vol III Development:

Chapter 2: Model

page 2: Fixed points

Aristotle thought that the mind was immaterial or spiritual because it could think a vast number of different things and hold a library of memories. He believed this to be impossible if the mind had a material organ. On the Soul - Wikipedia, Aristotle - On the Soul: 429 a 21 sqq.

He also believed that the human mind, part of the human soul, is immortal because it has no material parts to come apart. This line of thought also led to the idea of Angels, purely spiritual beings that had never been associated with a body like us humans. Angel - Wikipedia, On the Soul: 430 a 23

Aristotle's concept of mind is difficult to understand because it is difficult to understand how information can be represented without markers. The problem is more difficult when we come to the ultimate spiritual being, God, who is held to be absolutely simple (omnino simplex). How can God be both omniscient (knowing everything) and yet absolutely simple? The answer, I believe, lies in the mathematical theory of fixed points. Casti: Five Golden Rules

The writers of the Hebrew Bible concentrate on the personal features of their God Yahweh. The ancient Greeks also started with very human Gods whose deeds were widely celebrated in epic literature like the Iliad, written about 750 bce. Later, beginning about 500 bce, the Greeks moved toward a more scientific concept of deity. This approach developed from Parmenides through Plato to Aristotle and was then taken up more than a thousand years later by Aquinas to produce the definitive Christian model of God. Hebrew Bible - Wikipedia, Iliad - Wikipedia

Parmenides wanted to know how we can have certain knowledge of a changing world. He concluded that beneath the ephemeral appearances of everyday life there existed an unchanging core of being which is the subject of true knowledge. In the fragments we have of a poem that he wrote, this core is described as 'complete, immovable, and without end'. This heart of being imagined by Parmenides became a paradigm for much subsequent understanding of God. Parmenides, John Palmer - Parmenides

Plato took up Parmenides' approach to knowledge in his dialogue Parmenides written about 370 bce. The discussion revolves around the relationship between Parmenides' contention that all is one, and the apparent existence of many things. The background to the discussion is Plato's doctrine of forms, that for every thing on Earth there exists a perfect form or idea which determines te nature of the thing. Plato: Parmenides, Theory of Forms - Wikipedia

Since Plato expressed his ideas in dialogue form, putting them into the mouths of various actors, it is not easy to tell exactly what he thought. His student Aristotle clarified the matter by bringing some of Plato's forms down to Earth with his theory of matter (υλη) and form (μορφη), named (from the Greek) hylomorphism. This approach enabled Aristotle to explain change while maintaining the eternal nature of forms. He proposed that all material things comprised two elements, matter and form. Change occurred when some matter took on new form, as when a smith converted a bronze sword into a bronze ploughshare. Hylomorphism - Wikipedia

For Aristotle, hylomorphism applied to the interactions of physical bodies. He then went on to apply this idea to everything by developing the metaphysical ideas of potency (δυναμις) and act (εντελεχεια). Using this extended model, Aristotle defined motion as the the change from potency to act: in flight an arrow moves from actually in the bow and potentially in the target to actually in the target. He also established, as an axiom, that nothing could move from potentiality to actuality except through the agency of something already actual. Potentiality and actuality - Wikipedia

Aristotle used the axiomatic relationship between potency and act to prove the existence of the first unmoved mover which he believed was responsible for all the movement in the world. This entity was unmoved, since is had no potential to change, and was thus pure act. Aquinas used this same argument in his first proof that God is other than the Universe in his Summa Theologiae. Unmoved mover - Wikipedia, Aquinas 13: Whether God exists?

The Christian God is a living God.This raises a problem for Aquinas, who, following Aristotle, defines life as self motion. The problem arises because for Aristotle motion implies potency and there is no potency in God. Aquinas solves this problem in a simple ad hoc way by proposing another form of motion in God, from actuality to actuality. Aquinas 113, ad 3.

This solution is consistent with both modern physics and the hypothesis that the Universe is divine. Let us equate Aristotle's potency with modern potential energy and his actuality with modern kinetic energy. In the world described by modern physics, potential and kinetic energy are exactly equivalent as we can see in the operation of s simple harmonic oscillator like the pendulum, which converts kinetic energy into potential energy and back again. In the absence of friction, this process can continue forever. Pendulum - Wikipedia

God is pure action, that is motion within itself. From a mathematical point of view, this is to say that God maps onto itself. Fixed point theory tells us that under broad conditions, a set mapping onto itself has fixed points, that is points x such that the map f leaves x unchanged, ie f(x) = x. Fixed point theorem - Wikipedia

The application of fixed point theorems to the divine dynamics enables us to understand how God can be both absolutely simple and yet manifest the huge number of fixed points which define the structure of the Universe. The important point is that the fixed points of the divine Universe are not distinct from the dynamic life of God, but part of it. We see here where Parmenides and all who followed him to make God distinct from the world were mistaken: they though that stillness was necessarily distinct from motion, rather than part of it.

