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Mathematics rules and scientific representations


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

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Mathematics rules and scientific representations

  1. 1. Cover Page   Mathematics, Rules,  and Scientific  Representations  Author: Jeffrey G. Long ( 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   License This work is licensed under the Creative Commons Attribution‐NonCommercial 3.0 Unported License. To view a copy of this license, visit‐nc/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA.  Uploaded July 1, 2011 
  2. 2. Mathematics, Rules, andScientific RepresentationsThe Need for an Integrated, Multi- Multi Notational Approach to Science Jeffrey G. Long, September 12, 1998
  3. 3. Basic AB 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. 4. Basic Q tiB 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. 5. Background: N t tiB 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. 6. Background ( tiB 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. 7. Background ( tiB 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. 8. Background ( tiB 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. 9. Mathematics as the Language of ScienceM th ti th L fS i Equations represent behavior, not mechanism i b h i h i Offers conciseness of description Offers rigor
  10. 10. The Secret of th EffiTh 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. 11. Mathematics Deals Only With Certain yKinds 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. 12. Rules are a Broader Way of Describing y gThings 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. 13. Rules C Describe M h iR 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. 14. Any Notational Statement Can Be yReformulated 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. 15. Mathematical Statements Can BeReformulated into If-Then Rule Format y = ax + b d = 1/2 gt2 predator prey models predator-prey
  16. 16. Mechanism I li O t lM 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. 17. Rules Can be Represented in Place-Value pForm 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?)