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... to restore theology to the mainstream of science 

Is the Universe Divine?

No question is ever settled
Until it is settled right (Ella Wheeler Wilcox)

Many of the Catholic theologians insist on the services rendered by heretics in compelling the Church to bring out points that gave precision and support to the faith but could hardly have been understood except when the fatal effects of overlooking them had been developed by the heretics. (Wicksteed and Cornford)

In our times, when every day men are being drawn closer together and the ties between various peoples are being multiplied, the Church is giving deeper study to her relationship with non-Christian religions. In her task of fostering unity and love among men, and even among nations, she gives primary considerations in this document to what human beings have in common and to what promotes fellowship among them. (Nostra Aetate)

Structure of this proposal

1. The problem of global ecumenism
2. History

Divinity leaves the universe

Parmenides (fl c 480 bce)
Heracleitus (fl. c. 500 bce)
Plato (429-347 bce)
Aristotle (384-322 bce)
Thomas Aquinas (1224-1274)
Bernard Lonergan (1904-1984)

Divinity returns to the universe

Isaac Newton (1643-1727)
Charles Darwin (1809-1882)
George Boole (1815-1864)
Albert Einstein (1879-1955)
Quantum theory (1900 - ?)
Kurt Goedel (1906-1978)
Alan Turing (1912-1954)
Claude Shannon (1916- )
Norbert Wiener (1894-1964)
David Deutsch

3. Toward a position

4. Sources consulted

1. The problem of global ecumenism

The practical problem motivating this thesis is the search for global human cooperation.

At the Second Vatican Council, the Church used new language to restate its ancient aspiration to embrace all people (Mt 28:19): "For the first time in the history of Ecumenical Councils, a Council addresses itself to all men, not just to members of the Catholic Church." (Abbott, 3, note 2) The Church clearly aspires to a new dialogue with the world, although the documents of the council contain traces of old attitudes. The Fathers still see the world as fallen and in need of redemption (Dei Verbum) ; their Church as the only medium of salvation (Lumen Gentium); and separation from the Church as a sin (Unitatis redintegratio). Christians with "the gift of ultimate truth about human life" (John Paul II), stand fairly in the centre of the world. It is all for them (Lumen gentium).

It is essential to true dialogue that each side respects the position of the other. If its opening to the whole of humanity is to respect this feature of dialogue, the Church can no longer expect to constrain the whole of humanity to its own points of view. It must accept that from the human point of view, it is one religion among many.

If a notion of god is an essential element of religion, an important step toward global ecumenism is the development of a theory of god, that is a theology, that is acceptable to everyone. The global growth and practical utility of science suggests that such a theory must be purged of all linguistic, geographic, historical and political parochialism by using mathematics in its language and publicly available data for its foundation.

This project is an exploration of the relationship between god and the world taking the scientific route, aptly described by Popper as 'conjecture and refutation' (Popper 1972). The source of the conjecture is the classical exposition of the relationship between god and the world developed by Aquinas restated in the languages of formal logic, mathematics and computation (Aquinas, Luger 1993).

Measured relative the to complexity of the modern world, the body of information upon which the Church bases its judgments is tiny. The Church seems rather like a motorist trying to navigate heavy traffic viewed through a pinhole rather than a windscreen. The course of the Church through the traffic is correspondingly insecure and potentially erratic.

The ultimate source of this constraint on the Church is its implementation of the proposition that God is other than the visible universe. Related to this is the belief that God has revealed himself once for all in the person of Jesus the Christ. The resulting 'deposit of faith', encoded in various canonical texts, cannot be added to or subtracted from until the end of time.

The only degree of freedom available to the Church is reinterpretation of the deposit of faith in the light of changing circumstances. Such reinterpretation seems to be a very weak channel to the truth, since the vast cultural and linguistic differences between now and the 'era of canonical formation' render interpretations almost arbitrary: "There is no template against which one can measure the adequacy of a christological proposal, with the exception, perhaps, of some clear cases of extreme positions" (Haight, 47). For a Catholic, such extreme positions are excluded by the institutional "magisterium" of the Church, but to a citizen of the world this suppression of certain views may appear more a matter of corporate politics than scientific consensus.

This situation seems to be reflected in the vast array of theologies and reconstructions of the life of Jesus that have appeared in recent times.

An alternative source of information for corporate navigation is science. Scientific method confines its practitioners to dealing with observable entities, although the only limit on the creative ingenuity that may be involved in the establishment and interpretation of a given datum is consistency. If God were the same as the visible universe, the Church could enrich its stock of navigational information by sifting through the 'deposit of science'. A distinctive feature of science, in contrast to theology, is its global unity, based on the unity of the world it studies.