This point removes a major objection to identifying God and the world and so makes it easier for us to proceed on the assumption that the Universe is divine. Given this assumption God is no longer a mysterious other, totally beyond our ken. On the contrary, all our experience becomes experience of God and we can understand the local manifestations of God. Given that we can experience God, theology can become an evidence based science like all the other sciences. All science becomes science of God and so part of theology. Our hypothesis restores theology to its traditional position as the theory of everything, the Queen of the sciences. Fortun & Bernstein: Muddling Through, Theology - Wikipedia

(revised 14 August 2014)

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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!)

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

Fortun, Mike, and Herbert J Bernstein, Muddling Through: Pursuing Science and Truths in the Twenty-First Century, Counterpoint 1998 Amazon editorial review: 'Does science discover truths or create them? Does dioxin cause cancer or not? Is corporate-sponsored research valid or not? Although these questions reflect the way we're used to thinking, maybe they're not the best way to approach science and its place in our culture. Physicist Herbert J. Bernstein and science historian Mike Fortun, both of the Institute for Science and Interdisciplinary Studies (ISIS), suggest a third way of seeing, beyond taking one side or another, in Muddling Through: Pursuing Science and Truths in the 21st Century. While they deal with weighty issues and encourage us to completely rethink our beliefs about science and truth, they do so with such grace and humor that we follow with ease discussions of toxic-waste disposal, the Human Genome Project, and retooling our language to better fit the way science is actually done.'  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

Links

Angel - Wikipedia, Angel - Wikipedia, the free encyclopedia, 'An angel is a supernatural being or spirit, often depicted in humanoid form with feathered wings on their backs and halos around their heads, found in various religions and mythologies.' back

Aquinas 113, Summa I, 18, 3: Is life properly attributed to God?, Life is in the highest degree properly in God. In proof of which it must be considered that since a thing is said to live in so far as it operates of itself and not as moved by another, the more perfectly this power is found in anything, the more perfect is the life of that thing. ... back

Aquinas 13, Summa: I 2 3: Whether God exists?, I answer that the existence of God can be proved in five ways. The first and more manifest way is the argument from motion. . . . The second way is from the nature of the efficient cause. . . . The third way is taken from possibility and necessity . . . The fourth way is taken from the gradation to be found in things. . . . The fifth way is taken from the governance of the world. back

Aquinas 13 (Latin), Summa: I 2 3: Whether God exists?, 'Respondeo dicendum quod Deum esse quinque viis probari potest. Prima autem et manifestior via est, quae sumitur ex parte motus. Certum est enim, et sensu constat, aliqua moveri in hoc mundo. Omne autem quod movetur, ab alio movetur. Nihil enim movetur, nisi secundum quod est in potentia ad illud ad quod movetur, movet autem aliquid secundum quod est actu. Movere enim nihil aliud est quam educere aliquid de potentia in actum, de potentia autem non potest aliquid reduci in actum, nisi per aliquod ens in actu, sicut calidum in actu, ut ignis, facit lignum, quod est calidum in potentia, esse actu calidum, et per hoc movet et alterat ipsum. Non autem est possibile ut idem sit simul in actu et potentia secundum idem, sed solum secundum diversa, quod enim est calidum in actu, non potest simul esse calidum in potentia, sed est simul frigidum in potentia. Impossibile est ergo quod, secundum idem et eodem modo, aliquid sit movens et motum, vel quod moveat seipsum. Omne ergo quod movetur, oportet ab alio moveri. Si ergo id a quo movetur, moveatur, oportet et ipsum ab alio moveri et illud ab alio. Hic autem non est procedere in infinitum, quia sic non esset aliquod primum movens; et per consequens nec aliquod aliud movens, quia moventia secunda non movent nisi per hoc quod sunt mota a primo movente, sicut baculus non movet nisi per hoc quod est motus a manu. Ergo necesse est devenire ad aliquod primum movens, quod a nullo movetur, et hoc omnes intelligunt Deum.' back

Aristotle - On the Soul, On the Soul - The Internet Classics Archive, 'Holding as we do that, while knowledge of any kind is a thing to be honoured and prized, one kind of it may, either by reason of its greater exactness or of a higher dignity and greater wonderfulness in its objects, be more honourable and precious than another, on both accounts we should naturally be led to place in the front rank the study of the soul. The knowledge of the soul admittedly contributes greatly to the advance of truth in general, and, above all, to our understanding of Nature, for the soul is in some sense the principle of animal life. Our aim is to grasp and understand, first its essential nature, and secondly its properties; of these some are taught to be affections proper to the soul itself, while others are considered to attach to the animal owing to the presence within it of soul.' 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

Hebrew Bible - Wikipedia, Hebrew Bible - Wikipedia, The Hebrew Bible . . . is a term referring to the books of the Jewish Bible as originally written mostly in Biblical Hebrew with some Biblical Aramaic. The term closely corresponds to contents of the Jewish Tanakh and the Protestant Old Testament (see also Judeo-Christian) but does not include the deuterocanonical portions of the Roman Catholic or the Anagignoskomena portions of the Eastern Orthodox Old Testaments. The term does not imply naming, numbering or ordering of books, which varies (see also Biblical canon).' back

