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vol 6: Essays

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

2.1 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)

2.2 Divinity returns to the Universe

Isaac Newton (1643-1727)
Charles Darwin (1809-1882)
George Boole (1815-1864)
Georg Cantor (1845-1918)
Albert Einstein (1879-1955)
Quantum theory (1900 - )
Kurt Gödel (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." 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 ; their Church as the only medium of salvation; and separation from the Church as a sin. Christians with "the gift of ultimate truth about human life", stand fairly in the centre of the world. It is all for them. Abbott, 3, note 2), Dei Verbum, Lumen Gentium, Unitatis redintegratio, John Paul II.

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'. 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. Popper 1972, 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. Sacred tradition - Wikipedia

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". 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. Haight, 47

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

2.1 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. 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. Hussey, John 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. 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. Aristotle De Anima

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". Zeno's arguments were notconvincingly refuted until the mathematical developments of the nineteenth century. Hussey 1995, back

Heracleitus (fl. c. 500 bce)

Contrasting somewhat with Parmenides, we have the picture of the world developed by Heracleitus of Ephesus. About 100 sentences of his work survive. Burnet, Hussey 1995

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. 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". Plato appears to have felt, like the Hebrews, that things had come downhill from some past golden age. Popper 1966

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 1955

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 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 Philo's eternal, transcendent God became central to Christian doctrine. Fredriksen, 1988, von Campenhausen 1963, 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. Jaeger, studying the history of his development, writes that "Aristotle made himself out of the Platonic philosophy" Inwood, 1985, 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; . . .. " Aristotle, Physics, 185a12.

For Aristotle however, as for Plato, the study of Nature pointed beyond nature to '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, pp 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; Does God exist?

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: Insight, page 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, pp 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, page 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'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) representation. Lonergan 1997: Verbum: Word and Idea in Aquinas, back

2.2 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. 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. Newton 1966

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, Thorne & Wheeler 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 . . . Galileo: Il Saggiatore (The Assayer)

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. The standard model is not undisputed. Weinberg 1993, 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. Gregor Mendel - Wikipedia

The discovery of the genetic code is part of modern folklore . 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. Watson 1991

The process of establishing genetic texts and uncovering their meaning is proceeding at an increasing rate. 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. Cole 1998

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 Goddess) 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, Gaia hypothesis - Wikipedia, 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. It is also the practical foundation for the operation and design of computing machinery. Mendelson 1987

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. The introduction of calculus by Newton and Leibniz raised Zeno's problems anew, since calculus involves taking the ratios of infinitesimal quantities. Heath 1956, pp 349-369

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 and the discovery of transfinite structure within the real line while studying how faithfully a Fourier transformation may represent an arbitrary function. 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. Jech 1997, Jourdain 1955

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 furthergeneralized. Cantor 1955, p 117., Ordinal number - Wikipedia

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, Einstein, Podolsky and Rosen

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 Summa, 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. 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. Pais 1986

Quantum theory describes the world in terms of states and transformations between states. 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. Dirac 1983

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.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. Lo, Spiller & Popescu, 1998, Nielsen & Chuang, back

Kurt Gödel (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. Cantor's paradox - Wikipedia

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. 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. Gödel showed that small systems, like the propositional calculus, are indeed consistent and complete. Larger quantified calculi, (like that used by Whitehead and Russell to axiomatize mathematics) are not complete if they are consistent. Gödel 1929, Gödel 1931, Reid 1986, page 189, Whitehead & Russell 1997.

We might detect in Gödel's work the formal source of dynamics, the link between the formal world of Plato and the dynamic world of Aristotle. Gödel 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)

Gödel upset Hilbert's conjecture about completeness. Turing did the same for computability. 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. 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'. Turing 1937, Kneebone 1975, p 279)

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'. This theory provides an opening for a theory of computer networks. Davis 1982, p 20 sqq, Tanenbaum 1996, back

Claude Shannon (1916 - 2001)

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. 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. Khinchin 1957, 1-28)

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. Such encoding and decoding may be performed by a computer. Hill, 1986

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". 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. Wiener 1996

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 later expanded his insights into a comprehensive cosmology. 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 1997: The Fabric of Reality

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' 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. Tillich 1968, I, 14-18

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, Twenty Four Theses of Pius X

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, A Theory of Peace.

Although much modern theology finds its ground in personal experience, 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. Avis 1986

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.

back

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!)