Some might be inclined to see scientific method as alien to theology. This is true insofar as the data of theology are rather problematic interpretations of ancient traditions. If, however, god and the universe are one, we may observe the universe and so obtain sure data about god. Some might still argue that no interpretation of observed data can lead us to god. This is a matter of conjecture, and in the scientific way, may be answered by an alternative conjecture which fits both the data and the heuristic notions or models of god that have evolved through the history of religion. back

2. History

This skeletal history (annotated list of sources?) is intended to give roots to the formal core of this proposal: that we may see the universe as the incarnation of mathematics, an idea seen already in the Pythagoreans and Plato.

From this hypothesis, we may constrain the answer to the question is the universe divine? in two steps. First, show, by comparing it to the consensus of scientists, that the hypothesis is consistent with the known universe; and second, show that mathematics taken as a whole is big enough to represent any reasonable conception of god.

This history emphasizes two features: the distinction of God from the universe inherited through Christianity; and the union of god and the universe emerging from science, particularly through physicists, whose brush with mutual assured destruction may have turned their minds to god as an instrument of peace (Davies). back

Divinity leaves the universe

Parmenides (fl c 480 bce)

The dichotomy between god and the world enters western literature in fragments of a poem written by Parmenides of Elea (Hussey). The poem begins with the poet on a chariot journey from night to daylight. In the light he is welcomed by a goddess whose words complete the poem (Burnet). Feyerabend puts the story in a contemporary context:

... Parmenides claimed that the world was one, that change and subdivision did not exist, and that the lives of human beings that contained both were a chimaera. The proof (which he presents as being revealed by a goddess) rests on three assumptions said to be self evident: that Being is (estin ), that not-Being is not (ouk estin ), and that nothing is more fundamental than being. The argument then proceeds as follows: if change and difference exist, then there exists a transition from Being to not-Being (which is the only alternatives); not-Being is not, hence change and difference are not either. Here we have an early example of reductio ad absurdum - a kind of reasoning that extended the domain of demonstrable truths and separated it from intuition. The premise, estin, is the first explicit conservation law- it asserts the conservation of Being. Used in the form that nothing comes from nothing, it suggested more special conservation laws such as the conservation of matter (Antoine Lavoisier) or the conservation of energy (Robert von Mayer), who started a decisive paper with this very principle. The uniformity of Being survived as the idea that basic laws must be independent of space, time and circumstance. 'For us physicists', wrote Einstein, almost repeating Parmenides, 'the distinction between past, present and future has no other meaning than that of an illusion, though a tenacious one.' (Feyerabend 1995)

Although it seems that the goddess was proving a position, we may surmise that Parmenides really thought that the multiple and changing experiences of life lack reality, and the argument simply bolsters his position. In other words, the words of the goddess are a rationalization of faith.

Faith is a necessary element of culture, because, as Aristotle noted, the newborn human mind is like a slate yet to be written on, able to accept any information (De Anima). As soon as we are conceived, we begin to import information from our environment. This information, in conjunction with our genetic inheritance, shapes us from egg to adult. Because this information is given, it must be accepted. An infant can no more reject the language and culture of its family than it can reject its genotype.

This faith is accepted without conscious criticism, but not uncritically. Survival depends upon good information, and all our information processing systems are tuned to give reliable results from conception on. The faith that we receive depends upon where we were born. In most cases, it integrates the newborn into the community, underwriting its survival.

The rationalization of Parmenides position evolved steadily. Zeno of Elea (c 470 bce) provided arguments against the reality of plurality and motion. His arguments "... exploit properties of the infinite and use (perhaps for the first time) infinite regress as an argumentative device" (Hussey 1995). Zeno's arguments were not convincingly refuted until the mathematical developments of the nineteenth century. back

Heracleitus (fl. c. 500 bce)

Contrasting somewhat with Parmenides, we have the picture of the world developed by Heracleitus of Ephesus (Hussey 1995) About 100 sentences of his work survive (Burnet). Hussey summarizes Heracleitus' doctrine in five points:

1. The abstract notion of 'structure' is omnipresent, explicitly in the word harmonia, but mostly implicitly.

2. There is a parallelism or identity of structure between the operations of the mind, as expressed in thought and language, and those of the reality which it grasps.

3. In general the structure is that of 'unity in opposites' . This appears in many examples, static or dynamic, drawn from everyday life: 'People step into the same rivers, and different waters flow on to them'; ... These remarks and their generalizations are not meant to infringe the law of non-contradiction; rather they trade on it to point out a systematic ambivalence (between polar opposites) in the essential nature of things.

4. The parallelism of structure means that understanding the world is like grasping the meaning of a statement. The 'meaning of the world', like that of a statement in words, is not obvious, but yet is present in the statement and can be worked out, provided one 'knows the language'. Human reason has the power to know the language, precisely because its own operations are conducted in the very same or an analogous one. The word logos (basically 'story', 'account'; then 'calculation, proportion, reason') expresses this analogy or identity.

