Chapter 8: God's ideas, cybernetics and singularity

Synopsis

The traditional God created the world in six days. They had a plan in mind which we call God’s ideas, reminiscent of Plato. Since God is considered to be absolutely simple, this idea is inconsistent with the modern understanding of information. Instead we must begin from an ignorant omnipotent initial singularity, naked gravitation, analogous to the traditional God. We then explain, step by step, how evolution guided only by the impossibility of contradiction has led to the Universe whose life we now enjoy.

Table of contents

8.1: Plato's forms



8.2: A history of creation



8.3: Form, mathematics and matter



8.4: Cantor: the cardinal of the continuum



8.5: Gödel and uncertainty; Turing and incomputability



8.6: Chaitin, cybernetics and requisite variety

8.1: Plato's forms

The traditional story of creation assumes that God had a plan in mind for the world they were about to create. This plan is known as ideas. Plato first used this term for his invisible forms which serve both as patterns for the structure of the world and the source of our knowledge of the word. Aquinas, Summa I, 15, 1 Are there ideas in God?

Here, from a modern point of view, we run into a serious difficulty. We now understand information to be physical, represented by physical symbols, and there are no such physical symbols in the traditional God. God is absolutely simple, without any internal structure.

At the same time, we want to see the Universe as divine and we can see that it is full of physical symbols, like ourselves, all the other things that we see around us and this text. This leads us to the idea that the immensely complex structure of the Universe may be equivalent to the ideas in God, that is the Universe is the mind of God.

This raises our principal question: how does structure come to be from a structureless beginning? An analogous problem exists in our own mind: where do our ideas come from? In our conscious lives, we find that some of our ideas arise from sensation and some just seem to come from nowhere. At one moment we are completely puzzled. Moments later we have an idea which solves the puzzle with insight and understanding. We talk about working things out, but on reflection there is usually no clear link between the puzzle and the solution (which is why it is a puzzle in the first place). Bernard Lonergan (1992): Insight: A Study of Human Understanding

Darwin's theory of evolution shows how new species, that is new physical symbols, come to be. Darwin's theory does not start at the beginning, however. It assumes the existence of living creatures and the environment in which they live. Given this starting point, Darwin explains how new species arise through variation and selection. CharlesDarwin (1859): The Origin of Species: By Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life

We must therefore go deeper, since we assume that the initial singularity is identical to the traditional God, we must ask how ideas come to be.

8.2: A history of creation

We work our way through these problems step by step in the next few chapters. Beginning in Chapter 9 with the ideas developed by Darwin, we explore how random processes create variation and natural selection filters the variants to select the forms of life that survive in their environment. This process is not a matter of foresight or thought. It is simply a matter of fact that some variants live long enough to reproduce their kind.

We then turn on page 10 to explore the physiological processes in our brains that generate new ideas, which often appear to come from nowhere. Then on page 11 we Introduce the Hilbert space, the musical space in which quantum of quantum mechanics operates. In chapter 2 we have already seen the how the Christian doctrine of the Trinity explains how God the Father, through reflecting upon themself, gives rise to the Son. The Father and the Son, communicating through love, create the Spirit.

This ancient doctrine serves as a paradigm for the idea that a combination of quantum mechanics and gravitation creates new particles. From the point of view of quantum mechanics and communication theory, we see physical particles, including ourselves, as sources or personalities.

We finally make this analogy concrete by identifying the initial singularity as pure action, the definition of God that Aquinas drew from Aristotle. From the conclusion drawn from the theory of the Trinity, that the characteristic action of God is to create Gods, we trace a plausible route to the emergence of the Universe within the initial singularity. Where the Trinity stops at three divine persons, we assume that there is no limit to the emergence of new gods. Aristotle recorded that the ancient scientist Thales (may have) said, the world is full of Gods.

The key theological problem with the Trinity is to understand how the three divine persons can be both completely divine and yet be distinct from one another without sacrificing the unity of the divinity. The same problem arises with the distinction of all the elements that go to form the present Universe. The answers we will find lie in quantum mechanics, the theory of communication between discrete particles, persons or sources.

