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The Evolution of 

Abstractions  

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

Date: September 11, 1997 

Forum: Talk presented at a luncheon meeting of the Washington Evolutionary Systems Society.  

Contents 

Page 1: Proposal 

Pages 2‐22: Slides (but no text) for presentation 

 

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Title: The Evolution of Abstractions Speaker: Jeff Long, Director, GWU Notational Engineering Laboratory Date: September 11, 1997 at Noon Location: Faculty Club, The George Washington University (call ahead for lunch reservations please!) What is it that gives notational systems their power? Are they merely convenient collections of arbitrary tokens and rules that just happen to have a useful application in the real world? Or might there be a deeper connection between notational systems and reality? This talk will explore this question, and answer in the affirmative. We will discuss the conventional definitions of "abstraction" and their inadequacies, and seek a new definition. To do this we will sketch a theoretical framework -- a metaphysical system that attempts to account for the the law-abiding nature of physical objects, the nature of laws, and, ultimately, the nature of abstractions. The talk will discuss the notion of an "abstraction space" such as the field of numbers, and how three such spaces historically have been explored and tokenized ("settled"). The talk will end with a brief outline of a plan for improving the abstraction space settlement process. This plan is essentially an agenda for the proposed new field of "notational engineering".

The Evolution of Abstractions

Jeffrey G. LongGWU Notational Engineering Laboratory

Copyright 1997 © Jeffrey G. Long

My Work in Notational Engineering y g gInvolves Four Main Areas

What Does An Analytical Tool That WorksSay (If Anything) About Ontology?

OntologyNotational

OntologySystems

Any connection?

S ti f thi T lkSections of this Talk

h hi i l f l i b i1. The historical process of exploring abstractions2. An alternative metaphysical system 3 A general strategy for improving the correlation process3. A general strategy for improving the correlation process

Part OneThe Historical Process of Exploring Abstractions

Th A M D fi iti f ‘Ab t ti ’There Are Many Definitions of ‘Abstractions’

hi h i ll i bl (l Anything not concrete or physically perceivable (love, nations)

Ideal/perfect forms in the noumenal world (perfect justice Ideal/perfect forms in the noumenal world (perfect justice, perfect sphere)

Ideas or classifications formed by mental separation from particulars (rules, sets)

Entities lacking causal powers (universals, numbers, ideas) Referents of words that are not proper nouns (dogs cats) Referents of words that are not proper nouns (dogs, cats)

These have not been very useful distinctionsThese have not been very useful distinctions– they conflate things that must be distinguished

A T f Ab t tiA Taxonomy of Abstractions

Tokens & Operators Expressions composed of tokens, generated by operators Expressions referred to by other expressions Entities, classes & ideas named by expressions Expressions further delimited by their position in statements Variables acting as position-holders within statements Ruleforms composed of ordered sets of variables Ruleforms composed of ordered sets of variables Particular laws/rules are the resolution of ruleforms

E l i N Ab t ti S I V Diffi ltExploring a New Abstraction Space Is Very Difficult

i l i d i i f l k d Requires exploring and mapping into useful tokens and syntax

By definition entity was never before imagined By definition, entity was never before imagined (discoverer seems nuts)

There is no predefined language available for the concepts involved

Users require training and practice to “see” the entities (literacy)(literacy)

S ttli “Q tit S ” R i d C t iSettling “Quantity Space” Required Centuries

lli Tallies: 30,000 BP Accounting tokens: 8,000 BC Whole numbers: 1 900 BC Whole numbers: 1,900 BC Rational numbers: 500 BC Zero and real numbers: 200 Zero and real numbers: 200 Complex (imaginary) numbers: 1545 Transfinite numbers: c. 1900

S ttli “F S ” R i d C t iSettling “Form Space” Required Centuries

lid Euclidean geometry: c. 325 BC Non-Euclidean (hyperbolic, elliptic) geometries: c. 1850 Fractal geometry: c 1975 Fractal geometry: c. 1975

