AbstractThe notion of “abstractions” is used in many different ways. Before developing ataxonomy of abstractions it will be necessary to clarify the various kinds of entities thatare often subsumed under the rubric of “abstractions.” This paper makes an attempt atdefining the notion of abstraction, and distinguishing it from the many other kinds ofentities that are often called abstractions, by looking at several notational systems thatseem to reify or tokenize abstractions.
  4. 4. Page 3 Notational Systems and Abstractions IntroductionThere 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 anentity except one; e.g. abstract value takes away all properties of an object except itsvalue; abstract number takes away all properties of a class of objects except its numberof members. Another, “Platonic” viewpoint postulates that an “ideal” version ofeverything truly exists as an abstraction somewhere, and that the entities we seephysically 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). DefinitionsBy the term “notational system” I mean any system of tokens having a defined syntaxand 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 thenames of classes, which are the sense most people assume when considering thissubject. The class of “dogs,” for example, is indeed an abstraction, but it is merely aninstance of one category of abstractions, namely sets. Set theory is the notationalsystem that addresses this area. As important and fundamental as this area is, it ismerely one of several dozen major abstractions. Set theory can itself be reduced to thenotion of a “stroke” that divides the world in two (Spencer-Brown, 1972), and proceedsfrom there to create many other distinctions within distinctions.By the term “abstraction space” I mean an n-dimensional noumenal space that hasmany kinds of entities in it, which are all of the same class. An abstraction space mightexist for shape, for example, and be populated by shapes we are familiar with, such ascircles and triangles; it may also be populated by stranger entities such as fractalshapes, non-Euclidean spaces, etc. The entities in a single abstraction spaces followlocal rules. Human understanding makes a great step forward when it discovers a newabstraction space, although it usually requires centuries or more for the abstractionspace to be “settled” by explorers, i.e. fully mapped by a notational system. There areperhaps two dozen different abstraction spaces, of which we have settled maybe ten inthe past 50,000 years.
  5. 5. Page 4 BackgroundBorrowing from the foundations of mathematics, some people might say that notationalsystems are a mere formalism whose syntactical and semantic rules are socialconventions 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 notationalsystems 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 somethingequivalent to a “logicist” view, saying that the wide variety of species of notationalsystems can ultimately be reduced to but one, such as natural language perhaps, orlogic, from which the others may be derived. In contrast to all of these positions, I’vecome to believe a “(notational) realist” perspective that states that notational systemsmap portions of a noumenal world that I call abstraction spaces.To a notational realist, notational systems reify the noumenal world for us, assigningtokens 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 newnotational system via a process that fundamentally involves learning about how toperceive 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 classof intellectual and practical problems at once. Each different notational system is adifferent cognitive tool that must continually prove its worth in order to continue beingused. It must either evolve in the face of new challenges or be superseded by new andbetter notational systems.To a notational realist, notational systems are not merely useful formalisms; the fact thatany notational system is successful at all implies something very important the nature ofreality. 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 explorehow notational systems may shed light on the nature of reality, we must look at variousnotational systems. Examples from Different Notational SystemsMusical notation addresses the question of how a composer can communicate musicalideas 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, knowledgeablelisteners will be able to identify the name of the composition and may render an opinionon the performer’s success in following the composer’s instructions.The notational system of music wisely also gives performers some latitude in interpretinga composer’s instructions; I may decide that Rachmaninoff’s instructions allow me toplay 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 judgmentstoo. A music critic may not agree with any particular musical judgment made by aperformer, but so long as the instructions that do exist are followed the critic can notoffer any technical criticism. And if a composer wishes to define performance speed shemay include a word such as “allegro,” or even specify a metronome setting for definingthe duration of (say) each quarter-note in the piece.
