vol 3: Development
2 Model
page 6: Simplicity
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Simplicity
Having established the existence of something, the next question is how it exists in order that we learn its nature. But because we are unable to know the nature of God, but only what what god is not, we are not able to study how god exists, but rather how god does not exist. ...
We can show how god does not exist by removing from him inappropriate features such as composition, motion and other similar things. ... Aquinas 14, Latin
This is the famous via negativa, which we now proceed to apply.
Aquinas, following ancient tradition, finds that God is 'omnino simplex', altogether simple:
The absolute simplicity of God may be shown in many ways.
First, from the previous articles of this question. For there is neither composition of quantitative parts in God, since He is not a body; nor composition of matter and form; nor does His nature differ from His "suppositum"; nor His essence from His existence; neither is there in Him composition of genus and difference, nor of subject and accident. Therefore, it is clear that God is nowise composite, but is altogether simple. Aquinas 20
These conclusions are all derived from the premiss that God is 'pure act', ie the realization of all possibility.
The model of god proposed here seems to involve unlimited complexity, but we can restore simplicity if we interpret it in a dynamic rather than a static manner.
The central problem for physics and all the sciences that like to see their results written down is the contrast between the static nature of text and the dynamic nature of the universe. One can only write an eternally true text about eternal elements of reality. This problem was expressed in writing by Parmenides 2500 years ago. Parmenides Hussey
Physics does this. It seems that no matter when or where we look, F = ma (Newton), E = mc2 (Einstein) and S = k log W (Boltzmann). We are confident that these fragments of text, and millions like them, will always be mapped onto some constant features of the universe.
The central fact of dynamics as a science is that if we want to write it down, we must identify stationary points in the motion. Stationary points are not-motion, allowing us to describe a dynamic system in a static way, a form of via negativa. We require a minimum of two static symbols to a describe a motion, one representing some p (the 'before' state) and the other not-p (the 'after' state). We cannot say what happens between before and after, except that it is a process which transforms the initial state into the final state.
This insight is at the heart of calculus, the mathematics of motion. In the intuitive geometric picture we see the trajectory of a point through space as a line. A line is an ordered set of points which we may name with numbers. The natural numbers 0, 1, 2, ... may be used to name isolated points one unit apart. We can name the points in between the natural numbers using fractional and real numbers.
The intuitive view of calculus is described in the illustration, a two dimensional slice of spacetime relating before and after in time to before and after in space. As we noted above, we can represent the before and after in symbols, but the motion between is ineffable. Calculus uses a limiting process, bringing before and after closer together to arrive at a representation of change over an infinitesimal interval. This number, the ratio of two infinitesimal intervals is called the differential, written dx/dt. Landau, Calculus - Wikipedia
We can perform this mathematical limiting process in our imagination and represent it on paper. Much research has gone into determining which relationships between variables (ie, functions) are differentiable, and which are not. In general physicists have worked on the idea that a function which faithfully represents a physical process is continuous and differentiable because we assume that the universe itself is continuous (ie one) and differentiable (ie has stationary points).
Here, however, we run up against quantum physics which suggests that the universal process is not infinitely analyzable, but there is in fact some limit to the fineness of the universe measured by the quantum of action. At a certain level of resolution we discover that if we take action as the quantity to be represented by a line ('a line of action'), that line is not continuous, but comprises discrete points each one quantum of action apart.
Whatever happens between these points cannot, in principle, be observed. So a line of action comprises a set of discrete points just like the line of natural numbers. Any process that goes on between the natural numbers cannot be expressed in integers any more than what goes on between points one quantum of action apart. The static structure of the universe, that is the structure that we can sense and write down, has a fundamental graininess.
This is not to say that the universe itself is not continuous. Quantum mechanics envisages a continuous process (represented by a 'wave function') that 'fills in the gaps 'between the discrete observable events, but is not itself observable or recordable. Quantum mechanics supposes the existence of certain wave functions because they allow us to predict the frequencies of discrete events in the universe.
Now we can map this idea onto the Cantor universe. The natural numbers, of course, can be used to represent the discrete events, each event comprising the exchange of one or more quanta of action. The universe of discrete events has a different history for every possible permutation of these elementary events, and for every possible permutation of these permutations ... . So we imagine a universe based on aleph(0) elementary events, whose aleph(1), aleph(2) etc permutations provide an invisible 'wave function' which controls the actual sequence of events that we observe in our world.
It is but a small step from here to the transfinite network. Here we see each elementary event as a step in a universal computation involving communication between a transfinite hierarchy of machines. It is this invisible processing which gives meaning to the elementary events of the universe, connecting them together as parts of the universal process.
The theory of quantum communication suggests that this idea is not so far fetched. In a quantum computer, one elementary event is capable of doing the work of a Turing machine, performing, in effect, a computation with a matrix of aleph(0) numbers. (Lo, 3)
Physicists seek some sort of intuitive material continuity in the universe. Our model of the universe is based on 'logical continuity'. The steps forming the computational path of an isolated Turing machine are a logical continuum, each following deterministically from the step before. The universe may be a logical continuum and therefore a consistent unity which deserves the epithets 'simple', 'whole' or 'integral'.
Cognito de aliquo an sit, inquirendum restat quomodo sit, ut sciatur de eo quid sit. Sed quia de Deo scire non possumus quid sit, sed quid non sit, non possumus considerare de Deo quomodo sit, sed potius quomodo non sit. Primo ergo considerandum est quomodo non sit; ...