S ttli “Id tit S ” R i d C t iSettling “Identity Space” Required Centuries

h Speech: 100,000 BP? Pictograms: 3,400 BC Ideograms: 2 200 BC Ideograms: 2,200 BC

– Syllabic writing: 3,000 BC– Consonantal alphabet: 1,500 BC– Full alphabet: c. 776 BC

Stroke: 1969

B t W H D It I f ll M TiBut We Have Done It Informally Many Times

Part TwoAn Alternative Metaphysical System

for Exploring the Basic Issuesfor Exploring the Basic Issues

Th P ili (M t i li t) P diThe Prevailing (Materialist) Paradigm

Universe consists solely of matter/energy (physicalism) This substance follows certain laws, sought by science The universe is becoming more uniform over time (2nd Law) These laws and all such abstractions are useful fictions

( i li )(nominalism) Metaphysical questions are pseudo-questions (positivism)

But this paradigm leaves unanswered many questions viewed as non-scientific – why is the universe lawful?– what are laws/rules, really? Do they have component parts?

A Alt ti (Ult St t l) P diAn Alternative (Ultra-Structural) Paradigm

h i l ld d f ll l i l The material world doesn’t follow laws, it is laws– We perceive and define entities according to the laws they happen

to follow

A natural law is an ordered set of noumenal abstractions – e.g. identity & group & form, form&quantity & state

‘L ’ th i t th i t ti f ‘Laws’ are the name we give to the interaction of noumenal abstractions

Interaction of rules produces processes which generate p p g“events”– what we perceive to be the material world

t ll th i l d t l b t ti– eventually these include mental abstractions

Noumenal abstractions become more complex over time – they operate on themselves and evolve

E l f N l Ab t tiExamples of Noumenal Abstractions

ibl d i Possible Identity Possible Group Possible Relation Possible Relation Possible Form Possible Quantity Possible Quantity Possible State

Thi I li C t i F t f N l Ab t tiThis Implies Certain Features of Noumenal Abstractions

h i f d ll diff f i Each is a fundamentally different type of entity– Each has unique types of possible relations with other noumenal

abstractions– One cannot be fully translated into another

They are self-referential Th bi bl bl t h i t ti They are combinable or able to have interactions We can perceive them only by mind

– Similar to how we learn to perceive physical objectsSimilar to how we learn to perceive physical objects

They exist independently of any mind

hPart ThreeA General Strategy for

Improving the Correlation ProcessImproving the Correlation Process

St d R l ti N t ti l S tStudy Revolutionary Notational Systems

i f l b i Discovery of new noumenal abstractions– quantities, sets, infinitessimals, value, form, relation

Progressive exploration of noumenal abstractions Progressive exploration of noumenal abstractions– imaginary numbers, fractal geometry, fuzzy sets

Improved praxis with better tokens, media and teaching– Leibniz versus Newton’s tokenization, printing versus hand-

lettering, writing versus oral tradition

Develop Complete List of p pCurrent and Potential Noumenal Abstractions

d if ll i l ( ) Identify all current notational systems (20+) Determine uniqueness, i.e. inter-translatability (6+) Is there any pattern a la Mendeleev? (probably not!) Is there any pattern, a la Mendeleev? (probably not!) Are there practical and/or logical limitations for each

noumenal abstraction?

Improve Communication Among p gNotational Researchers

fi b i f bj Define scope, nature, basic concepts of subject Identify sources of information/participants

– people (maybe 1% of each group using a NS)people (maybe 1% of each group using a NS)– books, articles, Web sites (esp. foreign language)

Establish clearinghouse – Internet discussions (notation listserver)– conferences (NOTATE’97 at SSA)– publicationsp

C l iConclusion

l i di b d b i ili Alternative paradigm can be tested by its utility – an effective mental abstraction says something about noumenal

abstractions

Broaden the “linguistic turn” to be a “notational turn”– metaphysics is important after all

limitations are not just those of language but all NS– limitations are not just those of language, but all NS– language is not the only tool or reference point

We can speed up the process of settling abstractions– make it more of a regular discipline than an ad hoc event