  6. 6. Page 5Cartography is a notational systems that has had enormous practical effect on thedevelopment of civilization, for it defines relationships. Thus a globe may show thespatial relationships among cities, or terrains, or both. The same information, removedfrom a spherical surface, may be presented on a flat two-dimensional map or chart, butin moving from the three-dimensional medium to the two-dimensional medium someinformation is inevitably lost. How to make this transformation, using different mapprojections, has been as central to the evolution of cartography as have the facts thatmaps convey such as the borders of countries the locations of cities, and other mattersof fact. Even the presentation of so-called matters of fact by map-makers can bewarped, like any technology can be, by the powers that be, which may show one’s owncountry in disproportionate size to another “enemy” country, or show one’s own countryas the center of the map, and by subconscious implication, the center of the world.Not all maps or charts show geographic structures or relationships; mathematicalfunctions relate a given input to a given output, and mathematical graphs showinterconnections among nodes; they too thereby reify abstract relationships. If someonewere to perform a comparative study of the evolution of these three representations(maps, functions, and graphs) they might arrive at a fundamentally deeperunderstanding of the nature of relationships, and this could in theory be parlayed into anew notational systems that included and yet superseded maps, functions and graphs.Natural language is a particularly difficult notational system to discuss because we areso steeped in its everyday use that we almost cannot imagine how a mind might workbefore it was invented. It is the cognitive water that we fish swim in, and its existence isso 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 configurationsof sense data. While each notational system takes the continuous flux of sense dataand parses it according to the filters that it uses, language then takes these distinctionsand 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 higherepistemological status. Language cannot assign a sound pattern to anything that has notpreviously been parsed out of the flux of language by the rules of some notationalsystem; such things have no words to describe them and remain ineffable until a newdistinction is made, possibly by a new notational system. While notational systems giveus a framework in which to parse reality, language makes the resulting entities morevisible and communicable.Another difficult and fundamental notational system is money. Money is designed tonotate value, but it does a poor job of doing so, for it can assign value only to thosethings that can be traded in a marketplace. It thus does not work for anything that cannotby offered in a marketplace, such as clean air or friendship, resulting in the absurdity thatsuch things are formally valueless. It is dangerously flawed since it leads us to makecorporate and public policy decisions on issues in which non-market things are literallynot accounted for. We may try to bring these non-market items into a market, aseconomists 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 forvalue, based on abstractions yet to be discovered.
  7. 7. Page 6We may get an idea for some possibilities for the future of money by studying theevolution of the concept of number, in which the real number line was eventuallysupplemented with an imaginary number line. Both number lines are equally real andimportant, 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 asgovernmental or industrial standards. The net result might be a balance sheet that couldbe 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 thecompany.” ConclusionIn this brief paper we can’t review and discuss all major notational systems, althoughtime, chemistry, logic, software, architectural and engineering diagrams, and other areasare 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 ofone 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 freeto concentrate on more advanced problems, and in effect increases the mental power ofthe race." Understanding this, we must be prepared to greatly change and broaden ourconcept of the nature and reality of abstractions. ReferencesSpencer-Brown, George (1972). Laws of Form. New York, Julian PressWhitehead, Alfred North (1948): An Introduction to Mathematics. New York: OxfordUniversity Press
  8. 8. Notational Systems and y Abstractions Jeffrey G Long G. March 20, 2004
  9. 9. 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 particulars• Entities lacking causal powers• Referents of words that are not proper nounsThese have not been very useful distinctions – they conflate things that should be distinguished
  10. 10. Most 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• “nation” is “ ti ” i a possible value f a political status variable ibl l for liti l t t i bl• “125” and “” are possible values for a quantity variable These are not really fundamental
  11. 11. What Does an Analytical Tool that Works Say, if anything, About Ontology? Notational Ontology Systems Any connection?
  12. 12. We 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 communication convenience• Notation is incidental to perception• Notation is incidental to cognition• Notational evolution and revolution is incidental to civilization
  13. 13. What is a Notational System?
  14. 14. Examples of Notational Systems
  15. 15. If We Want to Understand Abstractions 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
  16. 16. Each Abstraction Space Contains Many Abstraction InstancesEntityhood: things, actions, eventsGrouphood: classes, setsRelationhood: graphs charts maps graphs,charts,Formhood: maps, geometriesQuantityhood: numbers
  17. 17. Notational 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 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 improved tokenization
  18. 18. Five Levels to Any Notational System
  19. 19. 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 new kind of ontology• There is a chasm between people having different ontologies
  20. 20. Develop 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!) pattern,• Are there practical and/or logical limitations for each abstraction space?
  21. 21. Study of Revolutionary Notational Systems can be Useful• Discovery of new abstraction spaces – quantities, sets, infinitessimals, value, form, relation• 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
  22. 22. 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)• There is no predefined language available for the concepts involved• 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 world
  23. 23. An Alternative PhenomenaConcept of SignsMindMi d Token T k Abstraction Space Cognitive Lens Reality
  24. 24. FidoAlternative View: Example Mind “Dog” Abstract Sets Ab t t S t Abstract ID
  25. 25. 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 on our perception• Revolutionary notational systems open up whole new worlds to us
  26. 26. 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 space• Abstraction spaces are the ontological dimensions of reality• Abstractions are not “forms”, a la Plato
  27. 27. Literacy 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 practice• Net result: user sees new abstraction space
  28. 28. Fundamental Hypothesis of Notational EngineeringMany problems in government, science, business, the performing arts, and engineering exist solely because of the way we currently represent them These them. 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 y intractable problems.
  29. 29. References• Long, J. (Guest Editor), Semiotica Special Issue on Notational Engineering, Volume 125-1/3 (1999)