vol 3: Development
chapter 2: Model
page 7: Constraint
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Constraint
Consistency
A constraint it something that limits or controls the motion of a system. Constraints may be external, like gravitation, or internal, resulting from the structure of the system in question. So the relationships between piston, cylinder, connecting rod and crankshaft in a reciprocating engine determine that the piston will move up and down the cylinder as the crankshaft rotates,
Traditionally, the only constraint on God is that it be consistent. When we ask ourselves questions like 'can an omnipotent god make a stone bigger than it can lift' we see that even omnipotence has to respect consistency.
Science recognizes consistency at two levels, which we might call formal and empirical. Formal consistency requires that the formal mathematical and logical models used in science be internally consistent. It has been shown that large formal systems are both incomplete and incomputable so that it is difficult to tell whether or not they are consistent. Empirical consistency requires that our formal models can be mapped consistently with observations, at least to the limits of observability.
The fundamental article of scientific faith is that the scientific method is leading us toward a consistent union of a formal 'theory of everything' and all available observations. In the end we would like to see that this is the only possible universe, ie a universe that embraces all possibilities.
Symmetry
The vast dynamic structure (space-time) outlined in previous pages may be big enough to model god, but seems far too big and shapeless to be of any practical use in the finite world of everyday life.
Closer study reveals that there are many propositions that are true for the whole system, or for various parts (ie subsets) of it. In physics such propositions are called symmetries. In mathematics, such symmetries are established by proofs. Pythagoras' theorem, for instance, tells us that in Euclidean spaces, the square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides. This symmetry is of inestimable value to all who have to navigate in Euclidian space. Euclidean space - Wikipedia
Our course is to move from general properties of the transfinite network towards more specific properties, until we eventually come to the full complexity of everyday life.
The real line
Cantor was led to set theory and the study of transfinite numbers by his work on the line, outwardly one of the simplest structures imaginable. Its mystery was already known to the ancients, however. They knew that line segments (like the side and diagonal of a square) could have irrational relationships to one another (i.e. relationships that cannot be expressed as the ratio of two integers). The proof of this fact is simple, and was known in the time of Euclid. Euclid, Elements, book X,
Euclid begins the Elements with 23 definitions, five postulates and five common notions. Of these, this history mainly concerns two, point and line. The difficult relationship between point and line becomes obvious in physics, where we have become accustomed since time immemorial to represent the position of an object in space by a point and the motion of an object from one point to another by a line or orbit.
This leads naturally to two propositions: 1. a line is a sequence of points (since we can imagine a physical body moving from point to point); and 2. a line is continuous (since we do not imagine the physical body ceasing to exist as it moves from one point to the next: so it must occupy the intermediate points between any two given points and so on ad infinitum.)
These two features of the physical (or natural) line are uneasy bedfellows. Zeno, supporting Parmenides, used the point like nature of the line to prove that motion was impossible. Aristotle, more inclined to accept the obvious, defined continuous as overlapping or sharing a common point. On this definition, a continuous line cannot be composed of points, since points are indivisible and cannot overlap or share.
This inconsistency between points and lines remains at the heart of modern physics. Weinberg Here we propose to bypass this inconsistency by modeling the universe by a logically consistent communication system whose operations are modelled by Turing machines. Infinitesimal points and continuous lines are taken to be fictions that overly constrain our model of the universe,
The initial singularity
From an empirical point of view, the basic constraint on the universe is the initial singularity. We see the history of the universe as a product of the laws of physics that were laid down at the beginning. Traditionally, this was the work of the creator, but here we like to think that the universe is divine and responsible for its own existence. We model the universe as the system which has no external constraints because there is, by definition, nothing outside it to constrain it.
If this is the case, the only constraints on the universe are the internal requirements of logical consistency. Can we devise a formal model of the creation of a universe like ours from an initial state of structurelessness? Formally, it seems that it must be so. Cantor's theorem guarantees that in a consistent formal system like that described by the axioms of set theory, any set generates a more complex set.
This 'Cantor symmetry' arising from Cantor's proof provides us with a path through the universe from the simplest to the most complex systems. Set theory makes the theory of transfinite numbers possible because sets themselves are symmetrical with respect to cardinality or complexity. When applying set theory to the observed world. we might expect all those features which are properties of sets as such to be common to everything. In this construction, smaller sets act as 'alphabets' from which to construct more complex sets.
