St Thomas Aquinas Meets Chaos Theory

8/10/2019 St Thomas Aquinas Meets Chaos Theory

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SAINT THOMAS AQUINAS MEETS CHAOS THEORYA paper by Frater Choronzon

first presented to Philos-o-Forum at Eccleston House

on Monday 8th July 1991

The title of this paper owes a debt of inspiration to Jim Henson's "Muppet Show". Fans with long memoriesmay recall the dauntless Gonzo attempting to play Franz Lizst's sensitive piece 'Liebestraum' on a grand pianowhile simultaneously fighting a rapier duel with a giant crab which STOCHASTICALLY manifests to disruptthe performance. That sketch was titled 'Classical Music Meets Seafood'; in this adaptation Aquinas mayrepresent the faltering classical rendition while the inevitabilities of Chaos play the role of the mauraudingcrustacean. 'Liebestraum' has never been the same again - any time one hears or even thinks of the piece, theimage of Gonzo in mortal combat with a piano sized crab takes control.

On that surreal note I will proceed to some introductory background of the protagonists in the philosophicalconfrontation which is the main event of the evening.

THOMAS AQUINAS

Born late in 1224 or early in 1225, Thomas was the seventh and youngest son of Landulfo, Count of Aquino(near Naples), and the Countess Teodora Carracciolo, who was of Norman descent. The family was heavilyinvolved in a squabble between the Holy Roman Emperor Frederick II and the Papacy, and in 1229 youngThomas' father and his elder brothers were involved in the plunder of the papal stronghold at Monte Cassino.In the peace settlement the following year the youngster was effectively offered as a hostage to the Abbeythere, and at five years old Thomas found himself compulsorily introduced to the delights of a mediaevalclerical education.

Throughout Western Europe the church had effectively established an intellectual and academic monopoly

which had been in place since Augustine's time some eight centuries earlier. Thomas was released fromMonte Cassino after another attack by the Imperial Army in 1239, and continued his education at the Universityof Naples. In 1244, against the wishes of his family, he joined the Dominicans and set off for Paris to studyTheology. His father, Landulfo, had died a short time previously, but his mother and older brothers felt sostrongly about his vocation that they actually seized him and held him prisoner for a year. The Dominicanspetitioned both the Pope and the Emperor, and eventually the family became convinced that nothing couldshake the young man's own determination; they relented, and Thomas Aquinas took up his place at theDominican convent in Paris as a pupil of Albertus Magnus.

The affairs of Christendom were in some disarray. Emperor Frederick was encommunicated twice, first forbeing insufficiently zealous in pursuing a crusade (the 6th), and then again some years later for joking that notonly Moses and Christ, but also Muhammad, were imposters who had themselves been "hoodwinked".

Frederick retaliated for the second excommunication by wrecking a Genoese fleet and capturing over 100Cardinals and Bishops on board who were in passage to a Synod at Rome. Matters were only defused by thetimely death of Pope Gregory IX and the serious distraction of an invasion of Europe by the Mongols. TheSaracens were starting to get the upper hand in the Middle East, perhaps assisted by covert connivancebetween the Knights Templar and the Assassins (who was "hoodwinking" who one wonders).

Nor were Islam's assaults on the Christian position confined to the battlefield. While Rome had been doingeverything it could to suppress and obliterate the ancient philosophies of the pagan era, the works of Aristotlein particular had gained some currency among the sages of the Arab world. Aristotelian rationalism had beenapplied, specifically by Averroes, in providing an intellectual basis for Islam, and Christianity found itself beingundermined within its own academic institutions which were becoming increasingly secular, where notdownright heretical.

The fightback was spearheaded by Thomas Aquinas who translated Aristotle into Latin and set about therestatement of Christian doctrine which was to become enshrined as the orthodox Roman position throughoutthe subsequent centuries of Dominican led Inquisition, and of which the central arguments provide theintellectual basis for much of Christian doctrine to this day. Aquinas was canonised in 1323, some 50 yearsafter his death, and, although Protestant denominations may reject much of the doctrine he expounded, they

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are nontheless happy to make use of some of his intellectual tricks in constructing their own versions of the"irrefutable hypothesis" of Christianity.

