The co evolution of symbol systems and society

Cover Page

The Co‐Evolution of
Symbol Systems and
Society

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

Date: January 18, 1995

Forum: Talk presented at the Scholars' Colloquium of the U.S. Library of Congress.

Contents
Page 1: Proposal

Pages 2‐34: Slides intermixed with text for presentation

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Uploaded June 22, 2011

Submitted for the
Library of Congress ‐‐ Scholars' Colloquia
September 1994

The Co‐Evolution of Symbol Systems and Society

Jeffrey G. Long
133‐1/2 11th Street, S.E.
Washington, DC 20003
(202) 547‐0268
jefflong@aol.com


There seems to be a general pattern in the structure of notational revolutions such as the move to the
alphabet, to hindu‐arabic numerals, and to staff musical notation. The Whorfian hypothesis has not
been generally accepted as applying to language, but it does seem to apply to notations.  This talk will
discuss the significance of these questions as they relate to the co‐evolution of society and notation, and
the social formation of mind.  The 35‐minute presentation will be followed by a 25‐minute audience
question‐and‐answer period.

The Co-Evolution of

Symbol Systems and Society
A Brief Overview

Jeffrey G. Long

voice: (202) 547‐0268
e‐mail: JeffLong@AOL.COM
letter: 133‐1/2 11th Street, S.E., Washington, DC 20003

Presented at the
Library of Congress Scholars’ Colloquium
January 18, 1995

Jeffrey G. Long [1/18/1995]

The CoEvolution of Symbol Systems and Society

Slide 1: Cover Page

I really appreciate this chance to share some ideas with you.

Twenty‐one years ago I got interested in complex systems analysis when I did simple neural net
simulations as an undergraduate at U.C. Berkeley.

Since then I've been looking at a wide range of issues in order to understand why our society doesn't
seem able to comprehend, create or control complex systems, whether the systems are natural or by‐
design.

I've concluded so far that mathematics and our other primary notational systems have FUNDAMENTAL
LIMITS in what they were designed to represent.

In other words, the NOTATION we use is a critical limitation on our ability to understand the world
around us.

Conversely, a new notation can resolve a large CLASS of problems all at once.

I summarize this belief with the phrase, "The notation is the limitation".

This afternoon I'll talk for 20 minutes about the NATURE of notation and about notational and societal
co‐evolution, using three examples of notational EVOLUTION and REVOLUTION to show the pattern.

We'll then have about 10 minutes for any questions or comments you may have.

Please hold your questions until the end.

If you'd like to talk more, I can be reached as shown on this slide.

I've also put my business card out, and some reprints of the slides.

Please feel free to contact me at any time with your thoughts and criticism.

Page 3 of 34



The use of a system of signs or symbols to represent

words, phrases, numbers, quantities, etc. 1

"For the purpose of determining logical structure it is, for instance, a matter of complete
indifference whether we represent certain features of states of affairs by spatial
arrangement rather than by sounds or shapes. Hence the unimportance in theory of
attempts to 'improve' symbolism: tokens of any properties whatsoever can be used as
the material for a complete language." 2

1 -- Webster's New World Dictionary, Second College Edition, 1984
2 -- Max Black, Language and Philosophy, 1949, page 160

Page 4 of 34



Slide 2: Standard Definition

First I would like to acknowledge that MANY people think notation is unimportant.

Looking at the dictionary definition, it's easy to see why they feel this way.

Under this definition, key concepts exist OUTSIDE of notation, presumably in LANGUAGE.

Notation is mere ABBREVIATION.

This premise is widely held, and is stated fairly clearly by the philosopher of mathematics Max Black.

MY OWN explorations of more than a dozen notational systems over the past ten years have convinced
me otherwise.

Page 5 of 34



Successful Notations Wannabe Notations

alphabet, chemical formulae,
Recognized algebra, dance,
Notations geometry, software design,
calculus, roadsigns
music,
logic,
cartography

Unrecognized Notations speech, votes,
time, emoticons,
money ???

Page 6 of 34



Slide 3: What Is a Notational System?

I believe we can get a better sense of the TRUE nature of notation by looking at examples of what I
CONSIDER to TRULY be notational systems.....

(Upper Left)

(Upper Right)

(Lower Left)

(Lower Right)

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A revolutionary notational system

does not merely symbolize abstractions ‐‐

it invents them,

it reifies them,

and it

provides a calculus for them.