Abbott, Walter M, and Joseph Gallagher (translation editor), The Documents of Vatican II: in a new and definitive translation, with notes and commentaries by Catholic, Protestant and Orthodox authorities, Geoffrey Chapman 1972 Jacket: 'All 16 Documents of Vatican II are here presented in a new and readable translation. Informed comments and appraisals by Catholics and non-Catholics make this book essential reading for anyone, of whatever shade of belief, who is interested in the changing climate of today's world.' 
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Aquinas, Thomas, Summa Theologica (translated by Fathers of the English Dominican Province), Tabor Publishing 1981 'Brother Thomas raised new problems in his teaching, invented a new method, used new systems of proof. To hear him teach a new doctrine, with new arguments, one could not doubt that God, by the irradiation of this new light and by the novelty of this inspiration, gave him the power to teach, by the spoken and written word, new opinions and new knowledge.' (William of Tocco, T's first biographer) 
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Aquinas 13, Thomas, Summa Theologica (translated by Fathers of the English Dominican Province), Tabor Publishing 1981 , I, 2, 3: "Does God Exist?", available from Amazon   back
Aquinas 14, Thomas, Summa Theologica (translated by Fathers of the English Dominican Province), Tabor Publishing 1981 , I, 3, 1: "Is God a Body?", available from Amazon   back
Aquinas 81, Thomas, Summa Theologica (translated by Fathers of the English Dominican Province), Tabor Publishing 1981 , I, 14, 2: Whether God understands himself?", available from Amazon   back
Aristotle, and (translated by P H Wickstead and F M Cornford), Physics books I-IV, Harvard University Press, William Heinemann 1980 Introduction: 'The title "Physics" is misleading. .. "Lectures on Nature" the alternative title found in editions of the Greek text, is more enlightening. ... The realm of Nature, for Aristotle, includes all things that move and change ... . Thus the ultimate "matter" which, according to Aristotle, underlies all the elementary substances must be studied, in its changes at least, by the Natural Philosopher. And so must the eternal heavenly spheres of the Aristotelean philosophy, insofar as they themselves move of are the cause of motion in the sublunary world.' 
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Aristotle, and (translated by W S Hett), On the Soul, Parva Naturalia, On Breath (translated by W S Hett) , Harvard University Press (USA) ; William Heinemann Ltd (UK) 1975 'What the mind thinks must be in it in the same sense as letters are on a tablet which bears no actual writing; this is just what happens in the case of the mind.' page 169 (Book III, chapter 4, 429b32) 
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Aristotle, and (translated by W S Hett), On the Soul, Parva Naturalia, On Breath, Harvard University Press (USA) ; William Heinemann Ltd (UK) 1975 Introduction: 'This collection of treatises belongs to subjects on the borderline between bodily and mental. Aristotle was the son of a doctor and himself a biologist, who believed in experiment and dissection as a means of collecting evidence. Thus his views on the soul are influenced by his physiology. Yet he never falls into the meshes of materialism, and appears quite certainn that the body cannot possibly explain the mind. . . .' 
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Aristotle, and (translated by W S Hett), On the Soul, Parva Naturalia, On Breath (translated by W S Hett) , Harvard University Press (USA) ; William Heinemann Ltd (UK) 1975 'What the mind thinks must be in it in the same sense as letters are on a tablet which bears no actual writing; this is just what happens in the case of the mind.' page 169 (Book III, chapter 4, 429b32) 
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Avis, Paul D L, The Methods of Modern Theology : the Dream of Reason , Marshall Pickering 1986  
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Avis, Paul D L, The Methods of Modern Theology : the Dream of Reason , Marshall Pickering 1986  
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Belaval, Yvon, "Leibniz, Gottfried Wilhelm" in Encyclopaedia Britannica (15th ed) , Helen Hemingway Benton 1981 'G W L, whose universal genius has influenced such fields as logic, mathematics, mechanics, geology, jurisprudence, history, linguistics and theology, dominated the intellectual life of Germany in the late 17th and early 18th centuries, influenced the German idealists Johann Gottlieb Fichte and G W F Hegel, reappeared as an intellectual force at the beginning of the 20th century, and became again in the 1970s, particularly through his scientific thought, one of the most relevant philosophers. ... '  
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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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Darwin, Charles, and Greg Suriano (editor), The Origin of Species, Gramercy 1998 Introduction: 'In considering the Origin of Species, it is quite conceivable that a naturalist, reflecting on the mutual affinities of organic beings, on their embryological relations, their geographical distribution, geological succession, and other such facts, might come to the conclusion that each species has not been independently created, but has descended, like varieties, from other species.' 
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Davies, Paul, The Mind of God: Science and the Search for Ultimate Meaning, Penguin Books 1992 'Paul Davies' "The Mind of God: Science and the Search for Ultimate Meaning" explores how modern science is beginning to shed light on the mysteries of our existence. Is the universe - and our place in it - the result of random chance, or is there an ultimate meaning to existence? Where did the laws of nature come from? Were they created by a higher force, or can they be explained in some other way? How, for example, could a mechanism as complex as an eye have evolved without a creator? Paul Davies argues that the achievement of science and mathematics in unlocking the secrets of nature mean that there must be a deep and significant link between the human mind and the organization of the physical world. . . . ' 
