Eternal return

Eternal return is a theory that the universe and all existence and energy has been recurring, will continue to recur, in a self-similar form an infinite number of times across infinite time or space. The theory is found in Indian philosophy and in ancient Egypt and was subsequently taken up by the Pythagoreans and Stoics. With the decline of antiquity and the spread of Christianity, the theory fell into disuse in the Western world, with the exception of Friedrich Nietzsche, who connected the thought to many of his other concepts, including amor fati. Eternal return relates to the philosophy of predeterminism in that people are predestined to continue repeating the same events over and over again; the basic premise proceeds from the assumption that the probability of a world coming into existence like our own is nonzero. If space and time are infinite it follows logically that our existence must recur an infinite number of times. In 1871 Louis Auguste Blanqui, assuming a Newtonian cosmology where time and space are infinite, claimed to have demonstrated eternal recurrence as a mathematical certainty.

In ancient Egypt, the scarab was viewed as a sign of eternal renewal and reemergence of life, a reminder of the life to come. The Mayans and Aztecs took a cyclical view of time. In ancient Greece, the concept of eternal return was connected with Empedocles, Zeno of Citium, most notably in Stoicism; the concept of cyclical patterns is prominent in Indian religions, such as Jainism, Hinduism and Buddhism among others. The important distinction is that events don't repeat endlessly but souls take birth until they attain salvation; the wheel of life represents an endless cycle of birth and death from which one seeks liberation. In Tantric Buddhism, a wheel of time concept known as the Kalachakra expresses the idea of an endless cycle of existence and knowledge; the concept of "eternal recurrence", the idea that with infinite time and a finite number of events, events will recur again and again infinitely, is central to the writings of Friedrich Nietzsche. As Heidegger points out in his lectures on Nietzsche, Nietzsche's first mention of eternal recurrence, in aphorism 341 of The Gay Science, presents this concept as a hypothetical question rather than postulating it as a fact.

According to Heidegger, it is the burden imposed by the question of eternal recurrence—whether or not such a thing could be true—that is so significant in modern thought: "The way Nietzsche here patterns the first communication of the thought of the'greatest burden' makes it clear that this'thought of thoughts' is at the same time'the most burdensome thought.' "The thought of eternal recurrence appears in a few of his works, in particular §285 and §341 of The Gay Science and in Thus Spoke Zarathustra. The most complete treatment of the subject appears in the work entitled Notes on the Eternal Recurrence, a work, published in 2007 alongside Søren Kierkegaard's own version of eternal return, which he calls'repetition'. Nietzsche sums up his thought most succinctly when he addresses the reader with: "Everything has returned. Sirius, the spider, thy thoughts at this moment, this last thought of thine that all things will return". However, he expresses his thought at greater length when he says to his reader: "Whoever thou mayest be, beloved stranger, whom I meet here for the first time, avail thyself of this happy hour and of the stillness around us, above us, let me tell thee something of the thought which has risen before me like a star which would fain shed down its rays upon thee and every one, as befits the nature of light.

- Fellow man! Your whole life, like a sandglass, will always be reversed and will run out again, - a long minute of time will elapse until all those conditions out of which you were evolved return in the wheel of the cosmic process, and you will find every pain and every pleasure, every friend and every enemy, every hope and every error, every blade of grass and every ray of sunshine once more, the whole fabric of things which make up your life. This ring in which you are but a grain will glitter afresh forever, and in every one of these cycles of human life there will be one hour where, for the first time one man, many, will perceive the mighty thought of the eternal recurrence of all things:- and for mankind this is always the hour of Noon". This thought is indeed noted in a posthumous fragment; the origin of this thought is dated by Nietzsche himself, via posthumous fragments, to August 1881, at Sils-Maria. In Ecce Homo, he wrote that he thought of the eternal return as the "fundamental conception" of Thus Spoke Zarathustra.

