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vol 6: Essays

On representation of the Word

Contents
1 Abstract
2 Introduction: Death
3 Revelation
4 Transmission and Reception
5 Linguistics I: The theory of information and computation
6 Linguistics II: Human Language
7 The Bible I: In itself
8 The Bible II: As an element of the Christian Religion
9 Doctrine and DNA
10 Evolution
11 Evolution of church in an ecumenical environment
12 Requisite variety
13 Mathematics as defined doctrine
14 Mathematical theology


For it is clear that whatever is received into something is received according to the condition of the recipient. Aquinas 366

 

The capacity for language is an essential part of the human mind; . . . and by the nature of this capacity languages can be changed and adapted as circumstances require, and only so can the central fact (and mystery!) of language can be explained: that speakers can make infinite use of the finite linguistic resources available to them at any time. Therefore no matter how much one analyses and describes a language, Something of its essential nature remains unsaid, . . . . Robins page 193.

1 Abstract

Modern critical approaches to linguistics and history cast doubt on the claim by Christianity to be founded on a specific divine intervention in human affairs. From this point of view, the doctrines that constitute Christianity are to a large extent arbitrary creations of the human spirit, rather than special communications from a divine creator existing outside the visible Universe. The doctrines which define Christianity are used to interpret the Bible, itself a creation of Christianity.

Is there any way to construct a religion which is not arbitrary? One possible approach is to see all events (rather than an arbitrary subset of events) as revelations of divinity, and to interpret these events using the assumption that history, insofar as it is knowable, is internally consistent. The requirement of consistency is mediated by mathematics, which is by definition the set of all consistent symbolic systems, although hypothetical. The method of interpreting events in terms of mathematical models is coterminous with science, and may be used to ground religion in observable events rather than human constructs. back

2 Introduction: Death

On the 20 January 2000 a group of local children went for a swim in the pond at the top of the Ellenborough Falls on the Ellenborough River, Elands, NSW. The falls are an almost straight drop of about 160 metres. One of them, a small boy with Downs syndrome, unable to swim, sat on the bank watching the others. At some point in the afternoon, the older children noticed that he was gone. A search followed, escalating over the next three hours to a major community, police and rescue operation. Toward evening his body was found lodged in rocks beneath the water at the foot of the falls.

This is an historical event typical of the events that motivate and are dealt with by religion. Reynolds and Tanner 211 sqq. Religion, language and mythology appear to be closely related elements of human existence. There is archaeological evidence for religious behaviour dating as far back as the Neanderthal period of human evolution. It has been conjectured that the recognition of mortality and the need to transcend it are a primary impulse toward mythology. Campbell, 5.

The belief that the dead live on in an invisible world is and was very common. In the case of Christianity, that other world has been given substantial and eternal reality. The ground for this position is held to be not just the common human belief in an afterlife, but special revelation from the God who created this world and ourselves. Christian belief is succinctly summarized in the Apostles Creed, which dates from the first century of the Christian era. Apostles Creed, Denzinger, 20 sqq.

The belief in eternal life is a great consolation to the bereaved and all of us who face death. Archer, 86. There is a strong incentive to hold this belief although all the evidence available to us by normal means seems to suggest that the dead are dead and live on only in human memories and other information storage systems. back

3 Revelation

It is basic to Christian belief that there is a personal God who has spoken to mankind. The receipt and continued possession of special revelation from this God is the essential property by which Christianity differentiates itself from other religions.

In its most abstract characterization, revelation is a body of information. Through our senses we collect information from our environment and find meaning in this data by the subjectively mysterious mental experience we call insight. Insight may be immediate and spontaneous, as when we immediately see the meaning of a familiar set of phenomena, or it may be the fruit of long contemplation. Individual insights have the property that they can combine seamlessly to form larger and larger insights, such as a cosmological vision of the world constructed from the shared and communicated experience of many generations of people. Lonergan. Insight, and revelation, are based on history, that is sequences of events or narratives.

The relationship between revelation and insight is not confined to the West. The word buddha comes from the Sanskrit root budh which carries the meanings to fathom a depth, to perceive, to know. Buddha became enlightened by seeing certain meaning in the world. We may not agree with his understanding of the world, but we might all agree that continued satisfactory life requires a good working understanding of one's environment.

From a physical point of view, the world may be seen as an ordered set of events. Quantum mechanics tells us that there are atomic, or smallest events, measured by the quantum of action. Larger events are constructed from smaller events. Sometimes the structural relationships between events are obvious. At other times they are more obscure. As physicists try to discern the structure behind physical events, historians seek structure in historical events, and all of us seek to know enough about the events of our lives to act effectively. In general, it seems that to act at random is to invite disaster.

Christianity is based on a subset of the history of the world which is given a special meaning within the Christian churches. For Christians, certain events are believed to reveal the intervention of God in human history. A recent definitive statement of the Catholic Church's understanding of this revelation has been provided by the second Vatican Council:

1 . . . Therefore, following in the footsteps of the Council of Trent and of the First Vatican Council, this present council wishes to set forth authentic doctrine on divine revelation and how it is handed on, so that by hearing the message of salvation the whole world may believe, by believing it may hope, and by hoping it may love. [cf. St. Augustine, "De Catechizandis Rudibus," C.IV 8: PL. 40, 316.] . . .

