October 29, 2005: “Notational Systems and Cognitive Evolution”. Presented at the 2005 Annual Conference of the American Society for Cybernetics. Paper published in conference proceedings.

1. Cover Page
Notational Systems and
Cognitive Evolution
Author: Jeffrey G. Long (jefflong@aol.com)
Date: October 29, 2005
Forum: Talk presented at the 2005 Annual Conference of the American Society for
Cybernetics. Paper published in conference proceedings.
Contents
Pages 1‐9: Preprint of paper
Pages 10‐28: 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
http://creativecommons.org/licenses/by‐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. Notational Systems and Cognitive Evolution
Jeffrey G. Long
jefflong@aol.com
2160 Leavenworth Street, #404
San Francisco, CA 94133
Abstract
For individual people, the process of acquiring literacy with a particular notational system
seems to result in significant new analytical, descriptive, and creative capabilities. For such
individuals, and for society as a whole, science must account for this apparent birth of new
cognitive abilities that arise by means of new and revolutionary notational systems. Just as
language is not “just another tool,” notational systems (which include language as an instance)
are not just another tool: they seem to affect what we can see and think about, as well as how
we calculate and communicate. The proper study of this subject will require a longitudinal and
comparative approach across multiple notational systems. The goal must be an understanding
of the nature of notational revolutions, and the creation of new tools allowing us to solve or
dissolve currently unsolvable problems.
Keywords
Symbol systems; cognition; literacy; mathematics; history
1. Background
There has been much recent discussion about how humanity’s future development may be affected by
genetics research and by computer science research in intelligent systems. Developments in these areas
will be very important, but we can also get more efficiency and effectiveness from existing human
biology and computers by means of improved notational systems. As of yet there has been very little
study of the evolution of cognition within our species, independent of genetic changes, that is evidently
caused by the discovery and development of new notational systems. This paper argues for systematic
study of this area, by trying to establish the basic importance of notational systems to cognition and to
civilization. Indeed the link between these areas is so strong that one may think of cognition and
civilization as co-evolving, based largely on the discovery of new notational systems, over the past fifty
thousand years.
The rate of this co-evolution has greatly increased in the last ten thousand years. While the human
genome may not have changed very much during that period, the way humans see the world and interact
with it has changed greatly because we have come to master new abstractions and formalized them into
notational systems. Speech, money, mathematics and music are but a few examples of things we are
familiar with that would have been incomprehensible 50,000 years ago to a hypothetical genetic
duplicate of ourselves. As humanity continues to discover new abstractions, it should be expected that
hypothetical genetic duplicates of ourselves of 500, 1000, or 50,000 years into the future will be utterly
incomprehensible to us.

3. This expansion of the Sapir-Whorf Hypothesis to include all notational systems rather than “just”
language asserts that the notational systems we use act as cognitive lenses that largely determine what
and how we see, think, and communicate about the world; that they are a critical interface between
higher forms of thought and reality; and that their study is urgently needed if we are to discover new
capabilities in science, the arts, economics, and many other areas of human activity. This paper
therefore advocates the systematic, comparative, and scientific study of notational systems by the
establishment of a new discipline perhaps to be called “Notational Engineering.” Only an integrative,
comparative and longitudinal study of notational systems will offer the insights that we need; the history
of any one notational system is not, by itself, adequate to bring the structure and importance of
notational revolutions into focus.
2. What is a Notational System?
In order to understand this argument better, it is important to address some of the common
misconceptions about notational systems.
Most people intuitively think of a notational system as being merely a set of symbols used for
abbreviating ideas that could just as well be expressed using other symbols. In this understanding the
particular notation used is not very important. While I agree that the particular symbols used are not
terribly important, I suggest that the tokens of a notational system are the least important feature of any
notational system; they are like the tip of the iceberg that one can see above the water, and the real
substance of the system lies out of everyday sight.
We use notational systems every day when we read or write, when we use a road map, when we
calculate using mathematics, and when we use money. These systems have been in use for centuries,
and we generally take them for granted as fixed and (for all practical purposes) immutable. But they
were created, and have evolved over hundreds and thousands of years, to address real and fundamental
human problems. They constitute a cognitive technology that in fact has been essential for the
development of modern civilization and the modern mind. Like any other technology, they have
strengths and weaknesses; their development is not yet finished. Recent examples of substantial
advances in existing notational systems can be found in fuzzy logic and fractal geometry, both of which
are still in their early stages of usage even decades after their introduction. Furthermore, there probably
are wholly new kinds of notational systems, equally as important as speech, writing, and mathematics,
that are yet to be developed.
We can perhaps gain a better understanding of their true nature as forces in the co-evolution of mind and
civilization by reviewing what people in very different fields have said about them. For example, the
mathematician and creator of modern logic Gottlob Frege (1972) wrote:
"Time and again, in the more abstract regions of science, the lack of a means to avoid misunderstandings
on the part of others, and also errors in one's own thought, makes itself felt. Both [shortcomings] have
their origin in the imperfection of language, for we do have to use sensible symbols to think.... Symbols
have the same importance for thought that discovering how to use the wind to sail against the wind had
for navigation. Thus, let no one despise symbols! A great deal depends upon choosing them
properly....And, without symbols, we would scarcely lift ourselves to conceptual thinking."