5. Hence the key to understanding the nature of the world is introspection. 'I went looking for myself'. The human self ('soul', psyche ) is variously occupied: it is combatively active, physically, emotionally and intellectually; it is reflectively self-discovering and self-extending; it is constantly self-reversing in the swings of circumstances or passion or thought. Yet it needs firm frameworks (objective truths, fixed rules of conduct) to be at all, or to make sense of its own existence. All this is true of the world too; here also there is no sharp line between what it is and what it means. ...

Difficulties with reconciling movement and rest, multitude and unity, knowledge and reality and the other dualities of our experience of the world remain to this day. back

Plato (429-347 bce)

Plato was, like Parmenides and Heracleitus, the child of a family involved in politics. Plato grew up in a time of political turmoil, during which he saw his uncles Critias and Charmides, leaders of the 'Thirty Tyrants', lose their lives in the restoration of democracy to Athens, and the execution of his friend Socrates by the democrats.

Popper sees this period as a transition from tribe to state. He finds echoes of this transition even in his present:

[This book] attempts to show that this civilization has not fully recovered from the shock of its birth - the transition from tribal or 'closed society', with its submission to magical forces, to the 'open society' which sets free the critical powers of man. (Popper 1966)

A similar idea is expressed by Jaynes, who places the origin of consciousness in the period between the composition of the Iliad and the Odyssey (Jaynes 1990). It seems certain that human critical self awareness took a giant leap forward in ancient Greece. Jaynes also sees the biblical history of the Hebrews in a similar light, as a metaphor for the new awareness that came to people as communications and the size of their social groupings increased (Jaynes 1990).

Plato, probably in harmony with his political party, "teaches that change is evil, and that rest is divine" (Popper 1966). Plato appears to have felt, like the Hebrews, that things had come downhill from some past golden age.

Most of Plato's writings are in the form of dialogues in which he does not personally appear. This makes it difficult to discern Plato's true opinions, if in fact he formed any firm conclusions about his experience of life.

To reconcile immobile underlying reality with the obvious flux of life, Plato uses his famous doctrine of forms or ideas, somewhat similar to Heracleitus' logos.

He believed that to every kind of ordinary or decaying thing there corresponds also a perfect thing that does not decay. This belief in perfect and unchanging things, usually called the Theory of Forms or Ideas, became the central doctrine of his philosophy. (Popper 1966)

Plato's Academy, founded in about 387 bce, lived on in various forms until it was finally destroyed by Justinian in 529. Plato's doctrine evolved in two directions, known now as Platonism and Aristotelianism (see below). In the hands of the Platonists, the forms became an hierarchy of invisible spiritual realities whose supreme member was the idea of the Good, destined to metamorphose into the Christian God.

Philo Judaeus of Alexandria (c. 25 bce - 50 ce) combined Platonism with Judaism, and although not a Christian was instrumental in introducing Platonism to Christianity (Fredriksen, 1988). Most of the ideas about god and the world found in the writings of the early Christian theologians show the influence of Plato and his successors (von Campenhausen 1963). Philo's eternal, transcendent God became central to Christian doctrine. back

Aristotle (384-322 bce)

Aristotle spent twenty years with Plato and was his most distinguished student. He did not inherit the Academy, probably because he was not qualified to own property in Athens (Inwood, 1985). Jaeger, studying the history of his development, writes that "Aristotle made himself out of the Platonic philosophy" (Jaeger, 1948).

Aristotle's concerns seem to have been more scientific and less political than Plato's. Where Plato was troubled by generation and corruption, motion and change, Aristotle was interested to observe and explain these phenomena. He sought dynamic rather than static explanations of the world: "Let us then start from the datum that the things of Nature, or (to put it at the lowest) some of them, do move and change, as is patent to observation; ..." (Physics, 185a12).

For Aristotle however, as for Plato, the study of Nature pointed beyond nature to the 'metanatural' reality. Aristotle explained change with his doctrine of four causes. Matter and form always correlative:

... matter does not exist as entirely undifferentiated; it passes through successive stages of differentiation, to each of which there is a corresponding form, until it merges as the proximate matter of the individual substance. (Tredennick 1980, xxvii)

Change at a particular level occurs when matter becomes associated with a new form, as when a bronze figure of a man is recast into the figure of a lion. In addition to matter and form, change also requires an agent and a purpose, the efficient and final causes (Tredennick 1980, 49, 81).

Like Plato and his predecessors, Aristotle was moved beyond physics to metaphysics by the universality of knowledge. The relationship of knowledge to mind is rather like form to matter. But ordinary matter is limited in the forms it can take; in particular, it cannot correlate to the catholicity of forms of intellectual knowledge. The human intellect therefore, and the soul which it occupies, must be non-material.

