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page 3: Logical continuity
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Logical continuity
Cantor invented (or discovered) set theory and transfinite numbers in his quest to determine the cardinal of the continuum, ie to count the points in the natural line? Cardinality of the continuum - Wikipedia
This is a very ancient question, as is the whole theory of continuity. Aristotle's common sense view is that anything that moves must follow a continuous path, and so the space in which it moves must also be continuous. Aristotle, Physics V, iii.
The concept of infinity is closely allied to that of continuity. On the one hand, to be continuous is to be featureless, as we expect the infinite to be. On the other hand, infinity is useful to bridge the gap between the definitions of a point ('something with position (identity) but no magnitude') and a line ('a breadthless length (ie magnitude)'. Heath The implicit equation is
Let us call this continuity by cardinality. This is effectively continuity by closeness, since the more magnitude free points we crowd into a given interval, the closer they must be.
By Aristotle's standards, this was not really continuity at all. His definition reads: I mean by one thing being continuous with another that those limiting extremes of the two things in virtue of which they touch eachother become one and the same thing and (as the very name 'syneches' indicates) are 'held together', 'which can only be if the two limits do not remain two but become one and the same. Physics 227a10-14 Let us call this (with quantum mechanics in mind) continuity by overlap.
It seems to me that continuity by overlap is much more reliable than continuity by contiguity. As a builder, I like to see the ends of beams overlapping their supports. Mere closeness is no guarantee of effective support.
One of the most fruitful relationships in the history of mathematics has been the relationship between arithmetic and geometry. Using a suitable system of coordinates, we may map arithmetic numbers onto geometric spaces. Such mappings allow us to use arithmetic to gain insight into geometry and vice versa. Descartes formalized this idea with his 'Cartesian coordinates' and the idea has spread into manifolds and function spaces to give us the mathematical foundations of physics. Peacock, Hobson
The Cantor universe described in the previous page is a recursive function space. There are aleph(1) different mappings of the set of aleph(0) natural numbers onto itself; aleph(2) mappings of this set of aleph(1) functions onto itself, and so on without end.
Cantor hoped that these huge numbers determined by set theory would lead him to the cardinal of the continuum. In 1963, Cohen proved that Cantor's continuum hypothesis is independent of set theory. Set theory can tell us nothing about it. Cohen Cohen's argument is a typical mathematical proof, a chain of logical steps leading from a set of axioms to a conclusion. Let us say that this chain, in a valid proof, is logically continuous.
Aristotle made a careful study of logic, and foreshadowed the concept of logical continuity with his understanding of the middle term in a logical syllogism. (Prior Analytics, I, iv). Aristotle The middle term is the area of overlap between the premisses of a syllogism which enable us to draw the conclusion. Cohen's proof uses logical continuity to tell us something about cardinal continuity. This is an example of the mathematical use of logical continuity to talk about continuity by cardinality.
The invention of calculus gave a new impetus to the study of continuity, and the immense success of calculus in physics led us to the idea that the physical universe itself is continuous, to be described by continuous functions. The requirement of continuity, however, is a very strong constraint on the space of functions. Continuous functions are an infinitesimal fraction of all functions. Ashby
The aim of this site is to build a model comprehensive enough to approach god and the universe. We therefore admit all functions, continuous or not. The only constraint we recognize on universal process is logical continuity. We will say that a logically continuous system is logically bound.
Logical binding is one of the traditional features of the divinity. We accept the hypothesis that there is no inconsistency in the divinity. This is also an article of scientific faith. Any scientist, finding inconsistencies within a model or between a model and observation automatically assumes that the trouble lies in the model.