Page 8 of 34



Slide 4: Criteria for a Notational System

So what criteria am I using to define a notational system?

In all of these cases, the notational SYSTEM does not merely ABBREVIATE anything; instead:

FIRST, it INVENTS or DISCOVERS a new abstraction that I call an ONTOLOGICAL INVENTION

An ontological invention is something we invent (or discover) and subsequently treat as "real", e.g.
numbers, truth , money, or time.

I suggest that what we call LITERACY is learning about the existence and properties of these
unobservable entities. And that when we learn about them, we literally see the world differently (a la
Vygotsky and Luria)

When I say invent or discover, I mean that these entities have ontological REALITY, but whether they
were ALWAYS there (a la Plato, and therefore DISCOVERIES) or whether they are products of mind
(therefore emergent, and therefore INVENTIONS) is unclear to me

SECOND, it REIFIES that abstraction with a SYMBOL inscribed on some MEDIUM (not always 2‐
dimensional, as we presume, but 1‐D, 3D, and n‐D also), and

FINALLY, it provides a CALCULUS or GRAMMAR of legal operations on those new symbols.

The difference between this and formal languages or formal systems is thus that it is fundamentally
exploratory and EMPIRICAL rather than AXIOMATIC; semantics are the BASIS of it, not an afterthought
or application as in model theory.

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Fashions of Use Style Hemmingway, Whitman...

"i before e except after c..."
Rules of Use Grammar
-- English, French, etc

words, letters, numbers...
Symbol Set Tokenization - Roman, Cyrillic, Hebrew, e

Abstraction Ontological Invention abstract truth, value,
quantity, relation, etc.

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Semiotic Notational Formal
Symbol Symbol Symbol
Systems Systems Systems
-- cars, clothes, haircut -- speech, writing -- formal languages
-- advertising, writing -- applied mathematics -- symbolic logic
-- religious symbolism -- money, music -- pure mathematics

Semantics: Semantics: Semantics:
Syntax: Syntax: Syntax:

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Logic Arithmetic

implication numbers
Cartography
& Charts Speech
Sets Quantity

maps naming
Relation Truth
Experience
Change Form
lines,
time
circles Geometry
Calendars Value Procedure
& Clocks
money
notes & rests

Accounting Musical Composition

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Slides 5‐7: Abstractions Lead to Ontological Inventions

So the key concept that I am advocating here is the notion of an ONTOLOGICAL INVENTION.

In this chart, we start with the undifferentiated mess that we call RAW EXPERIENCE.

Both societies and individuals, as they mature, develop and apply abstractions to this experience.....

From the abstractions we distill an ontological invention.....

Based on the ontological invention, we develop a notational system....

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Whorf's theses of linguistic relativity were summarized by the semanticist Stuart Chase as follows:

"First, that all higher levels of thinking are dependent upon language.
Second, that the structure of the language one habitually uses
influences the manner in which one understands his environment.
The picture of the universe shifts from tongue to tongue."3

Restating this in notational terms, we might say:

First, that all abstract thinking is dependent upon the existence or
invention of notational systems.  Second, that the underlying
ontological inventions of the notational systems one habitually uses
influences the manner in which one understands his environment.
Acquiring literacy in a major notation causes us to add a new
dimension to our picture of the universe.


3 -- John B. Carroll (Editor), Language, Thought, & Reality: Selected Writings of Benjamin Lee Whorf. Cambridge
MA: The M.I.T. Press, 1956. Page vi

Page 14 of 34



Page 15 of 34



Slide 8: The Notational Hypothesis

Looking at notational systems this way, we can broaden Whorf's hypothesis of linguistic relativity
regarding the effect of language.

I think he erred in making the theory too narrow, applying it to only one notation, namely speech.

If broadened to include all notations, not just language, I suggest that it is true.

And as with his hypothesis, it is true not just individually but societally.

Page 16 of 34



1.  Notation 1 is invented, and is revolutionary

2.  Notation 1 is developed and applied to its farthest limits

3.  Notation 1 hits a "complexity barrier" and progress stops

4.  Notation 2 is invented, and is revolutionary

5.  Notation 2 is developed and applied to its farthest limits

6.  Notation 2 hits a new "complexity barrier"; progress stops

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Notational Fulfillment & Societal Evolution

Slide 9: Notational Fulfillment & Societal Evolution

The revolutionary new ways of seeing, created and required by revolutionary new notations, have a big
impact on the cognitive capabilities and subsequent accomplishments of both individuals and societies.