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Davis, Martin, Computability and Unsolvability, Dover 1982 Preface: 'This book is an introduction to the theory of computability and non-computability ususally referred to as the theory of recursive functions. The subject is concerned with the existence of purely mechanical procedures for solving problems. . . . The existence of absolutely unsolvable problems and the Goedel incompleteness theorem are among the results in the theory of computability that have philosophical significance.' 
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Dei Verbum, Second Vatican Council, and Walter M Abbott and Joseph Gallagher (translation editors), Dogmatic Constitution on Divine Revelation The Documents of Vatican II, Geoffrey Chapman 1972 'God, who through the Word creates all things (cf. Jn. 1:3) and keeps them in existence, gives men an enduring witness to Himself in created realities (cf. Rom. 1:19-20). Planning to make known the way of heavenly salvation, He went further and from the start manifested Himself to our first parents. Then after their fall His promise of redemption aroused in them the hope of being saved (cf. Gen. 3:15), and from that time on he ceaselessly kept the human race in His care, in order to give eternal life to those who perseveringly do good in search of salvation (cf. Rom. 2:6-7).' para 3, page 112. 
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Denzinger, Henricus, and Adolphus Schoenmetzer, Enchiridion Symbolorum, Definitionum et Declarationum de Rebus Fidei et Morum, Herder 1963 Introduction: 'Dubium non est quin praeter s. Scripturam cuique theologo summe desiderandus sit etiam liber manualis quo contineantur edicta Magisterii ecclesiastici eaque saltem maioris momenti, et quo ope variorim indicum quaerenti aperiantur eorum materiae.' (3) 'There is no doubt that in addition to holy Scripture, every theologian also needs a handbook which contains at least the more important edicts of the Magisterium of the Church, indexed in a way which makes them easy to find.' back
Deutsch, David, The Fabric of Reality: The Science of Parallel Universes - and its Implications, Allen Lane Penguin Press 1997 Jacket: 'Quantum physics, evolution, computation and knowledge - these four strands of scientific theory and philosophy have, until now, remained incomplete explanations of the way the universe works. . . . Oxford scholar DD shows how they are so closely intertwined that we cannot properly understand any one of them without reference to the other three. . . .' 
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Dirac, P A M, The Principles of Quantum Mechanics (4th ed), Oxford UP/Clarendon 1983 Jacket: '[this] is the standard work in the fundamental principles of quantum mechanics, indispensible both to the advanced student and the mature research worker, who will always find it a fresh source of knowledge and stimulation.' (Nature)  
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Feyerabend, Paul K, "science, history of the philosophy of" in Ted Honderich (editor) The Oxford Companion to Philosophy, Oxford University Press 1995  
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Fredriksen, Paula, From Jesus to Christ: The Origins of the New Testament Images of Jesus, Yale University Press 1988 Jacket: 'How did Jesus of Nazareth become the Christs of the Christian tradition? And why did the early Christian communities develop different theological images of Jesus? In this exciting book, PF answers these questions by placing he various canonical images of Jesus within their historical context.' 
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Gödel, Kurt, and Solomon Feferman et al (eds), Kurt Gödel: Collected Works Volume 1 Publications 1929-1936, Oxford UP 1986 Jacket: 'Kurt Goedel was the most outstanding logician of the twentieth century, famous for his work on the completeness of logic, the incompleteness of number theory and the consistency of the axiom of choice and the continuum hypotheses. ... The first volume of a comprehensive edition of Goedel's works, this book makes available for the first time in a single source all his publications from 1929 to 1936, including his dissertation. ...' 
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Haight, Roger, Jesus Symbol of God, Orbis Books 1999 Jacket: 'This book is the flagship of the fleet of late twentieth century works that show American Catholic theology has indeed come of age. Deeply thoughtful in its exposition, lucid in its method, and by turns challenging and inspiring in its conclusions, this christology gives a new articulation of the saving "point" of it all. . . . Highly recommended for all who think about and study theology.' Elizabeth Johnson CSJ, Fordham University. 
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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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Heath, Thomas Little, Thirteen Books of Euclid's Elements (volume 1, I-II), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time - the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.' 
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Heath, Thomas Little, Thirteen Books of Euclid's Elements (volume 2, III-IX), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time - the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.' 
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Heath, Thomas Little, Thirteen Books of Euclid's Elements (volume 3, X-XIII), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time - the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.' 
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Hill, Raymond, A First Course in Coding Theory, Oxford University Press, USA 1990 Amazon Editorial Reviews Book Description: 'Algebraic coding theory is a new and rapidly developing subject, popular for its many practical applications and for its fascinatingly rich mathematical structure. This book provides an elementary yet rigorous introduction to the theory of error-correcting codes. Based on courses given by the author over several years to advanced undergraduates and first-year graduated students, this guide includes a large number of exercises, all with solutions, making the book highly suitable for individual study.' 
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Hussey, E L, "Parmenides" in Ted Honderich (editor) The Oxford Companion to Philosophy, Oxford University Press 1995  