[revised 19 November 2007)
Further reading
Books
| Aristotle, and P H Wickstead and F M Cornford, translators, Physics books V-VIII, Harvard University Press,William Heinemann 1980 Introduction: 'Simplicius tells us that Books I - IV of the Physics were referred to as the books Concerning the Principles, while Books V - VIII were called On Movement. The earlier books have, in fact, defined the things which are subject to movement (the contents of the physical world) and analyzed certain concepts - Time, Place and so forth - which are involved in the occurrence of movement.' Book V is a further intoduction to the detailed analysis in Books VI - VIII. Book VI deals with continuity, Book VII is an introductory study for Book VIII, which brings us to the conclusion that all change and motionin the unvierse are ultimately caused by a Prime Mover which is itself unchanging and unmoved and which has neither magnitude nor parts, but is spiritual and not in space. Amazon back |
| Aristotle, and H P Cooke (translator), Hugh Tredennick (translator), Aristotle: Categories. On Interpretation. Prior Analytics, Loeb Classical Library 2002 Amazon customer review: 'If you're not familiar with the Loeb's, this wonderful series aims to make accessible all important Greek and Latin literature in bilingual editions - English translations with the original text on the opposite page. These books can be of great value to students of classics as well as to professionals in other fields, e.g. philosophers that are not fluent in Greek, but need an accurate and dependable translation of the works of Plato or Aristotle. And in my experience, the Loeb's rarely fail to meet expectations. This volume contains Harold P. Cooke's translation of the Categories and De Interpretatione as well as Hugh Tredennick's translation of the Prior Analytics. I found Cooke's translations to be a little bit disapointing. The English translation often merely paraphrases Aristotle. This doesn't automatically make the translation a bad one, of course, for sometimes paraphrase is needed. But there are other translations available of these works, and, in my oppinion, Cooke's translation is inferior to J.L. Ackrill's translation of the Categories and De Interpretatione, which is both more accurate and relatively easy to read. Now, I assume that no one would buy a Loeb primarily for the Greek or Latin text - for that you would turn to the Oxford Classical Texts or other critical text editions. So if you're buying a Loeb it's either for the translation or to be able to compare an English translation with the original. If you need to compare an English translation of these particular works with the Greek text, then this volume will be useful to you. However, if you just want to read these works in translation, you might very well be satisfied with this one, but I still recommend other translations such as J.L. Ackrill's excellent translation of the Categories and De Interpretatione.' GT Amazon back |
| Ashby, W Ross, An Introduction to Cybernetics, Methuen 1964 'This book is intended to provide [an introduction to cybernetics]. It starts from common-place and well understood concepts, and proceeds step by step to show how these concepts can be made exact, and how they can be developed until they lead into such subjects as feedback, stability, regulation, ultrastability, information, coding, noise and other cybernetic topics' Amazon back |
| Boole, George, Investigation of the Laws of Thought, on which Are Founded the Mathematical Theories of Logic and Probabilities, Dover 1973 Paperback (June 1973) Dover Pubns; ISBN: 0486600289 Amazon back |
| Cohen, Paul J, Set Theory and the Continuum Hypothesis, Benjamin/Cummings 1966-1980 Preface: 'The notes that follow are based on a course given at Harvard University, Spring 1965. The main objective was to give the proof of the independence of the continuum hypothesis [from the Zermelo-Fraenkel axioms for set theory with the axiom of choice included]. To keep the course as self contained as possible we included background materials in logic and axiomatic set theory as well as an account of Goedel's proof of the consistency of the continuum hypothesis. ..' (i) Amazon back |
| Davis, Martin, Computability and Unsolvability, Dover 1982 Preface: 'This book is an introduction to the theory of computability and non-computability ususally referred to as the theory of recursive functions. The subject is concerned with the existence of purely mechanical procedures for solving problems. ... The existence of absolutely unsolvable problems and the Goedel incompleteness theorem are among the results in the theory of computability that have philosophical significance.' Amazon back |
| Davis, Phillip J, and Reuben Hersh, Descartes Dream: The World According to Mathematics, Penguin 1988 Preface: 'We are concerned with the impact mathematics makes when it is applied to the world that lies outside mathematics itself; when it is used in relation to the world of nature or of human activities. This is sometimes called applied mathematics. This activity has now become so extensive that we speak of the "mathematisation of the world." We want to know the conditions of civilisation that bring it about. We want to know when these applications are effective, when they are ineffective, when beneficial, dangerous or irrelevant. We want to know how they constrain our lives, how they transform our perception of reality.' Amazon back |
| Descartes, Rene, and (Translated by David Eugene Smith and Marcia L Latham) , Geometrie , Dover 1956 Jacket: ' ... With this volume, Descartes founded modern analytical geometry. Reducing geometry to algebra and analysis and, conversely, showing that analysis can be translated into geometry, it opened the way for modern mathematics. ... This edition contains the entire definitive Smith-Latham translation of Descartes three books: Problems the Construction of which requires Only Straight Lines and Circles; On the Nature of Curved Lines; On the Construction of Solid and Supersolid Problems. Interleaved page by page with the translation is a complete facsimile of the 1637 French text, together with Descartes' original illustrations. ...' Amazon back |
