An optimist is someone who believes this is the best of all possible worlds, and a pessimist is someone whoâ??s afraid he is right!
Annonymous

Paradox has different meanings depending upon the context in which we use the word and who you ask. Unfortunately people also have different ideas about exactly what reason and truth are which makes defining paradox all that much more difficult. Broadly paradox refers to the irrational, inexplicable, self-referential and self-contradictory, or merely contradictory but somehow true. It may be that, despite numerous definitions and widespread use of the term, paradox is ultimately ineffable. Thousands of years ago in ancient Greece formal logic and mathematics were first developed and, when this occurred, it began a heated conflict over paradox that is still going on to this day.

Much of western philosophy and this quarrel over paradox originated with Heraclitus, and the widespread use of the relatively new invention of writing. He compared life to a flowing river and wrote that existence was unified in some sense like the endless ebb and flow of a river. Within this unifying law of flux and change, he proposed that balanced opposites produced the laws of nature and moral laws as well.

Opposites such as up and down, he pointed out, were simultaneously one and the same thing. This is a variation of the paradoxical Asian principle of Yin and Yang, and Heraclitusâ?? philosophy resembles what is sometimes called Energetic Taoism. The motto of such philosophers can be summed up in the phrase, â??Change is the only constant.â? Heraclitus was well aware of this paradox at the heart of his philosophy and, like his Asian counterparts, contented himself with allegorical arguments and inferential logic. Those who admired his work, however, often took his thoughts and ideas to radically different conclusions.

One of these, Zeno of Elias, was the first to dramatically demonstrate that whatever explanation one put forward for existence led to a paradox. Zenoâ??s pantheism took the opposite view of existence from that of Heraclitus and asserted a yet again more radically paradoxical worldview. He argued that the universe is in reality indivisible, indestructible, eternal, and unchanging. Like his teacher before him, Parmenides, the thought of a constantly changing universe with all its messy complexity he felt was vulgar and undignified. To defend this belief in an unchanging universe, he invented several deceptively simple paradoxes demonstrating that the idea of motion and, therefore change, is absurd.

Despite this shocking denial of the constant ebb and flow Heraclitus had praised and we all observe in everyday life, Zenoâ??s paradoxes defied rigorous challenge until the invention of calculus millennia later and are still being debated to this day. Arguments such as Zeno presented revolve around what philosophers call reductio ad absurdum or what I shall refer to here as the â??backdoorâ? argument. Rather than directly proving something, you sneak in the backdoor and show the alternatives are patently ridiculous. For example, if I wish to prove its possible for me to argue in general, I could take the backdoor approach and demonstrate how ridiculous it is for me to argue I canâ??t argue.

This kind of backdoor approach of demonstrating that all the alternatives are at least equally absurd, is terrific for defending any number of beliefs you might have no matter how absurd and were the standard fare in ancient Greece. In other cases such as Zenoâ??s, whichever argument is the more entertaining can often win in the eyes of an audience. At other times, like children taunting each other on the playground, heated debate can ensue over just whoâ??s ideas are more absurd and snappy comebacks can sometimes settle the issue. This is precisely why Zeno was so successful defending his beliefs as Kurt Godel proved an eon later. The foundations of logic, Godel laboriously proved, are ultimately based on our faith in them, their emotional impact, and suspension of disbelief as much as anything else.

As difficult as it was to win an argument with Zeno, his philosophy was pretty useless at the time except for entertainment purposes and exasperating people wanting to prove alternative philosophies. Looking for a good challenge that might inspire new ideas, many made the attempt nonetheless. As a result it was mostly rebellious and argumentative young men who adopted Zenoâ??s ascetic lifestyle and absurd philosophy. After a while they usually rejoined society to focus on more constructive things.

Others like Socrates pointedly ignored Zenoâ??s backdoor challenge to common sense and, instead, championed logic as a useful idea for promoting common sense in addition to being entertaining. He became the first to popularize a system of inquiry and logic around the idea that paradoxes like Zenoâ??s are inherently wrong and, in doing so, originated the division between eastern and western philosophies. In the east paradox was commonly admired and revered while in the west it was to become increasingly viewed as predominantly counterproductive and, at times, downright dangerous and heretical.

Meanwhile the Pythagorean mystics used these evolving prejudices against certain paradoxes and their uses, to invent most of the foundations of mathematics and physics. Euclid as well put them to practical mathematical use to establish geometry and algebra. Later Plato combined this growing mathematical prowess with Socratesâ?? philosophy into an aesthetic and ethical spiritual philosophy of his own arguably still the most popular to ever come out of the west. However, he could not do so without also incorporating the paradoxical concept of infinity into his philosophy. It was he who first established infinity as the most valid spiritual conceptualization of paradox in the western tradition and the least opposed to the practical use of logic.

