August 1, 2007: "Ten Lessons from a Study of Ten Notational Systems". Presented at InterSymp 2007 Conference, sponsored by the International institute for Advanced Studies in Systems Research and Cybernetics (IIAS). Paper published in conference proceedings.

1. Cover Page
Ten Lessons from a
Study of Ten Notational
Systems
Authors: Jeffrey G. Long (jefflong@aol.com)
Date: August 1, 2007
Forum: Talk and preprint of paper presented at the InterSymp 2007 Conference
sponsored by the International institute for Advanced Studies in Systems
Research and Cybernetics (IIAS). Paper published in conference proceedings,
available at http://iias.info/pdf_general/Booklisting.pdf
Contents
Pages 1‐5: Preprint of paper
Pages 6‐20: 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. Ten Lessons from a Study of Ten Notational Systems
Jeffrey G. Long
Abstract
For the past 19 years, I’ve been doing a comparative and longitudinal study of the evolution of
ten different notational systems: (1) speech and alphabetic writing; (2) iconographic writing as
in Chinese writing or electrical engineering; (3) arithmetic and algebra; (4) geometry; (5)
cartography; (6) logic; (7) musical notation; (8) chemical notation; (9) time; and (10)
dance/movement notation. This paper will discuss ten important things I’ve learned in the study
thus far. In summary form, they are as follows: (1) notational systems are ubiquitous and non-
trivial; (2) notational systems are not about tokens, but about classes of abstract entities and
families of these classes; (3) each family of abstraction classes represents a different facet of
reality; different families are incommensurable (4) notational systems evolve over long periods
of time, as new abstract entity classes are discovered within abstraction families; (5) new
notational systems are very difficult to introduce for two main reasons; (6) revolutionary
notational systems arise under three conditions; (7) there is no discipline that studies notational
systems per se, although of course each subject area teaches its users about the notational
systems it uses; (8) civilization as we know it has been built on notational systems, ranging from
speech to writing, money, mathematics, voting, etc.; (9) we have not yet discovered all the
abstraction families there are, and new discoveries will affect and empower civilization as
greatly as past discoveries have; and (10) many of the most important problems we currently
face are notational, and will require a notational solution.
Keywords: notational systems, representation, abstraction, cognition, history
I’ve had a long-term interest in complex systems, dating back to about 1972, and more recently
an interest in notational systems, dating from 1988. To better understand notational systems I
decided to study the evolution of ten different notational systems, to see what I could learn about
them. Following are some important things I’ve learned from this study so far.
(1) Notational systems are ubiquitous and non-trivial
Notational systems such as speech, writing, maps, money, arithmetic, programming languages,
logic, mathematics, clocks, calendars, and voting, surround us. They define the mental
environment in which we live. Many people think of these systems as either trivial and
unworthy of study, or arcane and incomprehensible. They probably get this belief from the fact
that little children are taught the most widely used notational systems (reading, writing, and
arithmetic), and that advanced logic and mathematics do indeed utilize arcane abstractions.
Because most of these systems have worked for hundreds or even thousands of years, we believe
that they can do anything and everything we need, and we use them as blithely as any other
established technology. But notational systems are a special kind of technology – a cognitive
1

3. technology – that acts as a mental multiplier just as physical technologies multiply the effective
strength of our muscles and extend our grasp even beyond our home planet.
It is as critical to understand the limitations of cognitive technologies as it is with physical
technologies. Ignoring their limitations can have as disastrous of consequences as would occur
after building a bridge without understanding its materials and the forces they will be under.
One example is the use of money as a token for value. This has worked very well in the 4,500
years we have tried it, but inflation and hyper-inflation dating back to ancient Rome should have
taught us that more tokens does not mean more value; there is something real about value outside
of the notational system we have for it.
(2) Notational systems are not about tokens, but are about classes of
abstract entities and families of these classes
Most people think notational systems are merely the tokens we use, such as “1”, “a”, “≥”, or “+”.
The tokens (symbols) we’re all familiar with, important as they are, are the least important aspect
of notational systems; they are the tip of the iceberg. The tokens of first-order notational systems
denote abstract entities and operations. Acquiring literacy in a notational system requires much
training, and results in students really “seeing” these abstractions, and knowing that they have
great utility. These abstractions (e.g. 1, 2, “+”, “/”) may be grouped into classes (e.g., integers,
operations of arithmetic), and the classes may be grouped into families (e.g. abstract quantity).
There are higher orders of notational systems, which do not refer to abstract classes but instead
refer to other, lower-order notational systems. For example, speech (a first-order notational
system) reifies distinctions; the alphabet (a second-order notational system) represents speech;
and Unicode (a third-order notational system) represents the Roman alphabet as well as other
notational systems such as numbers, mathematical operators, other alphabets, etc.
(3) Each family of abstract entity classes represents a different facet
of reality; different families are incommensurable
What can be expressed using musical notation can not be equally well expressed said in any
other notational system, because the underlying abstractions of music are different than the
underlying abstractions of other notational systems. The same is true of mathematics, dance, etc.
This means that no new abstraction family is predictable based on existing abstraction families,
for each such family is by definition sui generis. Ontologically, this implies that reality is
comprised of many different and incommensurable dimensions, and cannot be represented in
fewer dimensions without losing critical information. Lastly, this means that there is absolutely
no substitute for finding the best notational system for a given problem.
2