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Hussey, E L, "Zeno of Elea" in Ted Honderich The Oxford Companion to Philosophy, Oxford University Press 1995  
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Hussey, E L, " Heraclitus of Ephesus" in Ted Honderich The Oxford Companion to Philosophy, Oxford University Press 1995  
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Inwood, M J, "Platonism" in Ted Honderich (editor) The Oxford Companion to Philosophy, Oxford University Press 1995  
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Jaeger, Werner, Aristotle: Fundamentals of the history of his development, Oxford University Press 1997 Jacket: '"Aristotle was the first thinker to set up along with his philosophy a conception of his own position in history; he thereby created a new kind of philosophical consciousness, more responsible and inwardly complex. He was the inventor of the notion of intellectual development in time . . . ." In this classic study, Professor Jaeger profoundly altered the general view of Aristotle among philosophers and classical scholars. He showed that Aristotle was not uncompromisingly opposed to Plato, that he developed gradually, applying step by step his particular genius to the problems of his age.' 
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Jaynes, Julian, The Origin of Consciousness in the Breakdown of the Bicameral Mind, Mariner Books 2000 Jacket: 'At the heart of this book is the revolutionary idea that human consciousness did not begin far back in animal evolution but is a learned process brought into being out of an earlier hallucinatory mentality by cataclysm and catastrophe only 3000 years ago and still developing.' 
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Jech, Thomas, Set Theory, Springer 1997 Jacket: 'This book covers major areas of modern set theory: cardinal arithmetic, constructible sets, forcing and Boolean-valued models, large cardinals and descriptive set theory. . . . It can be used as a textbook for a graduate course in set theory and can serve as a reference book.' 
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Jones, Steve, Almost like a Whale: The Origin of Species Updated, Doubleday 1999 An Historical Sketch: 'The Origin of Species is, without doubt, the book of the millennium. ... [This book] is, as far as is possible, an attempt to rewrite the Origin of Species. I use its plan, developing as it does from farms to fossils, from beehives to islands, as a framework, but my own Grand Facts ... are set firmly in the late twentieth century. Almost Like a Whale tries to read Charles Darwin's mind with the benefit of scientific hindsight and to show how the theory of evolution unites biology as his millenium draws to an end.' (xix)  
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Jourdain, Philip E B, "Introduction" in George Cantor Contributions to the Founding of the Theory of Transfinite Numbers , Dover 1955  
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Khinchin, Aleksandr Yakovlevich, 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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Kneebone, G T , Mathematical Logic and the Foundations of Mathematics, van Nostrand 1975 Preface: 'The present book . . . is designed to serve in the first instance, when supplemented by reference to original sources, as a comprehensive introduction to the earlier phases of the historical development of the philosophy of mathematics. p vi.back
Lo, Hoi-Kwong, and Tim Spiller, Sandra Popescu, Introduction to Quantum Computation and Information, World Scientific 1998 Jacket: 'This book provides a pedagogical introduction to the subjects of quantum information and computation. Topics include non-locality of quantum mechanics, quantum computation, quantum cryptography, quantum error correction, fault tolerant quantum computation, as well as some experimental aspects of quantum computation and quantum cryptography. A knowledge of basic quantum mechanics is assumed.' 
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Lonergan, Bernard J F, and Robert M. Doran, Frederick E. Crowe (eds), Verbum : Word and Idea in Aquinas (Collected Works of Bernard Lonergan volume 2) , University of Toronto Press 1997 Jacket: 'Verbum is a product of Lonergan's eleven years of study of the thought of Thomas Aquinas. The work is considered by many to be a breakthrough in the history of Lonergan's theology ... . Here he interprets aspects in the writing of Aquinas relevant to trinitarian theory and, as in most of Lonergan's work, one of the principal aims is to assist the reader in the search to understand the workings of the human mind.' 
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Lonergan, 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' 
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Lovelock, James, Ages of Gaia: A Biography of our Living Earth, W W Norton 1995 'This book describes a set of observations about the life of our planet which may, one day, be recognised as one of the major discontinuities in human thought. If Lovelock turns out to be right in his view of things, as I believe he is, we will be viewing the Earth as a coherent system of life, self regulating and self-changing, a sort of immense living organism.' Lewis Thomas 
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Luger, George F, and William A Stubblefield, Artificial Intelligence: Structures and Strategies for Complex Problem Solving, Addison Wesley 2004 Amazon Editorial Reviews Book Description 'Much has changed since the early editions of Artificial Intelligence were published. To reflect this the introductory material of this fifth edition has been substantially revised and rewritten to capture the excitement of the latest developments in AI work. Artificial intelligence is a diverse field. To ask the question "what is intelligence?" is to invite as many answers as there are approaches to the subject of artificial intelligence. These could be intelligent agents, logical reasoning, neural networks, expert systems, evolutionary computing and so on. This fifth edition covers all the main strategies used for creating computer systems that will behave in "intelligent" ways. It combines the broadest approach of any text in the marketplace with the practical information necessary to implement the strategies discussed, showing how to do this through Prolog or LISP programming.' 