It should be said that, as theological treatises go, the 'Summa Theologica' of Aquinas is well structured andclearly set out, and, although it is probably longer than the Bible, in my perception it makes much easierreading than, say, the rambling prose of Saint Augustine. The logical presentation, though, throws up aninherent weakness in the argument to the extent that the whole thesis seems to rest on a few crucial feats ofintellectual gymnastics in the opening pages.

In the first place, in establishing some need for "knowledge revealed by God" over and above what may bederived from philosophical reasoning, Aquinas argues that "it was necessary for the salvation of man thatcertain truths which exceed human reason should be made known to him by divine revelation". Since"salvation", in the sense intended by Aquinas, is a concept which is only significantly meaningful withinChristian (or related) paradigm(s), he seems to be assuming the validity of key elements within his paradigmbefore establishing that paradigm empirically, which is what he claims (in so many words) to be setting out todo.

The next crucial element in the thesis comes in 'Article 5' where Aquinas asks "Whether Sacred Doctrine IsNobler Than Other Sciences?" He concludes that it is "because other sciences derive their certitude from thenatural light of human reason, which can err, while this [sacred doctrine] derives its certitude from the light ofdivine knowledge, which cannot be deceived". The implication is one of reason being subordinate to somebaldly stated concept of divine infallibility.

In 'Article 8', where Aquinas appeals to "faith" in support of the above assertion, we find the followingremarkable passage:

"..... Sacred Scripture, since it has no science above itself, can dispute with one who denies its principlesonly if the opponent admits some at least of the truths obtained through divine revelation. Thus we canargue with heretics from texts in Holy Writ, and against those who deny one article of faith we can arguefrom another. But if our opponent believes nothing of divine revelation, there is no longer any means ofproving the articles of faith by reasoning, but only by arguing his objections - if he has any - against faith.Since faith rests on infallible truth, and since the contrary of a truth can never be demonstrated, it is clearthat proofs brought against faith cannot be demonstrations, but are arguments that can be answered."

By this means Aquinas seeks to provide a secure basis from which the rest of Christian doctrine can beexpounded. It is perhaps worth noting that in so doing he also provides the principal philosophical basis fromwhich any purportedly "infallible" revealed scripture can be presented. The argument seems to be capable ofapplication as much to the Christian Gospel as to the Torah, the Qu'ran, the Book of Mormon, and even to

Aleister Crowley's 'Liber Al' (the Book of the Law). The task for the proponent of whatever system of "faith" isbeing proposed remains simply to establish that the system which they are putting forward is itself thatinfallible ultimate truth which by definition defies contradiction.

The other protagonist in this evening's philosophical contest, Chaos Theory, takes issue with ThomasAquinas' position at this fundamental level.

CHAOS THEORY

The philosophical constructs of Chaos Theory are of comparatively recent provenance, albeit that a traditionalaphorism of the Assassins of Alamut and of the Illuminati of Bavaria, "Nothing is True: Everything is Permitted",has gained new currency in the modern paradigm.

Personally I prefer the paraphrase "There can be no Ultimate Truth : Everything is Permissible", and I wouldinterpose the statement "We exist in a STOCHASTIC Universe" to present a succinct encapsulation of thefundamental principles of Chaos Theory.

In my earlier expositions on the subject ('Liber Cyber') I have presented the statement "There can be no

Ultimate Truth" as a Philosophical Axiom, but in the present context, taking issue with no less an eminencethan Aquinas, a more rigorous analysis is called for.

One of the reasons I dislike the Illuminati formula "Nothing is True" is because that statement, to an InformationTechnologist, has no particular profundity - it is simply the corollary of another statement "+5 volts is False".The two statements together defining an information processing environment termed Negative Logic. For




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Positive Logic the analogous statements are "Nothing is False" and "+5 volts is True". Truth and Falsehoodare defined simply in terms of electrical voltage levels at semi-conductor outputs, or in the polarity of memorystorage elements, and whether a given information processing device uses positive or negative logic is entirelya function of the way in which the device is wired. The logic polarity is transparent to a user of the device; itwill give the same answers provided that polarity is consistent within the device, though the correctness ofthose answers is strictly independent of 'truth' and 'falsehood' as applied within the hard wired logic, and morea function of the validity of the input data and an absence of errors in the instruction path followed inprocessing that data.