The question of the motion of the visible universe also led Aristotle beyond nature toward unmoved immaterial beings which moved the world. Aristotle held that the material world was eternal, so that the first unmoved mover was not conceived as creator, an attribute that was added to Aristotle's ideas by Aquinas. back

Thomas Aquinas (1224-1274)

Aristotle played very little part in the formative years of the Church, but was at the heart of its medieval renaissance. Drawing on Aristotle's biological and psychological insight and the metaphysical doctrine of potency and act, Thomas was able to produce a comprehensive and attractive picture of the interactions of a living god with a living world.

Potency and act, in the guise of matter and form, explain motion in the world. Potency and act, in the guise of essence and existence, allow Thomas to postulate spiritual beings (angels) distinct both from the world and from God, whose essence and existence are identical. Thomas provides an Aristotelian ground for the distinction between God and the world that echoes back to Parmenides:

The existence of God can be proved in five ways.

The first and more manifest way is the argument from motion. It is certain, and evident to our senses, that in the world some things are in motion. Now whatever is in motion is put in motion by another, for nothing can be in motion except it is in potentiality to that towards which it is in motion; whereas a thing moves inasmuch as it is in act. For motion is nothing else than the reduction of something from potentiality to actuality. But nothing can be reduced from potentiality to actuality, except by something in a state of actuality. Thus that which is actually hot, as fire, makes wood, which is potentially hot, to be actually hot, and thereby moves and changes it. Now it is not possible that the same thing should be at once in actuality and potentiality in the same respect, but only in different respects. For what is actually hot cannot simultaneously be potentially hot; but it is simultaneously potentially cold. It is therefore impossible that in the same respect and in the same way a thing should be both mover and moved, i.e. that it should move itself. Therefore, whatever is in motion must be put in motion by another. If that by which it is put in motion be itself put in motion, then this also must needs be put in motion by another, and that by another again. But this cannot go on to infinity, because then there would be no first mover, and, consequently, no other mover; seeing that subsequent movers move only inasmuch as they are put in motion by the first mover; as the staff moves only because it is put in motion by the hand. Therefore it is necessary to arrive at a first mover, put in motion by no other; and this everyone understands to be God. (Aquinas 13)

Though the earth cannot operate of its own accord, God can and does:

Whether God understands Himself?

... God understands Himself through Himself. In proof whereof it must be known that although in operations which pass to an external effect, the object of the operation, which is taken as the term, exists outside the operator; nevertheless in operations that remain in the operator, the object signified as the term of operation, resides in the operator; and accordingly as it is in the operator, the operation is actual. Hence the Philosopher says (De Anima iii) that "the sensible in act is sense in act, and the intelligible in act is intellect in act." For the reason why we actually feel or know a thing is because our intellect or sense is actually informed by the sensible or intelligible species. And because of this only, it follows that sense or intellect is distinct from the sensible or intelligible object, since both are in potentiality.

Since therefore God has nothing in Him of potentiality, but is pure act, His intellect and its object are altogether the same; so that He neither is without the intelligible species, as is the case with our intellect when it understands potentially; nor does the intelligible species differ from the substance of the divine intellect, as it differs in our intellect when it understands actually; but the intelligible species itself is the divine intellect itself, and thus God understands Himself through Himself. Aquinas 81 back

Bernard Lonergan (1904-1984)

Aquinas' proofs for the existence of god (= the non-divinity of the universe) might be called physical, in that he starts each proof from a physical observation and uses his metaphysical theory of the world to show that this physical observation implies that the universe cannot account for itself. Lonergan moves the question into the psychological realm:

The existence of God ... is known as the conclusion to an argument, and while such arguments are many, all of them, I believe, are included in the following general form.

If the real is completely intelligible, God exists. But the real is completely intelligible. Therefore god exists. (Lonergan 1992, 695)

The universe, however, is not god, because it is not completely intelligible. This, Lonergan claims, is because there are positively given empirical data which lack intelligibility, the 'empirical residue' detected by 'inverse insight'. (Lonergan 1992, 43-56)

... the five ways in which Aquinas proves the existence of God are so many particular cases of the general statement that the proportionate universe is incompletely intelligible and that complete intelligibility is demanded. (Lonergan 1992, 700)

Lonergan's position is developed with subtlety and insight into the results of modern science. It is based on a careful study of the psychology of Thomas Aquinas as applied to understanding the possibility of the Trinity (Lonergan 1997). Lonergan's work was my personal historical starting point for this research. It falls down, I believe, in his affirmation of the empirical residue. It is an historical accident that we do not yet fully understand the universe, but this is no reason to assert that it is not fully intelligible. In particular, Lonergan seems to mistake scientific conjectures (such as the theory of relativity) for the realities of which they are an abstract (textual) representations. back

Divinity returns to the universe

Isaac Newton (1643-1727)

A decisive event, from the point of view of this proposal, was the publication of Isaac Newton's Principia (Newton 1966). Physics credits Newton with deriving the formal law of universal gravitation and using to unite the heavens and the earth in one magnificent system. Philosophy owes him credit also, since he advanced the Platonic dream into a new phase.