8.3: Form, mathematics and matter

Plato was among the first to take the idea of disembodied form seriously. It has remained central to theology ever since and has also become central to mathematics under the influence of Cantor and Hilbert. It is the foundation of the common idea that we have an immaterial spiritual soul which, because it has no material parts which can come apart, will live for eternity. Formalism (philosophy of mathematics) - Wikipedia

When Aristotle came to discussing human sensation and understanding he saw matter as an impediment. Senses function by receiving the form of the entity sensed. Their material construction constrains the ability of the senses to receive forms, but he felt that this difficulty is overcome to come extent by the organic structure of the sense organs.

When he came to intellectual knowledge, however, Aristotle seems to have felt that the presence of any matter in the intellect was too restrictive. It must therefore be separated from matter. This idea became central to human psychology.

Aquinas, following Aristotle, associated intellect with immateriality. He argued that since God is maximally immaterial, they are maximally knowing. Since the development of computation and information theory, however, we now understand that information is carried by marks, like the letters on this page. Information is stored in computers, discs and solid state memories by magnetic or electrical marks. We have found that matter is enormously complex, right down to the level of atoms and fundamental particles. It can represent huge amounts of information. Current technology hardly touches the surface of this ability. Aquinas, Summa: I, 14, 1: Is there knowledge in God?

8.4: Cantor: the cardinal of the continuum

George Cantor worked in a mathematical milieu where it is believed meaningful to create a continuum from closely spaced discrete points. He wanted to define a numeral to represent the cardinal of the continuum, ie the number of points required to make a continuous line. In mathematics, syntax enables us to construct numerals to represent any cardinal number by forming ordered sets of digits, as in decimal numbers.

A countably infinite string of decimal digits is considered adequate to represent any element of the set of real numbers. Toward the end of the nineteenth century, George Cantor set out to use syntactic methods to represent the cardinal of the continuum and realised that ordered sets of symbols could be used to represent anything representable well beyond the limit of real numbers and into the domain of infinite (or as he liked to say. transfinite) multi-dimensional structures.

Cantor was a very religious man, and felt that this theory of transfinite numbers brought him close to the mind of God. He sought support for his ideas in discussion with theologians because he thought that transfinite ideas could exist in the divine intellect:

. . . in the transfinite a vastly greater abundance of forms and of species numerorum is available, and in a certain sense stored up, than there is in the correspondingly small field of the unbounded infinite. Consequently these transfinite species were at the disposal of the intention of the Creator and his absolutely inestimable will power just as were the finite numbers. Michael Hallett (1984):Cantorian Set Theory and Limitation of Size

Obviously it makes no sense to create a continuum out of a set of discrete and isolated points so Cantor's search for the cardinal of the continuum ultimately failed. Infinity and geometric continuity are mathematical ideals that do not exist in reality. Paul Cohen (1980): Set Theory and the Continuum Hypothesis

Nevertheless he made great contributions to mathematics by making formally infinite objects finite and tractable by putting them in boxes or sets. The core of Cantor's discovery is the idea of one-to-one correspondence between infinite sets by matching their elements pairwise, so long as they can be uniquely identified. His set theory remains one of the foundations of modern mathematics.

Cantor's work led to renewed interest in formalism. Some theologians felt that Cantor's work verged on pantheism, but he was defended by Hilbert, who introduced an explicitly formalist approach to mathematics and declared Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können: From the paradise, that Cantor created for us, no-one shall be able to expel us.

If we are going to establish that the Universe is divine, it it necessary to unite physics and theology. At present physics is still bedevilled with problems arising from the attempts to unite quantum mechanics and the special theory of relativity. Attempts to quantize the general theory of relativity have produced many ideas, but progress is very slow. The principal source of these problems is the belief that this union implies that the foundations of the Universe envisaged by quantum field theory are infinite. It has therefore been necessary to develop a procedure known as renormalization to eliminate these infinities.

