This document provides an overview of notational systems and abstractions. It defines key terms like notational systems, abstraction spaces, and abstraction instances. It argues that notational systems reify abstractions by mapping abstraction spaces. Each notational system maps a different abstraction space, and a useful system implies something about the nature of reality and cognition. Studying notational systems can provide insights into the nature of abstractions.

1. Cover Page
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.
Contents
Pages 1‐6: Preprint of paper
Pages 7‐28: Slides (but no text) for presentation
License
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Uploaded June 26, 2011

2. 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

3. Page 2
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.

4. Page 3
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.

5. Page 4
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.

6. Page 5
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.

7. Page 6
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

8. Notational Systems and
y
Abstractions
Jeffrey G Long
G.
jefflong@aol.com
March 20, 2004

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 nouns
These have not been very useful distinctions
– they conflate things that should be distinguished

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. What Does an Analytical Tool that Works
Say, if anything, About Ontology?
Notational
Ontology
Systems
Any connection?

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. What is a Notational System?

14. Examples of Notational Systems

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. Each Abstraction Space Contains
Many Abstraction Instances
Entityhood: things, actions, events
Grouphood: classes, sets
Relationhood: graphs charts maps
graphs,charts,
Formhood: maps, geometries
Quantityhood: numbers

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. Five Levels to Any Notational System

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. 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. 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. 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. An Alternative Phenomena
Concept of Signs
Mind
Mi d Token
T k Abstraction
Space
Cognitive Lens Reality

24. Fido
Alternative View: Example
Mind “Dog” Abstract Sets
Ab t t S t
Abstract ID

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