vol III Development:
Chapter 2: Model
Introduction
Model and meaning
Understanding involves a duality, what we might call a mind and a thing. True understanding brings the mind into conformity with the thing. From one point of view the mind is a coordinate system by which we measure the thing. From the other point of view, the thing is a coordinate system by which we measure the mind.
The correspondence between the two facets of a duality can operate at various levels of detail. At one extreme is the mathematical idea of one-to one correspondence where every feature of one element of the duality corresponds exactly to a feature of the other element of the duality. The idea of one-to-one correspondence is a foundation of the set theory invented by Georg Cantor and consequently one of the foundations of mathematics. Bijection - Wikipedia, Cantor
At the other extreme are what we might call poetic similes, like Shakespeare's Shall I compare thee to a summers day. Here the relationship between the model and the reality is very loose and imaginative, summing up a lover in a few words. Within this spectrum of detail we run through all possible meaningful relationships including scale models, engineering plans, shopping dockets, art and literature. William Shakespeare
Modelling God
The starting point for this site is the model of God developed by Thomas Aquinas in his Summa Theologiae. Following Aristotle and the more mystical Christian thinkers, Aquinas defined God as pure action, in Latin actus purus. He concluded from this that God is absolutely simple ( omnino simplex), that is without any features that the human mind could grasp. From this he deduces what has come to be called the via negativa or apophatic theology: we cannot say anything positive about God, we can only say what God is not. Thomas Aquinas, Apophatic theology - Wikipedia
This approach to God is very similar to the the modern formalist approach to mathematics. From this point of view the only limit on mathematics is consistency. From a practical point of view, we also want or mathematical theories to be interesting and solve significant problems. Also, people do explore inconsistent mathematics. In theology, we can explore these ideas with questions like 'if God is omnipotent, can He make a stone bigger than He can lift' etc. Mortensen
The value of modelling
Here we set out to develop models to help us understand
and plan successful action in a divine world. The ancient religions teach that a
successful life is the result of pleasing an invisible God. Locally,
this God is often an abstract version of the reigning political power.
Here, given our assumption that the Universe is divine, we seek a model which embraces the physical and spiritual features of the the world.
Life is dynamic. My life changes from moment to moment. I am in
continual dialogue with my environment, be it a mountain, a power saw
or a loud and crowded party. My experience of life is embodied in a
fantastically complex physical system. I am a coalition of trillions
of cells; each cell comprising countless atoms and molecules. Physics
tells us that even when we magnify an atom trillions of times we
continue to uncover detailed processes that are part of its life.
Dynamics is central to all modern science, and its history
stretches back into the mists of time. The essence of survival is to
act appropriately at every moment of life: to avoid acts
leading to injury or death, and to perfect acts that lead to food and shelter, growth and reproduction. To do this, we must be able to predict
the behaviour of the world and the effects of our actions. This is
the role of knowledge. To know someone or something is to be able to
predict what they will do in certain circumstances. We do this by
creating within ourselves a model of the thing known, and using the
model to simulate its behaviour under different conditions.
Mathematical modelling
The theological and religious world is torn by sectarianism. If the Univese is divine, however, and God is one, we should be able to develop a universal theology and religion. The key to this development is mathematics, the only human language that loses nothing in translation,.
Mathematics is the most general and abstract form of language.
Mathematical modelling of the world begins with arithmetic. The
history of arithmetic is probably as long as the history of written
language. While we normally distinguish stars from sheep and atoms,
arithmetic pays attention only to the property of identifiable unity.
Counting applies equally to stars, sheep and atoms.
Arithmetic begins with numbers. The properties of numbers can be written down succinctly in the form of the Peano axioms. The natural numbers flower into the infinite hierarchy of transfinite
cardinal and ordinal numbers which define the space in which this
site operates. Peano axioms - Wikipedia, Transfinite numbers - Wikipedia
The specific strength of this model is that it may be used to
express the relationship between the matter and spirit. In this
application it would suggest that there is not an absolute divide
between the material and spiritual elements of the Universe. Rather,
there is an infinite layered spectrum of complexity running from 'pure
matter' to 'pure spirit'.