Thus it can be argued at the quantum information level of binary '1's and '0's that "there can be no UltimateTruth", just as "there can be no Ultimate Falsehood", the crucial factors in getting the right answers in the realworld are logical consistency, valid input data, and bug-free software. There is a school of thought which holdsthat in anything but the most trivially simple systems, completely bug-free software may be an unattainableideal, and therefore it may also be that "There can be no Ultimate Correctness" of answers under allcircumstances. On that basis, the best that can be said of any system attempting to model reality, or somesubset thereof, is "It appears to give correct answers most of the time".

For many centuries rational and irrational philosophers have turned to mathematics as a tool for modelling andmaking predictions not only about the real world but also concerning the abstract realms of the imaginary orunreal world. The reason being that mathematics appeared in many cases to give the correct answers most ofthe time, and moreover, at the most trivial level there were a number of absolute truisms or 'axioms' which

could be simply stated, and then used as a basis for the logical deduction and induction, or 'proof', of morecomplex statements or 'propositions'. When such propositions or 'theorems' have been proved with referenceto the accepted axioms, it is often found that the mathematical relationships established have some analoguein the behaviour of objective reality.

Until the present century it was generally supposed by mathematicians that all conceivable propositions aboutthe relationships between numbers or other mathematical entities might ultimately be proved to be either Trueor False, even if, in some cases, the procedure for actually accomplishing the proof or disproof might bedifficult in the extreme. For centuries there had been problems with whole number solutions for DiophantineEquations, and the generalisation of those problems in the 'Last Theorem' of Pierre de Fermat, but it wasgenerally felt that someone would come up with a proof or disproof of these propositions sooner or later.

During the first decade of this century the philosopher/mathematicians Bertrand Russell and Alfred NorthWhitehead embarked on a systematic exercise to codify the whole body of mathmatical knowledge, showingthat everything could be deduced from the most basic logical principles. The three volumes of their 'PrincipiaMathematica' were published between 1910 and 1913, and were profoundly influential, even though theauthors were forced to admit that their own ultimate objective had not been achieved, in that propositions likeFermat's Last Theorem still defied proof and disproof alike.

The Austrian mathematician Kurt Gdel took up this issue and generalised it, eventually in 1940 publishing hisown theorems which proved rigorously that there would always be propositions which could neither be provedor disproved. It is unsatisfactory in this context simply to state Gdel's conclusion, and although the tortuouslogic of the full proof is usually consigned to the more abstruse options available in an honours level universitymaths course, I shall nontheless now attempt a comprehensible rendition because of the importance of that

conclusion to a wider understanding of Chaos Theory.

A PROOF OF GDEL'S THEOREM

Gdel's Theorem is a theorem about theorems - a 'meta-theorem' if you like. It is derived in a mathematicallanguage or notation system known as 'Typographical Number Theory', or TNT for short. In TNT mathematicalsymbols representing theorems (and non-theorems) are manipulated using the procedures, transformationsand rules of 'Propositional Calculus', which also provides a means of determining the relationships betweentheorems in TNT.

Like a computer programming language, TNT has a restricted syntax, which means that mathematicalstatements have to be expressed in the most elementary way before they can be rendered into TNT. As an

example, one might wish in TNT to make the statement "7 is not a square number". The rules require that thissentence should be restated thus. "There does not exist any whole number 'b', greater than 1, such that 'b'times 'b' equals 7". The meaning is identical and moreover the concept of a square number is defined quiteconcisely. In the shorthand notation of TNT the sentence is written thus:

~Eb: (b.b) SSSSSSS0 (1)




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The notation decodes as follows:-

The 'Tilde' ~ denotes negation, like a NOT gate in a wiring diagram;The 'Upper Case E' is an 'assertion of existence' quantifier;The 'Colon' means 'such that';'b' is a 'free variable', like 'x' or 'y' in a normal equation;The 'Full Stop' (or 'period') indicates multiplication.The 'Equals Sign' has its usual meaning.The 'S' means 'the Successor of' or 'the next natural counting number up from'

'Zero' has its usual meaning.

In the notation, S0 means "the natural counting number which is the successor of 0", in ordinary notation '1'.

SS0 is "the natural counting number which is the successor of SO", commonly written as 2. Thus SSSSSSS0represents 7. All definite numerical values are written in this way in TNT.

If something can be expressed as a "well-formed" statement in TNT, then it is a rigorously provable theorem bydefinition, and, usefully, there are a number of rules to test for "well-formedness". These need not concern ushere, and they are set out in any relevant text-book.

