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Paraconsistent (or Inconsistent) Mathematics is a companion to Paraconsistent Logic in ways analogous to the relationship between classical mathematics and logic except that, in antiquity, Greek logic arose from Babylonian and Egyptian mathematics, "We must next speak of the origin of geometry in the present world cycle. For, as the remarkable Aristotle tells us, the same ideas have repeatedly come to men at various periods of the universe.  It is not, he goes on to say, in our time or in the time of those known to us that the sciences have first arisen, but they have appeared and again disappeared, and will continue to appear and disappear, in various cycles, of which the number both past and future is countless. But since we must speak of the origin of the arts and sciences with reference to the present world cycle, it was, we say, among the Egyptians that geometry is generally held to have been discovered. It owed its discovery to the practice of land measurement. For the Egyptians had to perform such measurements because the overflow of the Nile would cause the boundary of each person's land to disappear. Furthermore, it should occasion no surprise that the discovery both of this science and of the other sciences proceeded from utility, since everything that is in the process of becoming advances from the imperfect to the perfect. The progress, then, from sense perception to reason and from reason to understanding is a natural one. And so, just as the accurate knowledge of numbers originated with the Phoenicians through their commerce and their business transactions, so geometry was discovered by the Egyptians for the reason we have indicated. It was Thales, who, after a visit to Egypt, first brought this study to Greece. Not only did he make numerous discoveries himself, but laid the foundation for many other discoveries on the part of his successors, attacking some problems with greater generality and others more empirically. After him Mamercus the brother of the poet Stesichorus, is said to have embraced the study of geometry, and in fact Hippias of Elis writes that he achieved fame in that study. After these Pythagoras changed the study of geometry, giving it the form of a liberal discipline, seeking its first principles in ultimate ideas, and investigating its theorems abstractly and in a purely intellectual way. Plato, who lived after Hippocrates and Theodorus, stimulated to a very high degree the study of mathematics and of geometry in particular because of his zealous interest in these subjects. For he filled his works with mathematical discussions, as is well known, and everywhere sought to awaken admiration for mathematics in students of philosophy. Euclid composed Elements, putting in order many of the theorems of Eudoxus, perfecting many that had been worked out by Theatetus, and furnishing with rigorous proofs propositions that had been demonstrated less rigorously by his predecessors ... the Elements contain the complete and irrefutable guide to the scientific study of the subject of geometry.” – Proclus Diadochus, AD 410–485, Commentary on Euclid's Elements whereas, in the modern era, inconsistent mathematics is growing out considerations of paraconsistent logic. As opposed to classical applications, these two new specializations are grounded in Dialetheism in which functions both do and do not have a given value simultaneously, and both usually attempt to isolate the consistent aspects of a system to which conventional processes can apply and then try to handle the nasty contradictive portions with special care. They start to do so by overstepping the classical formal principle of ex contradictione quodlibet "from a contradiction every proposition may be deduced" (also called ECQ or, recently, Explosion). The issues that pertain to them are at the edge of contemporary considerations of reason and are difficult even for professionals in the specialties involved, so extra effort is required to insure that the causes of these difficulties are accessible to the general reader.

The extant texts that recorded the emergence of Greek logic from earlier Middle Eastern sources, occupying several centuries, are sparse and sporatic due to the ravages of time, whereas, the emergence of paraconsistent math from paraconsistent logic, although only a few decades old, has been so fertilely fast that the task of filtering relevant material from its explosive databases is daunting. But to clarify its significance for the Wiki public is a serious goal, since future centuries may bestow equivalent reverence to the intellectual events chronicled here to that we have bestowed on ancient Greece.

Section Structure
Editors: Read the Hidden REMarks.

Terms
It is common to confuse paraconsistency with dialetheism and equally so to confuse both with paradox, a concern which mandates a few philosophical term definitions. The roots and shared usages of these terms depicted in this section are to be elaborated and interelated in later sections, in concert with their matched linked Wikipedia articles.