6. Through divine revelation, God chose to show forth and communicate Himself and the eternal decisions of His will regarding the salvation of men. That is to say, He chose to share with them those divine treasures which totally transcend the understanding of the human mind. [cf. First Vatican Council, Dogmatic Constitution on the Catholic Faith, Chap. 2, "On Revelation:" Denzinger 1786 (3005)]

As a sacred synod has affirmed, God, the beginning and end of all things, can be known with certainty from created reality by the light of human reason (see Rom. 1:20); but teaches that it is through His revelation that those religious truths which are by their nature accessible to human reason can be known by all men with ease, with solid certitude and with no trace of error, even in this present state of the human race. [Ibid: Denzinger 1785 and 1786 (3004 and 3005).] Second Vatican Council, Dogmatic Constitution on Divine Revelation (Dei Verbum ) paras 1, 6. . First Vatican Council

Revelation begins in the mind of the prophet and makes its appearance through words, gestures, writings and other publicly visible output of the inspired person. Since both Jesus and the prophets are no longer with us, revelation is now, for practical purposes, encoded in various canonical and patristic texts known collectively as the deposit of faith. Dei Verbum, para 10. back

4 Transmission and reception

God, as conceived by Thomas Aquinas, is "absolutely simple". Aquinas 14. There is in him none of the various categories of differentiation observed in the world. Neither the human mind nor mathematics can grasp the idea of meaning mediated by something with no discernible structure. Since we cannot study the simplicity of God, we must turn to the complexity of the Universe.

To be heard by people, the word of God must be encoded in the physical Universe as we are. For Christianity, God is revealed in some way in all physical events, but most particularly in the event named Jesus. Responsibility for controlling and perpetuating the memory of Jesus lies with the Church:

Sacred tradition and sacred Scripture form one sacred deposit of the word of God which is committed to the Church. . . .

The task of authentically interpreting the world of God, whether written or handed on, has been entrusted exclusively to the living teaching office of the Church, whose authority is exercised in the name of Jesus Christ. Second Vatican Council, Dei Verbum , para 10.

As Aquinas noted, what is received is nuanced by the nature of the receiver. The meaning attributed to any text, then, depends both on the form of the text, determined in some degree by its source, and on the interaction of the text with the receiver, determined by the structure of the receiver. Modern communication theory and linguistics give us the tools to make practical use of Aquinas' insight.

The general process of determining the meaning of a certain text is called hermeneutics. The first hermeneutic step is to recover the original text from the potentially erroneous copies that presently exist. The second is to attribute meaning to the recovered text. back

5 Linguistics I: The theory of information and computation

Communication involves transferring information (that is a text or ordered set of symbols) from a source through a channel to a receiver. The first problem for the receiver is to determine whether the text received is true, that is whether it represents exactly what was transmitted. All real communication channels encode information in some physical process and so are affected by noise, because all physical processes are connected. Lightning may strike a phone line as easily as a tree. In the case of written texts, such noise may come about through copyists errors (deliberate or accidental), or by deterioration of the medium (eg papyrus, clay, stone) used to encode the text.

Shannon showed that it is possible, by using codes which exploit redundancy in the transmitted text, to transmit information without error over a noisy channel at a certain maximum rate. Shannon. Although the details require mathematical expression, the fundamental strategy behind Shannon's theorems is to encode simple symbols which are close together in a low dimensional space (and therefore easily confused) into complex symbols which are far apart in a high dimensional space, and therefore much less likely to be confused. In general, noise is overcome by 'digitization' that is by packaging information in discrete and easily differentiated units.

The actual encoding and decoding of messages is a task for a computer, silicon or biological. The error correcting power of modern computer networks is such that billions of symbols (representing perhaps a large computer program) may be transmitted over noisy telephone lines with complete fidelity. The relevant mathematical theorems are transcendental in the sense that they govern all symbolic communication without exception. We may see these theorems behind the 'particulate' nature of the Universe since it appears that the stability of all processes in the Universe depend upon error free communication

We can see the application of redundancy to combat error in the structure of natural languages. Certain words ("arphaught") and sentences ("John flies the brick") do not make sense, that is they occupy positions in the space of words or sentences that fall between legitimate words and sentences as determined by the coding scheme (language) shared by transmitter and receiver. Scholars have used redundant data in the form of multiple manuscripts, parallel texts and historical knowledge of the ancient world and its languages to recover a best estimate of the original Biblical texts. back

6 Linguistics II: Human language

Human languages are the special encodings that have evolved to transmit information between individual Homo sapiens. Nowak. We may surmise that there have been of the order of a million human languages in the course of the evolution of the genus Homo, and that they bear a genetic relationships to one another similar to the genetic relationships of the races and tribes of humans that have populated the earth.

A prehistoric triumph of human linguistics was the development of writing, that is the ability to represent the sounds of human speech and the information encoded in speech in enduring physical form. History as we know it depends on the surviving written records arising from this technology.

Consistent with the principle that all information is encoded in physical form, linguistics tends to fall into two sections. One, concerned with the sounds used in speech, and their representation in writing, known as phonetics, and the other concerned with the encoding of meaning in speech and writing, called semantics.

An enduring question in linguistics has been the relationship between phonetics and semantics. The overall conclusion after two and a half millennia of research seems to be that the connection is in general purely arbitrary, but that particular languages show an evolutionary relationship between sound, writing and meaning. The arbitrary nature of the connection allows meanings to evolve independently of sounds. The independent evolution of sound (and written form) and meaning may be seen by examining a dictionary such as the OED which provides a history of the meaning, spelling and pronunciation of each word. Oxford University Press

The independence of physical structure and meaning in language leads to us to mathematical linguistics. Peters. In its most general form, a language comprises a countable set of words (elements of meaning) which may combine in an uncountable number of ways to communicate more complex meanings. This broad understanding of language leads us to the " 'fundamental problem of linguistic theory', namely, that of accounting for the possibility of language acquisition. On the basis of exposure to a relatively small sample of language, any normal child is able to obtain mastery over the infinite set of sentences which constitute that language." Wasow.