4. The mathematician Philip E. B. Jourdain (1956) commented, “It is important to realize that the long and
strenuous work of the most gifted minds was necessary to provide us with simple and expressive
notation which, in nearly all parts of mathematics, enables even the less gifted of us to reproduce
theorems which needed the greatest genius to discover. Each improvement in notation seems, to the
uninitiated, but a small thing: and yet, in a calculation, the pen sometimes seems to be more intelligent
than the user. Our notation is an instance of that great spirit of economy which spares waste of labour
on what is already systematised, so that all our strength can be concentrated either upon what is known
but unsystematised, or upon what is unknown.”
Echoing this at a much later date, but more succinctly, the logician Alfred North Whitehead (1948)
stated, "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."
Historian Eric Havelock (1982) stated, "The Greek alphabet...is here introduced, when it impinges on
the Greek scene, as a piece of explosive technology, revolutionary in its effects on human culture, in a
way not precisely shared by any other invention." The historian James Breasted (1926) said, "The
invention of writing and of a convenient system of records on paper has had a greater influence in
uplifting the human race than any other intellectual achievement in the career of man. It was more
important than all the battles ever fought and all the constitutions ever devised."
Historian of mathematics Florian Cajori (1974) quoted from an 1800 text in which the French
mathematician Arbogast stated, "To form the [calculus], it becomes necessary to introduce new signs; I
have given this subject particular attention, being persuaded that the secret of the power of analysis
consists in the happy choice and use of signs..." The French philosopher Jean-Louis Le Moigne (1985)
notes, "It is, therefore, this process of production and recognition of symbols, codes, patterns, signs, or
combinations of signs that will show itself to be at the base of a process of modelization of complexity
by an intelligence."
In presenting a survey of chemical notations, the National Academy of Sciences' National Research
Council stated in 1964 that "Certainly the history of the first twenty years of chemical codes and
notations has been characterized by much original thinking and by many ingenious schemes for handling
chemical structures. There is great need for improved methods of handling the rapidly expanding fund
of chemical knowledge. Further developments in this area are awaited with great interest."
All of these thinkers in all of these fields are assuredly not talking about the shape of the letter “E”, or
the benefits of abbreviation. To understand what they are talking about, we must see that there is more
to notational systems than meets the eye. Examples of notational systems include the generally-
recognized notational systems of:
 Sign languages such as American Sign Language
 Spoken languages such as English, French, or Chinese
 Alphabetic and syllabic writing systems such as the Roman, Greek or Cyrillic alphabets
 Ideographic writing systems such as Chinese
 Computer languages such as Java, ‘C’, or Visual Basic
 Quantitative notational systems such as Hindu-Arabic numerals or Roman numerals
 Other kinds of mathematical systems such as geometry and calculus
 Chemical notation systems such as line-formula notation or Daltonian notation