Based on my study thus far of 12 notations, there seems to be a broad general PATTERN in notational
and societal co‐evolution.

First, a NEW notation is created, based on a revolutionary new ABSTRACTION that never existed before,
and this FIRST GENERATION notation is packaged as an ANALOGY of what it represents.

Next, this notation EVOLVES through improvement of PRAXIS, e.g.

 symbols are STREAMLINED for greater ease of use
 NEW symbols are introduced, e.g. lower case, punctuation
 there's a CONSENSUS on standards for USING the system
 new and better MEDIA is used [e.g. clay ‐> papyrus ‐> paper]
 new REFERENTIAL TIERS are created for the new media.

This process is generally what people think of when they think of notational evolution, e.g. the shape of
letters, the introduction of a new punctuation mark, etc..

But this is NOT where notation gets its enormous power.

Next, in spite of all refinements, the notation hits what I call a COMPLEXITY BARRIER.

No amount of effort seems to overcome the barrier, and progress comes, if at all, by random insight, not
systematic analysis.

The complexity barrier by its nature is very widespread, affecting entire fields of activity at a time.

Characteristics of a complexity barrier are that:

 there is a large class of problems that thwarts resolution
 increased money and effort produce FEW, if ANY, results.

Next, somehow, a NEW notation is created, based on a revolutionary new ontological invention.

This often leads to, or co‐exists with, a tremendous boom in cultural evolution.

Page 18 of 34



For example, the introduction of speech and writing, logic, money, time‐keeping (e.g. calendars and
clocks), all had a big effect on what society could do because suddenly a whole CLASS of problems was
resolved.

This is what I call a Second Generation Notation.

While the first generation was hindered by its preference for ANALOGY with common experience, the
second and subsequent generations are not so based and can be vastly stranger and more abstract.
Second generation notations are therefore based on far more powerful insights into the nature of what
can exist and be represented.  Its characteristics are that it:

 Solves a broad class of problems with far less effort
 Is accessible to more people
 Is accepted only grudgingly by the elite because it changes the power distribution in society

Finally, this notation, too, goes through the same KIND of evolutionary refinement that its predecessor
did, but eventually it hits its OWN complexity barrier.

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(Ideogram)

(Phonogram)

formant frequencies: letters:
MAN "M - A - N"

Page 20 of 34



Representing Names

Slide 10: Representing Names

The first example of this pattern is the development of technique for representing QUALITIES.

In this and the subsequent examples, the things that are in the "real world" are shown without boxes,
while the ONTOLOGICAL INVENTIONS (i.e. NOTATIONS) are in rounded‐edge boxes.

First, there's something in the "real world" that we want to represent, such as this man.

The first generation notation was SPEECH that represented by VOCAL SOUNDS, using AIR as a medium.

This is believed to have occurred with the first homo sapiens sapiens, around 100,000 years ago.

But speech has limitations, primarily because of its impermanence but also (as we discovered much
later) because of its imprecision.

So first CAVE ART was developed, then these were reduced and codified as PICTOGRAMS starting in
about 3400 BC.

After about 600 years, these evolved to be able to represent IDEAS and ACTIONS through the use of
IDEOGRAMS (where ideas are communicated through clever combinations of symbols) and
PHONOGRAMS (where concepts are hinted at by symbols that represent something that evokes a
particular sound) (circa 2800 BC).

This worked pretty well in ancient society, but eventually this ICONIC NOTATION hit its COMPLEXITY
BARRIER:  several thousand symbols are needed to convey the concepts of even a fairly simple culture.

Continuing to add new symbols or simplify existing symbols would have been fruitless: you can imagine
what a Shakespeare play might be like if every symbol was subject to personal interpretation.

A new Notational Revolution occurred when, about another 1,500 years later, someone noticed that
there were a limited number of SOUNDS we make in human speech, and they designed SYMBOLS to
represent our SPEECH (first alphabet, circa 1500 BC).

This was a 2nd TIER notation, i.e. a notational system that represents a notational system.

With this new approach, and after the invention of VOWELS by the Greeks about 750 years later (circa
750 BC), we were able to represent the >50,000 words known by the average adult with only 26 letters.

Thus the SCOPE of what could be represented was greatly increased, while the NUMBER of SYMBOLS
greatly decreased.

Page 21 of 34



This made the notation far more powerful and accessible to a far greater proportion of society.

