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SAINT THOMAS AQUINAS MEETS CHAOS THEORY |
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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 memories may recall the dauntless Gonzo attempting to play Franz Lizst's sensitive piece 'Liebestraum' on a grand piano while simultaneously fighting a rapier duel with a giant crab which stochastically manifests to disrupt the performance. That sketch was titled 'Classical Music Meets Seafood'; in this adaptation Aquinas may represent the faltering classical rendition while the inevitabilities of Chaos play the role of the marauding crustacean. 'Liebestraum' has never been the same again -= any time one hears or even thinks of the piece, the image 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 philosophical confrontation 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 heavily involved in a squabble between the Holy Roman Emperor Frederick II and the Papacy, and in 1229 young Thomas' 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 Abbey there, and at five years old Thomas found himself compulsorily introduced to the delights of a mediaeval clerical 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 from Monte Cassino after another attack by the Imperial Army in 1239, and continued his education at the University of Naples. In 1244, against the wishes of his family, he joined the Dominicans and set off for Paris to study Theology. His father, Landulfo, had died a short time previously, but his mother and older brothers felt so strongly about his vocation that they actually seized him and held him prisoner for a year. The Dominicans petitioned both the Pope and the Emperor, and eventually the family became convinced that nothing could shake the young man's own determination; they relented, and Thomas Aquinas took up his place at the Dominican convent in Paris as a pupil of Albertus Magnus. The affairs of Christendom were in some disarray. Emperor Frederick was encommunicated twice, first for being insufficiently zealous in pursuing a crusade (the 6th), and then again some years later for joking that not only 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 100 Cardinals and Bishops on board who were in passage to a Synod at Rome. Matters were only defused by the timely death of Pope Gregory IX and the serious distraction of an invasion of Europe by the Mongols. The Saracens were starting to get the upper hand in the Middle East, perhaps assisted by covert connivance between 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
doing everything it could to suppress and obliterate the ancient
philosophies of the pagan era, the works of Aristotle in particular had
gained some currency among the sages of the Arab world. Aristotelian
rationalism had been applied, specifically by Averroes, in providing an
intellectual basis for Islam, and Christianity found itself being
undermined within its own academic institutions which were becoming
increasingly secular, where not downright heretical. It should be said that, as theological treatises go, the 'Summa Theologica' of Aquinas is well structured and clearly set out, and, although it is probably longer than the Bible, in my perception it makes much easier reading than, say, the rambling prose of Saint Augustine. The logical presentation, though, throws up an inherent weakness in the argument to the extent that the whole thesis seems to rest on a few crucial feats of intellectual gymnastics in the opening pages. In the first place, in establishing some need for "knowledge revealed by God" over and above what may be derived from philosophical reasoning, Aquinas argues that "it was necessary for the salvation of man that certain 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 within Christian (or related) paradigm(s), he seems to be assuming the validity of key elements within his paradigm before establishing that paradigm empirically, which is what he claims (in so many words) to be setting out to do. The next crucial element in the thesis comes in 'Article 5' where Aquinas asks "Whether Sacred Doctrine Is Nobler Than Other Sciences?" He concludes that it is "because other sciences derive their certitude from the natural light of human reason, which can err, while this [sacred doctrine] derives its certitude from the light of divine knowledge, which cannot be deceived". The implication is one of reason being subordinate to some baldly stated concept of divine infallibility. In 'Article 8', where Aquinas appeals to "faith" in support of the above assertion, we find the following remarkable passage: "..... Sacred Scripture, since it has no science above itself, can dispute with one who denies its principles only if the opponent admits some at least of the truths obtained through divine revelation. Thus we can argue with heretics from texts in Holy Writ, and against those who deny one article of faith we can argue from another. But if our opponent believes nothing of divine revelation, there is no longer any means of proving 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 clear that 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 be expounded. It is perhaps worth noting that in so doing he also provides the principal philosophical basis from which any purportedly "infallible" revealed scripture can be presented. The argument seems to be capable of application 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" is being proposed remains simply to establish that the system which they are putting forward is itself that infallible ultimate truth which by definition defies contradiction. The other protagonist in this evening's philosophical contest, Chaos Theory, takes issue with Thomas Aquinas' position at this fundamental level. CHAOS THEORY The philosophical constructs of Chaos Theory are of comparatively recent provenance, albeit that a traditional aphorism 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 would interpose the statement "We exist in a Stochastic Universe" to present a succinct encapsulation of the fundamental 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 eminence than 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 Information Technologist, 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 Positive Logic the analogous statements are "Nothing is False" and "+5 volts is True". Truth and Falsehood are defined simply in terms of electrical voltage levels at semi-conductor outputs, or in the polarity of memory storage elements, and whether a given information processing device uses positive or negative logic is entirely a function of the way in which the device is wired. The logic polarity is transparent