A set is defined by its elements and so one may think of the elements of a set as constraints upon it. However, when we introduce ordered sets, any alphabet with two letters or more can be concatenated in an ordered string to represent any structure, no matter how complex. We see this in action in our digital communication networks which can represent anything representable with a string of bits. The 'alphabetic' constraint is, in effect, no constraint.
Using set theory as our model, we see the initial singularity as the empty set, and build up the rest of the structure of the universe from the empty set in the same way that set theory generates the finite and transfinite numbers. Cantor, Jech
Further reading
Books
| Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.' Amazon back |
| Davis, Philip J, and David Park (editors), No Way: The Nature of the Impossible, W H Freeman 1987 Introduction: 'Think about the miracles of religion: a virgin gives birth, a man raises the dead and walks on water. Are these possible or are they impossible? Decide. ... what really counts as impossible? ... You can prove logical impossibilities, but do they say anything about the real world? You can assert practical impossibilities, but are they really impossible? Why bother about the question? Because mankind is inspired by the challenge of the impossible ...' [pp xiv, xvi] Amazon back |
| Feynman, Richard Phillips, and Gerry Neugebauer (Preface), Roger Penrose (Introduction), Six Not-So-Easy Pieces: Einstein's Relativity, Symmetry and Space-Time, Perseus Press 1998 'No single breakthrough in twentieth-century physics (with the possible exception of quantum mechanics) changed our view of the world more than that of Einstein's discovery of relativity. The notions that the flow of time is not a constant, that the mass of an object depends on its velocity, and that the speed of light is a constant no matter what the motion of the observer, at first seemed shocking to scientists and laymen alike. But, as Feynman shows so clearly and so entertainingly in the lectures chosen for this volume, these crazy notions are no mere dry principles of physics, but are things of beauty and elegance. No one - not even Einstein himself - explained these difficult, anti-intuitive concepts more clearly, or with more verve and gusto, than Richard Feynman.' 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 |
| Heath, Thomas L, Thirteen Books of Euclid's Elements (volume 2, III-IX), 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 |
| Heath, Thomas L, Thirteen Books of Euclid's Elements (volume 3, X-XIII), 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 |
| Jech, Thomas, Set Theory, Springer 1997 Jacket: 'This book covers major areas of modern set theory: cardinal arithmetic, constructible sets, forcing and Boolean-valued models, large cardinals and descriptive set theory. ... It can be used as a textbook for a graduate course in set theory and can serve as a reference book.' Amazon back |
| Kuhn, Thomas S, The Structure of Scientific Revolutions, U of Chicago Press 1996 Introduction: 'a new theory, however special its range of application, is seldom just an increment to what is already known. Its assimilation requires the reconstruction of prior theory and the re-evaluation of prior fact, an intrinsically revolutionary process that is seldom completed by a single man, and never overnight.' [p 7] Amazon back |
| Man, John, Alpha Beta: How 26 Letters Shaped the Western World, John Wiley and Sons 2001 Jacket: ' The idea behind the alphabet - that language with all its wealth of meaning can be recorded with a few meaningless signs - is an extraordinary one. So extraordinary, in fact, that it occurrred only once in human history: in Egypt about 4000 years ago, newly discovered origins that this book is the first to detail. Apha Betas then follows the emergence of the western alphabet as it evolved into its present form, contributing vital elelemtns to our sense of identity along the way. The Israelites used it to define their God, the Greeks to capture their myths, the Romans to display their power. And today it seems on the verge of yet further expansion through the internet.' Amazon back |
| Newton, Isaac, and Julia Budenz, I. Bernard Cohen, Anne Whitman (Translators) , The Principia : Mathematical Principles of Natural Philosophy , University of California Press 1999 This completely new translation, the first in 270 years, is based on the third (1726) edition, the final revised version approved by Newton; it includes extracts from the earlier editions, corrects errors found in earlier versions, and replaces archaic English with contemporary prose and up-to-date mathematical forms. ... The illuminating Guide to the Principia by I. Bernard Cohen, along with his and Anne Whitman's translation, will make this preeminent work truly accessible for today's scientists, scholars, and students. Amazon 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 |
| Sato, Ryuzo, and Rama V Ramachandran, Conservation Laws and Symmetry: Application to Economics and Finance, Kluwer Academic 1990 Amazon back |