As the result of this ontological invention, society was able to create a collective memory that
superseded the fragile memory of the oral tradition that preceded it.

This was literally the beginning of "history", and is probably the most classic notational revolution.

Historian Eric Havelock said,  "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" [Havelock, 1982].

Historian James Breasted notes, "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" [Breasted, 1926, pages 53‐54].

The ontological invention behind writing are the abstract voice sounds that are more‐or‐less well‐
represented by LETTERS, which reify that ontological invention.  WRITING is a notational SYSTEM built
upon LETTERS representing ABSTRACT VOICE SOUNDS, and it defines a number of CONVENTIONS
regarding the proper use of this ontological invention.

Page 22 of 34



(Chemistry) 7.00

Page 23 of 34



Representing Quantities

Slide 11: Representing Quantities

Again, there's something in the "real world" that we want to represent, such as how many lions we saw
or how many enemy soldiers we saw.

Needless to say, getting the number right can be very important, although many societies had no words
for numbers greater than two or three.

The first generation of quantitative notation, introduced over 30,000 years ago, was TALLIES that
represented by ANALOGY (circa 30,000 BC).

These were based on the idea of a 1:1 CORRESPONDENCE.

Any objects ‐‐ stones, whatever‐‐ could be used as the basis for creating a 1:1 correspondence.

22,000 years later (circa 8,000 BC), this notation evolved so that "accounting tokens" represented both
the number of items AND their nature.

Each commodity, and each quantity of a commodity, could have its own clay token.

Impressions of these tokens on clay envelopes, and later on clay tablets, is believed to have become the
basis of writing.

This worked pretty well for the commercial requirements of the first cities, but eventually it hit a
COMPLEXITY BARRIER: it required too many separate and identifiable symbols.

Continuing to ADD new symbols or SIMPLIFY existing symbols would have been fruitless.

6,000 years later, a Notational Revolution occurred when someone noticed that there were
commonalities among certain groups of (say) seven things, if you eliminated everything about them but
the quantity of their members.

This "set of all sets of seven things", although it wasn't thought of that way at the time, was the FIRST
TIME that numerical concepts were represented BY THEMSELVES.

Examples of this are Egyptian hieroglyphic numerals (1900 BC), and ROMAN NUMERALS (circa 500 BC),
where (e.g.) a "V" could represent five "IIIII".

This was a revolution in SPECIFICATION, that moved us from E‐NUMERATION to NUMERATION.

This worked okay, but required a SECOND‐TIER notation called an ABACUS in order to facilitate
computations.

Page 24 of 34



The abacus worked very well for all PRACTICAL purposes up until recent times, but this notational
system was unable to represent irrational numbers, very large numbers, imaginary numbers, and so on.

Continuing to ADD new symbols or SIMPLIFY existing symbols would have been fruitless: we could never
send a man to the moon using what I call Abacus Notation.

About 500 years later, in India, another Notational Revolution occurred (circa 200 AD) when someone
noticed that the operations of the abacus (the computational device used up to then) could be
simulated by using ICONS for the counters of the abacus.

One of these was the icon for ZERO, which represented a line with no beads.

By defining the MEANING of each symbol absolutely by its LOCATION rather than by the symbols
SURROUNDING it (place‐value versus relative value), the groundwork was set for mathematics to move
beyond simple arithmetic.

The concepts of number line and more abstract operations on numbers could be contemplated.

This was not an easily accepted idea: for 400 years (1100 AD to 1500 AD), there was a long and
sometimes bitter fight between the "abacists" and the "algorists".

Eventually the algorists won, so since the 18th century few people in the West use an abacus for
calculations.

In 1525 the decimal point was introduced, and complex numbers were introduced in 1545.

Thus the SCOPE of what could be represented was greatly increased ‐‐ twelve basic symbols, including
the decimal point and "i", could represent all possible numbers.

This was a classic notational revolution.  The logician Alfred North Whitehead 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" (An Introduction to Mathematics).

So we again have created new entities in the world: NUMBERS.  MATHEMATICS began as a notational
SYSTEM built upon NUMBERS, and it defines a number of CONVENTIONS regarding the proper use of
this ontological invention.  Mathematics has evolved to deal with OTHER ENTITIES besides numbers,
such as angles and vectors.

Page 25 of 34



me di a vi ta ...

440

me di a vi ta ...