The Greek world that fed into Christianity saw the search for true guidance in introspection. This tendency was first formalized by Heracleitus (point 5 above) and became in time the introspective message of Christianity: "Jesus said, 'You must love the Lord your God with all your heart, with all your soul, and with all your mind" (Mt 22:37, cf Dt 6:5). This is not surprising if the shock of the origin of consciousness was the driving force behind the rapid evolution of Greek philosophy and Christianity.

The scientific spirit seeks to look out, not in; to divorce itself, insofar as possible and appropriate, from particular personality. Galileo's dicta set the tone:

In questions of science the authority of a thousand is not worth the humble reasoning of a single individual. (Misner 1973, 38)

Philosophy is written in this grand book - the universe, which stands continually open before our gaze. But the book cannot be understood unless one first learns to comprehend the language and to read the alphabet in which it is composed. It is written in the language of mathematics ... (Drake 1957, 238)

Newton's work ignited an explosion in the study of forms (mathematics) that continues today, particularly in the interpretation of the enormous texts (genotypes) that have evolved to guide the process of life. back

Charles Darwin (1809-1882)

Plato and all his successors assumed that forms were immutable, with the consequence that the world has always been as it is now, the flow of generation and corruption never deviating from a fixed set of species. The alternative flowered in the time of Darwin, who we remember for the classic exposition of descent with modification. (Darwin 1998, Jones 1999)

Features of our own physical and spiritual constitution can be traced back three billion years to the origins of life on earth. Reflecting on this fact, we may entertain the hypothesis that every current situation is the most recent link in an evolutionary chain that stretches back to the beginning. The standard model of the evolution of the universe (the 'big bang' theory) takes this history back very close to this initial singularity (Weinberg 1993) . The standard model is not undisputed (Burbridge 1999).

Although Darwin's observations demonstrated to him beyond reasonable doubt the fact of evolution, he knew nothing of genes and genetics, and so was without a plausible mechanism to explain it. The germ of the explanation lay in the work of Gregor Mendel, which after lying dormant for many years flowered in modern genetics.

The discovery of the genetic code is part of modern folklore (Watson 1991). The genotypes of many organisms have now been completely sequenced, and a project to sequence the human genotype is approximately one third complete. In the process we are becoming intensely aware of the vast complexity of life. The genotype of a bacterium is approximately the same length as the Bible.

The process of establishing genetic texts and uncovering their meaning is proceeding at an increasing rate (Cole 1998). This industry in molecular biology is in many ways a computerized repetition of the textual studies of the nineteenth century, where the principle of descent with modification was used in an attempt to recover the original form and meaning of ancient and much copied texts.

Darwin's theory of evolution which was a seedling in 1858 has expanded to embrace all information about living systems. This has been possible because the evolutionary process is essentially recursive, that is each generation builds on what has gone before, without any obvious limit. Geophysiology takes the view the earth is one organism (christened Gaia in deference to an ancient Greek god) which evolves as the product of the evolution of its parts. Beyond this is the view that the universe itself is a single living and evolving organism (Lovelock 1995). back

George Boole (1815-1864)

Boole was a largely self taught mathematician whose work on the laws of thought went a long way toward realizing Leibniz' dream of an 'ars combinatoria':

all reasoning, all discovery, verbal or not, is reducible to an ordered combination of elements, such as numbers, words, sounds or colours (Belaval 1981) .

Boolean algebra, as it is now known, is the foundation for the propositional calculus and the hierarchy of formal theories built on propositional calculus (Mendelson 1987). It is also the practical foundation for the operation and design of computing machinery.

Boole's formalization of the 'laws of thought' opened the way for the twentieth century expansion of formal theory, both abstract and implemented in physical machines. back

Georg Cantor (1845-1918)

The distinction between God and the world that began with Parmenides was entrenched formally by the arguments of Zeno. The scientific unification of god and the world must be consistent, meaning that the inconsistencies between motion and stillness noted by Zeno must be answered.

To the ancients, motion implied infinity. The Greeks, perhaps learning from the Egyptians or Babylonians, were aware of the existence of incommensurable magnitudes, which implied the existence of a degree of infinity beyond the natural numbers (Heath 1956, 349-369). The introduction of calculus by Newton and Leibniz raised Zeno's problems anew, since calculus involved taking the ratios of infinitesimal quantities.

Other difficulties in the application of calculus led to the development by Joseph Fourier of the theory of transformations through which a function which was difficult to handle was encoded in a sum (or superposition) of well behaved functions.