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Lumen Gentium, in Walter M Abbott and Joseph Gallagher (translation editor) The Documents of Vatican II: Dogmatic Constitution on the Church, Geoffrey Chapman 1972 'This sacred Synod turns its attention first to the Catholic faithful. Basing itself upon sacred Scripture and tradition, it teaches that the Church, now sojourning on earth as an exile, is necessary for salvation.' (para 14) 
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Lumen Gentium, in W M Abbott and Joseph Gallagher (translation editor) The Documents of Vatican II: Dogmatic Constitution on the Church, Geoffrey Chapman 1972 'By an utterly free and mysterious decree of His own wisdom and goodness, the eternal Father created the whole world. His plan was to dignify men with a participation in His own divine life.' para 2 
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Mendelson, Elliott, Introduction to Mathematical Logic, van Nostrand 1987 Preface: '... a compact introduction to some of the principal topics of mathematical logic. . . . In the belief that beginners should be exposed to the most natural and easiest proofs, free swinging set-theoretical methods have been used."  
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Misner, Charles W, and Kip S Thorne, John Archibald Wheeler, Gravitation, Freeman 1973 Jacket: 'Einstein's description of gravitation as curvature of spacetime led directly to that greatest of all predictions of his theory, that the universe itself is dynamic. Physics still has far to go to come to terms with this amazing fact and what it means for man and his relation to the universe. John Archibald Wheeler. . . . this is a book on Einstein's theory of gravity. . . . ' 
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Newton, Isaac, and Julia Budenz, I. Bernard Cohen, Anne Whitman (Translators), The Principia : Mathematica: l Principles of Natural Philosophy, University of California Press 1999 This completely new translation, the first in 270 years, is based on the third (1726) edition, the final revised version approved by Newton; it includes extracts from the earlier editions, corrects errors found in earlier versions, and replaces archaic English with contemporary prose and up-to-date mathematical forms. . . . The illuminating Guide to the Principia by I. Bernard Cohen, along with his and Anne Whitman's translation, will make this preeminent work truly accessible for today's scientists, scholars, and students. 
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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. 
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Nostra Aetate, in Walter M Abbott and Joseph Gallagher (translation editor) The Documents of Vatican II: Declaration on the Relationship of the Church to Non-Christian Religions, Geoffrey Chapman 1972 'Men look to the various religions for answers to those profound mysteries of the human condition which, today even as in olden times, deeply stir the human heart: what is man? What is the meaning and purpose of our life? What is goodness and what is sin? What gives rise to our sorrows and to what intent? What is the truth about death, judgement and retribution beyond the grave? What, finally, is that ultimate and unutterable mystery which engulfs our being, and whence we take our rise, and whither our journey leads us?' Article 1, page 661.  
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Nostra Aetate, in Walter M Abbott and Joseph Gallagher (translation editor) The Documents of Vatican II: Declaration on the Relationship of the Church to Non-Christian Religions, Geoffrey Chapman 1972 'Men look to the various religions for answers to those profound mysteries of the human condition which, today even as in olden times, deeply stir the human heart: what is man? What is the meaning and purpose of our life? What is goodness and what is sin? What gives rise to our sorrows and to what intent? What is the truth about death, judgement and retribution beyond the grave? What, finally, is that ultimate and unutterable mystery which engulfs our being, and whence we take our rise, and whither our journey leads us?' Article 1, page 661.  
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Pais, Abraham, 'Subtle is the Lord...': The Science and Life of Albert Einstein, Oxford UP 1982 Jacket: In this . . . major work Abraham Pais, himself an eminent physicist who worked alongside Einstein in the post-war years, traces the development of Einstein's entire ouvre. . . . Running through the book is a completely non-scientific biography . . . including many letters which appear in English for the first time, as well as other information not published before.' 
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Pais, Abraham, Inward Bound: Of Matter and Forces in the Physical World, Clarendon Press, Oxford University Press 1986 Preface: 'I will attempt to describe what has been discovered and understood about the constituents of matter, the laws to which they are subject and the forces that act on them [in the period 1895-1983]. . . . I will attempt to convey that these have been times of progress and stagnation, of order and chaos, of belief and incredulity, of the conventional and the bizarre; also of revolutionaries and conservatives, of science by individuals and by consortia, of little gadgets and big machines, and of modest funds and big moneys.' AP 
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Popper, Karl Raimund, The Open Society and its Enemies (volume 1) : The Spell of Plato, Routledge 1966 Introduction: 'This book ...attempts to show that [our civilisation] has not yet 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. ... It further tries to examine the application of the critical and rational methods of science to the problems of the open society.'  
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Popper, Karl Raimund, The Open Society and its Enemies (volume 1) : The Spell of Plato, Routledge 1966 , page 1, 
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Popper 1972, Karl Raimund, Conjectures and Refutations: The Growth of Scientific Knowledge, Routledge and Kegan Paul 1972 Preface: 'The way in which knowledge progresses, and expecially our scientific knowledge, is by unjustified (and unjustifiable) anticipations, by guesses, by tentative solutions to our problems, by conjectures. These conjectures are controlled by criticism; that is, by attempted refutations, which include severely critical tests.' [p viii]  