Page 26 of 34



Representing the Making of Sounds

Slide 12: Representing Musical Sounds

To understand musical notation we have to realize that a note played on an instrument does not just
generate ONE pitch or sound vibration, it generates a NUMBER of these.

These are called OVERTONES, and they form the TIMBRE or character of each different instrument.
These overtones are very IMPORTANT to music.

The first generation of Western musical notation that we really know about was developed only 1,000
years ago.

NEUMES represent the rising or lowering of the voice by an upward or downward line.

This was thus a first generation notation by ANALOGY (circa 900 AD).

These evolved over the next 600 years to a dead‐end.

They were changed to represent pitch better through the use of HEIGHTENED NEUMES (where relative
spacing indicated pitch) and LIGATURES (where the broader part of a line indicated pitch).

Later (circa 1260) MENSURAL NOTATION was better able to indicate the DURATION of each note, using
four symbols, each with 3x duration of the previous (perfect) or 2x (imperfect).

These were fine for simple music sung in unison, or even for the later organum music sung in fixed
intervals.

But they were unable to represent simultaneous different pitches, to coordinate the timing of diverse
themes, or to provide a reliable basis for tuning multiple instruments.  Their COMPLEXITY BARRIER was
their inability to represent the kind of complex POLYPHONIC MUSIC we are used to.

Continuing on that path of adding new symbols or simplifying existing symbols would have been
fruitless: no extension of that approach would have permitted polyphonic music such as Beethoven's
"5th Symphony".

A Notational Revolution occurred around the year 1,000, when Guido de Arrezo, a music teacher trying
to find a better way to teach music to his students, started a separate developmental path involving
LINES around the notation; defining separate NOTES and, equally important, RESTS; and "pinning" the
notes to a particular PITCH.

This essentially meant that the notation of music represent the INPUT to the instrument rather than its
OUTPUT.

Page 27 of 34



Like the mathematical revolution, this also represented a move from RELATIVE VALUE to PLACE‐VALUE
notation.

This tool settled down over the next 3 centuries to a 5‐line staff, permitting composers to write complex
musical ideas down, and then EDIT and REFINE them.

Thus the PRECISION and SCOPE of what could be represented was greatly increased.   Music could be
edited and COMPOSED before it was PERFORMED, unlike jazz (which is composed on the spot) or folk
music (which is memorized and traditional).

But it required the ontological invention of NOTES.  Musical COMPOSITION, the notational SYSTEM built
upon NOTES, defines a number of CONVENTIONS about the proper use of this ontological invention.

Page 28 of 34



Salt, Cows, Coin
Seashells,
Gold, Silver

Transferrable Receipts Paper Money

Check Credit Card

Page 29 of 34



Representing Value, Present and Future

Slide 6: Representing Value, Present and Future

Finally, I'd like to talk now about an UNRECOGNIZED NOTATION, namely MONEY.

This slide shows the PRE‐NOTATIONAL situation at the top.

If you want to trade your duck for my cat, we may agree on a BARTER arrangement.

A duck and a cat are roughly commensurable, partly because they're both animals and they both have
some real and obvious values to somebody; so it is fairly easy to make that trade.

But as you offer things that are less and less commensurable, it gets harder to make a trade.

The first generation of notation was COMMODITY MONEY that represented a certain REAL, PRACTICAL
VALUE.

Examples include cattle, salt, and tobacco.

This was widely used up until about 4,000 years ago; tobacco was the principal medium of exchange for
several CENTURIES in MARYLAND and VIRGINIA; and as recently as 1935, salt was still used in Ethiopia!
Even now, during its trying times, some people of the former Soviet Union are using vodka, Levis, and
other commodities as a preferred method of exchange.

Like all notations, commodity money evolved.   Any item having real value could be used, preferably
meeting the following criteria:

 known to many people
 recognizable in value
 scarce
 portable (at least not too bulky)
 physically stable over time (preferably imperishable)
 easily sub‐divided.

But eventually this basis for exchange hit a COMPLEXITY BARRIER: these items were logistically
inconvenient.

Commerce was thus still very DIFFICULT, and the more COMPLEX an economy got the more problems
were caused by commodity money.

Page 30 of 34



We can hardly imagine what the New York Stock Exchange or our economy in general might be like if
every transaction was paid for by weighing salt or some other physical commodity; continuing on that
path would have been fruitless.

A Notational Revolution occurred about 5,000 years ago (3000 B.C.) when people noticed that VALUE
could exist INDEPENDENTLY of an object, i.e. the ontological invention of ABSTRACT VALUE.