Cantor was led to the theory of sets (Jech 1997) and the discovery of transfinite structure within the real line while studying how faithfully a Fourier transformation may represent an arbitrary function (Jourdain 1955). Cantor saw that the infinite set of finite natural numbers (whose cardinal number he called aleph zero) could generate (through a not-inconsistent process) a set of strictly greater infinity (cardinal aleph one), and that this recursive generation of transfinite cardinals could continue without end, like the recursive generation of natural numbers by adding one. In the transfinite case, however, the one that was added at each step was a new structure comprising everything that could be constructed from the elements of the prior set.

The proof for the existence of the transfinite alephs rests on the concept of ordinal type. Cantor felt that

The concept of 'ordinal type' developed here, when it is transferred in like manner to 'multiply ordered aggregates' embraces, in conjunction with the concept of 'cardinal number' or 'power' ... , everything capable of being numbered (Anzahlmassige ) that is thinkable, and in this sense cannot be further generalized. (Cantor 1955, 117)

Cantor's ideas, which are important to mathematics, have their foundation in his theological views. Hallet writes:

It is clear that Cantor understands pure set theory as a quite general foundational theory which prepares the way for any theory which uses or relies on sets or numbers. But now we come back to theology and God, for this foundation, this understanding of what numbers are, or what sets etc. exist, is for Cantor intimately connected with the attempt to understand God's whole abstract creation and the nature of God himself. (Hallett 1984, 10) back

Albert Einstein (1879-1955)

It seems probable that Einstein changed our picture of the universe more than any other individual. To Planck, the quantization of action may have been merely a mathematical trick. Einstein took it seriously and in 1906 proposed that the light quantum (photon) was real. For this he won the Nobel prize in 1922. Although the atomic hypothesis had been with us since the time of Democritus, Einstein contributed definitively to the establishment of the reality of 'molecules' and to estimating their number. With Podolsky and Rosen, he laid the foundations of what is now known as 'quantum teleportation', a phenomenon which appears set to be an important element of future quantum computation. Yet he was never happy with quantum theory. (Pais 1982)

His best known contributions are the special and general theories of relativity. Among its many consequences, special relativity shows that space and time are in some sense inverses. Although the photons of cosmic background radiation that bathe us at this moment have travelled perhaps fifteen billion light years through space and fifteen billion years through time, special relativity predicts that the space time distance between their point of emission and point of absorption is precisely zero. The special theory also predicts that mass and energy are equivalent.

The general theory, built from the special theory with the addition of the principle of the equivalence of acceleration and gravitation, utterly changed our cosmological picture. Gravitation, he found, is a consequence of the geometric curvature of space. The curvature of space enables us to imagine a universe with no exterior, as we must expect of something properly named universe. Einstein's theory predicts that the universe is expanding, and astronomical measurements based on this expansion show that the universe is huge. One consequence of this size is that unless the probability of a planet like earth coming into existence is very low, there must be other 'human' creatures in the universe (Lissauer 1999).

Special relativity establishes that all the events which affect a particular event must lie in the past 'light cone' of that event, thus establishing an horizon of causality and knowledge. Such horizons are the 'event horizon' of a black hole, and the surface at which distant galaxies appear to be receding from us at the velocity of light. There is no particular evidence to suggest that the universe does not stretch boundlessly beyond these horizons.

It is important for this project that the Einstein picture of the universe seems to remove the force from many of the arguments against the divinity of the universe based on ancient understandings of the nature of space, time and motion (Aquinas 1962, I, 3, 1).

Einstein has shown us how to understand the whole of spacetime by looking at the local situation. By this he passes Tillich's test for a theologian:

He is a theologian in the degree to which his intuition of the universal logos of the structure of reality as a whole is formed by a particular logos which appears to him on his particular place and reveals to him the meaning of the whole. (Tillich 1968, I, 29) back

Quantum theory (1900 - ?)

Modern physics is based on relativity and quantum theory. Einstein established relativity almost single handedly. Quantum theory, on the other hand, has been a vast collaborative effort driven at a furious pace between the 1930's and 90's by the possibility of nuclear war (Pais 1986). Quantum theory is highly mathematical. From the point of view of this project, an important point is that the mathematical technique of quantum mechanics is in effect manipulation of text.

Quantum theory describes the world in terms of states and transformations between states (Dirac 1983). The states are represented as vectors in a 'Hilbert space' and the transformations of these vectors are represented by operators in this space. From an anthropomorphic point of view, we might say that states are represented by sentences (ordered sets of symbols or actions), and the transformation of vectors by translating one sentence expressing a certain idea (state) into another sentence expressing the same idea.

From a quantum mechanical point of view, a particular state (idea) may be represented by a superposition of vectors (sentences). When we observe (talk to) that state, it emits only one of the sentences in the superposition, and we have to question it repeatedly to gain an overall picture of the state, just as we may have to read many rephrasings of a complex idea to understand it properly.