What Gdel is seeking to do is to derive a "well-formed" TNT statement which, when deciphered, turns out not

to be a valid theorem in TNT; and to see how he does this it is necessary to introduce the property of self-reference, and a manipulation process known as 'Arithmoquining'.

To introduce self-reference, Gdel uses the concept of a 'Proof-Pair' and this involves effectively stepping outof the TNT paradigm and then back into it again. This is done using a procedure a bit like Gematria, known asGdel numbering. Each typographical symbol of TNT is assigned an arbitrary three digit numerical value,known as a 'Codon'. For example, 0 is 666, 5 is 123, the sign '-' is 111, and so forth (see Appendix). These arestrung together so, for example, the the TNT statement 0 - 0 can be represented by the numerical value666,111,666.

Entire proofs in TNT can be written out in this way, with a special Gdel gematria codon, 611, indicating a newline. If an entire proof is transposed in this way, then the Gdel number pertaining to the last line represents

the final outcome of the proof, usually denoted in TNT as a free variable, say, b' (referred to as "b-prime"),while the huge number corresponding to the entire working of the proof (complete with new line symbols) isdenoted as the corresponding free variable b.

The last line of the proof is a TNT theorem, and its Gdel gematria number may be referred to as a 'TNT-Theorem-Number'. Thus the proof-pair concept can be stated as follows: "There exists a Gdel Number bsuch that b and the Gdel Number of its last line, b', form a TNT-PROOF-PAIR". In the TNT notation this iswritten.

Eb: TNT-PROOF-PAIR { b, b' } (2)

This may also, by definition of the Proof-Pair property, be restated as "b' is a TNT-Theorem-Number".

Now to deal with 'Arithmoquining'. This is a numerical process deriving from a verbal self-reference techniqueknown as 'quining' in honour of Willard van Orman Quine, the American philosopher who developed it (born1908, and still alive, I think). An example of a quined sentence might be: {"Is the name of a Band not an Album"is the name of a Band not an Album.} Self-reference is achieved by preceding a sentence with its ownquotation.

In Arithmoquining the same sort of effect is achieved numerically by quoting a Gdel Number for an entireexpression as a substitution for a free variable in the rendition of a TNT statement. Thus, if we have a TNTstatement: ~b=SO where b is a free variable, by Gdel's gematria it has the number 223,262,111,123,666. Onarithmoquining, the number representing the whole TNT statement is substituted for the free variable b, giving:~223,262,111,123,666 = SO which can be rewritten in TNT notation as:

~SSSSSSS..............SSSSSS0 = S0 (3) | | 223,262,111,123,666 S's




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Although a bit nonsensical, this end product is also a True statement in the strict logical sense.

Of course the TNT expression at (3) above can itself be transcribed by the now familiar gematria process intoa Gdel Number:

223,123,123,123,123, .............. 123,123,123,666,111,123,666 (4) | | 223,262,111,123,666 copies of 123

This humungously large number, let's call it b'', is said to be the 'Arithmoquinification' of b, and the wholeprocess can be stated in TNT notation as

ARITHMOQUINE { b'', b } (5)

To achieve his objective, and to make the principle stick on meta-levels of 'theorems about theorems abouttheorems about ....', (imparting, incidentally, something akin to fractal self-replication permeating an infinity oftheorem levels), Gdel seeks to arithmoquine an expression which itself makes some statement aboutarithmoquining. Eventually he hit on the following formula, which we'll call "G's Fairy-God-Mother".

~Eb: Eb': ( TNT-PROOF-PAIR { b, b' } & ARITHMOQUINE { b", b'} ) (6).

This means: There do not exist numbers b and b' such that (1) b and b' form a TNT-Proof-Pair AND (2) b" isthe Arithmoquinification of b'.

This Fairy-God-Mother statement, let's call it 'f', can of course be transposed into a huge number by Gdel'sgematria:

f = 223,333,262,636,... etc ....etc ...,213 (7)

Now the whole thing is arithmoquined by substituting the huge number representing 'f' back into the Fairy-God-Mother in the place of the only free variable b'' (b and b' are the subjects of the initial 'assertions of existence',and as such are not considered 'free' in the context).

The result, in TNT notation, looks like this:

~Eb: Eb': (TNT-PROOF-PAIR {b, b'} & ARITHMOCUINE {SSSS...SSSO, b'}) (8) | | f S's

This expression, which we will call 'G', is sometimes referred to by irreverent mathematicians as Gdel's G-String.