5.  Musical notational systems such as staff notation or tablature notation
 Dance and movement notation systems such as Labanotation, Benesh notation, and Eshkol-
Wachman notation
 Other notational systems for engineering in fields such as computer science, electrical
engineering, or architecture.
I suggest that notational systems also include such unrecognized but ubiquitous systems as:
 Value representation notations such as money, checks, accounting systems, credit cards
 Opinion representation notations such as voting systems
 Change representation notations such as clocks and calendars (i.e., time)
3. The Foundations of Notational Systems
I suggest that what makes a notational system powerful is its ability to enable its users to see and utilize
facets of reality that they literally had not been able to see before. These systems do this by reifying and
accurately representing an abstraction space: they use physical tokens to represent a wide variety of
distinctions among a family of abstractions. Numbers, shapes, change, relationships, instructions, and
entityhood are all examples of different families of abstraction space.
For example, I have come to think of natural languages as the notational systems for representing
entityhood, and of musical notation and software notation as systems for representing instructions.
Whether these abstraction spaces are inventions or discoveries is debatable, although I think of them as
discoveries that any sufficiently high intelligence will eventually make, albeit using tokens that are best
suited to their anatomy and media. The mapping of these spaces into particular notational systems is not
obvious, partly because most of the notational systems we regularly use have components from other
notational systems, and partly because the mapping was almost always developed in an ad hoc, trial and
error manner rather than systematically.
Each abstraction space is reified by a different notational system. Competence in a notational system is
acquired through a process of learning how to see, work with, and apply the distinctions made within
that particular abstraction space (for reading and writing we call this literacy). While learning to see and
work with new abstractions is difficult, once learned the new way of seeing offers powerful new
capabilities to its users. This process of acquiring literacy can be both intellectually and culturally
revolutionary. People often feel threatened by change, however, especially when they are being asked to
see something they never thought was there before, so a particular notational system such as Hindu-
Arabic numerals, staff musical notation, or even the use of new calendars therefore often requires
decades or centuries for acceptance and general usage, even when in retrospect it is obvious that the new
notational system is far better than the old.
While it was stated above that notational systems map an abstraction space, these are only one (major)
kind of notational systems that I call “first-order” notational systems. Second- and higher-order
notational systems do not map an abstraction space; instead, they map a lower-order notational system.
Alphabetic writing systems are thus second-order notational systems that map a first-order notational
system such as English or another natural language. Morse Code, ASCII code (the American Standard
Code for Information Interchange), and Unicode (another encoding system for computers like ASCII but
including all major writing systems) are thus third-order notational systems. Encrypted text is a higher-

6. order notational system that has special features to make it readable only by those intended to read it,
who must somehow know the correct rules for decryption to a lower-order, readable notational system.
Abstraction spaces cannot be translated into one another; they are incommensurable. This means that
different types of notational systems cannot be translated into one another; for example, musical
notation cannot really be translated into mathematical notation, nor can chemical notation be translated
into movement notation. However, an instance of one type (say English as an instance of natural
language) can be more-or-less successfully translated into another instance of the same type (such as
French or Russian).
In addition to mapping an abstraction space or a lower-order notational system, fully- developed
(mature) notational systems also have the following critical components:
 rules for combining tokens to create statements having meanings that are more than the semantic
sum of the tokens (syntactical rules)
 a variety of styles of usage, which are consistent with the syntax and semantics of the notational
system but offer significant nuance of expression (e.g. Hemmingway vs. Shakespeare,
Beethoven vs. Bach)
 additional “aesthetic” rules for assessing the value of a given statement in a particular notational
system (the preferences and tastes of individual users and of particular time periods and
societies).
Any system not having all of these components is not a developed notational system. It usually requires
centuries for a nascent notational system to develop, and even then it will continue to evolve until it
reaches its useful limits.
4. The Limitations of Current Notational Systems
Like any technology, notational systems have limits within which they work quite well; indeed they
have enabled the creation of modern civilization. But beyond or outside those limits we cannot expect
them to be helpful. The way to tell that we have reached the limits of a notational system is when, in
using that system, we believe that the target system we are representing is “complex”. Complexity is
not an attribute of any target system, but is a euphemism for the perplexity of an observer or user of the
target system. It exists solely in the eye (mind) of the beholder and can be eliminated by use of a more-
powerful notational system. The target system may then appear to a user to be complicated, sometimes
having lengthy cause-and-effect chains, but not perplexing.
I call the limits of a notational system its “complexity barrier,” for that is where perplexity
masquerading as complexity arises. Overcoming this barrier requires either (a) a hunt for a new
abstraction space, or (b) finding and applying an existing notational system to the target system. An
example of the first case is Newton’s creation of the infinitesimal calculus to help describe motion; an
example of the second is Einstein’s application of non-Euclidean geometry to describe space-time. The
true wonder of mathematics is not that nature obeys mathematical rules, but that humans can create so
many notational systems that one can be found to fit almost any situation.
As a society we need to be able to recognize when we have reached a complexity barrier and need
something really new. If we have tried applying more power, more people, more money, or more
computational capability to solve a given problem, and have been unsuccessful (i.e. are still faced with