4. (4) Notational systems evolve over long periods of time, as new
abstract entity classes are discovered within abstraction families
Humans discovered the first member of the abstract quantity family -- integers -- in very ancient
times, before 28,000 years ago. At that time we tokenized it by tallies, then much later by clay
tokens, and finally by relative-value numeration systems such as Roman Numerals. It wasn’t
until the time of the ancient Greeks, about 2,500 years ago, that a new member of the abstract
quantity family was found -- rational numbers -- and tokenized with a new token that indicated
the ratio of two integers. This allowed numbers to be less than one, and to occupy many other
intervals between the integers. About seven hundred years later, in India, the concept of zero
was discovered, which permitted place-value as contrasted with relative-value representation of
numbers. Relative-value numeration systems such as Greek or Roman numerals had never been
intended to be used by themselves for any purposes other than recording a quantity; they were
intended for use in conjunction with an abacus that did the actual arithmetic, after which the
results might be written. Positional numeration allowed calculations without an abacus, and led
naturally to the notion of a number-line.
Abstract quantity has extended beyond rational numbers only in the last 400 years, to include the
ideas of infinite numbers, imaginary numbers, transfinite numbers, and even fuzzy numbers.
This same kind of ongoing discovery has occurred with every different notational family.
(5) New notational systems are very difficult to introduce for two main
reasons
Convincing others that a newly discovered abstraction really exists is very difficult, for the
abstraction cannot readily be shown, and it is very different from other, known abstractions.
Anyone trying to introduce a new abstraction risks being called crazy, and the only remedy is for
the new abstraction to demonstrate practical utility and thereby eventually (over decades or
centuries) gain acceptance. This is usually a difficult and sometimes even fatal venture, as with
Cantor’s despondence over the initial non-acceptance of transfinite numbers.
Another problem is that new notational systems threaten the current distribution of power within
society, and are usually intensely resisted by those who might lose power. A new notational
system may also make less useful or even obsolete the investment people have made in the
current system, even though the capabilities offered by the new notational system may be far
more powerful. A prime example of this is the change in Italy from Roman to Hindu-Arabic
numerals. This change required four centuries, with the “abacists” on one hand arguing for the
adequacy of Roman Numerals and an abacus, and the “algorists” proposing that the new
numeration system (with its bizarre concept of zero quantity) would be more useful and didn’t
require use of an abacus. It is only in retrospect that we can see that Hindu-Arabic numerals
were an obvious choice.
3

5. (6) Revolutionary notational systems arise under three conditions
The greatest notational revolutions occur when someone discovers the first of a wholly new
family of abstraction classes, as Aristotle did with logic, Euclid with geometry, or Newton and
Leibniz with calculus.
Discovering new kinds of abstractions within an existing family is also monumental. Frege did
this with his new symbolic logic, adding new concepts such as predication and quantification.
Riemann did this with a new geometry where two parallel lines can meet, thus later giving
Einstein a way to express his ideas of general relativity. Fractal geometry is the most recent
addition to the family of geometric abstraction classes.
Lastly, the invention of new media can change the economics and logistics, and hence the
practical utility, of a notational system. The move from clay tablets to papyrus changed the
economics and logistics of record-keeping. The subsequent move to paper did not make any real
difference until there was a realization by Gutenberg that wine presses could be used to make the
same imprint on different sheets of paper, and the resulting printing press changed civilization.
In our own time, there is a move from paper to electronic media, and it is truly changing the
economics and logistics of publishing. And fractal geometry would have very limited utility if it
could not be calculated cheaply by the computers that are part and parcel of electronic media.
(7) There is no discipline that studies notational systems per se,
although of course each subject area teaches its users about the
notational systems it uses
Students in any discipline are taught only what they need to know about the notational systems
and abstractions used by that discipline. They rarely study the evolution of that notational
system, nor are they led to realize that a new notational system developed tomorrow could
benefit and change their discipline again. Probably fewer than one percent of the practitioners in
a field pay any attention at all to the nature and limitations of the notational systems they use.
As a result of this, developers of new notational systems – people I call notational engineers –
have had to innovate from scratch. Guido d’Arrezo invented staff musical notation in a similar
quest to help his students learn hymns in weeks rather than the years that were customary in the
largely oral tradition of that time. Many of these notational engineers were teachers, looking for
a better way to codify knowledge about their subject. Many others, such as Newton and
Feynman, were leading-edge researchers who created new tools because the existing ones simply
could not express what needed to be expressed. Others, such as the blind teenager Louis Braille,
were working to improve the quality of their lives. Whatever their motivation, they had to do
this without the help of a general knowledge of the nature and characteristics of successful
notational systems.
A systematic, longitudinal and comparative study of the evolution of notational systems will help
to highlight the characteristics of revolutionary notational systems, and could help evaluate the
4

