vol III Development:
Chapter 2: Model
page 4: Logical continuity
It has been widely believed, since ancient times, that the world is continuous. 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. Over a few thousand years, mathematicians have devised apparently consistent means of representing continua in strings of discrete symbols. This has been necessary because the mathematical literature is represented by strings of discrete symbols. Aristotle, Physics V, iii, Linear continuum - Wikipedia
Motion looks continuous to us, and this observation in expressed by the mathematical formalism of classical dynamics which represents the relationships between variables in space and time as continuous functions. The only difficulty is that this mathematical model does not correspond to reality. Reality, as far as we can observe it, is not continuous but quantized. It comprises individual events and individual personalities. Continuous function - Wikipedia
This fact is obvious at the human scale, and Max Planck discovered that all processes proceed stepwise by quanta of action. The interface between continuous mathematics and a quantized Universe is problematic. What we have learnt is that the continuous functions which we believe describe the dynamics of the Universe yield only the probabilities of discrete events. On the other hand these functions appear predict the nature and probability of possible events with high precision. Precision tests of QED - Wikpedia, Planck constant - Wikipedia
Why is the Universe quantized? The answer proposed here is that the Universe is best modelled as a multilayered communication network. The strategy embodied in the mathematical theory of communication is to reduce the possibility of messages being confused by placing them as far apart as possible in the message space. This is tantamount to quantization, since valid messages are represented as discrete and isolated points in the space of all possible messages. Claude E Shannon: The mathematical theory of communication
Shannon's strategy to increase the distances between the messages in a message space depends on the fact that the size of a message space increases exponentially with the length of the messages occupying the space. The implementation of this encoding requires computation to convert the input text into the coded test, and a second computation, the inverse of the first, to convert the coded text back to the original text. These computations must be deterministic to ensure error free operation. The transmission process can be tested by the receiver transmitting the received message back to the transmitter for comparison with the original.
A universal Turing machine, named for Alan Turing, is a logical machine which can perform all possible deterministic calculations. Turing invented the machine to prove that there are incomputable functions, that is functions which cannot be computed by a machine that can do all possible computations. When it comes to implementing error free communication we can use only computable algorithms for encoding and decoding. Turing machine - Wikipedia
Let us say that a Turing machine establishes 'logical continuity' between its input and its output. We may contrast logical continuity so understood with classical, analytic or geometric continuity which became very important for mathematics with the discovery of calculus. Calculus has remained ever since the one of the main mathematical methods of physical science. Calculus - Wikipedia
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
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)'. The implicit equation is
no magnitude x infinite repetition = magnitude Heath
Classical continuity 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 each other 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 227 a 10-14 Let us call this 'continuity by overlap'. This approach to continuity seems more like logical continuity than to classical continuity. Aristotle - Physics
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 transfinite number space described in the previous page is a recursive
function space. There are ℵ1 different mappings of the
set of ℵ0 natural numbers onto itself; ℵ2 mappings of this set of ℵ1 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'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. Cohen
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). The middle term is the area of overlap between the premisses of a syllogism which enable us to draw the conclusion. We see a modern representation of the same idea in the Venn diagrams of set theory. 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. Aristotle: Prior Analytics, Robin Smith: Aristotle's Logic, Venn diagram - Wikipedia
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 same might be said of computable functions. Although there are ℵ1 mappings from the set of ℵ0 natural numbers to itself, there are only ℵ1 different Turing machines, so that the majority of these mappings are incomputable.
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. Since only a small fraction of functions are logically bound, there is ample scope for the uncertainty which we observe in the Universe
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. Often we find, as in quantum mechanics, that potential inconsistencies are avoided by uncertainties. Uncertainty principle - Wikipedia
(revised 26 February 2015)
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Further reading
Books
Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)
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 introduction 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 motion in the universe 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.'
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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'
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Boole, George, Investigation of the Laws of Thought, on which Are Founded the Mathematical Theories of Logic and Probabilities, Dover Publications 1958 First Sentence:
'1. THE design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method ; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities ; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.'
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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 Gödel's proof of the consistency of the continuum hypothesis. . . .'
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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.'
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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.'
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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. ...'
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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.
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Heath, Thomas Little, 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.'
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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.'
