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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

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Pages 6‐20: Slides (but no text) for presentation 

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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

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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.

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(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.

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(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

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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.

Ten Lessons from a Study yof Ten Notational Systems

Jeffrey G. Long

IIAS Conference August 2007IIAS Conference, August 2007

jefflong@aol.com

Notational Systems Studiedotat o a Syste s Stud ed

• speech and alphabetic writing• ideographic writing as in Chinese writing or electrical engineering diagrams• arithmetic and algebrag• geometry• cartography• logic• musical notation• chemical notation• time

dance/movement notation• dance/movement notation

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(1) N t ti l t bi it d(1) Notational systems are ubiquitous and non-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 l d th f h th k d h th i ht b i devolved thus far, why they work, and how they might be improved upon

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(2) N t ti l t b t l f(2) Notational systems are about classes of 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 notesthe 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 andNotational 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

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(3) E h f il f b t t tit t t(3) Each family of abstract entity types represents a different facet of reality; different families are incommensurableincommensurable

• An “abstraction family” is a set of abstract entity types that collectively map a particular facet of reality

I i i l “ t ” b t t tit t hil th b• In music, musical “notes” are one abstract entity type, while other members 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 prior, relative-value notational system p p , yof 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

h h i lsystems such as mathematics or 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

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to predict future abstraction families based on current ones

(3 ) Th ti f b t t titi t i(3a) The notion of abstract entities types requires 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 universali.e. universals; for example, he considered red to be a universal

• 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 f th b i f t th d l ti lform the basis of set theory and language respectively

• 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 gblocks for new AETs unique to that notational system (example: monetary abstractions utilize and depend upon arithmetic abstractions)

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(4) Notational systems evolve over long periods of(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 wellGeorg Cantor ( A set is the result of collecting together certain 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 pyears later by Lotfi Zadeh (where an element has degrees of membership in a given set, i.e. the set is not crisp)

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(4b) Notational systems evolve over long periods of(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 notational systems do not refer directly to AETs, but instead toHigher 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 behaviorIt took maybe 150,000 years to develop language and symbolic behavior (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 asIt took another 5,000 years to go to higher order notational systems such as Morse Code (1838)

• In contrast, it took “only” 600 years to go from neumatic musical notation to staff musical notation (c 1027- c 1600)

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staff musical notation (c. 1027 c. 1600)

(5) New notational systems are very difficult to(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 abstractionno easy test to determine the reality or usefulness of a new abstraction 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 powerlosing real power

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(6) R l ti t ti l t i d 3(6) Revolutionary notational systems arise under 3 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

( ) f ( )sets (1965), and Mandelbrot with 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 Gutenbergthe same abstractions but optimized for the new medium, e.g. Gutenberg 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.

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g y

(7) There is no discipline that studies notational(7) There is no discipline that studies notational systems per se, although of course each subject area teaches its users about the notationalarea 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)musical notation, Mendeleev with the Periodic Table)

• 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 t d f i t ti l t t ( ) h th l dstudy of various notational systems to (a) see how they evolve and co-

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?

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(8) Civilization as we know it has been built on(8) Civilization as we know it has been built on notational systems, ranging from speech to writing, money, mathematics, voting, etc.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 itsuch 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 was very difficult with the older neumatic musical notation (onlytiming, was very difficult with the older, neumatic musical notation (only fixed-interval melodies were common)

• Not everyone in a society has to be an abstract thinker, just a small percentage will suffice

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(9) We have not yet discovered all the abstraction(9) We have not yet discovered all the abstraction families there are, and new discoveries will affect and empower civilization as greatly as pastand 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 th i th li i i 1200 CE ld i itheir consequences, than a person living in 1200 CE could imagine a Beethoven symphony

• Such systems are, almost by definition, unimaginable in advance, except to their inventor/discoverer – but the needs and the clues are here nowW h i tit ti t h l f l ti d• We have no institution, technology or process for evaluating proposed new notational systems

• Nevertheless we ought to have at least a clearinghouse for those interested in new notational systems and families of notational systems

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(10) M f th t i t t bl(10) Many of the most important problems we currently face are notational, and will require a notational solutionnotational solution.

• Money as a token of value works only when there is a system for establishing value; otherwise things have no value for purposes ofestablishing 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

l b dl h l AET b di dvalue more broadly, or whole new AETs may be discovered • 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 explicitlyrespect 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 or economics from a systems perspective

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medicine, ecology, climatology, or economics from a systems perspective.

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• Such research and collaboration does not require much capital, as might a 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 gLaboratory, attended by 55 people from 10 countries; published in Semioticaas Special Issue on Notational Engineering, Vol. 125-1/3, 1999 )

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