Hylomorphism - Wikipedia, Hylomorphism - Wikipedia, the free encyclopedia, 'Hylomorphism (Greek ὑλο- hylo-, "wood, matter" + -morphism < Greek μορφή, morphē, "form") is a philosophical theory developed by Aristotle, which analyzes substance into matter and form. Substances are conceived of as compounds of form and matter.' back

On the Soul - Wikipedia, On the Soul - Wikipedia, the free encyclopedia, 'On the Soul (Greek Περὶ Ψυχῆς, Perì Psūchês; Latin De Anima) is a major treatise by Aristotle on the nature of living things. His discussion centres on the kinds of souls possessed by different kinds of living things, distinguished by their different operations. Thus plants have the capacity for nourishment and reproduction, the minimum that must be possessed by any kind of living organism. Lower animals have, in addition, the powers of sense-perception and self-motion (action). Humans have all these as well as intellect.; back

Iliad - Wikipedia, Iliad - Wikipedia, the free encyclopedia, 'The Iliad (sometimes referred to as the Song of Ilion or Song of Ilium) is an ancient Greek epic poem in dactylic hexameter, traditionally attributed to Homer. Set during the Trojan War, the ten-year siege of the city of Troy (Ilium) by a coalition of Greek states, it tells of the battles and events during the weeks of a quarrel between King Agamemnon and the warrior Achilles.' 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

Parmenides, Fragements 1-19, Burnet's English translation, '8 One path only is left for us to speak of, namely, that It is. In it are very many tokens that what is is uncreated and indestructible; for it is complete, immovable, and without end.' back

Pendulum - Wikipedia, Pendulum - Wikipedia, the free encyclopedia, 'A pendulum is a weight suspended from a pivot so that it can swing freely.[1] When a pendulum is displaced sideways from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth.' back

Plato, Parmenides, 'Parmenides By Plato Written 370 B.C.E Translated by Benjamin Jowett Persons of the Dialogue CEPHALUS ADEIMANTUS GLAUCON ANTIPHON PYTHODORUS SOCRATES ZENO PARMENIDES ARISTOTELES Scene Cephalus rehearses a dialogue which is supposed to have been narrated in his presence by Antiphon, the half-brother of Adeimantus and Glaucon, to certain Clazomenians. back

Potentiality and actuality - Wikipedia, Potentiality and actuality - Wikipedia, the free encyclopedia, 'In philosophy, Potentiality and Actualit are principles of a dichotomy which Aristotle used to analyze motion, causality, ethics, and physiology in his Physics, Metaphysics, Ethics and De Anima (which is about the human psyche). The concept of potentiality, in this context, generally refers to any "possibility" that a thing can be said to have. Aristotle did not consider all possibilities the same, and emphasized the importance of those that become real of their own accord when conditions are right and nothing stops them.[3] Actuality, in contrast to potentiality, is the motion, change or activity that represents an exercise or fulfillment of a possibility, when a possibility becomes real in the fullest sense. back

Theology - Wikipedia, Theology - Wikipedia, the free encyclopedia, Theology is the systematic and rational study of concepts of God and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university, seminary or school of divinity. . . . 'During the High Middle Ages, theology was therefore the ultimate subject at universities, being named "The Queen of the Sciences" and serving as the capstone to the Trivium and Quadrivium that young men were expected to study. This meant that the other subjects (including Philosophy) existed primarily to help with theological thought.' back

Theory of Forms - Wikipedia, Theory of Forms - Wikipedia, the free encyclopedia, 'Plato's theory of Forms or theory of Ideas asserts that non-material abstract (but substantial) forms (or ideas), and not the material world of change known to us through sensation, possess the highest and most fundamental kind of reality. When used in this sense, the word form or idea is often capitalized. Plato speaks of these entities only through the characters (primarily Socrates) of his dialogues who sometimes suggest that these Forms are the only true objects of study that can provide us with genuine knowledge; thus even apart from the very controversial status of the theory, Plato's own views are much in doubt. Plato spoke of Forms in formulating a possible solution to the problem of universals.' back

Unmoved mover - Wikipedia, Unmoved mover - Wikipedia, the free encyclopedia, 'The unmoved mover (ού κινούμενον κινεῖ oú kinoúmenon kineῖ) is a philosophical concept described by Aristotle as a primary cause or "mover" of all the motion in the universe. As is implicit in the name, the "unmoved mover" is not moved by any prior action. In Book 12 (Greek "Λ") of his Metaphysics, Aristotle describes the unmoved mover as being perfectly beautiful, indivisible, and contemplating only the perfect contemplation: itself contemplating. He equates this concept also with the Active Intellect. This Aristotelian concept had its roots in cosmological speculations of the earliest Greek "Pre-Socratic" philosophers and became highly influential and widely drawn upon in medieval philosophy and theology. St. Thomas Aquinas, for example, elaborated on the Unmoved Mover in the quinque viae.' back