Mathematics rules and scientific representations
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The document discusses the inadequacies of existing notational systems in understanding complex systems and proposes the need for an integrated multi-notational approach to science. It highlights that notational systems influence perception and reasoning and that mathematics, while effective for describing certain behaviors, has limitations in addressing the complexities of real-world systems. The author advocates for the development of systematic ways to explore and define abstraction spaces through various notational systems and rules.
Mathematics rules and scientific representations
- 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 License This work is licensed under the Creative Commons Attribution‐NonCommercial 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by‐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. 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?)