vol 3: Development
chapter 3: Physics
page 9: Fixed points
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Fixed Points
The game of life
Energy, we might say, is the spacetime representation of action. Let this representation be described mathematically by the quantum mechanics of an isolated system, described in page 4: Quantum mechanics. The next step is to find out how the space and time in which we and all particles live arises from the communication of isolated quantum systems.
From a physical point of view, space-time is the stage in which the game of existence is played. The game of existence, like all games is a game of communication, played by passing messages (balls, cards, looks, words) to one another. Intuitively space is something that remains remains the same (the playing field, the rules of the game) while other things move around in it (players, messages).
Space orders the positions of the players while time orders their motions. In computing terms, space corresponds to memory, time to processing. In relativistic physics, space corresponds to momentum, time to energy. Space - Wikipedia, Time - Wikipedia, Momentum - Wikipedia, Energy - Wikipedia
Fixed point theorems
A mathematical key to the relationship between motion and stillness seems to be the fixed point theorems. These theorems are a line of mathematical development which started with Brouwer, who proved Brouwer's fixed point theorem: Casti, Brouwer fixed point theorem - Wikipedia, Fixed point theorem - Wikipedia
The content of fixed point theorems is similar to the content of quantum mechanics: the fixed points in the range of a motion induce further fixed points in the motions. Thus the fixed end points in a vibrating string induce further nodes in the vibrating string corresponding to the overtones of the vibration. The fixed points induced by the nucleus of an atom induce an infinite complex of stationary states in the motion of the atomic electrons. Atomic nucleus - Wikipedia, Atomic orbital - Wikipedia
We may think of the rules of a game as the fixed points of its motion. The players may bounce around within the rules but (at least in theory) can never step outside them. The rules are not isolated from the game, however, they are part of it.
Because they are fixed points in the dynamics of the game, they can be committed to writing, which is also fixed. Further, they can be committed to memory by referees, players and spectators and give meaning to the motions of the players as they manoeuvre for advantage within the rules.
From this point of view, fixed points are messages, outside time, like momentum, text and all purely formal systems. They are 'revelations' of the dynamics.
God and the initial singularity
The primary fixed point for the Christian world is called God, the eternal immobile creator of the world. God provides a fixed frame of reference within which to understand everything. In other words, the theological study of the image of God, sometimes called the Word, is the traditional theory of everything. Christ the Logos - Wikipedia
From the point of view of modern cosmology, the primary fixed point in the universe, predicted by the general relativity, is the initial singularity. Hawking & Ellis, Gravitational singularity - Wikipedia God controls and determines everything that happens in the world. The initial singularity, however, is indeterminate, so anything can happen. The resulting universe, constrained only by having an empty starting point, realizes all of possibilities, that is all consistent events, and so naturally deserves the name God.
Conclusion
Fixed points theorems tell us that whenever a compact convex set is mapped onto itself (by any motion) there will be fixed points in this motion. Such fixed points can act as durable symbols, points effectively outside time, which form the foundations for memory and space.
The unconstrained motion that we associate with energy can be expected to develop fixed points as soon as we constrain it in some way. The constraint considered here is communication. The effect of communication is to correlate the communicants. Using the theory of probability, we can parametrize this correlation by real number lying between 0 (complete independence) and 1 (identity).
The fixed points in the universe make knowledge and survival possible. Hunters and gatherers learn the habits and seasons of their prey. Whenever we make something we rely for success on our knowledge of the fixed properties of our materials.
(revised 20 November 2008)
Further reading
Books
| Casti, John L, Five Golden Rules: Great Theories of 20th-Century Mathematics - and Why They Matter, John Wiley and Sons 1996 Preface: '[this book] is intended to tell the general reader about mathematics by showcasing five of the finest achievements of the mathematician's art in this [20th] century.' p ix. Treats the Minimax theorem (game theory), the Brouwer Fixed-Point theorem (topology), Morse's theorem (singularity theory), the Halting theorem (theory of computation) and the Simplex method (optimisation theory). Amazon back |
| Hawking, Steven W, and G F R Ellis, The Large Scale Structure of Space-Time , Cambridge UP 1975 Preface: Einstein's General Theory of Relativity ... leads to two remarkable predictions about the universe: first that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity; and secondly that there is a singularity in our past which constitutes, in some sense, a beginning to our universe. Our discussion is principally aimed at developing these two results.' Amazon back |
Links
| Simple harmonic motion - Wikipedia Simple harmonic motion - Wikipedia, the free encyclopedia 'Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. The motion is periodic, as it repeats itself at standard intervals in a specific manner - described as being sinusoidal, with constant amplitude. It is characterized by its amplitude (which is always positive), its period which is the time for a single oscillation, its frequency which is the number of cycles per unit time, and its phase, which determines the starting point on the sine wave. The period, and its inverse the frequency, are constants determined by the overall system, while the amplitude and phase are determined by the initial conditions (position and velocity) of that system.' back |