Chomsky concluded from this ability of children that we are all equipped with a language learning faculty, and that there must be a certain amount of structure and consistency in language that relates to this faculty. We do not yet know the structure of this faculty, but we can surmise that whatever it is, we can mathematically model it with a Turing machine or a network of Turing machines. Bach.

The independence of symbol and meaning makes possible the existence of a vast array of languages. Each language is a shared code allowing a certain population to communicate. Each of us has the ability to communicate in many different languages, realized if we grow up in a multilingual environment. Secret languages and deceptive languages fall into this category, since the independence of symbol and meaning goes both ways. One may use different symbols to communicate the same meaning, or use the same symbols to communicate different meanings. back

7 The Bible I: In itself

Detailed linguistic study of the Bible began to make a significant contribution to Christian theology in the nineteenth century as part of general trend toward scientific treatment of ancient texts. Sir William Jones' discovery of the kinship of Sanskrit with Latin, Greek and Germanic languages (1786), was a formative event in the development of modern linguistics. Robins, 149. Sanskrit was not just another language. It came with a history of linguistic scholarship complementary to western work.

Robins writes:

So far as we can tell, the original inspiration for linguistics in India was the need that was felt to preserve certain ritual and religious orally transmitted texts coming from the Vedic period (c. 1200-1000 B. C.), the oldest known stage of Sanskrit literature, from the effects of time and what was feared as dialectal contamination. Robins, 151.

A similar oral tradition is known to have preceded many books of the Bible. The Bible as we know it was committed to writing in the period stretching from about 500 bce to 100 ce. The Reformation led to renewed interest in the Bible, since it, rather than the established church, was seen by the reformers as the foundation of their religion. This interest ultimately led to the careful scientific study of the Biblical texts.

This open scholarship had two effects. First, it laid the foundation for Christian ecumenism, since Christian scholars using the same critical methods moved toward a consensus on the content of Bible; and second, it raised critical questions for Christianity, since there seemed to be no objective criteria upon which to judge the content of the books of the Bible to be closer to the truth of the human condition that many other ancient texts, including the sacred literature in Sanskrit. This pressure led in turn to a flowering of theologies as Christianity began to explore the space of possibilities consistent with the Biblical text.

As Ford sees the position,

The themes of suspicion, doubt and radical critique are constantly present in modern thought, raising most sharply the issues of authority and reliability. For many the very discipline of theology has disintegrated and lost its intellectual integrity in the face of all this. So most theologians discussed in these volumes [sic] are engaged in a recovery of Christianity in the face of unprecedentedly devastating, sophisticated and widely disseminated dismissals of both Christianity and theology. Ford, 6. back

8 The Bible II: As an element of the Christian Religion

Considered in the mathematical space of all possible texts, the Bible is a particular text from a particular era. Because we can only partially reconstruct the human mental space of that era, we can only partially reconstruct the meaning intended in that era. Because it is so underdetermined, the meaning of the Bible is, from a general linguistic point of view, a variable.

The Bible also serves as the fundamental text of the Christian Churches. From an abstract point of view a Church is a corporation like other corporations, a legal person, able to own property, to enter into contracts, and to be subject to the processes of justice.

From an even more abstract point of view, a corporation is a particle like any other in the Universe. A particle, from the physical point of view, is a centre of coordinated activity, a definition that applies as well to a Church as to an electron. The nature of each particle has a static element, which we always associate with the particle, and a dynamic element which varies through time.

The invariant element of a particle can be represented in text, itself an invariant means of representation. The invariant element that defines a particular legal corporation is its name and its constitution (articles of association, mission statement, ethical statement, quality statement, environmental statement, etc etc). The actual constitution of a particle may be revealed to us (as members for instance) or may need to be surmised (as in particle physics) from the behaviour of the particle.

The Christian Churches are founded on the Bible, but there is much more to them than that. One may gain a general perspective on the defined doctrines of the Catholic Church from Denzinger. For much of its life, particularly after it became established as a political power in the fourth century ce, the Church has used its doctrinal definitions as a criterion for the detection of heresy.

The critical revolution mentioned by Ford (above) has found that very little Christian belief is unambiguously attested by the Bible. The recent scholarship of Thiering, which suggests that much of the New Testament has a secret meaning not always in harmony with its more obvious meaning, makes it even more difficult to correlate the Christian Churches as we find them with their Biblical roots. Thiering 1993, Thiering 1995.

In the face of the critical attack on its position, the Roman Catholic Church, at the First Vatican Council, made a claim to infallibility. This fundamental doctrinal definition serves as an archetype for all such definitions:

Therefore faithfully adhering to the tradition received from the beginning of the Christian faith, for the glory of God Our Saviour, the exaltation of the Catholic Religion, and the salvation of Christian people, the Sacred Council approving, We teach and define that it is a dogma divinely revealed: that the Roman Pontiff, when he speaks ex cathedra, that is, when in discharge of the office of Pastor and Doctor of all Christians, by virtue of his supreme Apostolic authority he defines a doctrine regarding faith or morals to be held by the Universal Church, by the divine assistance promised to him in blessed Peter, is possessed of that infallibility with which the divine Redeemer willed that His Church should be endowed for defining doctrine regarding faith or morals: and that therefore such definitions of the Roman Pontiff are irreformable of themselves, and not from the consent of the Church. [The First Vatican Council, Session 4, Chapter 4, Dogmatic Constitution on the Catholic Faith] Manning p 239.