to a user of the device; it will give the same answers provided that polarity is consistent within the device, though the correctness of those answers is strictly independent of 'truth' and 'falsehood' as applied within the hard wired logic, and more a function of the validity of the input data and an absence of errors in the instruction path followed in processing that data. Thus it can be argued at the quantum information level of binary '1's and '0's that "there can be no Ultimate Truth", just as "there can be no Ultimate Falsehood", the crucial factors in getting the right answers in the real world are logical consistency, valid input data, and bug-free software. There is a school of thought which holds that in anything but the most trivially simple systems, completely bug-free software may be an unattainable ideal, and therefore it may also be that "There can be no Ultimate Correctness" of answers under all circumstances. On that basis, the best that can be said of any system attempting to model reality, or some subset 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 modeling and making predictions not only about the real world but also concerning the abstract realms of the imaginary or unreal world. The reason being that mathematics appeared in many cases to give the correct answers most of the 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 more complex statements or 'propositions'. When such propositions or 'theorems' have been proved with reference to the accepted axioms, it is often found that the mathematical relationships established have some analogue in the behavior of objective reality. Until the present century it was generally supposed by mathematicians that all conceivable propositions about the relationships between numbers or other mathematical entities might ultimately be proved to be either True or False, even if, in some cases, the procedure for actually accomplishing the proof or disproof might be difficult in the extreme. For centuries there had been problems with whole number solutions for Diophantine Equations, and the generalization of those problems in the 'Last Theorem' of Pierre de Fermat, but it was generally 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 North Whitehead embarked on a systematic exercise to codify the whole body of mathematical knowledge, showing that everything could be deduced from the most basic logical principles. The three volumes of their 'Principia Mathematica' were published between 1910 and 1913, and were profoundly influential, even though the authors were forced to admit that their own ultimate objective had not been achieved, in that propositions like Fermat's Last Theorem still defied proof and disproof alike. The Austrian mathematician Kurt Gödel took up this issue and generalised it, eventually in 1940 publishing his own theorems which proved rigorously that there would always be propositions which could neither be proved or disproved. It is unsatisfactory in this context simply to state Gödel's conclusion, and although the tortuous logic of the full proof is usually consigned to the more abstruse options available in an honours level university maths course, I shall nonetheless now attempt a comprehensible rendition because of the importance of that conclusion to a wider understanding of Chaos Theory. A PROOF OF GÖDEL'S THEOREM Gödel's Theorem is a theorem about theorems - a 'meta-theorem' if you like. It is derived in a mathematical language or notation system known as 'Typographical Number Theory', or TNT for short. In TNT mathematical symbols representing theorems (and non-theorems) are manipulated using the procedures, transformations and rules of 'Propositional Calculus', which also provides a means of determining the relationships between theorems in TNT. Like a computer programming language, TNT has a restricted syntax, which means that mathematical statements 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 this sentence 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 quite concisely. In the shorthand notation of TNT the sentence is written thus: ~Eb: (b.b) SSSSSSS0 (1) The notation decodes as follows:- The 'Tilde' ~ denotes
negation, like a NOT gate in a wiring diagram; 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 SSSSSSS0 represents 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 by definition, and, usefully, there are a number of rules to test for "well-formedness". These need not concern us here, and they are set out in any relevant text-book. What Gödel 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, Gödel uses the concept of a 'Proof-Pair' and this involves effectively stepping out of the TNT paradigm and then back into it again. This is done using a procedure a bit like Gematria, known as Gödel 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 are strung together so, for example, the the TNT statement 0 - 0 can be represented by the numerical value 666,111,666. Entire proofs in TNT can be written out in this way, with a special Gödel gematria codon, 611, indicating a new line. If an entire proof is transposed in this way, then the Gödel 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) is denoted as the corresponding free variable b. The last line of the proof is a TNT theorem, and its Gödel gematria number may be referred to as a 'TNT-Theorem-Number'. Thus the proof-pair concept can be stated as follows: "There exists a Gödel Number b such that b and the Gödel Number of its last line, b', form a TNT-PROOF-PAIR". In the TNT notation this is written. 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 technique known as 'quining' in honour of Willard van Orman Quine, the American philosopher who developed it (born 1908, 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 own quotation. In Arithmoquining the same sort of effect is achieved numerically by quoting a Gödel Number for an entire expression as a substitution for a free variable in the rendition of a TNT statement. Thus, if we have a TNT statement: ~b=SO where b is a free variable, by Gödel's gematria it has the number 223,262,111,123,666. On arithmoquining, 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) 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 into a Gödel Number: 223,123,123,123,123,