Several authors have pointed out other occurrences of this hypothesis in contemporary thought. Rudolf Steiner, who revised the first catalogue of Nietzsche's personal library in January 1896, pointed out that Nietzsche would have read something similar in Eugen Dühring's Courses on philosophy, which Nietzsche criticized. Lou Andreas-Salomé pointed out that Nietzsche referred to ancient cyclical conceptions of time, in particular by the Pythagoreans, in the Untimely Meditations. Henri Lichtenberger and Charles Andler have pinpointed three works contemporary to Nietzsche which carried on the same hypothesis: J. G. Vogt, Die Kraft. Eine real-monistische Weltanschauung, Auguste Blanqui, L'éternité par les astres and Gustave Le Bon, L'homme et les sociétés. Walter Benjamin juxtaposes Blanqui and Nietzsche's discussion of eternal recurrence in his unfinished, monumental work The Arcades Project. However, Gustave Le Bon is not quoted anywhere in Nietzsche's manuscripts.

Radial recurrent artery

The radial recurrent artery arises from the radial artery below the elbow. It ascends between the branches of the radial nerve, lying on the supinator muscle and between the brachioradialis muscle and the brachialis muscle, supplying these muscles and the elbow-joint, anastomosing with the terminal part of the profunda brachii; this article incorporates text in the public domain from page 594 of the 20th edition of Gray's Anatomy lesson4arteriesofarm at The Anatomy Lesson by Wesley Norman lesson4artofforearm at The Anatomy Lesson by Wesley Norman

Mathematical induction

Mathematical induction is a mathematical proof technique. It is used to prove that a property P holds for every natural number n, i.e. for n = 0, 1, 2, 3, so on. Metaphors can be informally used to understand the concept of mathematical induction, such as the metaphor of falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung and that from each rung we can climb up to the next one; the method of induction requires two cases to be proved. The first case, called the base case, proves that the property holds for the number 0; the second case, called the induction step, proves that, if the property holds for one natural number n it holds for the next natural number n + 1. These two steps establish the property P for every natural number n = 0, 1, 2, 3... The base step need not begin with zero, it begins with the number one, it can begin with any natural number, establishing the truth of the property for all natural numbers greater than or equal to the starting number.

The method can be extended to prove statements about more general well-founded structures, such as trees. Mathematical induction in this extended sense is related to recursion. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs. Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning as used in philosophy. Mathematical induction is an inference rule used in formal proofs. Proofs by mathematical induction are, in fact, examples of deductive reasoning. In 370 BC, Plato's Parmenides may have contained an early example of an implicit inductive proof; the earliest implicit traces of mathematical induction may be found in Euclid's proof that the number of primes is infinite and in Bhaskara's "cyclic method". An opposite iterated technique, counting down rather than up, is found in the Sorites paradox, where it was argued that if 1,000,000 grains of sand formed a heap, removing one grain from a heap left it a heap a single grain of sand forms a heap.

An implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem and properties of Pascal's triangle. None of these ancient mathematicians, explicitly stated the induction hypothesis. Another similar case was that of Francesco Maurolico in his Arithmeticorum libri duo, who used the technique to prove that the sum of the first n odd integers is n2; the first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique. Another Frenchman, made ample use of a related principle, indirect proof by infinite descent; the induction hypothesis was employed by the Swiss Jakob Bernoulli, from on it became more or less well known. The modern rigorous and systematic treatment of the principle came only in the 19th century, with George Boole, Augustus de Morgan, Charles Sanders Peirce, Giuseppe Peano, Richard Dedekind; the simplest and most common form of mathematical induction infers that a statement involving a natural number n holds for all values of n.