7. great complexity), then we need to consider the possibility that our notational technology has reached its
natural limits. To not do this is wasteful and ultimately futile: if our ancestors had chosen to build a
steam-powered abacus rather than switch from Roman to Hindu-Arabic numerals, modern mathematics
and technology would not exist. Problems that have this characteristic may be thought of as primarily
“representational problems,” as contrasted with those caused by lack of data, lack of theory, or lack of
effort.
In spite of the great success of our existing notational systems, many examples can be found where we
have seemingly reached a complexity barrier. Unfortunately for all of us, many of these areas have
important scientific, commercial, artistic, and/or public policy ramifications, so our inability to address
them is more than a mere annoyance. Examples of areas that seem to qualify as essentially
representational or notational problems include the following:
(1) In software engineering we have the requirement for both substantial functionality and substantial
flexibility of functionality at the same time. We can create multi-functional systems that don’t change,
or highly changeable systems that are simple (i.e. not multi-functional). We don’t know how to create
systems that have the characteristics of both functionality and flexibility, so we settle for systems that
are moderately functional (and moderately dysfunctional) and that can be changed only with great
difficulty and expense. The real problem here is that we are increasingly dependent upon these software
systems for all aspects of our life and safety.
(2) In determining corporate and public policy we are faced with the use of money as the only tokens of
value. But price, and therefore monetary amounts, can only be set for those things which have a
marketplace; and the most important things – family, friends, clean air, drinkable water, stable climate,
ecological diversity, etc – have no marketplace and therefore, under our current system of accounting,
have no value. How can we make wise decisions in such a situation?
(3) In trying to understand complex man-made and natural systems such as we find in medicine,
economics, and climatology, we are forced to make numerous simplifying assumptions. We know these
assumptions are not really valid but without them our mathematical representations become unsolvable,
so we use the limited models, and often need to make grave decisions that can affect many people. We
achieve simplicity through over-simplification, when what we really need is simplicity without
simplification.
True solutions in these areas will not be a matter of trying harder, spending more money, building faster
tools, or punishing those who fail to manage the problems. No amount of effort would have allowed us
to send a man to the moon if we were still using Roman numerals; no level of effort would have
permitted Beethoven to write his symphonies if there had not already existed a tool for him to express
sophisticated and beautiful musical ideas.
Notational revolutions happen when (a) wholly new abstraction spaces are discovered, (b) major new
areas of an existing abstraction space are discovered and reified by a new or extended notational system,
or (c) a new notational order is developed, usually to make fuller use of new media as in printing or the
Internet. By opening up more of reality to study, notational revolutions can cause intellectual
revolutions. They may also be culturally revolutionary in two distinct ways: by empowering new groups
of people, and by constituting and permitting new kinds of understandings.

8. Contrasting with these rare revolutions, notational systems undergo evolution when their tokens and/or
rules change and become easier to use and clearer in their representations. This may result in cultural
evolution, as when reading, voting, or the use of money became more widespread and people’s lives
changed.
5. Notational Engineering as a New Discipline
Unfortunately there is no field that studies notational systems per se. Instead, each field that uses
notational systems has a few (maybe 1%) of its practitioners who care about the nature and limitations
of the notational systems used in that field; the rest of the professionals in that field are generally
uninterested in this area and are often unaware of the limitations imposed by the notational systems they
use.
One might think that philosophy, which is concerned (among other things) with the nature of
metaphysics, mind, mathematical objects, and truth, would be the proper home for a study of notation.
But modern American and British philosophy is focused largely on language, to the general exclusion of
other notational systems. Having taken a “linguistic turn” in the 20th century, perhaps it will yet make a
broader “notational turn” in the 21st.
Mathematics is the home of many distinct notational systems such as arithmetic, geometry, graph
theory, topology, and calculus. But mathematicians are interested in mathematical objects and do not
often become involved with objects perceived to be inherently non-mathematical such as those reified
by musical notation or chemical notation. Perhaps this is because these latter notational systems, unlike
many in mathematics, have not been systematized to the degree that most mathematical systems have
been.
One might think that semiotics, as the study of sign systems, would be the proper home for a study of
notation. But modern American semiotics is focused largely on what I call informal systems, to the
general exclusion of formal systems (i.e. those having syntax but no semantics, such as pure
mathematics, formal language theory, and pure logic) and/or notational systems (which have both syntax
and semantics). These informal systems have great meaning (semantics) but no syntax with which to
express larger statements. Examples of such systems are flags, trademarks, religious symbols, coats of
arms, etc.
Cognitive science, as the study of intelligent systems, may seem to be the proper home for a study of
notation. But cognitive science sees the problem only from the mind side of the reality/mind link. If the
practical success of any notational system tells us something about cognition but also sheds light on the
nature of reality, then notational engineering must involve many facets of cognitive science but also
include physics and metaphysics as critical facets of the problem space.
Efforts since the 1980s have focused on complexity as a subject in its own right, across many kinds of
systems. I believe complexity is a euphemism for perplexity and can be resolved (dissolved) by the use
of more capable notational systems. The study of complexity was aided by the new mathematical
concept of fractional dimensions (“fractals”), as well as the use of cellular automata. While both led to
interesting results, the problem of complexity is clearly not dissolved.