.............. 123,123,123,666,111,123,666 (4) This humungously large number, let's call it b'', is said to be the 'Arithmoquinification' of b, and the whole process 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 about theorems about ....', (imparting, incidentally, something akin to fractal self-replication permeating an infinity of theorem levels), Gödel seeks to arithmoquine an expression which itself makes some statement about arithmoquining. 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" is the Arithmoquinification of b'. This Fairy-God-Mother statement, let's call it 'f', can of course be transposed into a huge number by Gödel's gematria: 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'}) ( This expression, which we will call 'G', is sometimes referred to by irreverent mathematicians as Gödel's G-String. There are two things to say about it. Firstly, G's own Gödel 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 the arithmoquinification 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 number b that forms a TNT-Proof-Pair with the arithmoquinification of the Fairy-God-Mother". This is the same as saying "The statement whose Gödel Number is the arithmoquinification of the Fairy-God-Mother is not a theorem of TNT"; but "the statement whose Gödel Number is the arithmoquinification of the Fairy-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 a theorem of TNT". It is effectively a mathematical TRUTH asserting its own FALSITY. A sublime contradiction at the meta-level of 'theorems about theorems about theorems about...'. 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 a theorem of TNT". So what we wind up with is a valid theorem which states that it is invalid; and its negation, an expression without validity in a strict formal sense, making the assertion that it is a valid theorem. What we have is an undecidable 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 an ultimate undecidable proposition, it follows that "there can be no ultimate truth". QED. INTERPRETING GÖDEL'S THEOREM IN REALITY Because Gödel's Theorem is a multi-level meta-theorem about theorems, there is a whole class, an infinity even, of mathematically valid propositions which will defy both proof and disproof. Such propositions permeate the structure of the Cosmos. They describe the ultimate uncertainty about the expansion of the universe itself, which is perceived to be expanding at exactly the critical rate where is undecidable 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 are described by Werner Heisenberg in his statement of the 'Uncertainty Principle'. Yet other Gödelian propositions may model the uncertainties attending supposedly causal processes on a terrestrial or human scale. Although a butterfly flapping its wings on a Caribbean island may initiate the process which causes a hurricane in London, as determined by Chaos Maths, it is undecidable which particular butterfly was responsible. Alternatively, we can conduct a magical operation with the intent of abolishing an unpopular tax, but, when that very tax is then abolished within the specified time period, it is nonetheless undecidable whether the magic caused the result, or whether it would have happened anyway. The class of mathematical expressions which satisfy the criteria of Gödel's G-String are intermingled at all levels with the processes of the real world. In lyrical terms, they seem to open a window from reality onto some void of undecidability through which Chaos can become manifest. More than anything, this mathematics confirms to us rigorously that "there can be no ultimate truth". Not on a sub-atomic scale, not on a human scale, not on a cosmic scale, and not in the domain of mathematical theorems expressing philosophical concepts. If "there can be no ultimate truth", the concept of "infallibility" becomes meaningless, and to maintain a belief system based on such a concept is no different to maintaining a belief that the Sun goes round the Earth, even after 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 Gödel's Proof, for the simple reason that it provides a rigorous underpinning for the philosophical position of Chaos Theory. In so doing it raises a challenge, not only to the philosophical and doctrinal basis of Christianity as set out by Aquinas, but also, by implication, to every other religion or philosophy which depends for its authority on the purported infallibility of some individual 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 denominations of 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 by Gödel's Proof. In my view, the only cogent response left to the protagonists of Infallibility and Absolute Truth is the memorable utterance of The Koresh, Dr Cyrus Teed, who taught during the early years of this century that we are all living on the interior surface of a concave sphere. "Believe ME, not Mathematics" cried The Koresh, "When will you learn that equations are useless?" Such proclamations from proponents sharing the standpoint of Aquinas are not unfamiliar; but even in the days of their tyrannical efforts to maintain an intellectual monopoly, neither the recantation of the aging Galileo under threat of the red-hot poker and the rack, nor burning Giordano Bruno, would make the Sun go round the Earth. Faith and belief may be modified by torturing adherents of unfashionable philosophies, but the realities of mathematics and physics cannot be placed under arrest or put to death by Church or by State. Few people today have even heard of Gödel'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) are shattered 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 Gödel's Proof itself is invalid, and in that endeavour they would find support among many traditional mathematicians who do not like it, and who have been trying to refute it for 50 years. All "infallible" belief
systems find themselves in the same boat; driven before the Storms of
Chaos onto the rocks which are the rigours of Gödel's Theorem. The
defender of Faith, Saint Thomas Aquinas, like Gonzo, plays 'Liebestraum'
among the wreckage, awaiting the approach of So, can there be any meaningful religion? Perhaps not, I would suggest, unless it is one where God is called Chaos, in short for Universal Undecidability, and where "Salvation" for humankind is ultimate freedom from the frauds 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 Gödel Codons as used in this paper. (After D Hofstadter) SYMBOL CODON REMARKS AND MNEMONIC HINTS
O 666 Zero as the Illuminati
Sol symbol REFERENCES AND FURTHER READING. AQUINAS, St T Summa
Theologica 1273 (trans Fathers of the English Dominican Province) Ed. Univ
of Chicago 1952 HOFSTADTER, D Gödel, Escher,
Bach (Penguin) 1979 ACKNOWLEDGEMENTS. I am indebted to Peter Carroll for use of the phrase "There can be no Ultimate 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 the permitting. Finally I am grateful to Douglas Hofstadter for his clear exposition of the proof of Gödel's Theorem, parts of which I have abstracted from his book 'Gödel, Escher'
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