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Reid, Constance, Hilbert-Courant, Springer Verlag 1986 Jacket: '[Hilbert] is woven out of three distinct themes. It presents a sensitive portrait of a great human being. It describes accurately and intelligibly on a non-technical level the world of mathematical ideas in which Hilbert created his masterpieces. And it illuminates the background of German social history against which the drama of Hilbert's life was played. ... Beyond this, it is a poem in praise of mathematics.' Science 
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Schmemann, Alexander, "A response (to the decree on the Eastern Catholic Churches)" in Walter M Abbott and Joseph Gallagher (translation editor) The Documents of Vatican II: Dogmatic Constitution on the Church, pp 387-388, Geoffrey Chapman 1972 "There can be no doubt as to the positive, irenic and constructive intentions of the Decree as a whole. .. Certain important reservations, must, however, be made. First of all, the Decree seems to "take for granted" and to perpetuate the reduction of the differences between the East and the West to the sole area of rites, discipline and "way of life". But it is preceisly this reduction which forms the basis of "uniatism" that the Orthodox reject, for they affirm that the liturgical and canonical tradition of the East cannot be isolated from the sdoctrinal principles which it implies and which constitute the real issue between Roman Catholicim and Eastern Orthodoxy.' 
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Tanenbaum, Andrew S, Computer Networks, Prentice Hall International 1996 Preface: 'The key to designing a computer network was first enunciated by Julius Caesar: Divide and Conquer. The idea is to design a network as a sequence of layers, or abstract machines, each one based upon the previous one. . . . This book uses a model in which networks are divided into seven layers. The structure of the book follows the structure of the model to a considerable extent.'  
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Tillich, Paul, Systematic Theology, James Nisbet and Company Limited 1968 Preface: 'It has always been impossible for me to think theologically in any other than a systematic way. The smallest problem, if taken seriously and radically, drove me to all the other problems and to the anticipation of a whole in which they could find their solution. ... My purpose ... has been to present the method and structure of a theological system written from an apologetic point of view and carried through in a continuous correlation with philosophy."back
Tredennick, H, Aristotle: Metaphysics I-IX, Harvard University Press, William Heinemann 1980 Introduction: 'The theory of universal science, as sketched by Plato in the Republic, was unsatisfactory to Aristotle's analytical mind. He felt that there must be a regular system of sciences, each concerned with a different aspect of reality. At the same time it was quite reasonable to suppose that there is a supreme science, which is more ultimate, exact, more truly Wisdom than any of the others. The discussion of this science -- Wisdom, Primary Philosophy or Theology, as it is variously called -- and of its scope forms the subject of the Metaphysics .... It is from the consideration of change and motion that Aristotle proceeds to develop his theology. The continuity of the processes in the universe presupposes a moving cause by which they are eternally maintained. This cause, or Prime Mover, must itself be eternal and immutable, and must therefore be entirely immaterial. It is pure form and actuality; and this is mind or god.' pp xxv-xxix 
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Unitatis redintegratio, in Walter M Abbott and Joseph Gallagher (translation editor)The Documents of Vatican II: Decree on Ecumenism, Geoffrey Chapman 1972 'But in subsequent centuries more widespread disagreements appeared and quite large Communities became separated from full communion with the Catholic Church - developments for which, at times, men of both sides were to blame. However, one cannot impute the sin of separation to those who at present are born into these Communities and are instilled therein with Christ's faith.' para 3, p 145. 
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von Campenhausen, Hans, and (English translation revised by L A Garrard), The Fathers of the Greek Church, Adam and Charles Black 1963 Professor of Ecclesiastical History, Heidelberg Universityback
Watson, James D, and Lawrence Bragg (Preface), The Double Helix : A Personal Account of the Discovery of the Structure of DNA , Touchstone 2001 Jacket: 'By identifying the structure of DNA ... Francis Crick and James Watcon revolutionized biochemistry and won themselves a Nobel Prize. At the time, Watson was only twenty-four, a young scientist hungry to make his mark. His uncompromisingly honest account of the heady days of their thrilling sprint against other world-class researchers to solve one of science's greatest mystries give a dazzlingly clear picture of a world of brilliant scientists with great gifts, very human ambitions and bitter rivalries. With humility unspoiled by false modesty, Watson relates his and Crick's desperate efforts to beat Linus Pauling to the Holy Grail of life sciences, the identification of the basic building blocks of life. Never has a scientist been so truthful in capturing in words the flavor of his work.' 
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Weinberg, Steven, The First Three Minutes: a modern view of the origin of the universe, Basic Books 1993 Preface: 'The present book is concerned with the early unvierse, and in particular with the new understanding of the early universe that has grown out of the discovery of the cosmic microwave radiation background in 1965.'  