| Schattschneider, Doris, Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher, W H Freeman and Company 1990 Jacket: 'VISIONS OF SYMMETRY narrates and beautifully illustrates M. C. Escher's lifelong passion for symmetry, one that underlies all his works. The book tells the fascinating story of the artict's discovery of the world of geometry and how he used this knowledge to create the intriguing interlocking figures and patterns that are his graphic signature. Amazon back |
| Stewart, Ian, Life's Other Secret: The new mathematics of the living world, Allen Lane 1998 Preface: 'There is more to life than genes. ... Life operates within the rich texture of the physical universe and its deep laws, patterns, forms, structures, processes and systems. ... Genes nudge the physical universe in specific directions ... . The mathematical control of the growing organism is the other secret ... . Without it we will never solve the deeper mysteries of the living world - for life is a partnership between genes and mathematics, and we must take proper account of the role of both partners.' (xi) Amazon back |
| Stewart, Ian, Why Beauty is Truth: A History of Symmetry, Basic Books/Perseus 2007 Jacket: ' ... Symmetry has been a key idea for artists, architects and musicians for centuries but within mathematics it remained, until very recently ,an arcane pursuit. In the twentieth century, however, symmetry emerged as central to the most fundamental ideas in physics and cosmology. Why beauty is truth tells its history, from ancient Babylon to twenty-first century physics.' Amazon back |
| Zee, Anthony, Quantum Field Theory in a Nutshell, Princeton University Press 2003 Amazon book description: 'An esteemed researcher and acclaimed popular author takes up the challenge of providing a clear, relatively brief, and fully up-to-date introduction to one of the most vital but notoriously difficult subjects in theoretical physics. A quantum field theory text for the twenty-first century, this book makes the essential tool of modern theoretical physics available to any student who has completed a course on quantum mechanics and is eager to go on. Quantum field theory was invented to deal simultaneously with special relativity and quantum mechanics, the two greatest discoveries of early twentieth-century physics, but it has become increasingly important to many areas of physics. These days, physicists turn to quantum field theory to describe a multitude of phenomena. Stressing critical ideas and insights, Zee uses numerous examples to lead students to a true conceptual understanding of quantum field theory--what it means and what it can do. He covers an unusually diverse range of topics, including various contemporary developments,while guiding readers through thoughtfully designed problems. In contrast to previous texts, Zee incorporates gravity from the outset and discusses the innovative use of quantum field theory in modern condensed matter theory. Without a solid understanding of quantum field theory, no student can claim to have mastered contemporary theoretical physics. Offering a remarkably accessible conceptual introduction, this text will be widely welcomed and used. Amazon back |
| Zee, Anthony, Fearful Symmetry: The Search for Beauty in Modern Physics, Macmillan Publishing Company 1986 Jacket: 'Fearful Symmetry invites us to view physics not merely as a body of theories and facts, but as a spirited quest to to fathom the workings of the universe It is the first book to offer an accessible explanation of how symmetry forms the intellectual and aesthetic foundation of modern physics. We go from the left-right symmetry of living forms to the deep, abstract symmetries of the fundamental laws of Nature. Zee shows how symmetry guides Nature's grand design in such diverse phenomena as the longevity of stars, the magic of light, the expansion of the cosmos, and the life and death of particles.' Amazon back |
Papers
| Weinberg, Steven, "The cosmological constant problem", Reviews of Modern Physics, 61, , 1989, page 1-23. 'Astronomical observations indicate that the cosmological constant is many orders of magnitude smaller than estimated in modern theories of elementary particles. After a brief review of the history of this problem, five different approaches to its solution are described.'. back |
Links
| Euclidean space - Wikipedia Euclidean space - Wikipedia, the free encyclopedia 'An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean. For example, the surface of a sphere is not; a triangle on a sphere (suitably defined) will have angles that sum to something greater than 180 degrees. In fact, there is essentially only one Euclidean space of each dimension, while there are many non-Euclidean spaces of each dimension. Often these other spaces are constructed by systematically deforming Euclidean space.' back |
| Pythagorean theorem - Wikipedia Pythagorean theorem - Wikipedia, the free encyclopedia 'In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). This is usually summarized as: The square on the hypotenuse is equal to the sum of the squares on the other two sides.' back |
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