| Descartes, Rene, Rules for the direction of the mind: Discourse on the method, Encyclopaedia BritannicaB0006AU8ZG 1955 Amazon back |
| Hallett, Michael, Cantorian set theory and limitation of size, Oxford UP 1984 Jacket: 'This book will be of use to a wide audience, from beginning students of set theory (who can gain from it a sense of how the subject reached its present form), to mathematical set theorists (who will find an expert guide to the early literature), and for anyone concerned with the philosophy of mathematics (who will be interested by the extensive and perceptive discussion of the set concept).' Daniel Isaacson. Amazon back |
| Heath, Thomas L, Thirteen Books of Euclid's Elements (volume 1, I-II), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time - the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.' Amazon back |
| Hobson, M P, and G. P. Efstathiou, A. N. Lasenby, General Relativity: An Introduction for Physicists, Cambridge University Press 2006 Amazon Editorial Reviews Book Description 'After reviewing the basic concept of general relativity, this introduction discusses its mathematical background, including the necessary tools of tensor calculus and differential geometry. These tools are used to develop the topic of special relativity and to discuss electromagnetism in Minkowski spacetime. Gravitation as spacetime curvature is introduced and the field equations of general relativity derived. After applying the theory to a wide range of physical situations, the book concludes with a brief discussion of classical field theory and the derivation of general relativity from a variational principle.' Amazon back |
| Kneebone, G T , Mathematical Logic and the Foundations of Mathematics, van Nostrand 1975 Preface: 'The present book ... is designed to serve in the first instance, when supplemented by reference to original sources, as a comprehensive introduction to the earlier phases of the historical development of the philosophy of mathematics. p vi.back |
| Peacock, John A, Cosmological Physics, Cambridge University Press 1999 Nature Book Review: 'The intermingling of observational detail and fundamental theory has made cosmology an exceptionally rich, exciting and controversial science. Students in the field -- whether observers or particle theorists -- are expected to be acquainted with matters ranging from the Supernova Ia distance scale, Big Bang nucleosynthesis theory, scale-free quantum fluctuations during inflation, the galaxy two-point correlation function, particle theory candidates for the dark matter, and the star formation history of the Universe. Several general science books, conference proceedings and specialized monographs have addressed these issues. Peacock's Cosmological Physics ambitiously fills the void for introducing students with a strong undergraduate background in physics to the entire world of current physical cosmology. The majestic sweep of his discussion of this vast terrain is awesome, and is bound to capture the imagination of most students.' Ray Carlberg, Nature 399:322 Amazon back |
| van Heijenoort, Jean, From Frege to Goedel: A Source Book in Mathematical Logic 1879 - 1931. , iUniverse.com 1999 Amazon book description: 'Collected here in one volume are some thirty-six high quality translations into English of the most important foreign-language works in mathematical logic, as well as articles and letters by Whitehead, Russell, Norbert Weiner and Post...This book is, in effect, the record of an important chapter in the history of thought. No serious student of logic or foundations of mathematics will want to be without it.' Amazon back |
| von Neumann, John, and Robert T Beyer (translator), Mathematical Foundations of Quantum Mechanics, Princeton University Press 1983 Jacket: '... a revolutionary book that caused a sea change in theoretical physics. ... JvN begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which vN regards as the definitive form of quantum mechanics. ... Regarded as a tour de force at the time of its publication, this book is still indispensible for those interested in the fundamental issues of quantum mechanics.' Amazon back |
Papers
| Goedel, Kurt, "On the completeness of the calculus of logic", in Solomon Fefferman et al (eds), Kurt Goedel: Collected Works Volume 1 Publications 1929-1936, New York, OUP 1986, , , , , page 61-101. '1. Introduction The main object of the following is the proof of the completeness of the axiom system for what is clled the restricted funcitonal calculus, namely the system given in Whitehead and Russel 1910 part 1 *1 ns *10 ... Here 'completeness' is to mean that every valid formula expressible in the restricted functional calculus ... can be derived from the axioms by means of a finite sequence of formal inferences. ...'. back |
Links
| Cardinality of the continuum - Wikipedia Cardinality of the continuum - Wikipedia, the free encyclopedia 'In mathematics, the cardinality of the continuum (sometimes also called the power of the continuum) is the cardinal number of the set of real numbers R (sometimes called the continuum). This cardinal number is often denoted by c, so c = R.' back |
| Peter Suber Logical Systems ' Logical Systems. A second course in logic, on the properties of formal systems and the metatheory of standard first-order logic. The course focuses on proof theory, model theory, and the important limitative results of the 20th century, including Gödel's theorems. Topics sometimes covered in addition include recursive function theory, computability, paradoxes, non-standard logics, and the nature and limits of formalism.' back |
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