6. technical benefits of proposed new notational systems, thereby perhaps greatly speeding up the
rate of discovery of new notational systems. This is not a subject area that needs cyclotrons or
orbiting telescopes; it can be done on a very low budget. But its benefits could be enormous if it
helped even one new notational system get established that solved even one major problem of
modern civilization, or created one major new art form.
(8) Civilization as we know it has been built on notational systems,
ranging from speech to writing, money, mathematics, voting, etc.
Abstract thinking is fundamental to civilization, and tokenized abstractions (i.e., notational
systems) are essential to abstract thinking. Not everybody in a civilization must engage in
abstract thinking in order to benefit from abstractions; we fly in airplanes that were designed by
engineers using mathematics that few of us might understand, yet we benefit from their use of
abstractions. At every stage in the development of human society, dating back some 50,000
years or more to the origins of speech, the discovery and tokenization of various families of
abstractions has been fundamental to permitting the next stage of a civilization to arise. Key
moments include the development of the first writing, arithmetic, money, and voting.
(9) We have not yet discovered all the abstraction families there are,
and new discoveries will affect and empower civilization as greatly as
past discoveries have
We have identified and started to settle maybe fifteen families of abstraction classes, but there
are surely many others to find. Just as no one prior to the advent of staff music notation could
have imagined a Beethoven symphony, people in the future will have ideas that we cannot begin
to imagine, based on the discovery of new abstraction classes and the creation of new notational
systems. While philosophy in the last century took a “linguistic turn” when it realized that many
of its problems were caused by language itself, hopefully someday it will take a “notational turn”
to help us better understand abstraction families and notational systems besides language.
(10) Many of the most important problems we currently face are
notational, and will require a notational solution
The limitations of our current notational systems for representing value, intentions (voting),
complex systems, and many other areas, will require solutions based upon vastly better
representations. These better representations must await the discovery of deeper understandings,
i.e. new abstraction classes. Our economic systems, for example, currently assign monetary
value (price) only to things that can be bought and sold; everything else either has no price, or is
assigned value on a per-case basis in courtrooms. Yet we make commercial and public policy
decisions every day based on this limited system for representing value. Surely there is room for
improvement. Of course there is far more to this than can be presented in this brief overview.
But we must at least begin to seek, and ask the right questions, before we can start finding new
kinds of notational solutions.
5

7. Ten Lessons from a Studyy
of Ten Notational Systems
Jeffrey G. Long
IIAS Conference August 2007
Conference,
jefflong@aol.com

8. Notational Syste s Stud ed
otat o a Systems Studied
• speech and alphabetic writing
• ideographic writing as in Chinese writing or electrical engineering diagrams
• g
arithmetic and algebra
• geometry
• cartography
• logic
• musical notation
• chemical notation
• time
• dance/movement notation
6/9/2011 2

9. (1) N t ti
Notational systems are ubiquitous and non-
l t bi it d
trivial
• We use them every day with maps, arithmetic, speech, writing, timekeeping,
logic, music, etc.
g
• They provide our primary cognitive toolset
• They extend our mental capabilities just as physical toolsets do, by
leveraging our natural abilities
• As with physical toolsets, cognitive toolsets have limitations that we must be
aware of if we are to avoid profound mistakes
• As with physical toolsets there are benefits to understanding how they have
evolved th far, why they work, and how they might b i
l d thus f h th k dh th i ht be improved upon
d
6/9/2011 3