Dialetheism
Though dialetheism is not a new vision, the word itself is. It was coined by Graham Priest and Richard Routley in 1981, from Ancient Greek di (di, “two”) + ἀλήθεια (alētheia, “truth”). The inspiration for the name was a passage in Wittgenstein's Remarks on the Foundations of Mathematics, where he describes the Liar Paradox sentence (‘This sentence is not true’) as a Janus-headed figure facing both truth and falsity. Hence a di-aletheia is a two(-way) truth.

From the Wiki Article on Dialetheism: "Dialetheism is the view that there are statements which are both true and false." From the Stanford Encyclopedia: "One can define a contradiction as a couple of sentences, one of which is the negation of the other, or as a conjunction of such sentences. Therefore, dialetheism amounts to the claim that there are true contradictions."

A universal acceptance of self-contradition has pervaded Hindu and Buddhist thought for millenia as the mutually-inclusive Yin/Yang of Taoism, in sharp opposition to the exclusionistic law of the Excluded Middle of Dualism that has dominated the West. In fact, the fundamental cultural assumptions of one half of Earth revolve as a rotating mirror opposite of the other.

The term, Dialetheism, which arose in response to the exotic and paradoxical demands of ultra-modern paraconsistent logic, has, to date, been applied primarily within its narrow confines. In fact, Graham Priest himself stated : "...These are called paraconsistent logics, and there is now a very robust theory of such logics. In fact, the mathematical details of these logics are absolutely essential in articulating dialetheism in any but a relatively superficial way. But the details are, perhaps, best left for consenting logicians behind closed doors."

A further serious issue is the confusion over the conditions that underlie the elemental state of opposition. Is it simply the negation of true vs. false (as is assumed by both mathematicians and logicians), or is darkness not the absence of light but instead the medium upon which it is projected? What singular balancing characteristic distinguishes Odd from Even? How about clockwise vs. counterwise motion? Imagine a top spinning both ways at once, which is logical but impossible, thereby aligning dialetheism with paradox.

ECQ
In the West, from Aristotle forward, it has been rigidly sustained that something and its negation cannot coexist. The proof of this states that if A does not equal A then everything is both true and false, which is absurd.

In the beginning of the Wiki ECQ article: "The Principle of Explosion (Latin: ex falso (sequitur) quodlibet (EFQ), "from falsehood, anything (follows)", or ex contradictione (sequitur) quodlibet (ECQ), "from contradiction, anything (follows)") is a basic law of classical and intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition (including its negation) can be inferred from it."

Apparently, its first formal proof was provided by William of Soissons, living in Paris in the 12th century as a member of a school of logicians called the Parvipontians: 'It is raining (P) and it is not raining (¬P), so you may infer that there are trees on the moon (or whatever else)(E). In symbolic language: P & ¬P → E."

More from Wiki's Explosive article: "If P and its negation ¬P are both assumed to be true, then P is assumed to be true, from which it follows that at least one of the claims P and some other (arbitrary) claim A is true. However, if we know that either P or A is true, and also that P is not true (that ¬P is true) we can conclude that A, which could be anything, is true. Thus if a theory contains a single inconsistency, it is trivial—that is, it has every sentence as a theorem.

Due to the principle of explosion, the existence of a contradiction (inconsistency) in a classical axiomatic system is disastrous; since any statement can be proved true it trivializes the concepts of truth and falsity. Around the turn of the 20th century, the discovery of contradictions such as Russell's paradox at the foundations of mathematics thus threatened the entire structure of mathematics. Mathematicians such as Gottlob Frege, Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem put much effort into revising set theory to eliminate these contradictions, resulting in the modern Zermelo–Fraenkel set theory."

All formal proofs of explosive theory are fallacies because, despite the clever intricacies of their arguments, in assuming that "A is not A is illogical" they ultimately turn into a vicious circle that assumes from the beginning what they are trying to prove. Stripped of that defense, a classical logician must now become resigned to the fact that his belief in the impossibility of self-contradiction is a value he holds onto with no securer grip than that of a believer in the existence of true inconsistencies.