It is perhaps not surprising, given the linguistic structure of quantum mechanics, that there is now a strongly based hope that we will be able to use quantum phenomena to build computers which are in some sense infinitely more powerful than current 'classical' computers (Lo 1998). Since quantum theory seems to have tapped into a natural language of the universe, the alliance between quantum theory and computing suggests a model for the universe. This model is a space of unbounded size (the Cantor universe) populated by aleph zero computers in communication with one another. back

Kurt Goedel (1906-1978)

Cantor's theory of transfinite numbers had a difficult birth and turned out to be dangerous, in that it led to paradoxes (such as Cantor's paradox). The effort to overcome such paradoxes led to the revitalization of the axiomatic method pioneered by Euclid in the Elements. One axiomatizes a mathematical entity by trying to capture the essence of it in a few succinct statements from from which the structure of the entity may be derived by logical argument. Different axiomatisations of the same entity may lead to different expressions of its structure.

An early champion of axiomatisation, David Hilbert, felt that the method could solve all mathematical problems. He conjectured that if mathematics was consistent, it would be both complete and computable (Reid 1986 , 189). A mathematical theory is complete if one can decide, for every legal statement in the theory, that it is or is not consistent with the axioms of the theory. Goedel showed that small systems, like the propositional calculus, are indeed consistent and complete (Goedel 1929 ). Larger quantified calculi, (like that used by Whitehead and Russell to axiomatize mathematics (Whitehead and Russell 1997)), are not complete if they are consistent (Goedel 1931).

We might detect in Goedel's work the formal source of dynamics, the link between the formal world of Plato and the dynamic world of Aristotle. Goedel tells us that a suitably large consistent system in incomplete. We may suspect from this conclusion that a large complete system may be inconsistent. Inconsistency creates a potential for change, to eliminate the inconsistency. Large complete systems are thereby moved to act. back

Alan Turing (1912-1954)

Goedel upset Hilbert's conjecture about completeness. Turing did the same for computability. (Turing 1937) The entscheidungsproblem or 'decision problem' proposed by Hilbert asks whether, given a formal system F, there is a definite procedure with a finite number of steps which can decide whether any arbitrary formula f of F is derivable in F (Kneebone 1975, 279) Such a derivation may be called a computation. Turing devised a formal 'machine' (ie deterministic automaton) now known as a Turing machine which could perform any activity that might reasonably be called a computation, and showed that such a machine could not in general solve the decision problem for Hilbert and Ackermann's restricted calculus of predicates. A similar result was obtained by Church through a mechanism known as 'lambda-definability'.

These results depend on the 'Turing-Church hypothesis' (or principle, aka Church's thesis) that a number theoretic (digital) function is computable if there is a definite mechanical procedure or algorithm that computes it, and not otherwise.

This idea lies at the foundation of the modern computing revolution. Turing himself became deeply involved in practical computing through his work on breaking codes during the second world war, and soon after the war practical digital computers began to appear and evolve rapidly toward the machines we have now. The theory of computing gives us a formal definition of operation or action.

A Turing machine may be programmed to halt and ask for advice from outside. This allows for the development of a theory of 'relatively computable functions' (Davis 1982, 20 sqq). This theory provides an opening for a theory of computer networks (Tanenbaum 1996). back

Claude Shannon

One of the most striking features of our universe, and a source of endless fascination for philosophers, is the existence of knowledge. Most philosophy seems to be based on the notion that the existence of human knowledge imposes certain constraints on reality.

Knowledge allows us to encode and communicate a complex structure using a simpler structure. When I say "Bessie is in the top paddock" the simple object in inverted commas conveys information about an enormously complex piece of landscape, and helps my hearer to perform a definite set of actions which might be encoded "milk Bessie".

Shannon was concerned with the prevention of misunderstanding in communication arising from the corruption of text during transmission. He devised a mathematical measure of information called entropy, modelled on the entropy of nineteenth century thermodynamics (Khinchin 1957, 1-28). He was then able to show that (given certain plausible statistical properties of messages) there exist encodings which allow messages to be transmitted without error over error prone channels.

His theory did not produce any actual encodings, but knowledge of their possibility let to rapid discovery of a large range or error detecting and error correcting codes which have made structures such as the internet possible (Hill, 1986) . Such encoding and decoding may be performed by a computer.

Shannon's theory shows that error is resisted by a combination of redundancy and complex coding. These theorems appear to have a strong bearing on the direction of evolution of life. back

Norbert Wiener (1894-1964)

Wiener contributed significantly to the foundations of cybernetics: "control and communication in the animal and the machine" (Wiener 1996). A cybernetic system has sensor(s) to detect the state of its environment, a computer to decide what (if any) action is necessary to cope with the state of the environment, and actuator(s) to execute its decision. Some systems are merely reactive, like the governor on an engine, which increases the energy supply when the engine slows below its desired speed and decreases the supply when the engine exceeds the desired speed. This is called feedback.