There are two things to say about it. Firstly, G's own Gdel Number is the arithmoquinification of the Fairy-God-Mother expression. Secondly, G has an interpretation which, in a literal translation, runs as follows:

"There do NOT exist numbers b and b' such that BOTH (1) they form a TNT-Proof-Pair AND (2) b' is thearithmoquinification of the Fairy-God-Mother".

What can be made of this? We know that there is a number b' which is the arithmoquinification of the Fairy-God-Mother, so the second half of G checks out OK, allowing us to restate G as follows: "There is no numberb that forms a TNT-Proof-Pair with the arithmoquinification of the Fairy-God-Mother".

This is the same as saying "The statement whose Gdel Number is the arithmoquinification of the Fairy-God-Mother is not a theorem of TNT"; but "the statement whose Gdel Number is the arithmoquinification of theFairy-God-Mother" is none other than G itself, so the ultimate translation becomes:

"G is not a theorem of TNT".

So, having been constructed as a "well-formed" TNT statement, in the final analysis G says "I am not atheorem of TNT". It is effectively a mathematical TRUTH asserting its own FALSITY. A sublime contradiction atthe meta-level of 'theorems about theorems about theorems about...'.




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Now let's consider ~G, the negation of G.

Since G was constructed as a "well-formed" statement of TNT, its negation, by definition, cannot be "well-formed", yet the interpretation of ~G is going to be the negation of the interpretation of G. That is: "I am atheorem of TNT".

So what we wind up with is a valid theorem which states that it is invalid; and its negation, an expressionwithout validity in a strict formal sense, making the assertion that it is a valid theorem. What we have is anundecidable proposition - effectively a 'hole' in the system, or, since we are dealing with the natural counting

numbers, a hole in the reality of the universe. Clearly, with a hole in the reality of the universe revealing anultimate undecidable proposition, it follows that "there can be no ultimate truth". QED.

INTERPRETING GDEL'S THEOREM IN REALITY

Because Gdel's Theorem is a multi-level meta-theorem about theorems, there is a whole class, an infinityeven, of mathematically valid propositions which will defy both proof and disproof.

Such propositions permeate tke structure of the Cosmos. They describe the ultimate uncertainty about theexpansion of the universe itself, which is perceived to be expanding at exactly the critical rate where isundecidable whether the expansion will continue for ever, or whether everything will eventually start to

contract again under the influence of gravity, terminating in a 'Big Crunch'.Similar propositions apply to the uncertainties in the observable behaviour of sub-atomic particles which aredescribed by Werner Heisenberg in his statement of the 'Uncertainty Principle'.

Yet other Gdelian propositions may model the uncertainties attending supposedly causal processes on aterrestrial or human scale. Although a butterfly flapping its wings on a Carribean island may initiate the processwhich causes a hurricane in London, as determined by Chaos Maths, it is undecidable which particularbutterfly was responsible.

Alternatively, we can conduct a magical operation with the intent of abolishing an unpopular tax, but, when thatvery tax is then abolished within the specified time period, it is nontheless undecidable whether the magic

caused the result, or whether it would have happened anyway.

The class of mathematical expressions which satisfy the criteria of Gdel's G-String are intermingled at alllevels with the processes of the real world. In lyrical terms, they seem to open a window from reality onto somevoid of undecidability through which Chaos can become manifest. More than anything, this mathematicsconfirms to us rigorously that "there can be no ultimate truth". Not on a sub-atomic scale, not on a humanscale, not on a cosmic scale, and not in the domain of mathematical theorems expressing philosophicalconcepts.

If "there can be no ultimate truth", the concept of "infallibility" becomes meaningless, and to maintain a beliefsystem based on such a concept is no different to maintaining a belief that the Sun goes round the Earth, evenafter Copernicus, Kepler, Galileo, Giordano Bruno and Newton had argued and proved that the exact opposite

was the case.

CONCLUSION

I have devoted almost half of this paper to an exposition of Gdel's Proof, for the simple reason that it providesa rigorous underpinning for the philosophical position of Chaos Theory. In so doing it raises a challenge, notonly to the philosophical and doctrinal basis of Christianity as set out by Aquinas, but also, by implication, toevery other religion or philosophy which depends for its authority on the purported infallibility of someindividual and/or of some text bearing the description of Holy Writ.