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Whitehead, Alfred North, and Bertrand Arthur Russel, Principia Mathematica to *56 , Cambridge University Press 1997 Amazon editorial review: 'Editorial Reviews Amazon.com Could it be true that Whitehead and Russell's Principia Mathematica is the most influential book written in the 20th century? Ask any mathematician or philosopher--or anyone who understands the impact these fields have had on modern thinking--and you'll get a short answer: yes. Their goal, to set mathematics on a firm logical foundation, was revolutionary, and their tools and rigor continue to influence modern professionals. Using Peano's symbolic logic, they formalized axioms and produced theorems (including the famous "1 + 1 = 2") in orderings, continuous functions, and other areas of mathematics. Although the Principia is far from comprehensive, Whitehead and Russell's method and program captivate their readers. The audacity to hope to formalize all of mathematics logically was inspirational and helped to give great boosts to math and logical philosophy. Though Gödel proved in 1931 that any such program is doomed to incompleteness, the tools found in and developed from the three volumes helped build the atomic bomb and the Internet. It may not be summer vacation reading (for most), but Principia Mathematica will reward the dedicated student with a deeper understanding of how we got here.' --Rob Lightner  
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Whitehead, Alfred North, and Bertrand Arthur Russel, Principia Mathematica to *56 , Cambridge University Press 1997 Amazon editorial review: 'Editorial Reviews Amazon.com Could it be true that Whitehead and Russell's Principia Mathematica is the most influential book written in the 20th century? Ask any mathematician or philosopher--or anyone who understands the impact these fields have had on modern thinking--and you'll get a short answer: yes. Their goal, to set mathematics on a firm logical foundation, was revolutionary, and their tools and rigor continue to influence modern professionals. Using Peano's symbolic logic, they formalized axioms and produced theorems (including the famous "1 + 1 = 2") in orderings, continuous functions, and other areas of mathematics. Although the Principia is far from comprehensive, Whitehead and Russell's method and program captivate their readers. The audacity to hope to formalize all of mathematics logically was inspirational and helped to give great boosts to math and logical philosophy. Though Gödel proved in 1931 that any such program is doomed to incompleteness, the tools found in and developed from the three volumes helped build the atomic bomb and the Internet. It may not be summer vacation reading (for most), but Principia Mathematica will reward the dedicated student with a deeper understanding of how we got here.' --Rob Lightner  
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Wicksteed, P H, and F M Cornford, (translators), Aristotle, Physics books I-IV, Harvard University Press, William Heinemann 1986 Amazon customer review: 'Like most volumes in the Loeb series, the emphasis is not on word-for-word precision in the translation, but on acheiving greater readability in broader terms. Since the original text in ancient Greek is provided on the facing page, the editors assume that anyone with a little knowledge of Greek can supplement the looseness of the translation by referring to the original. And in general, the compromises made in this way are good ones throughout the series. In this case, perhaps, the translation may be a little too loose, and also given over to some unfortunate jargon that can distort Aristotle's meaning. But even so, this is still a very useful text for the specialist or the student.' J. Duvoisin "politeia" (Santa Fe, NM United States) 
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Wiener, Norbert, Cybernetics or control and communication in the animal and the machine, MIT Press 1996 The classic founding text of cybernetics. 
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Wilcox, Ella Wheeler, "Settle the Question Right" in John Daintith et al. (editors) Bloomsbury Thematic Dictionary of Quotations, Bloomsbury Publishing Limited 1989 back
Papers
Burbridge, Geoffrey, Fred Hoyle and Jayant V Narlikar, "A different approach to cosmology", Physics Today, 52, 4, April 1999, page 38-44. "In this unorthodox assault on mainstream cosmology, three venerable stalwarts argue for a quasi-steady-state universe, with some quasars quite nearby and no Big Bang.'. back
Cole, S T, et al, "Deciphering the Biology of Mycobacterium tuberculosis from the complete genome sequence", Nature, 393, 6685, 11 June 1998, page 537-544. back
Deutsch, David, "Quantum theory, the Church-Turing principle and the universal quantum computer", Proceedings of the Royal Society of London, , A 400, 1985, page 97-117. 'It is argued that underlying the Church-Turing hypothesis there is an implicit physical assertion. Here this assertion is presented explicitly as a physical principle: 'every finitely realizible physical system can be perfectly simulated by a universal model computing machine operating by finite means'.'. back
Goedel, Kurt, "On formally undecidable problems of Principia Mathematica and related systems I", Monatshefte fur Mathematik und Physik, 38, , 1931, page 173-198. Reprinted in Goedel, Kurt, Kurt Goedel: Collected Works Volume 1 Publications 1929-1936, Oxford UP 1986 pp 144-195.   Amazon. back
Gottesman, Daniel, Isaac L. Chuang, "Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations", Nature, 402, 6760, 23 November 1999, page 390-393. back
Lissauer, Jack J, "How common are habitable planets", Nature, 402 , 6761 (supplement), 2 December 1999, page C11-C14. back
Turing, Alan, "On Computable Numbers, with an application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, 2, 42, 12 November 1937, page 230-265. 'The "computable" numbers maybe described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost as easy to define and investigate computable functions of an integrable variable or a real or computable variable, computable predicates and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique. I hope shortly to give an account of the rewlations of the computable numbers, functions and so forth to one another. This will include a development of the theory of functions of a real variable expressed in terms of computable numbers. According to my definition, a number is computable if its decimal can be written down by a machine'. back
Links
Aquinas 1, Summa: I 1 1: Is God a body? , 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
Aquinas 14 (Introduction), Summa: I 3: Via Negativa, 'I answer that, It is absolutely true that God is not a body; and this can be shown in three ways. First, because no body is in motion unless it be put in motion, as is evident from induction. Now it has been already proved (2, 3), that God is the First Mover, and is Himself unmoved. Therefore it is clear that God is not a body. .. .' back