The resulting theory is a bit of a mess. Meinard Kuhlman, writes:

In conclusion one has to recall that one reason why the ontological interpretation of QFT is so difficult is the fact that it is exceptionally unclear which parts of the formalism should be taken to represent anything physical in the first place. And it looks as if that problem will persist for quite some time. Meinard Kuhlmann: Quantum Field Theory

Hopefully the discussion of the creation of the Universe that begins in Chapter 11: The axioms of Hilbert space will provide some ideas to establish a fruitful union of quantum physics and evidence based theology. It may help to eliminate the misuse of infinity and other unphysical mathematical and theological ideals in our descriptions of the Universe we inhabit. The idea that carries us from the theory of the physical electron to the theology of the Universe is explained in Chapter 28.7: Symmetry with respect to complexity.

8.6: Gödel and uncertainty; Turing and incomputability

Cantor's introduction of set theory not only clarified the foundations of mathematics, but also introduced paradoxes that led to careful reevaluation of mathematical proofs. Hilbert believed that a proper application of formalist methods would eventually solve all these problems. This was not to be.

The search for the roots of mathematics emphasized the importance of logic and gradually brought symbolic logic and mathematics closer together. Whitehead and Russell set out to use logic to develop mathematics and built on the work of logicians to write Principia Mathematica. Their idea was to deduce all the fundamental propositions of logic and mathematics from a small number of logical premisses and primitive ideas, and so to prove that mathematics is a development of logic. Whitehead and Russell (1910): Principia Mathematica

One of the most significant consequences of Whitehead and Russell's work was to lead Kurt Gödel and Alan Turing to discover snakes in Hilbert's paradise. Hilbert defined what he considered to be the ideal features of formal mathematics. It should be consistent, complete and computable.

Consistency means that no argument could arrive at the conclusion P ≡ not-P. Completeness means that any properly formed mathematical statement can be proved either true or false. Computable means that there exists definite algorithms capable of proving consistency and completeness.

Kurt Gödel upset the first of Hilbert's expectations by establishing that consistent mathematics is incomplete. Soon afterwords Alan Turing established the existence of incomputable functions.

Thomas Aquinas established a limits to God's omnipotence: God cannot make a contradiction or inconsistency exist, eg that Socrates should be simultaneously sitting and standing. From this it has traditionally been concluded that God has absolute deterministic control over everything. The theorems of Gödel and Turing show, however, that from a formal point of view, absolute consistency does not always imply absolute control. Even in the mind of a Divinity totally constrained by consistency, (as we understand to be the case with our Universe) there may still be still room for incompleteness and incomputability, in other words, uncertainty.

5.7: Chaitin, cybernetics and requisite variety

Apart from the theory of information and communication, there is a second modern development about which the ancients knew very little even though it actually governed their efforts at controlling the world: cybernetics. One of its founders, Norbert Wiener, defined it as the science of control and communication in the animal and the machine. W. Ross Ashby (1964): An Introduction to Cybernetics

Gödel found that logically consistent formal systems are not completely determined and Chaitin interpreted Gödel's work to be an expression of the limits to control known as the cybernetic principle of requisite variety: One system can control another only if it has equal or greater entropy than the system to be controlled. This principle suggests that a completely structureless initial singularity has no power to control anything, even itself. Insofar as the initial singularity acts, its action must be random, like the toss of a coin or the throw of a die. God does play dice.

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Notes and references

Further reading

Books

Ashby (1964), W Ross, An Introduction to Cybernetics, Methuen 1956, 1964 'This book is intended to provide [an introduction to cybernetics]. It starts from common-place and well understood concepts, and proceeds step by step to show how these concepts can be made exact, and how they can be developed until they lead into such subjects as feedback, stability, regulation, ultrastability, information, coding, noise and other cybernetic topics.'  Amazon   back