Although this definition may be reassuring from the point of view of the internal structure of the Church, it carries no weight in the scientific world, since it can only be considered true if the Church is in fact infallible, something which can at best be known the end of time if it becomes clear that the Church has not in fact been responsible for any error. What then, are we to make of this definition, and of all the other constitutional claims of the Church, including the fundamental claim that it is custodian of special revelation from God? back

9 Doctrine and DNA

The way forward here seems to have been pioneered by Descartes and his famous assertion cogito ergo sum. Gaukroger. It seems clear that human self awareness is something which has evolved like all other features of our nature. There was a time when people were not particularly conscious of their individual personality and destiny, but rather functioned as components of the wider world and accepted fate as it came. The Renaissance, the Reformation, and the Enlightenment are all characterized by a heightened awareness of human individuality. This has led to the spread of political demands for democracy, and to the concepts of human rights, personal spiritual growth and personal control over one's destiny. The Church as a corporate person, participates in this search for identity by attempting to clearly define its nature and role in the world.

Every organism has fixed and variable elements. In the case of a living creature, the fixed formal element is encoded in its genome, written in the chemical molecule DNA. Apart from minor errors, the DNA of an organism remains unchanged throughout its lifetime and determines the constitution of that organism. An organism may unconditionally assert the truth of its own genome, as Churches may assert the truth of their chosen constitutions. The Catholic doctrine of infallibility appears to do just this, establishing in particular the role of the Papacy in the constitution of the Church. back

10 Evolution

The work of defining the modern Church continued at the Second Vatican Council. The documents of the Council, while still reflecting some of the Church's ancient habits of absolute monarchy, reveal a new ecumenical personality. While holding fast to its mission to all people, the Church conceded that it is one organization among many with the same general brief, and so takes its place in the world as a religion, rather than the religion.

Ecumenism falls naturally into two movements: first, healing the rifts that have appeared within the Christian churches over the last two millennia; and second, uniting all the religions of mankind in a common global religious framework. Both movements require change in churches, religions and people. Such changes hold the same risk of death or injury for Churches as for any other organisms.

On the one hand, too much change may lead the Church to voluntarily abdicate its essential nature, so that it becomes in effect a different organism. On the other, too little change may deprive the Church of the resources that it needs for survival, so that it dies. A middle way must be found.

Since this problem has been faced by living creatures many times before, we can find the answer in the history of life, which is encapsulated in the biological theory of evolution. Evolution has two facets: the first is descent with modification; the second is selection.

In the case of Christianity, descent with modification is illustrated by the portfolio of theologies assembled by Ford. Ford op. cit. To ensure the survival of Christianity, it remains only to select the theology or theologies from this group best fitted to survive. The question is, which? In the case of living creatures, the critical judgment as to who will reproduce (and so pass on their genome) and who will not is made by interaction with the environment. We might surmise from this that the theology selected to survive and reproduce itself is the theology which best fits the modern world. Is such a theology to be found within the Christian range of variation? back

11 Evolution of church in an ecumenical environment

The ecumenical dialogue between the Christian Churches has progressed most effectively in the relationship between the Roman Catholic Church and the Church of England. This is perhaps because the differences between them had more to do with Henry VIII's political needs than doctrine, in contrast to the Continental Reformation, where difficult doctrinal questions were more important. At the other extreme is the relationship between the Church and people whose critical view of the world rejects much of traditional Christianity and traditional religion in general. In the context of human equality, such people have no lesser claim to spirituality, humanity or virtue than those who profess to be members of Christian churches.

Religion exists in the natural world and we must of necessity interpret the content of any religious belief in terms expressible in this world. The structure of the world therefore constrains the interpretation of revelation in general in a manner reminiscent in the way that the structure of the Church constrains the interpretation of the deposit of faith.

It is clear that we can model the world as a structure of particles within particles exactly analogous to set theory. At one extreme we may imagine a particle so small that it has nothing inside it, modelled by the empty set. On the other we may imagine a particle so large that it has nothing outside it. Set theory itself tells us that there is no satisfactory way to represent this particle since the set of all sets is a concept that leads to paradox. Mendelson, page 2. There is no largest set.

For practical purposes, we imagine a set large enough to contain all the objects of interest and do not concern ourselves with objects outside this set. In the physical Universe, the largest possible set of objects that we can deal with is constrained by an event horizon, that is the limit of causality and observation. We need not be concerned with things we can neither know nor influence. The Christian notion of God seems consistent with the physical notion of 'all things outside my event horizon', whatever they may be.

Between these extremes we find all the other particles in the Universe grouped into various sets. One particle may be in many sets, and a set may itself be considered a particle (or element) in a larger set. Set theory gives us a neat formal way to model the Universe which would be the envy of the Pythagoreans and Plato. All the specific mathematical structures that are used to model particular features of the world can be described in terms of set theory.

When we come to map set theory onto the world of action, the natural feature best corresponding to the empty set is the quantum of action. Every event, such as my life, or the life of a church may be modelled as an ordered set of actions, so building up the world as the infinite structure of mathematical theory may be built from ordered sets of empty sets. Jech, page 10. The only practical constraint on the creation of mathematical structures is that they be consistent. It seems that consistency, too, is the only constraint on the physical world.

[It is important, when considering the application of this model to the world, to be aware of the vast size and complexity of human-scale natural processes when measured in terms of the quantum of action. The measure of the quantum of action, Planck's constant, h, is approximately 6 x 10-34 Joule.sec. A typical human mass is 80 kg, lifetime say 100 years, so the number of quanta of action constituting such a life = mc2.t = 80 x 300 000 0002 x 100 x 31 500 000 /h= 4 x 1060 approximately.]

The only constraint on truly ecumenical theology, then, is that it be consistent with the world we know. back

12 Requisite variety

The purpose of defined doctrine appears to be to control the interpretation of revelation. In the Christian case, revelation is contained in a certain 'deposit of faith', representing a certain sacred subset of human history. In the more general case, all history may be considered sacred, and every action in the world may be considered as revealing God.