10. (2) N t ti
Notational systems are about classes of
l t b t l f
abstract entities, and families of these classes
• People think that notational systems consist merely of their tokens, such as
the letters of the alphabet, or numbers, or musical notes
• Focusing on, say, the evolution of the shapes of notational system tokens
misses the essence of how the tools work, although ease of use of tokens is
important
• Notational systems are fundamentally about reifying abstract entities and
classes of abstract entities, called Abstract Entity Types (“AETs”)
• Reifying (tokenizing) AETs allow us to “see” the abstractions (as tokens),
store them externally, remember them, think or calculate with them, and
communicate them to others; learning this is via the process of “literacy”,
; g p y,
which is different than ordinary learning
6/9/2011 4

11. (3) E h f il of abstract entity t
Each family f b t t tit types represents t
a different facet of reality; different families are
incommensurable
• An “abstraction family” is a set of abstract entity types that collectively map
a particular facet of reality
• In
I music, musical “ t ” are one abstract entity t
i i l “notes” b t t tit type, while other members
hil th b
of that family include the notions of sharp, flat, relative timing, harmony, etc.;
these abstract entity types collectively allow us to understand and represent
music at a deeper level than did the p , relative-value notational system
p prior, y
of neumes
• No other family of abstract entities can deal equally well with musical ideas,
and musical abstractions cannot be reduced or converted to other notational
systems such as mathematics or l
h h i language
• The same is true for every first-order notational system; there is thus no
good substitute for having the right abstraction family, and there is no way
to predict future abstraction families based on current ones
6/9/2011 5

12. (3a) The ti
(3 ) Th notion of abstract entities types requires
f b t t titi t i
careful analysis
• We have notions dating back to Plato of the nature of abstractions; Plato
defined them as those aspects of things that were common to many entities,
i.e. universals; for example, he considered “red” to be a universal
red
• But “red” is merely the name of a particular color or set of colors
• While “red” is the name of an abstraction, the more fundamental and
important ideas are the notions of “set” and “naming”, which are AETs that
form the basis of set theory and l
f th b i f t th d language respectively
ti l
• Each AET is a member of a broader family that constitutes a notational
system
• Notational systems often use AETs of other notational systems as building
y y g
blocks for new AETs unique to that notational system (example: monetary
abstractions utilize and depend upon arithmetic abstractions)
6/9/2011 6

13. (4) Notational systems evolve over long periods of
time, as new abstract entity types are discovered
within abstraction families.
• Sometime between 50,000 - 200,000 years ago, humans developed the
notion of groupings of things having attributes that distinguished them from
other groupings (e.g. edible vs. inedible, dangerous vs. not dangerous)
• It took perhaps another 50,000 years for set theory to be developed by
Georg Cantor (“A set is the result of collecting together certain well
(A well-
determined objects of our perception or our thinking into a single whole;
these objects are called the elements of the set.”)
• The controversial new abstraction of fuzzy sets was developed about 100
y p
years later by Lotfi Zadeh (where an element has degrees of membership in
a given set, i.e. the set is not crisp)
6/9/2011 7

14. (4b) Notational systems evolve over long periods of
time, as higher orders of notational system are
developed that refer to lower orders (instead of
AETs)
• Higher order
Higher-order notational systems do not refer directly to AETs, but instead to
lower-order notational systems
• Example: the alphabet is a 2nd-order notational system that represents the
sounds of speech, which in turn represent the AET of entityhood)
• It took maybe 150,000 years to develop language and symbolic behavior
150 000
(e.g. burial rituals)
• It took another 50,000 years to go from language to writing (around 1,600
BCE)
• It took another 5,000 years to go to higher-order notational systems such as
5 000 higher order
Morse Code (1838)
• In contrast, it took “only” 600 years to go from neumatic musical notation to
staff musical notation (c. 1027 c 1600)
(c 1027- c.
6/9/2011 8

15. (5) New notational systems are very difficult to
introduce for two main reasons
1. New abstractions are, by definition, invisible and unseen by those who
have not been taught how to see them by a process of literacy; there is
no easy test to determine the “reality” or usefulness of a new abstraction
reality
except years of usage to demonstrate it; therefore the presumption is that
the developer is crazy
2. Existing abstractions, reified into various notational systems, are heavily
used by the established classes of society; new notational systems pose
a threat to them, not just in having to learn new ideas but also in terms of
losing real power
6/9/2011 9