The existence of this spectrum suggests a new way to look at the
nature and existence of God. Beginning with the mathematical model
constructed in this part of the site, we go on to develop a
theological model of the whole of existence. This model will serve as
a foundation for the construction and interpretation of religion.
religion.
(revised 12 August 2014)
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Copyright:
You may copy this material freely provided only that you quote fairly and provide a link (or reference) to your source.
Further reading
Books
Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)
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.'
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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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Hazewinkel, Michiel, and (managing editor), Encyclopaedia of Mathematics (6 volumes), Kluwer Academic and Toppan 1995 'The Encyclopaedia of mathematics aims to be a reference work for all parts of mathematics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-85.'
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Hazewinkel, Michiel, and (managing editor), Encyclopaedia of Mathematics (6 volumes), Kluwer Academic and Toppan 1995 'The Encyclopaedia of mathematics aims to be a reference work for all parts of mathematics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-85.'
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Hodges, Andrew, Alan Turing: The Enigma, Burnett 1983 Author's note: '... modern papers often employ the usage turing machine. Sinking without a capital letter into the collective mathematical consciousness (as with the abelian group, or the riemannian manifold) is probably the best that science can offer in the way of canonisation.' (530)
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Kauffman, Stuart, At Home in the Universe: The Search for Laws of Complexity, Oxford University Press 1995 Preface: 'As I will argue in this book, natural selection is important, but it has not laboured alone to craft the fine architectures of the biosphere . . . The order of the biological world, I have come to believe . . . arises naturally and spontaneously because of the principles of self organisation - laws of complexity that we are just beginning to uncover and understand.'
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Khinchin, A I, Mathematical Foundations of Information Theory (translated by P A Silvermann and M D Friedman), Dover 1957 Jacket: 'The first comprehensive introduction to information theory, this book places the work begun by Shannon and continued by McMillan, Feinstein and Khinchin on a rigorous mathematical basis. For the first time, mathematicians, statisticians, physicists, cyberneticists and communications engineers are offered a lucid, comprehensive introduction to this rapidly growing field.'
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Mortensen, Chris, Inconsistent Mathematics, Kluwer Academic 1995 'The argument from pure mathematics for studying inconsistency is the best of reasons: because it is there. . . . It is always dangerous to think that a physical use will never be found for a given piece of mathematics. Nor is present-day mathematical physics anomaly free: witness the singularities at the beginning of time or in black holes, delta functions in elementary quantum theory, or renormalisation in quantum field theory.' p 8-9.
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Smolin, Lee, The Life of the Cosmos, Oxford University Pres 1997 Jacket: 'Smolin posits that a process of self-organisation like that of biological evolution shapes the universe, as it develops and eventually reproduces through black holes, each of which may result in a big bang and a new universe. Natural selection may guide the appearance of the laws of physics, favouring those universes which best reproduce. . . . Smolin is one of the leading cosmologists at work today, and he writes with an expertise and a force of argument that will command attention throughout the world of physics.'