Aquinas 14: Is God a body?, Summa: I 3 1: Is God a body? , 'I answer that, It is absolutely true that God is not a body; and this can be shown in three ways. First, because no body is in motion unless it be put in motion, as is evident from induction. Now it has been already proved (2, 3), that God is the First Mover, and is Himself unmoved. Therefore it is clear that God is not a body. .. .' back
Aquinas 81, Does God understand himself?, 'Since ...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..' back
Cantor's paradox - Wikipedia, Cantor's paradox - Wikipedia, the free encyclopedia, 'In set theory, Cantor's paradox is derivable from the theorem that there is no greatest cardinal number, so that the collection of "infinite sizes" is itself infinite. The difficulty is handled in axiomatic set theory by declaring that this collection is not a set but a proper class; in von Neumann–Bernays–Gödel set theory it follows from this and the axiom of limitation of size that this proper class must be in bijection with the class of all sets. Thus, not only are there infinitely many infinities, but this infinity is larger than any of the infinities it enumerates.' back
Einstein, Podolsky and Rosen, Can the Quantum Mechanical Description of Physical Reality be Considered Complete?, A PDF of the classic paper. 'In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false, One is thus led to conclude that the description of reality given by the wave function is not complete.' back
Gaia hypothesis - Wikipedia, Gaia hypothesis - Wikipedia, the free encyclopedia, 'The Gaia hypothesis, also known as Gaia theory or Gaia principle, proposes that organisms interact with their inorganic surroundings on Earth to form a self-regulating, complex system that contributes to maintaining the conditions for life on the planet.. . . The hypothesis was formulated by the scientist James Lovelock and co-developed by the microbiologist Lynn Margulis in the 1970s. While early versions of the hypothesis were criticized for being teleological and contradicting principles of natural selection, later refinements have resulted in ideas highlighted by the Gaia Hypothesis being used in subjects such as geophysiology, Earth system science, biogeochemistry, systems ecology, and climate science.' back
Galileo, Il Saggiatore (The Assayer), 'The pages of The Assayer contain Galileo's famous affirmation that Nature, though "deaf and inexorable to our vain desires", though producing its effects "in a manner unthinkable for us" has within it a harmonic structure and an order which is essentially geometrical: "Philosophy is written in this great book of the Universe which is continually open before our eyes but we cannot read it without having first learnt the language and the characters in which it is written. It is written in the language of mathematics and the characters are triangles, circles and other geometrical shapes without the means of which it is humanly impossible to decipher a single word; without which we are wandering in vain through a dark labyrinth."' back
Gregor Mendel - Wikipedia, Gregor Mendel - Wikipedia, the free encyclopedia, 'Gregor Johann Mendel (July 20, 1822 – January 6, 1884) was an Austrian scientist and Augustinian friar who gained posthumous fame as the founder of the new science of genetics. Mendel demonstrated that the inheritance of certain traits in pea plants follows particular patterns, now referred to as the laws of Mendelian inheritance. Although the significance of Mendel's work was not recognized until the turn of the 20th century, the independent rediscovery of these laws formed the foundation of the modern science of genetics.' back
John Burnet, Early Greek Philosophy: Heracleitus of Ephesus: Fragments, back
John Burnet, John Burnet's Early Greek Philosophy: chapter IV, Parmenides of Elea: 85: The Poem, back
John Burnet, Parmenides of Elea: The Poem, 'The Poem Parmenides was the first philosopher to expound his system in metrical language. His predecessors, Anaximander, Anaximenes, and Herakleitos, wrote in prose, and the only Greeks who ever wrote philosophy in verse at all were just these two, Parmenides and Empedokles; for Xenophanes was not a philosopher any more than Epicharmos. Empedokles copied Parmenides; and he, no doubt, was influenced by the Orphics. But the thing was an innovation, and one that did not maintain itself. The fragments of Parmenides are preserved for the most part by Simplicius, who fortunately inserted them in his commentary, because in his time the original work was already rare. I follow the arrangement of Diels.' back
John Paul II, Fides et Ratio: On the relationship between faith and reason. , para 2: 'The Church is no stranger to this journey of discovery, nor could she ever be. From the moment when, through the Paschal Mystery, she received the gift of the ultimate truth about human life, the Church has made her pilgrim way along the paths of the world to proclaim that Jesus Christ is “the way, and the truth, and the life” (Jn 14:6).' back
Nicholls, A theory of Peace, back
Ordinal number - Wikipedia, Ordinal number - Wikipedia, the free encyclopedia, 'Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are called well-ordered (so intimately linked, in fact, that some mathematicians make no distinction between the two concepts). A well-ordered set is a totally ordered set (given any two elements one defines a smaller and a larger one in a coherent way) in which there is no infinite decreasing sequence (however, there may be infinite increasing sequences); equivalently, every non-empty subset of the set has a least element. Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labelled 0, the one after that 1, the next one 2, "and so on") and to measure the "length" of the whole set by the least ordinal that is not a label for an element of the set. This "length" is called the order type of the set.' back
Sacred tradition - Wikipedia, Sacred tradition - Wikipedia, the free encyclopedia, 'Sacred tradition or holy tradition is a theological term used in some Christian traditions, primarily in the Catholic, Anglican, Eastern Orthodox and Oriental Orthodox traditions, to refer to the fundamental basis of church authority. back

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