September 12, 1998: "Mathematics, Rules, and Scientific Representations". Presented at a symposium of the Washington Evolutionary Systems Society.

1. Cover Page
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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2. Mathematics, Rules, and
Scientific Representations
The Need for an Integrated, Multi-
Multi
Notational Approach to Science
Jeffrey G. Long, September 12, 1998
jefflong@aol.com

3. Basic A
B i Assertions
ti
 In spite of all progress to date, we still don’t “understand”
i f ll d ill d d d
complex systems
 This is not because of the nature of the systems, but rather
systems
because our notational systems are inadequate

4. Basic Q ti
B i Questions
 Why do we use the notational systems we use?
h d h i l
 What are their fundamental 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
science?

5. Background: N t ti
B k d Notational H
l Hypotheses
th
 There are f
h four ki d of sign systems
kinds f i
– Formal: syntax only
– Informal: semantics only
– Notational: syntax and semantics
– Subsymbolic: neither syntax nor semantics
 Of these, notational systems are the least-explored

6. Background ( ti
B k d (continued)
d)
 Each primary notational system maps a different
h i i l diff
“abstraction space”
– Abstraction spaces are incommensurable
p
– Perceiving these is a unique human ability
 Abstraction spaces are discoveries, not inventions
– Ab
Abstraction spaces are real
i l
– Their interactions are the basis of physical law

7. Background ( ti
B k d (continued)
d)
 Acquiring literacy in a notation is learning how to see a
i i li i i i l i h
new abstraction space
– This is one of many ways we manage p
y y g perception (
p (“intellinomics”)
)
 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
p
picture of the universe shifts from notation to notation

8. Background ( ti
B k d (continued)
d)
 Notational systems have been central to the evolution of
i l h b l h l i f
civilization
 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 spaces

9. Mathematics as the Language of Science
M th ti th L fS i
 Equations represent behavior, not mechanism
i b h i h i
 Offers conciseness of description
 Offers rigor

10. The Secret of th Effi
Th S t f the Efficacy of M th
f Math
 Many f
formal models are created
l d l d
 Applied mathematics uses only those that apply!
 Shorthand operations obscure mechanism (e.g.
(e g
exponentiation)
 Other formal models may exist and apply also
y y

11. Mathematics Deals Only With Certain
y
Kinds of Entities
 Entities capable of being the subject of theorems
ii bl f b i h bj f h
 Entities that behave additively, without emergent
properties

12. Rules are a Broader Way of Describing
y g
Things
 Can b multi-notational
be li i l
 Can describe both mechanism and behavior
 Thousands can be assembled and acted upon by computer
 Can shed light on ontology or basic nature of systems

13. Rules C Describe M h i
R l Can D ib Mechanism
 Causality
li
 Discreteness/quanta
 Probability even if 1.00
Probability, 1 00
 Qualities of all kinds
 Fuzziness of relationships

14. Any Notational Statement Can Be
y
Reformulated into If-Then Rule Format
 natural language assertions
ll i
 musical instructions
 process descriptions e.g. business processes
descriptions, e g
 structural descriptions, e.g. chemical
 relational descriptions, e.g. linguistic ontologies

15. Mathematical Statements Can Be
Reformulated into If-Then Rule Format
 y = ax + b
 d = 1/2 gt2
 predator prey models
predator-prey

16. Mechanism I li O t l
M h i Implies Ontology
 What is common among all systems of type A?
h i ll f
 What is the fundamental nature of systems of type A?
 What makes systems of type A different from systems of
type B??

17. Rules Can be Represented in Place-Value
p
Form
 Place value assigns meaning based on content and location
l l i i b d dl i
– In Hindu-Arabic numerals, this is column position
– In ruleforms, this is column p
, position
 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?)