People and, later, governments, designated arbitrary objects as commonly accepted SYMBOLS of this
abstract value.

The intrinsic PRACTICAL VALUE of these objects was nowhere near their DECLARED VALUE, and was
often basically zero.

Examples include seashells, beads (strung together = "wampum"),  gold, silver, and copper.

Eventually, the precious metals (particularly silver) won this contest, especially with the help of
government LAWS that regulated their availability and use.

But those tokens of value were subject to dilution, unfair scales, and other problems, so they evolved
into other equally value‐less forms.

After another 750 years (2250 B.C.), COINS having a declared FACE VALUE were introduced.

These dominated for over 4000 years, until  recently, although TRANSFER‐RABLE RECEIPTS were
introduced in the Middle Ages, and then FIDUCIARY MONEY in the form of BANK NOTES was introduced
by the government of Sweden in 1656.

From 1825 through 1875 in the United States there was a major political debate between the "paper
money men" and the "gold bugs" about how abstract value should be represented in America.

America ended up being the birthplace of widespread use of paper money in the Western world.

This paper money was initially backed by gold  ‐‐ an equally worthless commodity until its electrical
properties began to have real value in electronics.

Eventually we ended up with FIAT MONEY (circa 1934), not based on the gold standard, to enable
governments to print money as desired, independently of their actual gold reserves, and thereby control
aspects of their economy through monetary policy.

A second‐tier advance over that is checks, where actual money does not even change hands during a
transaction, but waits until a later and more convenient time.

This too was limiting, as people could only spend what they had.

Page 31 of 34



So a Notational Revolution occurred with the invention of FUTURE EARNING POWER, as represented by
credit in general and credit cards in particular. ,

Thus VALUE‐IN‐THE‐ABSTRACT came to be REAL, and could be traded like a real duck for my cat.

Since in principle anything could be traded for this symbol, the BREADTH of what could be readily traded
was greatly increased, and this EASE OF USE encouraged more commercial activity.

Once governments understood the power of this notation, they regulated it and then completely took it
over so THEY could control its abuse.

Again we created a new entity in the world: DOLLARS (or their equivalent). ACCOUNTING is the
notational SYSTEM built upon DOLLARS as ontological inventions, provides RULES for the proper use of
this notation.

As the result of this ontological invention, society was able to divide work more readily into specialized
categories, for there was now a common denominator that could be used in any commercial activity.

This was the beginning of "commerce" as we know it today.

I wasn't able to find a quotation about the impact of money on societal evolution, but I think you'd all
agree that money is important in society!

Page 32 of 34



Conclusions
 There are limitations to what any notation can express, based on what domain
that notation was designed to represent and consequently what abstractions it
embodies.  This is true even for the "language of science", mathematics.
 A key factor in the evolution of society has been the introduction and
fulfillment of revolutionary new notational systems such as speech, logic, time,
writing, mathematics, music, and money.
 Our society currently faces a "complexity barrier" in dealing with so‐called
"complex systems" such as medicine, ecology, economics, psychology, and
public policy. But complexity is in the eye of the beholder and can be
eliminated by better representation.  Larger computers, more data, and harder
work will not by themselves overcome this complexity barrier.
 We need to perform research to develop at least one wholly new notation,
using abstractions beyond fractals or other fundamentally quantitative
constructs. One key area of research should be the study of complicated
conditional rules, and higher‐level abstractions of classes of rules. More
generally, we need to develop the study of notational engineering so that
society does not have to wait centuries between notational revolutions.

To forward this effort I have started a thing called the Notational Engineering
Laboratory (NEL) at George Washington University.  We have a biweekly series of
lunchtime seminars, to which the public is invited.

NEL has a World Wide Web site that can be accessed through the Internet at:

http://www.seas.gwu.edu/seas/nel

Page 33 of 34



Slide 7: Conclusions

So much for the past; Janus‐like, we must also look to the future.

The history of notation shows ongoing evolution punctuated by periodic revolution.

Have we reached the end of the road?

Do we already have all the notations we need?

I don't think so.

Particularly in the area of complex systems, as one example, we have poor empirical results.

I believe we have hit a complexity barrier that no amount of money or effort will overcome.

We need a new Notational Revolution.

Towards this end I am currently trying to get funding for a Notational Engineering Laboratory at a local
university.

Thank you for your attention.

Are there any questions?

Page 34 of 34

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