The proof consists of two steps: The base case: prove that the statement holds for the first natural number n0. N0 = 0 or n0 = 1; the step case or inductive step: prove that if the statement holds for any n ≥ n0, it holds for n+1. In other words, assume the statement holds for some arbitrary natural number n ≥ n0, prove that the statement holds for n + 1; the hypothesis in the inductive step, that the statement holds for some n, is called the induction hypothesis or inductive hypothesis. To prove the inductive step, one assumes the induction hypothesis and uses this assumption, involving n, to prove the statement for n + 1. Whether n = 0 or n = 1 is taken as the standard base case depends on the preferred definition of the natural numbers. In the fields of combinatorics and mathematical logic it is common to consider 0 as a natural number. Mathematical induction can be used to prove that the following statement, P, holds for all natural numbers n. 0 + 1 + 2 + ⋯ + n = n 2. P equal to number n; the proof that P is true for each natural number n proceeds as follows.

Base case: Show that the statement holds for n = 0. P is seen to be true: 0 = 0 ⋅ 2. Inductive step: Show that if P holds also P holds; this can be done. Assume P holds, it must be shown that P holds, that is

Historic recurrence

Historic recurrence is the repetition of similar events in history. The concept of historic recurrence has variously been applied to the overall history of the world, to repetitive patterns in the history of a given polity, to any two specific events which bear a striking similarity. Hypothetically, in the extreme, the concept of historic recurrence assumes the form of the Doctrine of Eternal Recurrence, written about in various forms since antiquity and was described in the 19th century by Heinrich Heine and Friedrich Nietzsche. While it is remarked that "History repeats itself", in cycles of less than cosmological duration this cannot be true. In this interpretation of recurrence, as opposed to the Nietzschean interpretation, there is no metaphysics. Recurrences take place due to ascertainable chains of causality. An example of the mechanism is the ubiquitous phenomenon of multiple independent discovery in science and technology, described by Robert K. Merton and Harriet Zuckerman. G. W. Trompf, in his book The Idea of Historical Recurrence in Western Thought, traces recurring patterns of political thought and behavior in the west since antiquity.

If history has lessons to impart, they are to be found par excellence in such recurring patterns. Historic recurrences of the "striking-similarity" type can sometimes induce a sense of "convergence", "resonance" or déjà vu. Prior to the theory of historic recurrence, offered by Polybius, a Greek Hellenistic historian, ancient western thinkers who had thought about recurrence had been concerned with cosmological rather than historic recurrence. Western philosophers and historians who have discussed various concepts of historic recurrence include Polybius, the Greek historian and rhetorician Dionysius of Halicarnassus, Luke the Evangelist, Niccolò Machiavelli, Giambattista Vico, Arnold J. Toynbee. An eastern concept that bears a kinship to western concepts of historic recurrence is the Chinese concept of the Mandate of Heaven, by which an unjust ruler will lose the support of Heaven and be overthrown. G. W. Trompf describes various historic paradigms of historic recurrence, including paradigms that view types of large-scale historic phenomena variously as "cyclical".

He notes "he view proceeding from a belief in the uniformity of human nature. It holds that because human nature does not change, the same sort of events can recur at any time." "Other minor cases of recurrence thinking," he writes, "include the isolation of any two specific events which bear a striking similarity, the preoccupation with parallelism, that is, with resemblances, both general and precise, between separate sets of historical phenomena." G. W. Trompf notes that most western concepts of historic recurrence imply that "the past teaches lessons for... future action"—that "the same... sorts of events which have happened before... will recur..." One such recurring theme was early offered by Poseidonius, who argued that dissipation of the old Roman virtues had followed the removal of the Carthaginian challenge to Rome's supremacy in the Mediterranean world. The theme that civilizations flourish or fail according to their responses to the human and environmental challenges that they face, would be picked up two thousand years by Toynbee.