Armed with the model of the world proposed above, what can we say about the control of interpretation? For an answer, we turn to cybernetics, the science (or art) of control and communication in animals and machines. Wiener. From a cybernetic point of view, the limits of control are determined by the principle of requisite variety: If one system is to control another (that is determine its outcome exactly), the controlling system must have equal or greater variety than the controlled system. Ashby, pp 202-216.

The principle of requisite variety may be proven by relating information theory to Gödel's theorem as Chaitin has done, to get the result that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be proved from the axioms. Chaitin, pp 55-69. To prove (that is control or constrain) a theorem, the axioms must contain at least as much information as the theorem.

A couple of applications illustrate this principle. First, the relationship between human mind and body. Each of our bodies is defined by our genome, approximately 100 000 genes each carrying about a kilobit of information, for a total of (say) 100 megabits. Our central nervous system comprises some 100 billion neurons, each with about a thousand synapses, each capable of storing at least one bit of information, for a total of (say) 100 terabits. From this we can conclude that our minds cannot be controlled by our genes, but have instead an enormous amount of unconstrained entropy open, like Aristotle's tabula rasa, to determination by interaction with our environment.

Second the relationship between a teacher and students. Given that we all have the same variety, many people have more variety than one. One person therefore cannot control many unless the variety of the many is in some way reduced. The traditional method of variety reduction in this case is to reduce the variety of the children by making them all sit still and listen to the words of the teacher rather than pursuing their own thoughts.

Thirdly, the relationship between government and governed has a similar structure. In traditional societies the monarch can control the people because the variety of the people is reduced by culture and convention. When people begin to think for themselves and go their own ways, monarchy must break down, giving way to democracy, government of the people by the people, a situation in which the principle of requisite variety may be honoured. back

13 Mathematics as defined doctrine

What is the relationship between defined doctrine and revelation in a Universe constrained only by consistency and requisite variety?

Mathematics has played a key role in the interpretation of the world since time immemorial. Already in Euclid's Elements (c 300 bce) we have a compilation of ancient mathematical results that anticipate some of the deepest and most interesting problems and techniques of modern times. Heath. For more than 2000 years the Elements has been a text immense practical importance. Its influence on human affairs is comparable to the Bible.

The bifurcation of theology and science is marked by Galileo's mathematical manifesto:

Philosophy is written in that great book which lies forever before our eyes - I mean the Universe - but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Yourgrau and Mandelstam page 8.

Modern developments, particularly the theory of computation, have moved mathematics from a consideration of number and geometric form squarely into the area of linguistics and communication. Although the artificial intelligence community started out optimistically fifty years ago with the hope that computers would soon be speaking and performing intelligent tasks, the sheer complexity of the world has slowed them. It seems clear in principle, however, that there is no feature of human intelligence and linguistic ability that cannot be modelled by a machine as fast and complex as the human central nervous system.

It has also become clear that mathematics of itself tells us very little about the real world. Mathematics may be considered as a set of formal proofs, each beginning from a set of hypotheses and leading to a set of conclusions by a process (proof) that is believed to preserve formal consistency. It remains for the scientist to decide whether the hypotheses are realized in nature and whether the mathematical proof is reflected in natural process.

From a general point of view, mathematics is an explicit realization of the demand for consistency in the world. It is an article of scientific faith, much tried by quantum mechanics, that the world is consistent. Quantum mechanics showed us that the demand for consistency is stronger even than the demand for full knowledge, since the consistency of quantum mechanics is closely related to the uncertainty principle.

A fundamental feature of quantum mechanics is the concept of superposition illustrated by "Schrödinger 's Cat". Superposition means that one system may be in many states at one time (as the cat may be both dead and alive at the same time). Dirac, pp 1 sqq. This appears to contradict the ancient principle of non-contradiction (a thing cannot be both p and not-p), but the situation is saved by the fact that we cannot observe a superposition of states directly, but only through various events resulting from the superposition. The principle of superposition is in effect the physical realization of the ancient idea that words are inadequate to express reality. In the quantum theory, an infinity of words (state descriptions) is necessary to define (for instance) the internal state of an atom.

In science, therefore, mathematics plays the role of a defined doctrine, making explicit the requirement of consistency in our attempts to interpret the world. back

Mathematical theology

Christian theology as we know it is not a science. And yet the modern world is inclined to expect scientific levels of certainty from the agents which fulfill human needs. Thus we expect drug companies to spend millions on testing their product to prove that it is both efficacious and entails an acceptably low level of risk to the consumer. Similar scientifically based standards are being extend into all classes of consumer goods. What about religion? If theology is not a science, can religions (whose knowledge base is tested by theology) offer reliable truths (and meaningful consolations) to those who contribute their resources to a religion in the hope of some real benefit?

Avis concluding his survey of the methods of modern theology asks "Is theology a science?" and replies

There are . . . two respects in which theology resists assimilation to natural science: the problematic nature of its data and the particular reference of its statements. . . .

The centrality of hermeneutical questions in theology and the singular and unrepeatable character of the phenomena with which it deals both suggest a stronger affinity of theology with the human sciences than with the natural sciences. In the human sciences, as the German term Geisteswissenschaften suggests, our objects are the spiritual ones of human life and thought, culture and development, and require accordingly 'spiritual' methods involving imagination and insight, empathy and indwelling. Avis, pp 210, 215.

In other words, yes and no. We might shine a little more light on this issue by using the principle of requisite variety to reinterpret Naturwissenschaften and Geisteswissenschaften not as a simple opposition but as well separated points on a spectrum whose parameter is certainty or probability of truth.

To measure something is to compare it to some fixed standard, which might be a simple thing like a tape measure, or a complex (but well defined) model. All scientific assertions come with a tolerance: "from our measurements we can assert with 90% confidence that x is greater that a and less than b." This tolerance is calculated by mathematically modelling measurements and relationships between measurements. The general message of the theory of probability is the same correlation between redundancy and precision as we find in information theory (an application of the theory of probability): repeated measurement of essentially similar systems gives us a better estimate of the truth than a single measurement.