| Quantum harmonic oscillator - Wikipedia Quantum harmonic oscillator - Wikipedia, the free encyclopedia 'The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point. Furthermore, it is one of the few quantum mechanical systems for which a simple exact solution is known.' back |
| Brouwer fixed point theorem - Wikipedia Brouwer fixed point theorem - Wikipedia, the free encyclopedia 'In mathematics, the Brouwer fixed point theorem is an important fixed point theorem that applies to finite-dimensional spaces and forms the basis for several more general fixed point theorems. It is named after Dutch mathematician L. E. J. Brouwer.' back |
| Fixed point theorem - Wikipedia Fixed point theorem - Wikipedia, the free encyclopedia 'In mathematics, a fixed point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. Results of this kind are amongst the most generally useful in mathematics. The Banach fixed point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. By contrast, the Brouwer fixed point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).' back |
| FIFA FIFA - Laws of the Game 'On 1 July 2006 the new Laws of the Game, modified at the 119th International Football Association Board (IFAB) meeting in London, England on 26 February 2005, came into force. You can check out the Laws of your favourite game by clicking on the image to the right. Authorised by the International Football Association Board This booklet may not be reproduced or translated in whole or in part in any manner without the permission of FIFA. Published by FIFA Federation Internationale de Football Association FIFA-Strasse 20, P.O. Box 8044 Zurich, Switzerland' back |
| Space - Wikipedia Space - Wikipedia, the free encyclopedia 'The concept of space has been of interest for philosophers and scientists for much of human history. The term is used somewhat differently in different fields of study, hence it is difficult to provide an uncontroversial and clear definition outside of specific defined contexts. Disagreement also exists on whether space itself can be measured or is part of the measuring system. (See Space in philosophy.) Science considers space to be a fundamental quantity (a quantity which can not be defined via other quantities because other quantities -- like force and energy -- are already defined via space). Thus an operational definition is used in which the procedure of measurement of space intervals (distances) and the units of measurement are defined.' back |
| Time - Wikipedia Time - Wikipedia, the free encyclopedia 'Time is a basic component of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects. Time has been a major subject of religion, philosophy, and science, but defining time in a non-controversial manner applicable to all fields of study has consistently eluded the greatest scholars.' back |
| Momentum - Wikipedia Momentum - Wikipedia, the free encyclopedia 'In classical mechanics, momentum (pl. momenta; SI unit kg·m/s, or, equivalently, N·s) is the product of the mass and velocity of an object (p=mv). For more accurate measures of momentum, see the section "modern definitions of momentum" on this page.' back |
| Energy - Wikipedia Energy - Wikipedia, the free encyclopedia 'In physics and other sciences, energy (from the Greek energos, 'active, working) is a scalar physical quantity that is a property of objects and systems which is conserved by nature. Energy is often defined as the ability to do work. back |
| Atomic nucleus - Wikipedia Atomic nucleus - Wikipedia, the free encyclopedia 'The nucleus of an atom is the very small dense region of an atom, in its centre consisting of nucleons (protons and neutrons). The size (diameter) of the nucleus is in the range of 1.6 fm (10-15 m) (for a proton in light hydrogen) to about 15 fm (for the heaviest atoms, such as uranium). These dimensions are much smaller than the size of the atom itself by a factor of about 23,000 (uranium) to about 145,000 (hydrogen). Almost all of the mass in an atom is made up from the protons and neutrons in the nucleus with a very small contribution from the orbiting electrons.' back |
| Atomic orbital - Wikipedia Atomic orbital - Wikipedia, the free encyclopedia 'An atomic orbital is a mathematical function that describes the wave-like behavior of an electron in an atom. The region in which an electron may be found around a single atom in a particular energy state can be calculated from this function. The term "orbital" has become known as either the "mathematical function" or the "region" generated with the function.[1] Specifically, atomic orbitals are the possible quantum states of an individual electron in the electron cloud around a single atom, as described by the function.' back |
| Christ the Logos - Wikipedia Christ the Logos - Wikipedia, the free encyclopedia 'In Christology, the conception that the Christ is the Logos (the Greek for "word", "wisdom", or "reason") has been important in establishing the doctrine of the divinity of Jesus Christ and his position as God the Son in the Trinity as set forth in the Chalcedonian Creed. The conception derives from the opening of the Gospel of John: "In the beginning was the Word, and the Word was with God, and the Word was God." In the original Greek, Logos is used, and in theological discourse, this is often left untranslated. Word and related terms in earlier Jewish tradition prepared the way for its use here to denote Jesus as revealer of the unseen God (see Wisdom 9:1-4, 9, 17-18; Ecclesiasticus 24:1-12)' back |
| Gravitational singularity - Wikipedia Gravitational singularity - Wikipedia, the free encyclopedia 'A gravitational singularity (sometimes spacetime singularity) is, approximately, a place where quantities which are used to measure the gravitational field become infinite. Such quantities include the curvature of spacetime or the density of matter. More accurately, a spacetime with a singularity contains geodesics which cannot be completed in a smooth manner. The limit of such a geodesic is the singularity.' back |
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