9. I therefore have proposed (Long, 1999) a new interdisciplinary study called “notational engineering,”
whose object of study is notational systems, and whose goal is to develop new and/or significantly
improved notational systems able to dissolve entire classes of problems.
This proposed discipline presupposes that an expert in (say) music who is concerned about the
limitations imposed by modern musical notation could usefully speak to an expert in (say) chemistry or
logic about the common areas that are representational in nature rather than subject-related. I believe this
to be the case, but with the caveat that a common framework for discussing problems in notation be built
as soon as possible.
A “Notational Engineering Laboratory” could also add value by working on some or all of the following
fourteen areas:
(1) acting as a clearinghouse of information and resources for people with an interest in any notational
system in any field
(2) performing research into the structure of notational revolutions by studying the history of various
notational systems, utilizing a comparative approach to highlight what is essential, and what is
incidental, about each notational system
(3) determining the limitations imposed upon their users by existing and proposed notational systems
(4) studying the philosophical foundations of various notational systems and their associated
abstractions (corresponding in many ways to studies of the foundations of mathematics)
(5) helping to establish criteria for adequate new notational systems in various fields
(6) interviewing living creators of notational systems to learn more about how and why they did that,
and what the reactions were to their work, so that future “notational engineers” might travel a somewhat
easier road
(7) developing experimental new notational systems for various fields of knowledge
(8) developing scientifically well-grounded test problems, test data and test procedures for proposed
notational systems, and carrying out such tests on selected proposed notational systems
(9) organizing conferences and seminars pertaining to notational research and engineering
(10) publishing a new Journal of Notational Engineering to discuss issues pertaining to notational
systems
(11) creating and maintaining an Internet web site that offers educational information about notational
engineering goals and activities, a comprehensive bibliography of materials related to notational
systems, an online “Encyclopedia of Notation”, and online meetings
(12) creating and maintaining a research library of reference materials related to notational systems

10. (13) facilitating development of a television series on the history and impact of various major notational
systems for the general public
(14) creating and maintaining a Notational Museum of notational systems throughout history and pre-
history, and describing their role in the continuing evolution of the human mind, so that the public may
better understand and appreciate the value and importance of notational systems.
By creating new and improved notational systems we create new and improved ways to see, think and
communicate about the world. We thus transform ourselves. In the future, people will use notational
systems that we can’t imagine today; these systems will enable them to see and do things we cannot
currently conceive of, just as we can see and do things that people 1,000 or even 100 years ago could not
imagine. The missing link is a deeper appreciation of the nature and role of all notational systems in
human cognition and civilization. Doing this work is hard and offers no guarantees of immediate
success, but it may be the only way to successfully address a wide variety of problems in today’s and
tomorrow’s world. All we require is the will to investigate.
6. References
Breasted, James H. (1926). The Conquest of Civilization. New York: Harper & Brothers
Cajori, Florian (1974). A History of Mathematical Notations. LaSalle, IL: Open Court Publishing
Company
Frege, Gottlob (1972). Conceptual Notation, And Related Articles. Oxford: Clarendon Press
Havelock, Eric A. (1982). The Literate Revolution in Greece and its Cultural Consequences. Princeton:
Princeton University Press
Jourdain, Philip E. B. (1956). The Nature of Mathematics. In James R. Newman, The World of
Mathematics, Volume I. New York: Simon and Schuster
Le Moigne, Jean-Louis (1985). The intelligence of complexity. In The Science and Praxis of
Complexity. Tokyo: The United Nations University
Long, Jeffrey G. (1999). “How could the notation be the limitation?” Semiotica Special Issue on
Notational Engineering, Vol. 125-1/3
National Research Council (1964). Survey of Chemical Notation Systems. Washington DC: National
Academy of Sciences Publication 1150
Whitehead, Alfred North (1948). An Introduction to Mathematics. New York: Oxford University Press