The Christian story of salvation is a model with which we measure certain events in history. Since these events are believed to have occurred just once, our tolerance limits must be wide. It is very difficult to decide whether Jesus was in fact a special event, or whether his life is typical of the millions of people who have been so convinced of their point of view that they are prepared to die rather than change. Of these millions Jesus may have been blessed by being simply the right person at the right time to initiate a revolution in human relationships. From a formal point of view, we might see Jesus as the nucleation point of a phase change.

From a scientific point of view, Christianity, by restricting its constitutional data to a tiny remote event (the life of one person among billions at one period among thousands of years of human history) condemns itself to uncertainty. Consequently, insofar as Christianity defines itself by a definite suite of doctrines, its choice of doctrine is arbitrary. Here it finds itself identical to any other corporation, whose promoters choose an arbitrary product or an arbitrary technology as the foundation of their endeavour.

The Christian churches are of course free to choose any constitution whatever for themselves. An ethical question arises, however, when they take resources from people on the basis of certain promises. Like a vendor marketing an inert substance as an effective drug, Christianity is open to question by those who expect to be told the truth. The consolation of life after death, in this context, is an arbitrary consolation rooted not in the nature of reality, but in the nature of Christianity.

How can we change this situation? The only route to certainty is more data, and the only source of data is the present. The life and work of Jesus is effectively invisible, and can not be the foundation for certainty. It is an arbitrary Christian belief that God revealed all he was going to reveal in the person of Jesus. One might equally suppose that God's revelation is continuous, and that we can learn the reality of true religion by looking at the world around us.

Christianity and science share the faith that the world is consistent. Mathematics shows us, in a hypothetical manner, the structure of consistency. Mathematics thus reveals to us the features of the world that cannot be otherwise. In a scientifically developed theology, then, mathematics is the defined doctrine that shows us the way to correctly interpret the continual flow of history that we witness in our lives, a history that reveals the divinity to us in its full complexity. Theology can become a science if it opens itself to sufficient data to constrain its hypotheses. back

[revised 20 December 2015]