Finally, physicists now estimate that there may be ten to the tenth to the sixteenth power number of "different universes that appear locally uniform," which means that any statement imaginable may exist somewhere in one of them. Therefore, ECQ may be true in reverse; because everything is true and not true, A does not equal A.

Paraconsistency
Citation from the beginning of the Wiki Paraconsistent Logic article: "Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of Aristotle); however, the term paraconsistent ("beside the consistent") was not coined until 1976, by the Peruvian philosopher Francisco Miró Quesada at the Third Latin America Conference on Mathematical Logic.[11] The prefix ‘para’ in English has two general meanings: ‘quasi’ (similar to, modelled on’) or ‘beyond' (beside).  Quesada seems to have had the first meaning in mind; however, many paraconsistentists have taken on the second, which has provided quite differing reasons for its development."

Look at other definitions of the prefix, resulting in a plethora of periphrasis :

alongside of : aside from : faulty : abnormal : associated in a subsidiary or accessory capacity : closely resembling : almost -- all derived from the Greek, 'from'; akin to Greek pro 'before'.

Here's one complete list of para-prefixed words:

This list of "para" prefix meanings (including many that don't apply here) are presented in order to underline the problems that can arise because a cloud of differing term definitions can confuse issues for a novice, especially those as innovative and complex as the ones under question. Even professional mathematicians might be subject to vagaries (see above: "Quesada seems to have had the first meaning in mind")

What all these these words have in common is the fact that they tend to vary or deviate from some normative state defined by the suffix. In other words, they are heterodox to some orthodox. Whether the variance from norm is positive, neutral or negative depends in many cases on the context or circumstances in which the term is used.

This complexity of deviance in the backgground of paraconsistency may prove to be a benefit to professional paraconsistentists because it provides them with a rationale for treating the specific traits of inconsistency that entangle their particular logic or mathematics with individual concern.

This means that not only can logic bugs be separated from the mainstream system for treatment with special procedures as they are now, but that eventually, in reverse, like an infection that spreads back outward from its isolated source, the techniques devised to individually treat the malady may evenually infiltrate the whole system leading to a "paraversion" of classic logic and math until a complete paralogos (see term below) has been constructed.

Paralogos then could become the rich, robust and powerful partner of the mathematics that fueled the scientific revolution resulting in our modern world, with unforseeable consequences.

Paradox
530-40AD; Latin paradoxum, Greek parádoxon, noun use of neuter of parádoxos "unbelievable", literally, "beyond belief". From start of Wiki Paradox article: ""A paradox is a statement that, despite apparently sound reasoning from true premises, leads to an apparently self-contradictory or logically unacceptable conclusion."

Paralogic
From the start of the Wiki article on Faulty Logic (a term which has expropriated the term Paralogic: "A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves" in the construction of an argument. A fallacious argument may be deceptive by appearing to be better than it really is. Some fallacies are committed intentionally to manipulate or persuade by deception, while others are committed unintentionally due to carelessness or ignorance. The soundness of legal arguments depends on the context in which the arguments are made."

Paraloque
A new term for the Logic of Paradox that is needed to highlight that, unlike "Paralogic" (which has come to mean "faulty logic"), the logic of paradox, not as a state but as a process, is not illogical.

Defined
The current crypticities arising from the "paraprecision" of the above term definitions may gain perspective by the introduction of 'Continuity' as a term that wil companion 'Consistency.'  This new philosophical concept will slide all present "para" considerations to one side making room for the future development of a comprehensive theoretical system grounded in the idea of paradox as, not a state, but a higher order process.