Dionysius of Halicarnassus, while praising Rome at the expense of her predecessors—Assyria, Media and Macedonia—anticipated Rome's eventual decay. He thus implied the idea of recurring decay in the history of world empires—an idea, to be developed by the Greek historian Diodorus Siculus and by Pompeius Trogus, a 1st-century BCE Roman historian from a Celtic tribe in Gallia Narbonensis. By the late 5th century, Zosimus could see the writing on the Roman wall, asserted that empires fell due to internal disunity, he gave examples from the histories of Macedonia. In the case of each empire, growth had resulted from consolidation against an external enemy. With Rome's world dominion, aristocracy had been supplanted by a monarchy, which in turn tended to decay into tyranny; the Roman Empire, in its western and eastern sectors, had become a contending ground between contestants for power, while outside powers acquired an advantage. In Rome's decay, Zosimus saw history repeating itself in its general movements.

The ancients developed an enduring metaphor for a polity's evolution: they drew an analogy between an individual human's life cycle, developments undergone by a body politic. This metaphor was offered, in varying iterations, by Cicero, Seneca and Ammianus Marcellinus; this social-organism metaphor would recur centuries in the works of Émile Durkheim and Herbert Spencer. Niccolò Machiavelli, about to analyze the vicissitudes of Florentine and Italian politics between 1434 and 1494, described recurrent oscillations between "order" and "disorder" within states: when states have arrived at their greatest perfection, they soon begin to decline. In the same manner, having been

Recursion

Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic; the most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this defines an infinite number of instances, it is done in such a way that no loop or infinite chain of references can occur. In mathematics and computer science, a class of objects or methods exhibit recursive behavior when they can be defined by two properties: A simple base case —a terminating scenario that does not use recursion to produce an answer A set of rules that reduce all other cases toward the base caseFor example, the following is a recursive definition of a person's ancestors: One's parents are one's ancestors; the ancestors of one's ancestors are one's ancestors. The Fibonacci sequence is a classic example of recursion: Fib = 0 as base case 1, Fib = 1 as base case 2, For all integers n > 1, Fib:= Fib + Fib.

Many mathematical axioms are based upon recursive rules. For example, the formal definition of the natural numbers by the Peano axioms can be described as: 0 is a natural number, each natural number has a successor, a natural number. By this base case and recursive rule, one can generate the set of all natural numbers. Recursively defined mathematical objects include functions and fractals. There are various more tongue-in-cheek "definitions" of recursion. Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be'recursive'. To understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules; the running of a procedure involves following the rules and performing the steps. An analogy: a procedure is like a written recipe. Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure.

For instance, a recipe might refer to cooking vegetables, another procedure that in turn requires heating water, so forth. However, a recursive procedure is where one of its steps calls for a new instance of the same procedure, like a sourdough recipe calling for some dough left over from the last time the same recipe was made; this creates the possibility of an endless loop. If properly defined, a recursive procedure is not easy for humans to perform, as it requires distinguishing the new from the old invocation of the procedure. For this reason recursive definitions are rare in everyday situations. An example could be the following procedure to find a way through a maze. Proceed forward until reaching either an exit or a branching point. If the point reached is an exit, terminate. Otherwise try each branch in turn, using the procedure recursively. Whether this defines a terminating procedure depends on the nature of the maze: it must not allow loops. In any case, executing the procedure requires recording all explored branching points, which of their branches have been exhaustively tried.

Linguist Noam Chomsky among many others has argued that the lack of an upper bound on the number of grammatical sentences in a language, the lack of an upper bound on grammatical sentence length, can be explained as the consequence of recursion in natural language. This can be understood in terms of a recursive definition of a syntactic category, such as a sentence. A sentence can have a structure in which what follows the verb is another sentence: Dorothy thinks witches are dangerous, in which the sentence witches are dangerous occurs in the larger one. So a sentence can be defined recursively as something with a structure that includes a noun phrase, a verb, optionally another sentence; this is just a special case of the mathematical definition of recursion. This provides a way of understanding the creativity of language—the unbounded number of grammatical sentences—because it predicts that sentences can be of arbitrary length: Dorothy thinks that Toto suspects that Tin Man said that.... There are many structures apart from sentences that can be defined recursively, therefore many ways in which a sentence can embed instances of one