Platoâ??s student, Aristotle, then took this process of relegating various paradoxes to the sidelines to the next obvious level and significantly limited its use for academic practical purposes. Applying Zenoâ??s emotionally persuasive backdoor approach to the subject of logic itself, Aristotle demonstrated that unless we assume whatever we are talking about is either true or false, the result would always lead to contradictions and paradoxes. Ironically, by appealing to the increasing faith among people that some absurdities are contestable, Aristotle created a proof that the genuinely absurd and contestable cannot exist.

Aristotle did this knowing full well that earlier two apparently genuine paradoxes had already been discovered by academics that could not be described as either true or false as his logic demanded: The so-called Cretan Liarâ??s Paradox and Sorites Heap Paradox which, have not been resolved to this day. The Sorties Paradox is related to Zenoâ??s paradoxes on motion and was relatively easy to brush off as merely a problem with semantics or logic that would eventually be resolved, but the Cretan Liarâ??s Paradox presented a direct contradiction to the validity of Aristotleâ??s logic. The assertion, â??This statement is falseâ? is widely considered today to quite likely be a genuine paradox that defies Aristotleâ??s black and white view of the world. If true it must be false, but if false then it must be true and so on.

Thus by examining both the basic assumptions and conclusions of Aristotleâ??s logic we are lead inexorably back to the use of the backdoor argument and Zenoâ??s proof of the paradox of existence. The Cretan Liarâ??s Paradox can be restated as, â??This Statement cannot be proven to be true.â? By then broadening its scope to include the existence we can make absurd statements such as, â??Existence cannot be proven to be true,â? â??Existence is false,â? and so on. Likewise, the Sorites Heap Paradox implies at the very least the concept of existence requires a more accurate definition and, from an extreme interpretation, that existence is ultimately genuinely paradoxical and indefinable. That both paradoxes can be so easily applied to existence in a large variety of ways indicates the depth of the paradox.

Despite these pervasive absurdities, Aristotleâ??s logic proved incredibly useful and he went on to combine it with shrewd observations of nature that allowed him to reorganize the various branches of the sciences. In the process of accomplishing this monumental feat, he successfully banished a huge number of additional paradoxes from academic use, most notably many of the uses for the concept of infinity. His logic and reorganizations of the sciences were so compatible with the already well established direction western thought was moving in, that despite a number of other weak arguments they contained no one seriously attempted to correct them.

Formal western logic then, can be clearly and concisely described as a science of the absurd. A practical way of conceptualizing and organizing absurdities into abstract hierarchies we call logical that reflect to varying degrees commonalties in our perception of existence. Like so many good ideas, however, Aristotle and others took this wonderfully practical tool to tautological extremes and promoted it as a kind of fundamentalist belief reflecting an absolute reality. Like many forms of fundamentalism, they then proceeded to attack other philosophies that threatened to undermine their own.

Caught in the middle of this mounting battle over paradox was Democritus, the most famous of the Atomists. He argued that existence is random, an idea similar to what Quantum Mechanics proposes today. First Zeno pounded away at the inherent paradoxes of the Atomists and, later, Plato used his political influence with the Romans to have all of Democritusâ?? books burned as â??ugly and demeaning.â? Rigidly circumscribed concepts of beauty, ethics, observation, logical inquiry, and human dignity steadily began to dominate academia while all else was thoroughly condemned. Zenoâ??s ideas subsequently lost their popular following as his absurdities became increasingly viewed as crude humor rather than the subject of serious academic inquiry. When the early Christians then burned down the library of Alexandria all but a few of the remaining copies of the seventy books Democritus had written were lost forever.

It was another of these rational philosophers, Augustine, who systematically integrated these biases against the chaotic, ugly, amoral, and paradoxical into the Catholic Church and, in so doing, firmly established the pursuit of truth, justice, and aesthetics as the western tradition mandated not only by the hard realities of life but God himself. Thus, Christian fundamentalism was born. For the next millennia western philosophy, arts, and sciences largely ignored paradox as the Islamic world also adopted the new logic and its own fundamentalist attitudes and religious emphasis on infinity and more rational ideas.

Eventually, the arrival of Newtonâ??s theory of motion and his new calculus drew the attention of western academia back to the subject of paradox and the absurd. Newton succeeded in profoundly unifying vast panoramas of natural phenomena and mathematics believed at the time by the fundamentalist west to be entirely unrelated. He successfully incorporated not only paradoxical infinities into his mathematics, but also an ethereal vision of space and time, and an almost magical view of the action-at-a-distance of gravity. From these quasi-supernatural ingredients he fashioned an incontestably powerful yet starkly mechanical science.