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Further reading

Books

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Archer, John , The Nature of Grief : The Evolution and Psychology of Reactions to Loss, Routledge 1999 Jacket: 'The Nature of Grief is an innovative and provocative new synthesis of material from evolutionary psychology, ethology and experimental psychology on the process of grief. It argues that grief is not an illness or a disorder but a natural reaction to losses of many kinds.' 
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Ashby, W Ross, An Introduction to Cybernetics, Methuen 1964 'This book is intended to provide [an introduction to cybernetics]. It starts from common-place and well understood concepts, and proceeds step by step to show how these concepts can be made exact, and how they can be developed until they lead into such subjects as feedback, stability, regulation, ultrastability, information, coding, noise and other cybernetic topics' 
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Avis, Paul D L, The Methods of Modern Theology : the Dream of Reason , Marshall Pickering 1986 'The purpose of this book is to give an in depth critical introduction to the methods of modern theology.' [xi] Discusses Barth, Lonergan, Pannenberg, Rahner, Ritschl, Schleiermacher, Tennant and Tillich . 
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Avis, Paul D L, The Methods of Modern Theology : the Dream of Reason , Marshall Pickering 1986 'The purpose of this book is to give an in depth critical introduction to the methods of modern theology.' [xi] Discusses Barth, Lonergan, Pannenberg, Rahner, Ritschl, Schleiermacher, Tennant and Tillich . 
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Bach, Emmon , and William Marsh, "An elementary proof of the Peters-Ritchie Theorem" in Walter J Savitch et al, The Formal Complexity of Natural Language, page 41-55, D Reidl 1987 'Peters and Ritchie proved that transformational grammars of the 'standard' sort with a context sensitive base were equivalent to unrestricted rewriting systems (equivalently, Turing machines) in their weak generative capacity, that is there was such a grammar for every recursively enumerable language. ...' 
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Campbell, Joseph, Myths to Live By: how we re-create ancient legends in our daily lives to release human potential, Penguin/Arkana 1993 Amazon customer review: 'I read this book while on a cruise, and found myself spending a lot of time reading. Of all his works, this is the most down to earth. The others are too 'professorial' as if intended to impress, while this one simply lays it on the line. Psychology and mythology relate to each other very nicely, as Mr. Campbell realized when asked to share his concepts with those of a Psychologist. Jung was a favorite because of his concept of Universal Mind. Contrary to what might be thought, the book is not anti-religious, but it does explode particular Christian beliefs. Rather, it reveals the Universal meaning of 'life' which each community resolves in its own way, frequently as not, in similar ways. Boil away the variety of customs, etc, and you have the essence of Joseph Campbell's work and a better appreciation of man's universal mind.'Kenneth G. Ramey 
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Chaitin, Gregory J, "Goedel's Theorem and Information" in Information, Randomness & Incompleteness: Papers on Algorithmic Information Theory, Reprinted from the International Journal of Theoretical Physics (1982) 22, 941-954., World Scientific 1987 Abstract: 'Goedel's theorem may be demonstrated using arguments having an information-theoretic flavour. In such an approach, it is possible to argue that if a theorem contains more information than any given set of axoms, then it is impossible for the theorem to have been derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual.' 
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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
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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First Vatican Council, , We may compare this text with the parallel passage from the Firt Vatican Council: 'The same Holy Mother Church holds and teaches that God, the beginning and end of all things, may be certainly known by the natural light of human reason, by means of created things; "for the invisible things of Him from the creation of the world are clearly seen, being understood by the things that are made" (Romans i. 20), but that it pleased His wisdom and bounty to reveal Himself, and the eternal decrees of His will, to mankind by another and a supernatural way : as the Apostle says, "God, having spoken on divers occasions, and in many ways, in times past, to the fathers by the prophets; last of all in these days, hath spoken to us by His Son" (Hebrews i. 1, 2) It is to be ascribed to this divine revelation, that such truths among things divine as of themselves are not beyond human reason, can, even in the present condition of mankind, be known by every one with facility, with firm assurance and with no admixture of error. This, however, is not the reason why revelation is to be called absolutely necessary; but because God is His infinite goodness has ordained man to a supernatural end, viz to be a sharer of divine blessings which utterly exceed the intelligence of the human mind; for "eye hath not seen, nor ear heard, neither hath it entered the heart of man, what things God hath prepared got them that love him" (1 Cor. ii. 9). [Henry Edward Manning, The Vatican Council and its Definitions: A Pastoral Letter to the Clergy (New York: Excelsior Catholic Publishing House, 1905), 210] In Latin: 'Eadem sancta mater Ecclesia tenet et docet, Deum, rerum ommnium principium et finem, naturali humanae rationis e rebus creatis certo cognosci posse; "invisibilium enim ipsius, a creatura mundi, per ea quse facta sunt, intellecta conspiciuntur" [Rom 1, 20 ]: attamen placuisse eius sapientiae et bonitati, alia eaquae supernaturali via se ipsum ac aeterna voluntatis suae decreta humano generi revelare, dicente Apostolo: "Multifariam multisque modis olim Deus loquens patribus in Prophetis: novissime diebus istis locustus est nobis in Filio" [Hebr 1, 1 s; canon 1 ] 'Huic divinae revelationi tribuendem quidem est, ut ea quae in rebus divinis humanae rationi per se impervia non sunt, in praesenti quoque generis humani condicione ab omnibus expedite, firma certitudine et nullo admixto errore cognosci possint. [S Thomas, Summa theol 1, q. 1, a. 1.] Non haec tamen de causa revelatio absolute necessaria dicenda est, sed quia Deus ex infinita bonitate sua ordinavit hominem ad finem supernaturalem, ad participanda scilicet bona divina, quae humanae mentis intelligentiam omnino superant; siquidem "oculus non vidit, nec auris audivit, nec in cor hominis ascendit, quae preparavit Deus iis, qui dilligunt illum' [1 Cor 2, 9; canon 2 et 3 ] [ H Denzinger, Enchiridion, 3004, 3005]back
Ford, David, The Modern Theologians : An Introduction to Christian Theology in the Twentieth Century, Blackwell 1997 Preface: 'The main aim of this volume is to introduce the theology of most leading twentieth-century Christian theologians and movements in theology. . . . The contributors are mostly based in Europe of North America and come from a wide range of institutions, denominational backgrounds, and countries. Most are themselves constructively engaged in modern theology, and their purpose has been to produce a scholarly account of their subject and also carry further the theological dialogue in each case.'  
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Gaukroger, Stephen, Descartes: An Intellectual Biography, Clarendon Press 1995 Jacket: 'Rene Descartes (1596-1650) is the father of modern philsoophy and one of the greatest of all thinkers. This is the first intellectual biography of Descartes in English; it offers a fundamental reassessment of all aspects of his life and work. ... Descares' early work in mathematics and science produced ground-breaking theories, methods and tools still in use today. This book gives the first full acount of how this work infomred and influenced the later phisosophical studies for which, above all, Descartes is renowned.... [It] offers for the first time a full understanding of how Descartes developed his revolutionary ideas. It will be a landmark publication, welcomed by all readers interested in the origins of modern thought.' 
