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Mathematics, Rules, 

and Scientific 

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

Date: September 12, 1998 

Forum: Talk presented at a symposium sponsored by the Washington Evolutionary Systems Society.  

Contents 

Pages 1‐16: Slides (but no text) for presentation 

 

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Mathematics, Rules, and Scientific RepresentationsScientific RepresentationsThe Need for an Integrated, Multi-The Need for an Integrated, Multi

Notational Approach to Science

Jeffrey G. Long, September 12, 1998jefflong@aol.com

B i A tiBasic Assertions

i f ll d ill d d d In spite of all progress to date, we still don’t “understand” complex systems

This is not because of the nature of the systems but rather This is not because of the nature of the systems, but rather because our notational systems are inadequate

B i Q tiBasic Questions

h d h i l Why do we use the notational systems we use? What are their fundamental limitations? Are there ways to get around these limitations? Are there ways to get around these limitations? What is the objective of scientific description? Is there a level of formal understanding beyond current Is there a level of formal understanding beyond current

science?

B k d N t ti l H thBackground: Notational Hypotheses

h f ki d f i There are four kinds of sign systems– Formal: syntax only

Informal: semantics only– Informal: semantics only– Notational: syntax and semantics– Subsymbolic: neither syntax nor semanticsSubsymbolic: neither syntax nor semantics

Of these, notational systems are the least-explored

B k d ( ti d)Background (continued)

h i i l diff Each primary notational system maps a different “abstraction space”– Abstraction spaces are incommensurablep– Perceiving these is a unique human ability

Abstraction spaces are discoveries, not inventionsAb i l– Abstraction spaces are real

– Their interactions are the basis of physical law

B k d ( ti d)Background (continued)

i i li i i i l i h Acquiring literacy in a notation is learning how to see a new abstraction space– This is one of many ways we manage perception (“intellinomics”)y y g p p ( )

All higher forms of thinking are dependent upon the use of one or more notational systems

The notational systems one habitually uses influences the manner in which one perceives his environment: the picture of the universe shifts from notation to notationp

B k d ( ti d)Background (continued)

i l h b l h l i f Notational systems have been central to the evolution of civilization

Every notational system has limitations: a complexity Every notational system has limitations: a complexity barrier

The problems we face now as a civilization are, in many cases, notational

We need a more systematic way to develop and settle abstraction spacesabstraction spaces

M th ti th L f S iMathematics as the Language of Science

i b h i h i Equations represent behavior, not mechanism Offers conciseness of description Offers rigor Offers rigor

Th S t f th Effi f M thThe Secret of the Efficacy of Math

f l d l d Many formal models are created Applied mathematics uses only those that apply! Shorthand operations obscure mechanism (e g Shorthand operations obscure mechanism (e.g.

exponentiation) Other formal models may exist and apply alsoy y

Mathematics Deals Only With Certain yKinds of Entities

i i bl f b i h bj f h Entities capable of being the subject of theorems Entities that behave additively, without emergent

propertiesproperties

Rules are a Broader Way of Describing y gThings

b l i i l Can be multi-notational Can describe both mechanism and behavior Thousands can be assembled and acted upon by computer Thousands can be assembled and acted upon by computer Can shed light on ontology or basic nature of systems

R l C D ib M h iRules Can Describe Mechanism

li Causality Discreteness/quanta Probability even if 1 00 Probability, even if 1.00 Qualities of all kinds Fuzziness of relationships Fuzziness of relationships

Any Notational Statement Can Be yReformulated into If-Then Rule Format

l l i natural language assertions musical instructions process descriptions e g business processes process descriptions, e.g. business processes structural descriptions, e.g. chemical relational descriptions, e.g. linguistic ontologies relational descriptions, e.g. linguistic ontologies

Mathematical Statements Can Be Reformulated into If-Then Rule Format

b y = ax + b d = 1/2 gt2

predator prey models predator-prey models

M h i I li O t lMechanism Implies Ontology

h i ll f What is common among all systems of type A? What is the fundamental nature of systems of type A? What makes systems of type A different from systems of What makes systems of type A different from systems of

type B??

Rules Can be Represented in Place-Value pForm

l l i i b d d l i Place value assigns meaning based on content and location– In Hindu-Arabic numerals, this is column position– In ruleforms, this is column position, p

Thousands of rules can fit in same ruleform There are multiple basic ruleforms, not just one (as in

math) – But the total number is still small (<100?)