Cohen (1980), Paul J, Set Theory and the Continuum Hypothesis, Benjamin/Cummings 1966-1980 Preface: 'The notes that follow are based on a course given at Harvard University, Spring 1965. The main objective was to give the proof of the independence of the continuum hypothesis [from the Zermelo-Fraenkel axioms for set theory with the axiom of choice included]. To keep the course as self contained as possible we included background materials in logic and axiomatic set theory as well as an account of Gödel's proof of the consistency of the continuum hypothesis. . . .'   Amazon   back

Darwin (1859), Charles, The Origin of Species: By Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life, Cambridge University Press 1859, 2009 ' It's hard to talk about The Origin of Species without making statements that seem overwrought and fulsome. But it's true: this is indeed one of the most important and influential books ever written, and it is one of the very few groundbreaking works of science that is truly readable. . . . Darwin's friend and "bulldog" T.H. Huxley said upon reading the Origin, "How extremely stupid of me not to have thought of that." Alfred Russel Wallace had thought of the same theory of evolution Darwin did, but it was Darwin who gathered the mass of supporting evidence—on domestic animals and plants, on variability, on sexual selection, on dispersal—that swept most scientists before it.' Mary Ellen Curtin  Amazon   back

Hallett (1984), 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

Lonergan (1992), Bernard J F, Insight: A Study of Human Understanding (Collected Works of Bernard Lonergan : Volume 3), University of Toronto Press 1992 '. . . Bernard Lonergan's masterwork. Its aim is nothing less than insight into insight itself, an understanding of understanding'  Amazon   back

Links

Aquinas, Summa I, 15, 1, Are there ideas in God?, ' . . . in other agents (the form of the thing to be made pre-exists) according to intelligible being, as in those that act by the intellect; and thus the likeness of a house pre-exists in the mind of the builder. And this may be called the idea of the house, since the builder intends to build his house like to the form conceived in his mind. As then the world was not made by chance, but by God acting by His intellect, as will appear later, there must exist in the divine mind a form to the likeness of which the world was made. And in this the notion of an idea consists.' back

Aquinas, Summa: I, 14, 1, Is there knowledge in God?, ' I answer that, In God there exists the most perfect knowledge. . . . it is clear that the immateriality of a thing is the reason why it is cognitive; and according to the mode of immateriality is the mode of knowledge. Hence it is said in De Anima ii that plants do not know, because they are wholly material. But sense is cognitive because it can receive images free from matter, and the intellect is still further cognitive, because it is more separated from matter and unmixed, as said in De Anima iii. Since therefore God is in the highest degree of immateriality as stated above (Question 7, Article 1), it follows that He occupies the highest place in knowledge.' back

Formalism (philosophy of mathematics) - Wikipedia, Formalism (philosophy of mathematics) - Wikipedia, the free encyclopedia, 'In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be considered to be statements about the consequences of certain string manipulation rules. . . . Formalism stresses axiomatic proofs using theorems, specifically associated with David Hilbert. A formalist is an individual who belongs to the school of formalism, which is a certain mathematical-philosophical doctrine descending from Hilbert.' back

Meinard Kuhlmann (Stanford Encyclopedia of Philosophy), Quantum Field Theory, ' Quantum Field Theory (QFT) is the mathematical and conceptual framework for contemporary elementary particle physics. In a rather informal sense QFT is the extension of quantum mechanics (QM), dealing with particles, over to fields, i.e. systems with an infinite number of degrees of freedom. (See the entry on quantum mechanics.) In the last few years QFT has become a more widely discussed topic in philosophy of science, with questions ranging from methodology and semantics to ontology. QFT taken seriously in its metaphysical implications seems to give a picture of the world which is at variance with central classical conceptions of particles and fields, and even with some features of QM.' back

Whitehead and Russell (1910), Principia Mathematica, Jacket: 'Principia Mathematica was first published in 1910-1913; this is the fifth impression of the second edition of 1925-7. The Principia has long been recognized as one of the intellectual landmarks of the century. It was the first book to show clearly the close relationship between mathematics and formal logic. Starting with a minimal number of axioms, Whitehead and Russell display the structure of both kinds of thought. No other book has had such an influence on the subsequent history of mathematical philosophy .' back