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| 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
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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.'
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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 JvN regards as the definitive form of quantum mechanics. . . . Regarded as a tour de force at the time of its publication, this book is still indispensable for those interested in the fundamental issues of quantum mechanics.'
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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 functional 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
| Aristotle, Prior Analytics, 'We must first state the subject of our inquiry and the faculty to which it belongs: its subject is demonstration and the faculty that carries it out demonstrative science. We must next define a premiss, a term, and a syllogism, and the nature of a perfect and of an imperfect syllogism; and after that, the inclusion or noninclusion of one term in another as in a whole, and what we mean by predicating one term of all, or none, of another.' back |
| Aristotle - Physics, The Internet Classic Archive | Physics Aristotle, Written 350 B.C.E
Translated by R. P. Hardie and R. K. Gaye back |
| Aristotle: Prior Analytics, Prior Analytics: The Internet Classics Archive, 'We must first state the subject of our inquiry and the faculty to which it belongs: its subject is demonstration and the faculty that carries it out demonstrative science. We must next define a premiss, a term, and a syllogism, and the nature of a perfect and of an imperfect syllogism; and after that, the inclusion or noninclusion of one term in another as in a whole, and what we mean by predicating one term of all, or none, of another.' back |
| Calculus - Wikipedia, Calculus - Wikipedia, the free encyclopedia, 'Calculus (Latin, calculus, a small stone used for counting) is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern university education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of equations.' back |
| 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 |
| Claude E Shannon, A Mathematical Theory of Communication, 'The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages.' back |
| Continuous function - Wikipedia, Continuous function - Wikipedia, the free encyclopedia, 'IIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be a "discontinuous function". A continuous function with a continuous inverse function is called "bicontinuous".' back |
| Linear continuum - Wikipedia, Linear continuum - Wikipedia, the free encyclopedia, 'In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line.
Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two members there is another, and which "lacks gaps" in the sense that every non-empty subset with an upper bound has a least upper bound.' 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 |
| Planck constant - Wikipedia, Planck constant - Wikipedia, the free encyclopedia, 'The Planck constant (denoted h), also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1900. . . . Planck discovered that physical action could not take on any indiscriminate value. Instead, the action must be some multiple of a very small quantity (later to be named the "quantum of action" and now called Planck's constant).' back |
| Precision tests of QED - Wikpedia, Precision tests of QED - Wikpedia, the free encyclopedia, 'Quantum electrodynamics (QED), a relativistic quantum field theory of electrodynamics, is among the most stringently tested theories in physics.
The most precise and specific tests of QED consist of measurements of the electromagnetic fine structure constant, α, in various physical systems. Checking the consistency of such measurements tests the theory.' back |
| Robin Smith, Aristotle's Logic (Stanford Encyclopedia of Philosophy), 'Aristotle's logic, especially his theory of the syllogism, has had an unparalleled influence on the history of Western thought. It did not always hold this position: in the Hellenistic period, Stoic logic, and in particular the work of Chrysippus, took pride of place. However, in later antiquity, following the work of Aristotelian Commentators, Aristotle's logic became dominant, and Aristotelian logic was what was transmitted to the Arabic and the Latin medieval traditions, while the works of Chrysippus have not survived.' back |
| Turing machine - Wikipedia, Turing machine - Wikipedia, the free encyclopedia, 'Turing machines are extremely basic abstract symbol-manipulating devices which, despite their simplicity, can be adapted to simulate the logic of any computer algorithm (as we understand them). They were described in 1936 by Alan Turing. Though they were intended to be technically feasible, Turing machines were not meant to be a practical computing technology, but a thought experiment about the limits of mechanical computation; thus they were not actually constructed. Studying their abstract properties yields many insights into computer science and complexity theory.' back |
| Uncertainty principle - Wikipedia, Uncertainty principle - Wikipedia, the free encyclopedia, 'In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle known as complementary variables, such as position x and momentum p, can be known simultaneously.' back |
| Venn diagram - Wikipedia, Venn diagram - Wikipedia, the free encyclopedia, 'A Venn diagram or set diagram is a diagram that shows all possible logical relations between a finite collection of different sets. Venn diagrams were conceived around 1880 by John Venn. They are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics and computer science.' back |
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