Potest autem ostendi de Deo quomodo non sit, removendo ab eo ea quae ei non conveniunt, utpote compositionem, motum et alia huiusmodi. ... (back)
Further reading
Books
| Cummins, Denise Dellarosa, and Colin Allen (editors), The Evolution of Mind, Oxford University Press 1998 Introduction: This book is an interdisciplinary endeavour, a collection of essays by ethologists, psychologists, anthropologists and philosophers united in the common goal of explaining cognition. ... the chief challenge is to make evolutionary psychology into an experimental science. Several of the chapters in this volume describe experimental techniues and results consistent with this aim; our hope and intention is that they lead by example in the development of evolutionary psychology from the realm of speculation to that of established research program' 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 |
| Hussey, E L, "Parmenides" in Ted Honderich (editor) The Oxford Companion to Philosophy, Oxford University Press 1995 Amazon back |
| Landau, Edmund, and Melvin Hausner, Martin Davis (translators), Differential and Integral Calculus, American Mathematical Society 2001 Amazon editorial review: 'Editorial Reviews Product Description After completing his famous Foundations of Analysis . . . Landau turned his attention to this book on calculus. The approach is that of an unrepentant analyst, with an emphasis on functions rather than on geometric or physical applications. The book is another example of Landau's formidable skill as an expositor. It is a masterpiece of rigor and clarity.' Amazon back |
| Lo, Hoi-Kwong, and Tim Spiller, Sandra Popescu, Introduction to Quantum Computation and Information, World Scientific 1998 Jacket: 'This book provides a pedagogical introduction to the subjects of quantum information and computation. Topics include non-locality of quantum mechanics, quantum computation, quantum cryptography, quantum error correction, fault tolerant quantum computation, as well as some experimental aspects of quantum computation and quantum cryptography. A knowledge of basic quantum mechanics is assumed.' Amazon back |
| Lonergan, Bernard J F, Method in Theology, University of Toronto Press for Lonergan Research Institute 1996 Introduction: 'A theology mediates between a cultural matrix and the signifcance and role of religion in that matrix. ... When the classicist notion of culture prevails, theology is conceived as a permanent achievement, and then one discourses on its nature. When culture is conceived empirically, theology is known to be an ongoing process, and then one writes on its method. Method ... is a framework for collaborative creativity.' Amazon back |
| Nielsen, Michael A, and Isaac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press 2000 Review: A rigorous, comprehensive text on quantum information is timely. The study of quantum information and computation represents a particularly direct route to understanding quantum mechanics. Unlike the traditional route to quantum mechanics via Schroedinger's equation and the hydrogen atom, the study of quantum information requires no calculus, merely a knowledge of complex numbers and matrix multiplication. In addition, quantum information processing gives direct access to the traditionally advanced topics of measurement of quantum systems and decoherence.' Seth Lloyd, Department of Quantum Mechanical Engineering, MIT, Nature 6876: vol 416 page 19, 7 March 2002. Amazon back |
| Suzuki, Daisetz Teitaro, Mysticism, Christian and Buddhist, George Allen and Unwin 1976 Jacket: 'In this clear account of the essentials of mysticism, Mr. Suzuki has taken as examples the Zen Buddhism of the East and the reflections of Meister Eckhart. With a wealth of illustration and explanation, he shows how the Chinese sage and the German philosopher meet on common ground.' Amazon back |
Links
| Aquinas 14 Summa: I 3 1: Is God a body? 'I answer that, It is absolutely true that God is not a body; and this can be shown in three ways. First, because no body is in motion unless it be put in motion, as is evident from induction. Now it has been already proved (2, 3), that God is the First Mover, and is Himself unmoved. Therefore it is clear that God is not a body. ...' back |
| Aquinas 20 Summa: I 3 7: Whether God is altogether simple?? 'I answer that, The absolute simplicity of God may be shown in many ways. First, from the previous articles of this question. For there is neither composition of quantitative parts in God, since He is not a body; nor composition of matter and form; nor does His nature differ from His "suppositum"; nor His essence from His existence; neither is there in Him composition of genus and difference, nor of subject and accident. Therefore, it is clear that God is nowise composite, but is altogether simple. . . . ' back |
| Calculus - Wikipedia Calculus - Wikipedia, the free encyclopedia 'Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. Historically, it was sometimes referred to as "the calculus", but that usage is seldom seen today. Calculus has widespread applications in science and engineering and is used to solve complicated problems for which algebra alone is insufficient. Calculus builds on algebra, trigonometry, and analytic geometry and includes two major branches, differential calculus and integral calculus, that are related by the fundamental theorem of calculus. In more advanced mathematics, calculus is usually called analysis and is defined as the study of functions.' back |
| John Burnet John Burnet's Early Greek Philosophy:, chapter IV, Parmenides of Elea: 85: The Poem back |
| Burnet Parmenides of Elea: The Poem 'The Poem Parmenides was the first philosopher to expound his system in metrical language. His predecessors, Anaximander, Anaximenes, and Herakleitos, wrote in prose, and the only Greeks who ever wrote philosophy in verse at all were just these two, Parmenides and Empedokles; for Xenophanes was not a philosopher any more than Epicharmos. Empedokles copied Parmenides; and he, no doubt, was influenced by the Orphics. But the thing was an innovation, and one that did not maintain itself. The fragments of Parmenides are preserved for the most part by Simplicius, who fortunately inserted them in his commentary, because in his time the original work was already rare.11 I follow the arrangement of Diels.' back |
| Math Archives Mathematics Archives WWW Server 'The goal of the Mathematics Archives is to provide organized Internet access to a wide variety of mathematical resources. The primary emphasis is on materials which are used in the teaching of mathematics. Currently the Archives is particularly strong in its collection of educational software.' back |
| Parmenides Pdf file of Greek text and English tanslationof extant fragments. back |
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