A Note on Paradoxes and Some Applications Ram Prasad At times thoughts in prints, dialogues, conversations and the likes create illusion among people. There may be one reason or the other that causes fallacies. Whenever one attempts to clear the illusion to get the logical end and is unable to, one may slip into the domain of paradoxes. A paradox seemingly may appear absurd or self contradictory that may have in fact a high sense of thought. Here a wide meaning of it including its shades is taken. There is a group of similar sensing words each of which challenges the wit of an onlooker. A paradox sometimes surfaces as and when one is in deep immersion of thought. Unprinted or oral thoughts including paradoxes can rarely survive. Some paradoxes always stay folded to gaily mock on. In deep immersion of thought W S Gilbert remarks on it in the following poetic form - How quaint the ways of paradox At common sense she gaily mocks1 The first student to expect great things of Philosophy only to suffer disillusionment was Socrates (Sokratez) -'what hopes I had formed and how grievously was I disappointed'. In the beginning of the twentieth century mathematicians and logicians rigidly argued on topics which appear possessing intuitively valid but apparently contrary statements. At times when no logical end is seen around and the topic felt hot, more on lookers would enter into these entanglements with argumentative approach. May be, but some 'wise' souls would manage to escape. Zeno's wraths - the Dichotomy, the Achilles, the Arrow and the Stadium made thinkers very uncomfortable all along. The recorded beginning of such nuggets existed in the later Latin thought process. In English language the word paradox, with the same meaning, is derived from the Greek usage paradoxon. Variants of this word with retaining same sense and colorful new meanings may be in uses in languages and dialects written and spoken throughout the world. Within the domain of English language spoken in most parts of the world some of the variants of paradox appear to be antinomy, labyrinth, intertwine, braid, bewilderment, conflict, perplexity, fallacy, illusion, deception et al. F P Ramsey had categorized paradoxes of known shades in to two classes in the year 1926. Madhya Bharti-75, July-December, 2018, ISSN 0974-0066, pp. 177-189 178@e/; Hkkjrh (a) Mathematical or logical paradoxes (b) Semantic or Linguistic paradoxes2 Categorization of paradoxes by Ramsey some hundred years ago was the need of the subject. Today the domain appears to be so hyper complex that a sane attempt in this direction would engulf any onlooker. Among the logical paradoxes Zeno's antinomies are considered be the most disturbing tools. These complexes have been hounding but enlightening philosophers since their appearance. Beside these teasers, some other gems are produced by Galileo Galilei, Burali - Forti, Georg Cantor, Bertrand Russell. Among the semantic paradoxes, the paradox of Epiminides, Gödel's Incompleteness theorems, the paradoxes of Berry, Richards, Grelling - Nelson, poetic riddles, quizzes, pictorial expressions etc aghast people. When one wades through the writing spaces, one appears delighted on seeing many shades of paradoxes some of which may lean on any of the above pluralities. For example the intertwines presented in Upnishads (Upanishads), the Ulatabansiyaan (Writings, which common people feel contrary to mean in their messages) of Kabir, Mira's pains and pleasures ( Mira ke pâdâ) the world of M C Escher3, the Koans of Zen Buddhism and many more sketches & spaces. The entanglements of Zen Buddhism are in tight loops of high order. One such loop is: 'Some subjects cannot be expressed in words and the same cannot be expressed without words'. These hyper flights of imaginations of thinkers, poets, painter, prose writer, et al open spaces for the onlookers to enter in to this fascinated world. Take an example of hyper imagination of thinker - the gossipers. It may amuse to those who dip in such flights: Two monks were arguing on seeing a flag hoisted on a house top. One said, 'The flag is moving'. The other said, 'Not the flag, wind is moving.' A patriarch who just was around corrected them 'Mind is moving - neither the flag, nor the wind'4. The author of these lines notes that how ideas enter from finiteness to seemingly infiniteness or from the physical world into abstract world. There are some more strings that entangle onlookers in messy entanglements of labyrinths. The strings of religious & cultural impacts, tradition bounds, illusions and writing - reflections, inappropriate amalgamation of different subjects, common strings in subjects such as the concept of infinite in logic and mathematics, set paradoxes, language of madness to some extent5, poetic convergences & divergences, sketch spaces, the cartoon escapes and many more parallels including the transliteration effects push sane minds into deceptions. The