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Papers
| Imamizu, Hiroshi, et al, "Human cerebellar activity reflecting an acquired internal model of a new tool", Nature, 403, 6766, 13 January 2000, page 192-195. Letters to Nature: 'Theories of motor control postulate that the brain uses internal models of the body to control movements accurately. ... Previous studies have shown that the cerebellar cortex can acquire internal models through motor learning. Because the human cerebellum is involved in higher cognitive function as well as motor control, we propose a coherent computational theory in which the phylogenetically newer part of the cerebellum similarly acquires internal models of objects in the external world.'. back |
| Schnellnhuber, H J, "'Earth system' analysis and the second Copernican revolution", Nature, 402 supplement, 6761, 2 December 1999, page C19-C23. Impacts of forseeable science: 'Optical magnification instruments once brought about the Copernical revolution that put the Earth in its correct astrophyscial context. Sophisticated information compression techniques including simulation modelling are now ushering in a second 'Copernican' revolution. The latter strives to understand the 'Earth system' as a whole, and to develop, on this cognitive basis, concepts for global environmental management.'. back |
| West, Geoffrey B, James H Brown and Brian J Enquist, "A general model for the structure and allometry of plant vascular systems", Nature, 400, 6745, 12 August 1999, page 664-667. Letters to Nature: 'Vascular plants vary in size by about twelve orders of magnitude, and a single individual sequoia spans nearly this entire range as it grows from a seedling to a mature tree. .. Here we present an integrated model for the hydrodynamics, biomechanics and brnching geometry of plants, based on the application of a general theory of resource distribution through hierarchical branching networks to the case of vascular plants.'. back |
Links
| Apophatic theology - Wikipedia, Apophatic theology - Wikipedia, the free encyclopedia, 'Apophatic theology (from Greek ἀπόφασις from ἀπόφημι - apophēmi, "to deny")—also known as negative theology or via negativa (Latin for "negative way")—is a theology that attempts to describe God, the Divine Good, by negation, to speak only in terms of what may not be said about the perfect goodness that is God. It stands in contrast with cataphatic theology.' back |
| Argonne National Laboratory, The QED Project - Home Page, The aim of the QED project is to build a single, distributed, computerized repository that rigorously represents all important, established mathematical knowledge.' back |
| Bijection - Wikipedia, Bijection - Wikipedia, the free encyclopedia, 'In mathematics, a bijection (or bijective function or one-to-one correspondence) is a function between the elements of two sets, where every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set. There are no unpaired elements.' back |
| Brouwer fixed point theorem - Wikipedia, Brouwer fixed point theorem - Wikipedia, the free encyclopedia, 'Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f(x0) = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to itself. A more general form is for continuous functions from a convex compact subset K of Euclidean space to itself. back |
| Dynamics - Wikipedia, Dynamics - Wikipedia, the free encyclopedia, '. . . in physics, dynamics refers to time evolution of physical processes' back |
God - Catholic Encyclopedia, God - Catholic Encyclopedia, 'God
Etymology of the Word "God"
Discusses the root-meaning of the name "God", which is derived from Gothic and Sanskrit roots.
Existence of God
Formal dogmatic Atheism is self-refuting, and has never won the reasoned assent of any considerable number of men. Nor can Polytheism ever satisfy the mind of a philosopher. But there are several varieties of what may be described as virtual Atheism which cannot be dismissed so quickly.
Nature and Attributes of God
In this article, we proceed by deductive analysis to examine the nature and attributes of God to the extent required by our limited philosophical scope. We will treat accordingly of the infinity, unity, and simplicity of God, adding some remarks on Divine personality.
Relation of God to the Universe
The world is essentially dependent on God, and this dependence implies (1) that God is the Creator of the world — the producer of its whole substance; and (2) that its continuance in being at every moment is due to His sustaining power.
The Blessed Trinity
The Trinity is the term employed to signify the central doctrine of the Christian religion — the truth that in the unity of the Godhead there are Three truly distinct Persons: the Father, the Son, and the Holy Spirit. back |
| Heraclitus - Wikipedia, Heraclitus - Wikipedia, the free encyclopedia, 'Heraclitus of Ephesus (Greek: Ἡράκλειτος ὁ Ἐφέσιος—Hērákleitos ho Ephésios; c. 535 – c. 475 BCE) was a pre-Socratic Greek philosopher, a native of the Greek city Ephesus, Ionia, on the coast of Asia Minor. . . .
Heraclitus is famous for his insistence on ever-present change in the universe, as stated in his famous saying, "No man ever steps in the same river twice" (see panta rhei, below). He believed in the unity of opposites, stating that "the path up and down are one and the same", all existing entities being characterized by pairs of contrary properties. His cryptic utterance that "all entities come to be in accordance with this Logos" (literally, "word", "reason", or "account") has been the subject of numerous interpretations.'