11. Notational systems and
cognitive evolution
g
Jeffrey G. Long
October 29, 2005
jefflong@aol.com
American Society for Cybernetics

12. Current analysis and design methods work well
y g
only under certain conditions
October 29, 2005 2

13. W * ll * need t understand complex
We *really* d to d t d l
dynamic systems much better
Modern society may have competence in using certain kinds of
complex systems but we still don’t understand them
 climate and weather
 economics, fi
i finance, markets
k t
 medicine, physiology, biology, ecology, epidemiology
This is not inherent in the nature of the systems, but rather
because our notational systems – our abstractions -- are
inadequate
October 29, 2005 3

14. We therefore face “complexity barriers”
complexity barriers
What we don’t understand we call complex. But complexity is
not a property of systems; rather perplexity is a property of the
rather,
observer. Systems that are understood can only be complicated.
Complexity barrier problems cannot be solved by working
harder, using faster computers, or gathering more data
Complexity barrier problems are fundamentally representational
problems. Using the wrong, or too-limited, a representational
(notational) system is inescapably self-defeating.
October 29, 2005 4

15. Other system types have long been studied
Formal: human-assigned syntax only, e.g., formal logic, formal
language theory, pure mathematics
theory
Informal: human-assigned semantics only, e.g., art, advertising,
p
politics, religious symbols
, g y
Subsymbolic: no human-assigned syntax or semantics, but
“natural” syntax and semantics (i.e., physically necessary
interactions), e.g.,
interactions) e g natural systems DNA neural networks
systems, DNA,
Notational: human-assigned syntax and semantics, e.g.,
natural language, musical notation, money, cartography
October 29, 2005 5

16. G. Frege (1972) noted…
Time and again, in the more abstract regions of science, the
lack f
l k of a means to avoid misunderstandings on the part of
id i d di h f
others, and also errors in one's own thought, makes itself felt.
Both have their origin in the imperfection of language, for we do
have to use sensible symbols to think.... Symbols have the
same importance for thought that discovering how to use the
wind to sail against the wind had for navigation. Thus, let no
one despise symbols! A great deal depends upon choosing
them properly... A d without symbols, we would scarcely lift
th l And, ith t b l ld l
ourselves to conceptual thinking.
October 29, 2005 6

17. P E B Jourdain (1956) noted
P. E. B. noted…
It is important to realize that the long and strenuous work of the
most gifted minds was necessary to provide us with simple and
expressive notation which, in nearly all parts of mathematics,
enables even the less gifted of us to reproduce theorems which
needed th greatest genius t discover. E h i
d d the t t i to di Each improvement i
t in
notation seems, to the uninitiated, but a small thing: and yet, in a
calculation, the pen sometimes seems to be more intelligent
than the user.
October 29, 2005 7

18. A. N. Whitehead (1948) noted…
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.
October 29, 2005 8

19. E. Havilock (1982) noted…
The Greek alphabet...is here introduced, when it impinges on
the Greek scene, as a piece of explosive technology,
revolutionary in its effects on human culture, in a way not
precisely shared b any other i
i l h d by th invention."
ti "
October 29, 2005 9

20. Notational systems give us a cognitive
lens with which to see anew
Each major notational system maps a different “abstraction
s stem
space”
Abstraction spaces are
 incommensurable
 discoveries, not inventions
 real in some sense
Acquiring literacy in a notation is learning how to see anew
Perceiving these is a uniquely human capability
October 29, 2005 10

21. So far we have settled maybe
12 major abstraction spaces
October 29, 2005 11

22. The notational hypothesis
All higher forms of thinking require the use of one or more
notational systems
The notational systems we use influence the way we perceive
our environment: our analysis of events changes as we acquire
i t l i f t h i
literacy in new notational systems
Notational systems have been central to the co-evolution of
co evolution
mind and civilization
October 29, 2005 12