Had Henry the Eighth not previously kicked the Catholic Church out of England and his new inventions not proven so overpoweringly successful, he might have been imprisoned on the spot for such heresy. Newton himself was not pleased with having to use concepts that could be interpreted as supernatural and absurd, but could think of no viable alternatives. As his predecessors had, he simply denied their irrational and poorly defined nature had any validity and insisted these difficulties would be cleared up someday.

At the same time, Descartes combined the geometry and algebra of the early Greeks to promote an entirely mechanistic philosophy of existence. Of course, Descartes also ignored the inherent paradox of existence undermining the foundations of his philosophy and was forced to flee the unforgiving wrath of the Catholic Church. Fundamentalist religion and fundamentalist science began to diverge and go their separate ways. The result was that once again the more useful rational philosophies progressively dominated academia for the next three hundred years, despite their increasing resemblance to a giant house of cards built on quicksand, and began to replace religion as the dominant force shaping western society. Although these newly emerging rational philosophies no longer forbade the use of paradoxical infinities, ironically they asserted more stalwartly than ever before that paradox was intrinsically false and, perhaps, infinity wasnâ??t as irrational as they had assumed.

Another philosopher, Spinoza, then proposed an elegant and ethical philosophy that proved extremely compatible with Newtonian Mechanics and logic, but had a strikingly paradoxical Asian flavor similar to those of Zeno and Heraclitus. Spinoza became the first public example of the social upheaval and conflict this renewed interest in infinity, paradox, and science were leading up to. Pantheism had become so thoroughly alien and unacceptable in the fundamentalist west after all this time that among others Newtonâ??s close compatriot, Leibnitz, publicly lambasted poor Spinoza ruthlessly. As a result of all these difficulties, Pantheism languished for three hundred years.

One philosopher who did not ignore Zenoâ??s proof or the dominant cultural imperative was Hegel. Capitalizing on the newly increased stature of both logic and infinity, Hegel created a scientific philosophy more compatible with Christianity than Spinozaâ??s Pantheism. In great detail he knowingly put forth the tautological argument that, â??What is rational is real and what is real is rational.â? Thus he restored the scope of respectable philosophy once again to include existence itself and blurred the lines between the academic use of potential infinities and the religious use of actual infinities.

Hegel successfully adapted Heraclitusâ?? argument of complementary opposites into a synergistic dialectical logic. One idea, he asserted, generates its own opposite, which then together generate entirely new ideas. Within this infinitely expanding field of ideas our job, he believed, is to understand the whole, which he defined as a rational but infinite and absolute reality or, in other words, God. Towards accomplishing that purpose, Hegel invented a slight modification to Aristotleâ??s logic, which the west had tacitly accepted virtually unchallenged, that everything is either true or false.

True or false, Hegel insisted in all seriousness, is a false dichotomy in some sense and between them lies another creative realm of infinite possibility. Instead of focusing only on what can be measured and observed, we should be focusing on ideas and idealism. Rather than merely denying the validity of Aristotleâ??s logic it expanded upon it in a way the fundamentalist Christian, Islamic, and academic worlds could support, but in the process it opened a whole new paradoxical can of worms.

Three hundred years after Newton, Einsteinâ??s pantheistic theory of Relativity brought hope to many that the controversy over paradox and existence would finally be laid to rest as philosophy became increasingly integrated with the quantitative sciences. Concurrent with the discovery of Relativity, however, another more powerful physical theory began taking shape that was outrageously paradoxical, culturally unacceptable, and entirely irrational and inconceivable. The introduction of Quantum Mechanics threw a colossal monkey wrench into the western passion for mechanistic and geometric philosophies of any sort. In the tradition of his Greek forefathers, Einstein criticized the theory as so irrational, ugly, and meaningless it just couldnâ??t be true. Others retorted with comparisons of the theory to pursuits like jazz and the avant-garde, which obviously not everyone is capable of appreciating.

To this day exactly what constitutes a paradox remains a controvercial subject in western culture. Being devoted to strict definitions, logicians define paradox as the demonstrably self-referential and self-contradictory. However, logic itself has been based on natural language, the language of everyday people, which has as I mentioned in the beginning a much broader definition of paradox.

Thus, a race has ensued between the linguistic and logistic approaches to once and for all define exactly what is and is not a paradox. It may be that we will forever have to settle for constant revisions of our definition, but currently it is the subject of not only academic debate, but national security as well. As we refine our definitions of paradox both logistically and linguistically we clarify the boundaries of known and unknown. Often, what we don't know is of more intense interest than what we do know and, paradoxically, because we are ignorant we may learn.