More complex ('intelligent') systems use stored information, perhaps derived from previous experience, as well as current data, to tune their interaction with their environment. This is sometimes called feed forward. Cybernetics provides a formalism for linking formal systems and action, and an overall mathematical framework for understanding the behaviour of all organisms in the world. back

David Deutsch

Deutsch was one of the first to propose that the formal systems devised by Turing are implemented by quantum processes, which might therefore be exploited to construct computers (Deutsch 1985). Deutsch later expanded his insights into a comprehensive cosmology (Deutsch 1997). The general effect of this new work is to move our conceptions of the universe further from the notion of 'inert matter' which fuelled the speculation of the ancients toward the view that human life and consciousness is a 'virtual reality' arising from a living thinking universe.

Deutsch's advocacy of the 'many world's hypothesis' seems difficult to cope with. I would like to replace the many worlds by one world with a 'law' that conserves the flow of action in the universe (actus purus). This law is the assumption that the total activity of the universe is measured (in units of the quantum of action) by aleph zero; and that this flow realizes a particular course through the 'phase space' symbolized by the many worlds. The many worlds, described by the 'wave function of the universe' is here to be described by the transfinite numbers whose cardinal is greater than aleph zero.

The field of quantum computation is growing rapidly with significant discoveries still flowing freely (Gottesman and Chuang 1999). back

3. Toward a position

The research to be reported in this thesis has been conducted intermittently over thirty years since I was asked to leave the Order of Preachers. The day I found myself on the street in a new suit with a few dollars in my pocket was the worst shock of my life and set me thinking about truth, justice and the Catholic Church.

My dismissal was partly my own fault. In the brief 'Prague Spring' that accompanied the Second Vatican Council, I said far too much about academic freedom, scientific method, democratic government and practical religion based on current reality rather than ancient dreams. Subsequently I 'lost my faith' but now have a new faith which is not so different from the old, and remain a stakeholder in the human condition. My 'ultimate concern' (Tillich 1968 I, 14-18) remains peace inside and outside myself. I firmly believe that such peace can be achieved by a proper understanding of my situation and proper action in the light of that knowledge.

Inquiries in 1988 revealed that the Order held no documentation whatever which might explain why and how I was dismissed, only a copy of a my petition to the Pope for dispensation of my solemn vows. I subsequently received an academic record which indicated good results in all subjects. I am left with the vague memory that I was held not to conform with certain of the Twenty Four Theses of Pius X (Denzinger 1963, 3601-3624).

The sudden end to my vocation was followed by difficult years in which I became aware of the depth to which faith acquired in childhood is embedded in the mind. After twenty years I was stable enough to express a public opinion on the roles of religion and theology in the world (Nicholls 1987).

Although much modern theology finds its ground in personal experience, (Avis 1986) the conjecture outlined here attempts to abstract from my personal background and treat only the question of whether the proposition "the universe is divine" is to be judged true or false.

The following is a tentative outline of the thesis:

Proposed title: Is the universe divine?

Preface

General mise en scene: condensed from the material above

Acknowledgements
Table of contents
Introduction

Summary presentation of the model, evidence of the operation of the model and consequences from the establishment of the model.

Chapter 1. The problem:

See above "1. The problem of global ecumenism".

Chapter 2. Methodology

Theological method: a search for the meaning of canonical text.
Scientific method: the meaning of observable events.
Cybernetics: model and action.
A definition of method: everything we have learned up to this moment.

Chapter 3. The model

Outline of a mathematical model of unlimited size and detail:

Set theory.
Application of set theory to describe Cantor universe.
Application of set theory to describe computer network.
The model: a transfinite network.

Chapter 4. A Fit

A mapping between the model and features of the universe observed and modelled by the sciences:

The universe is both discrete and continuous;
Quantum mechanics, quantum computation and relativity;
The universe both changes and remains the same;
Not all possibilities are realized at any point in spacetime;
The universe has cardinal number;
The universe has order;
The universe has a smallest element;
The universe has no maximum size;
Entropy tends to increase;
Knowledge is possible;
etc etc.

Chapter 5. Is the universe divine?

If the universe can be modelled with a model of god, we have a ground for calling the universe divine. Having outlined a fit between model and observation in the previous chapter, we here ask if, according to traditional criteria, the model may be called a model of god:

Actus purus is modelled as a transfinite set of quanta of action;Simplicity arises because two actions may blend seamlessly to form one more complex action, as we feel in music. (This observation may be supported by mathematical consideration of the product of Hilbert spaces);
Perfection ... ;
Infinity ... ;
Eternity ... ;
Unity ... ;
Knowledge of God ... ;
Will of God ... ;
Omnipotence ... ;
Providence ... .

Chapter 6. Some answers

What about eternal life?
What about sin and redemption?
etc

Chapter 7. A prediction

If the universe is properly called divine, it may be that heaven for human beings can be found on earth:

Aquinas on beatitude;
Correspondence between model and Aquinas.

Bibliography
Index

Further reading

Books

Papers

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