23. Corollaries…
Where would human culture be without language, writing,
musical notation chemical notation mathematics maps or
notation, notation, mathematics, maps,
logic notation?
To address the problems we currently face as a civilization, we
need new perceptual, cognitive, and communication t l such
d t l iti d i ti tools, h
as…
 measures of value far more sophisticated than money
 the ability to represent millions of complex rules driving a
rule-based system such as a climate or a cell biology model
 semantic tools that allow us to mechanically integrate
scientific hypotheses so as to automatically generate new
conclusions from the integrated arguments
Why rely, as we have so far, on pure chance to develop such
tools when we can attempt to design them deliberately?
p g y
October 29, 2005 13

24. We can and should systematically,
y y,
comparatively, and longitudinally study
notational systems to facilitate the
discovery of new abstractions on which
f
to base new or greatly enhanced
notational systems. Even if only one
systems
new notational system ever came out of
that effort, it would repay society many
effort
times over.
October 29, 2005 14

25. Notational engineering tasks
(1) provide a clearinghouse of information and resources for people
with an interest in any kind of notational system in any field
(2) perform research into the structure of notational revolutions by
studying the history of various notational systems, utilizing a
comparative approach t determine what i essential, and what
ti h to d t i h t is ti l d h t
is incidental, about each notational system
(3) determine the limitations imposed upon their users by existing
and proposed notational systems
(4) study the philosophical foundations of various notational
systems and their associated abstractions (lik studies of the
t d th i i t d b t ti (like t di f th
foundations of mathematics)
October 29, 2005 15

26. (5) help to establish criteria and desiderata for new notational
systems in various fields
(6) interview living creators of notational systems to learn more
about how and why they did that, and what the reactions were to
their
th i work, so th t future “notational engineers” might t
k that f t “ t ti l i ” i ht travel a
l
somewhat easier road
(7) help to develop experimental new notational systems for various
fields of knowledge
(8) develop well-grounded test problems and test procedures for
proposed notational systems, and carry out such t t on
d t ti l t d t h tests
selected notational systems
October 29, 2005 16

27. (9) organize conferences and seminars pertaining to the history,
philosophy, psychology, sociology, engineering,
philosophy psychology sociology engineering and applications
of notational systems
(10) publish a Journal of Notational Engineering to discuss issues
pertaining to notational systems
t i i t t ti l t
(11) create and maintain a website that offers educational information
about notational engineering goals and activities an online
activities,
“Encyclopedia of Notation”, and specialized online listservers
(12) create and maintain a comprehensive bibliography of materials
related t notational systems, and a lib
l t d to t ti l t d library of reference materials
f f t i l
October 29, 2005 17

28. References
Frege, Gottlob (1972). Conceptual Notation, And Related
Articles.
Articles Oxford: Clarendon Press
Havelock, Eric A. (1982). The Literate Revolution in Greece and
its Cultural Consequences Princeton: Princeton University
Consequences.
Press
Jourdain,
Jourdain Philip E B (1956) The Nature of Mathematics In
E. B. (1956). Mathematics.
James R. Newman, The World of Mathematics, Volume I. New
York: Simon and Schuster
Whitehead, Alfred North (1948). An Introduction to Mathematics.
New York: Oxford University Press
October 29, 2005 18

29. Further Reading
Long, J., and Denning, D., “Ultra-Structure: A design theory for
complex systems and processes.” I C
l d ” In Communications of th
i ti f the
ACM (January 1995)
Long, J., “Representing emergence with rules: The limits of
addition.
addition ” In Lasker G E and Farre G. L. (editors), Advances
Lasker, G. E. Farre, G L (editors)
in Synergetics, Volume I: Systems Research on Emergence.
(1996)
Long, J., “A new notation for representing business and other
g, , p g
rules.” In Long, J. (guest editor), Semiotica Special Issue:
Notational Engineering, Volume 125-1/3 (1999)
Long, J., “How could the notation be the limitation?” In Long, J.
(guest editor), S i ti S
( t dit ) Semiotica Special Ii l Issue: N t ti
Notational E i
l Engineering,
i
Volume 125-1/3 (1999)
October 29, 2005 19