' back |
| John O'Connor and Edmund F Robertson, MacTutor History of Mathematics, The MacTutor History of Mathematics archive ... developed initially as part of the Mathematical MacTutor system for learning and experimenting with mathematics. back |
| John Palmer - Parmenides, Parmenides (Stanford Encyclopedia of Philosophy), First published Fri Feb 8, 2008
'Parmenides of Elea, active in the earlier part of the 5th c. BCE., authored a difficult metaphysical poem that has earned him a reputation as early Greek philosophy's most profound and challenging thinker. His philosophical stance has typically been understood as at once extremely paradoxical and yet crucial for the broader development of Greek natural philosophy and metaphysics. He has been seen as a metaphysical monist (of one stripe or another) who so challenged the naïve cosmological theories of his predecessors that his major successors among the Presocratics were all driven to develop more sophisticated physical theories in response to his arguments.' back |
| London Mathematical Society, Bulletin of the London Mathematical Society, You must register to enter the Cambridge Journals Online site. One may then browse abstracts of articles in journals published by CUP, and purchase copies of the articles themselves. back |
| Molecular biology - Wikipedia, Molecular biology - Wikipedia, the free encyclopedia, 'Molecular biology (pronounced /məˈlɛkjʊlər .../) is the branch of biology that deals with the molecular basis of biological activity. This field overlaps with other areas of biology and chemistry, particularly genetics and biochemistry. Molecular biology chiefly concerns itself with understanding the interactions between the various systems of a cell, including the interactions between the different types of DNA, RNA and protein biosynthesis as well as learning how these interactions are regulated.' back |
| Peano axioms - Wikipedia, Peano axioms - Wikipedia - the free encyclopedia, 'In mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of consistency and completeness of number theory.' back |
| Richard Kraut - Plato, Plato (Stanford Encyclopedia of Philosophy), First published Sat Mar 20, 2004; substantive revision Thu Sep 17, 2009
'Plato (429–347 B.C.E.) is, by any reckoning, one of the most dazzling writers in the Western literary tradition and one of the most penetrating, wide-ranging, and influential authors in the history of philosophy. . . . Few other authors in the history of philosophy approximate him in depth and range: perhaps only Aristotle (who studied with him), Aquinas, and Kant would be generally agreed to be of the same rank.' back |
| Thomas Aquinas, Summa Theologica, Thomas Aquinas: The medieval theological classic online : 'Because the doctor of Catholic truth ought not only to teach the proficient, but also to instruct beginners (according to the Apostle: As unto little ones in Christ, I gave you milk to drink, not meat -- 1 Cor. 3:1-2), we purpose in this book to treat of whatever belongs to the Christian religion, in such a way as may tend to the instruction of beginners. We have considered that students in this doctrine have not seldom been hampered by what they have found written by other authors, partly on account of the multiplication of useless questions, articles, and arguments, partly also because those things that are needful for them to know are not taught according to the order of the subject matter, but according as the plan of the book might require, or the occasion of the argument offer, partly, too, because frequent repetition brought weariness and confusion to the minds of readers.'
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| Transfinite numbers - Wikipedia, Transfinite numbers - Wikipedia, the free encyclopedia, 'Transfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite. Few contemporary workers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". However, the term "transfinite" also remains in use.' back |
| University of Tennessee, Knoxville, 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. . . . A second strength of the Archives is its extensive collection of links to other sites that are of interest to mathematicians. back |
| W3C, W3C Document Object Model, 'The Document Object Model is a platform- and language-neutral interface that will allow programs and scripts to dynamically access and update the content, structure and style of documents. The document can be further processed and the results of that processing can be incorporated back into the presented page. This is an overview of DOM-related materials here at W3C and around the web.' back |
| William Shakespeare, Shall I compare thee to a summer's day (Sonnet 18), 'Shall I compare thee to a summer’s day?
Thou art more lovely and more temperate.
Rough winds do shake the darling buds of May,
And summer’s lease hath all too short a date.
Sometime too hot the eye of heaven shines,
And often is his gold complexion dimmed;
And every fair from fair sometime declines,
By chance, or nature’s changing course, untrimmed;
But thy eternal summer shall not fade,
Nor lose possession of that fair thou ow’st,
Nor shall death brag thou wand’rest in his shade,
When in eternal lines to Time thou grow’st.
So long as men can breathe, or eyes can see,
So long lives this, and this gives life to thee.' back |
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