In that sense it is strange to find Aquinas in the position of presenting the case, not only for all denominationsof Christianity, but also for Islam, for Judaism, for Crowley's Thelema and perhaps also for the discredited

mechanical models of deterministic science. All of these absolutist belief systems are equally challenged byGdel's Proof.

In my view, the only cogent response left to the protagonists of Infallibility and Absolute Truth is the memorableutterance of The Koresh, Dr Cyrus Teed, who taught during the early years of this century that we are all livingon the interior surface of a concave sphere. "Believe ME, not Mathematics" cried The Koresh, "When will you




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learn that equations are useless?"

Such proclamations from proponents sharing the standpoint of Aquinas are not unfamiliar; but even in the daysof their tyrannical efforts to maintain an intellectual monopoly, neither the recantation of the aging Galileounder threat of the red-hot poker and the rack, nor burning Giordano Bruno, would make the Sun go round theEarth. Faith and belief may be modified by torturing adherents of unfashionable philosophies, but the realitiesof mathematics and physics cannot be placed under arrest or put to death by Church or by State.

Few people today have even heard of Gdel's Proof, and even fewer have given much thought to its

philosophical implications, but the monolithic infallibilities of the Belief Systems of the Book (any Book) areshattered by it. Forced recantations, or burning books, or burning me or anyone else will not alter that fact.Their only recourse would be in seeking to show that Gdel's Proof itself is invalid, and in that endeavour theywould find support among many traditional mathematicians who do not like it, and who have been trying torefute it for 50 years.

All "infallible" belief systems find themselves in the same boat; driven before the Storms of Chaos onto therocks which are the rigours of Gdel's Theorem. The defender of Faith, Saint Thomas Aquinas, like Gonzo,plays 'Liebestraum' among the wreckage, awaiting the approach ofSeafood, the crustacean devourer of infallible philosophies.

So, can there be any meaningful religion? Perhaps not, I would suggest, unless it is one where God is calledChaos, in short for Universal Undecidability, and where "Salvation" for humankind is ultimate freedom from thefrauds of infallibilities and from the laws which legitimize them.

Remember: The proposition "There Can Be No Ultimate Truth" cannot itself be an ultimate truth.

You can prove that yourself!

APPENDIX- Some examples of Gdel Codons as used in this paper. (After D Hofstadter)

SYMBOL CODON REMARKS AND MNEMONIC HINTS

O 666 Zero as the Illuminati Sol symbolS 123 Denotes numerical successorship- 111 Visual resemblance+ 112 Suggests additionu 1 + 1 = 2. 236 Suggests multiplication. 2 x 3 - 6b 262 This codon is conventionally used for 'a' as a free variable, but in this paper 'b' is substituted for ease of legibility.' 163 The 'prime' symbol{ 212} 213( 362) 323

& 161 Logical AND| 616 Logical OR~ 223 'Tilde' Logical NOTE 333 Existential Qualifier usually a 'Backwards E'... $: 636 611 Codon for new line in a proof sequence.

REFERENCES AND FURTHER READING.

AQUINAS, St T Summa Theologica 1273 (trans Fathers of the English Dominican Province) Ed. Univ of Chicago 1952BREWSTER, C G Liber Cyber (Ecliptica) 1990

CARROLL, P J Liber Kaos, The Psychonomicon (lOT) 1991ENCYCLOPAEDIA Various references including. Ed 1988BRITANNICA Saint Thomas Aquinas; Emperor Frederick II; Gdel, Kurt; Quine, Willard van Orman; Doctrine and Dogma, Religious; Russell, Bertrand

HOFSTADTER, D Gdel, Escher, Bach (Penguin) 1979




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MICHELL, J Eccentric Lives and Peculiar Notions 1984 (Thames & Hudson)ROSA, P de Vicars of Christ (Corgi) 1988

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ACKNOWLEDGEMENTS. I am indebted to Peter Carroll for use of the phrase "There can be noUltimate Truth", and to Hilary Hayes for her perception that "Everything is Permissible" is a

preferable usage to "Everything is Permitted" which implies that there is something to do thepermitting. Finally I am grateful to Douglas Hofstadter for his clear exposition of the proof of Gdel'sTheorem, parts of which I have abstracted from his book 'Gdel, Escher, ~

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St Thomas Aquinas Meets Chaos Theory