concept of infiniteness has been a favored attraction among the theologians, dialecticians, mystics, transcendentalist and more scientific bent of minds. It permeates in all mathematics. It is difficult to realize the illusions of infinite that when it had begun to surface. In Vedic sermons there are Richyas in some of the Upanishads which convey deep philosophical quotients. The Eeshava shyopanishad begins and ends with the Shanti Path6. The fathom of the Path can be sensed by seers only. Å¡ iw.kZen% iw.kZfenaiw.kkZRiw.kZeqP;rsA iw.kZL;iw.kZeknk;iw.kZesokof'k";rsA A Note on Paradoxes and Some Applications@179 In this poetic convergence no individual word directly represents the meaning of infinite, however, the sage intrinsically sermons on the concept of infinite. The ParBramha and the KaryaBramha, the sage conveys, are intertwined. This needs to be summed up with some deeper quotient. The divinity of this entanglement can be expressed up to some extent in Hindi language as: Å¡ og ¼ijczã½ iw.kZ gS vkSj ;g ¼dk;Zczã½ Hkh iw.kZ gS( D;ksafd iw.kZ ls iw.kZ dh mRifÙk gksrh gS rFkk og ¼izy; dky esa½ iw.kZ ¼dk;Zczã½ dk iw.kZRo ysdj iw.kZ ¼ijczã½ gh lp jgrk gSA Similarly, in Kenopanishad of the Sam Ved Incomprehensibility of the Bramha is visualized in one such poetic convergence as stated below - ;L;kera rL; era era rL; u osn l%A vfoKkra fotkurka foKkrefotkurke~AA The metric is unfolded as: By him, who thinks Bramha is not comprehended, Bramha is comprehended. That who thinks Bramha is comprehended, does not know Him, so not comprehended. Bramha is unknown to those who think they know Him and is known to those who do not think to know Him. The infinite cannot be comprehended.7 Some groups of Greek philosophers had formed a common opinion that the world of their imagination was formed on the edge of apeiron - the original total disorder that had swept the nature. Long after the apeiron-belief, the Pythagoreans and the Platonics hypothesized that things that existed in the nature can be ordered on the basis of integrals (the numbers 1, 2, 3, ...). In the same land much before the beginning of Christian era, a prodigy who lived in the city named Elea in western Italy, named Zeno, the Zeno of Elea, observed that motion is inherent. It is the same Zeno who rejected the tendencies of Pythagorean, Xenophanean and Perminidean groups. There had been some sixteen known tendencies established by groups of twinkling sages of that land. Zeno felt that motion pervades everywhere so it needs to be understood faultlessly. He established the basic notion of motions which are commonly known as antinomies. In support of his thesis he posed the following arguments - (Z-1): The Dichotomy : There is no motion. Because that which moves must arrive at the middle before it arrives at the end. And it must traverse the half of the half before it reaches the middle and so on (ad infinimum). (Z-2): The Achilles : The slower when running will never be overtaken by the quicker. For that one who is pursuing must first reach the point from which the slower started so that the slower must necessarily always be at some distance ahead always. (Z-3): The Arrow : If everything is either at rest or in motion in space equals to itself and if what moves is away in the instant, the moving arrow is unmoved. The tip of the arrow is in one and only one position at each and every instant of time. In other words, at every instant of time it is at rest. Hence it never moves. The motion is illusory, irregular. (Z-4): The Stadium : Two rows composed of equal number of bodies of equal size pass one another on a race course as they proceed with equal velocity in opposite directions, one arrow starting from the end of the course and the other from the middle. Thus a given line equals its double. 180@e/; Hkkjrh The Zeno's antinomies8 give some mental image of the young prodigy. Saint Thomas Aquinas, Rene Descartes, Leibniz, Spinoza, among others, had given serious thoughts on the disturbing facts placed by Zeno. Most of them concluded that Zeno objects the infinite divisibility of a finite time - space domain and that discreteness exists not. It opines that the world is complex continuous event which opposes the doctrine of multiplicity of Pythagoreans. The European medieval thinkers had no way to deal with the infinitude of entities. The famous puzzle of the time that 'how many angels can dance on the top of a pin?' could be viewed as a question about the relationship between the infinite creator and the finite world. This mindset continued among their successive generations. Later on establishing & realizing that a line segment may contain infinite number of points, the next question came to the mind of thinkers, if two circles concentric or not, are such that one contains the other then the number of points on the circumference of the outer circle must be more than the number of points that are in the circumference of the inner circle.