16. (6) R
Revolutionary notational systems arise under 3
l ti t ti l t i d
conditions
1. When a person (and it is just a single person in many cases) discovers a
new family of abstract entity types, e.g. Aristotle with logic (c. 350 BCE),
Newton and Leibniz (c. 1665) with calculus
2. When a person finds new AETs within an existing family; Frege (1895)
did this with logic, and other recent examples include Zadeh with fuzzy
sets (
(1965), and Mandelbrot with f
) fractal geometry (1975)
( )
3. When a new medium is utilized, often requiring different tokenizations of
the same abstractions but optimized for the new medium e g Gutenberg
medium, e.g.
with the printing press (1448), Morse with Morse Code (1838), the
Unicode Consortium with Unicode (1991). These can change the
economics and logistics of use of a notational system.
g y
6/9/2011 10

17. (7) There is no discipline that studies notational
systems per se, although of course each subject
area teaches its users about the notational
system(s) it uses.
• People who create new notational systems do so completely on their own,
p y p y ,
without any guidance from the collective wisdom of others who have done
so in the past
• Many were teachers, motivated to help their students (Guido d’Arrezo with
musical notation Mendeleev with the Periodic Table)
notation,
• Many were trying to solve immediate problems that required such
inventiveness (Newton and Feynman in physics, Braille as a blind teenager)
• Why not make such work easier by doing a longitudinal, cross-disciplinary
study of various notational systems to (a) see how they evolve and co-
t d f i t ti l t t ( ) h th l d
evolve with society and each other, (b) better understand AETs, (c) create
and/or test new notational systems, and (d) understand what makes a
revolutionary notational system?
6/9/2011 11

18. (8) Civilization as we know it has been built on
notational systems, ranging from speech to writing,
money, mathematics, voting, etc.
• Abstract thinking is fundamental to civilization, and tokenized abstractions
(i.e., notational systems) are essential to abstract thinking.
• Each new notational system allows the society using it to grow beyond its
current limitations; conversely, the absence of an appropriate notational
system shows up as a “complexity barrier”, where the work is considered
not just complicated but complex (complexity is a euphemism for perplexity)
such that few if any could do it
• Example: in music, staff musical notation and polyphonic music co-evolved
to the point where operas and symphony orchestras were possible in the
early 1600s; polyphonic music, requiring multiple musicians and careful
timing,
timing was very difficult with the older neumatic musical notation (only
older,
fixed-interval melodies were common)
• Not everyone in a society has to be an abstract thinker, just a small
percentage will suffice
6/9/2011 12

19. (9) We have not yet discovered all the abstraction
families there are, and new discoveries will affect
and empower civilization as greatly as past
discoveries have.
• We currently have maybe 15 notational system families; in a thousand
years there might be another 15! (or in 100 years, as the rate of change
increases!)
• We can no more imagine notational systems that will exist in the future, and
their
th i consequences, than a person living in 1200 CE could i
th li i i ld imagine a
i
Beethoven symphony
• Such systems are, almost by definition, unimaginable in advance, except to
their inventor/discoverer – but the needs and the clues are here now
• We h
W have no institution, t h l
i tit ti technology or process f evaluating proposed new
for l ti d
notational systems
• Nevertheless we ought to have at least a clearinghouse for those interested
in new notational systems and families of notational systems
6/9/2011 13

20. (10) M
Many of th most i
f the t important problems we
t t bl
currently face are notational, and will require a
notational solution
solution.
• Money as a token of value works only when there is a system for
establishing value; otherwise things have no value for purposes of
accounting, planning, or preservation. Value has traditionally been
established by markets (whether free or controlled), and by courts; now with
ideas such as cap-and-trade markets for carbon dioxide output, or paying
the cost of disposition up front, new “marketplaces” may emerge to define
value more b dl or whole new AET may b di
l broadly, h l AETs be discoveredd
• English Language has an unstated metaphysics based on objects and their
attributes, with distinctly second place given to actions and processes; but it
can be argued that everything is a process and is highly relativistic with
respect to the perceiver; perhaps this could be incorporated explicitly
someday
• We need a good notational system for large systems of complicated,
contingent rules if we are ever to understand complex systems such as
medicine, ecology, climatology,
medicine ecology climatology or economics from a systems perspective
perspective.
6/9/2011 14

21. Suggestions for Next Step
• This work requires expertise of people in many different disciplines; my study
is very preliminary
• Such research and collaboration does not require much capital as might a
capital,
conventional physics or biology lab
• I’m willing to set up a foundation for notational engineering if others are
interested, and will donate $10,000 to start it
• Will need volunteer Board, Officers, members
• Could develop website, online journal, and/or have periodic conferences on
notational engineering (had one conference in 1996 (Notate ’96) under
auspices of George Washington University Notational Engineering
p g g y g g
Laboratory, attended by 55 people from 10 countries; published in Semiotica
as Special Issue on Notational Engineering, Vol. 125-1/3, 1999 )
6/9/2011 15