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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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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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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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Manning, Henry Edward, The Vatican Council and its Definitions: A Pastoral Letter to the Clergy, Excelsior Catholic Publishing House 1905 Latin version of this quotation: 'Itaque Nos traditioni a fidei Christianae exordio perceptae fideliter inhaerendo, ad Dei Salvatoris nostri gloriam, religionis catholicae exaltationem et christianorum populorum salutem, sacro approbanto Concilio, docemus et divinitus revelatum dogma esse definimus; Romanem Pontificem, cum ex cathedra loquitur, id est, cum omnium Christianorum pastoris et doctoris munere fungens, pro suprema sua Apostolica auctoritate doctrinam de fide vel moribus ab universa Ecclesia tenendam definit, per assistentiam divinam, ipsi in Beato Petro promissam, ea infallibilitate pollere, qua divinus Redemptor Ecclesiam suam in definienda doctrina de fide vel moribus instructam esse voluit; ideoque euismodi Romani Pontificis definitiones ex sese, non autem ex consensu Ecclesiae, irreformabiles esse.' Henricus Denzinger and Adolphus Schoenmetzer, Enchiridion Symbolorum, Definitionum et Declarationum de Rebus Fidei et Morum (Freiburg im Breisgau: Herder, 1963), §3073, p 601. back
Manning, Henry Edward, The Vatican Council and its Definitions: A Pastoral Letter to the Clergy, Excelsior Catholic Publishing House 1905 Latin version of this quotation: 'Itaque Nos traditioni a fidei Christianae exordio perceptae fideliter inhaerendo, ad Dei Salvatoris nostri gloriam, religionis catholicae exaltationem et christianorum populorum salutem, sacro approbanto Concilio, docemus et divinitus revelatum dogma esse definimus; Romanem Pontificem, cum ex cathedra loquitur, id est, cum omnium Christianorum pastoris et doctoris munere fungens, pro suprema sua Apostolica auctoritate doctrinam de fide vel moribus ab universa Ecclesia tenendam definit, per assistentiam divinam, ipsi in Beato Petro promissam, ea infallibilitate pollere, qua divinus Redemptor Ecclesiam suam in definienda doctrina de fide vel moribus instructam esse voluit; ideoque euismodi Romani Pontificis definitiones ex sese, non autem ex consensu Ecclesiae, irreformabiles esse.' Henricus Denzinger and Adolphus Schoenmetzer, Enchiridion Symbolorum, Definitionum et Declarationum de Rebus Fidei et Morum (Freiburg im Breisgau: Herder, 1963), §3073, p 601. back
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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Peters, Stanley, "What is mathematical linguistics" in Walter J Savitch et al, The Formal Complexity of Natural Language, pp 1-18., D Reidl 1987 'How, one may reasonably wonder, could mathematics be applied to anything so unquantifiable as language? ... The source of such confusiuon is the common but incorrect idea that all mathematics deals with numbers. Virtually any paper in generative linguistics illustrate (sic) how one can make precise statements of a nonquantitative nature about language. Mathematical linguistics involves studying this sort of statement by applying mathematics. ... ' 
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Reynolds, Vernon, and Ralph Tanner, The Social Ecology of Religion, Oxford University Press 1995 Jacket: 'No society exists in which religion does not play a significant part in the lives of ordinary people. Yet the functions of the world's diverse religions have never been fully described and analyzed, nor has the impact of adherence to those religions on the health and survival of the populations that practice them. . . . this extraordinary text reveals how religions in all parts of the world meet the needs of ordinary people and frequently play an important part in helping them to manage their affairs.' 
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Robins, R F, A Short History of Linguistics, Longman 1990 R H Robins is Professor Emeritus of General Linguistics in the University of London. back
Thiering, Barbara, Jesus the Man: A new interpretation from the Dead Sea scrolls, Acacia Press 1993 'Some will see her as an anti-Christ, a mischievous scholar determined to destroy Christianity. To others she will be a source of comfort and peace, enabling them to live Christian lives without having to accept as fact Jesus's divinity, his miracles, the virgin birth and resurrection.' The Australian Magazine  
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Thiering, Barbara, Jesus of the Apocalypse: The life of Jesus after the crucifixion, Doubleday 1996 Introduction: 'It is now possible to show that ... the bizarre images of the [Book of Revelation] were deliberately constructed ... to read like fantastic images but to convey through this form actual historical information. ... Above all the Book of Revelation contains evidence, supplied by the early Christians themselves, that Jesus survived the crucifixion and remained active for many years afterwards. ... " vi 
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Wasow, Thomas, "On constraining the class of transformational languages" in Walter J Savitch et al, The Formal Complexity of Natural Language, pp 56-86., D Reidl 1987 'The key distinction I want to discuss in this paper is the difference between restricting the class of grammars made available by a theory of grammar and restricting the class of languages that can be generated by the grammars the theory makes available. ... ' 
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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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Yourgrau, Wolfgang, and Stanley Mandelstam, Variational Principles in Dynamics and Quantum Theory, Dover 1979 Variational principles serve as filters for parititioning the set of dynamic possibilities of a system into a high probability and a low probability set. The method derives from De Maupertuis (1698-1759) who formulated the principle of least action, which states that physical laws include a rule of economy, the principle of least action. This principle states that in a mathematically described dynamic system will move so as to minimise action. Yourgrau and andelstam explains the application of this principle to a variety of physical systems.  
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Papers
Nowak, Martin A, Joshua B Plotkin and Vincent A A Jansen, "The evolution of syntactic communication", Nature, 404, 6777, 30 March 2000, page 495-498. Letters to Nature: 'Animal communication is typically non-syntactic, which means that signals refer to whole situations. Human language is syntactic, and signals consist of discrete components that have their own meaning. Syntax is requisite for taking advantage of combinatorics, that is 'making infinite use of finite means'. ... Here we present a model for the population dynamics of language evolution, define the basic reproductive ratio of words and calculate the maximum size of a lexicon.'. back
Shannon, Claude E, "The mathematical theory of communication", Bell System Technical Journal, 27, , July and October, 1948, page 379-423, 623-656. 'A Note on the Edition Claude Shannon's ``A mathematical theory of communication'' was first published in two parts in the July and October 1948 editions of the Bell System Technical Journal [1]. The paper has appeared in a number of republications since: • The original 1948 version was reproduced in the collection Key Papers in the Development of Information Theory [2]. The paper also appears in Claude Elwood Shannon: Collected Papers [3]. The text of the latter is a reproduction from the Bell Telephone System Technical Publications, a series of monographs by engineers and scientists of the Bell System published in the BSTJ and elsewhere. This version has correct section numbering (the BSTJ version has two sections numbered 21), and as far as we can tell, this is the only difference from the BSTJ version. • Prefaced by Warren Weaver's introduction, ``Recent contributions to the mathematical theory of communication,'' the paper was included in The Mathematical Theory of Communication, published by the University of Illinois Press in 1949 [4]. The text in this book differs from the original mainly in the following points: • the title is changed to ``The mathematical theory of communication'' and some sections have new headings, • Appendix 4 is rewritten, • the references to unpublished material have been updated to refer to the published material. The text we present here is based on the BSTJ version with a number of corrections.. back
Links
Aquinas 14, 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 366, Summa I 75, 5: , Corpus, Latin 'Manifestum est enim quod omne quod recipitur in aliquo, recipitur in eo per modum recipientis' back
Dei Verbum, Dei verbum, Dogmatic Constitution on Divine Revelation Dei Verbum solemnly promulgated by his Holiness Pope Paul VI on November 18, 1965. back
Second Vatican Council, Dei verbum, Dogmatic Constitution on Divine Revelation Dei Verbum solemnly promulgated by his Holiness Pope Paul VI on November 18, 1965. back

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