'Continuity,' from Google Query [continuity defined]: "forming an unbroken whole; without interruption." Synonyms: unceasing, uninterrupted, unbroken, constant, ceaseless, incessant, steady, sustained, solid, continuing, ongoing, without a break, nonstop, around/round-the-clock, persistent, unremitting, relentless, unrelenting, unabating, unrelieved, without respite, endless, unending, never-ending, perpetual, everlasting, eternal, interminable; More

From Meriam-Webster: uninterrupted extension in space, time, or sequence : continuing without intermission or recurring regularly after minute interruptions.

‎ Cambridge itself: without a pause or interruption: 2. without a pause or break.

Dictionary>com: uninterrupted in time; without cessation.

Oxford: forming an unbroken whole; without interruption,

'Con·sist·ency' from Google Query [consistency defined] : "acting or done in the same way over time, especially so as to be fair or accurate." synonyms: constant, regular, uniform, steady, stable, even, unchanging, undeviating, unfluctuating, compatible with, congruous with, consonant with, in tune with, in line with, reconcilable with."

From Merriam-Webster: always acting or behaving in the same way, of the same quality, good each time.

And Cambridge: always behaving or happening in a similar, especially positive, way.

One term not listed above among the plethora of definitions and synonyms (not only for Continuity and Consistency but for Dialetheism, ECQ, Paraconsistency, Paradox, Paralogic and Paraloque) that could serve to cover them all is "seamless." Among some of the first known human texts that contain cognative content is a Himalayan Vedic script that states "Being is a rip in the perfect fabric of Nonbeing." This rip "seems" to have its own list of synonyms: "Crack, Fracture, Fissure, Rip, Rent, Tear, Break, Discontinuity, Split, Rift, Rive, Slit, Gap, Chasm, Flaw, Fault, Defect, Rupture,  Separation, Breach, Splinter, Crevice, Cleft, Cranny, Chink, Gash, Slash, Opening, Divide, Defect, Error, Blemish and Border," all of which can be placed in a set called "Schizm," a term normally limited to the breaks in Christian tradition between Eastern and Western Orthodoxy as well as the more recent cleft between Catholicism and Protestantism.

Simply stated, the governing principle behind all the terms in the Schizm Set is that there is something slightly Wrong with Everything. How can that be? What is imperfection? What is an error? What is at fault? Who's to blame? To declare that something is not right is to say that it is not what it should be. It is other than its self. That we can stand apart and criticize ourselves indicates that we are divided from ourselves. We are not we.

Therefore, the premise that A does not equal A is valid in all cases, and the types of case are myriad. Consistency has to do with Space; Continuity with Time. Lumps in a soup are Paraconsistencies. A break in Time is a Paracontinuity. Since it is possible to view dimensions of Space as projections of Time, an entire philosophy of fault can arise based on dynamic Time rather than the present considerations of Paraconsistency, which are solely devoted to static Space. Our computers now only uses the Boolean alphabet of eight logic gates; "If Input A is On and Input B is OFF then turn On Output C." But what about "If Input A has Changed and Input B has not then Reverse Output C."

The invasion of the new logical world of Paraconsistency that is at root here by its mirror reflection, Paracontinuity, will pervade the rest of this article as a larger context, framework and perspective than is now available within which to grapple with the ubiquity of anomaly.

Anomaly' from Google Query [anomaly defined] : something that deviates from what is standard, normal, or expected. synonyms: oddity, peculiarity, abnormality, irregularity, inconsistency, incongruity, aberration, quirk, rarity

At Dictionary.com [uh-nom-uh-lee] noun, plural a·nom·a·lies. a deviation from the common rule, type, arrangement, or form. an anomalous person or thing; one that is abnormal or does not fit in: With his quiet nature, he was an anomaly in his exuberant family. an odd, peculiar, or strange condition, situation, quality, etc.

The Cambridge English Dictionary: a person or thing that is different from what is usual, or not in agreement with something else and therefore not satisfactory.

By Merriam-Webster: something different, abnormal, peculiar, or not easily classified.

Schools of Paraconsistent Logic
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