Infinity

Infinity (symbol: ∞ {displaystyle infty } ) is a concept describing something without any bound, or something larger than any natural number. Philosophers have speculated about the nature of the infinite, for example Zeno of Elea, who proposed many paradoxes involving infinity, and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion. This idea is also at the basis of infinitesimal calculus.Mathematics is the science of the infinite. Infinity (symbol: ∞ {displaystyle infty } ) is a concept describing something without any bound, or something larger than any natural number. Philosophers have speculated about the nature of the infinite, for example Zeno of Elea, who proposed many paradoxes involving infinity, and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion. This idea is also at the basis of infinitesimal calculus. At the end of 19th century, Georg Cantor introduced and studied infinite sets and infinite numbers, which are now an essential part of the foundation of mathematics. For example, in modern mathematics, a line is commonly viewed as the set of all its points, and their infinite number (the cardinality of the line) is larger than the number of integers. Thus the mathematical concept of infinity refines and extends the old philosophical concept. It is used everywhere in mathematics, even in areas such as combinatorics and number theory that may seem to have nothing to do with it. For example, Wiles's proof of Fermat's Last Theorem uses the existence of very large infinite sets. The concept of infinity is also used in physics and the other sciences. Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept. The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus. He used the word apeiron which means infinite or limitless. However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea (born c. 490 BCE), a pre-Socratic Greek philosopher of southern Italy and member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, described by Bertrand Russell as 'immeasurably subtle and profound'. In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the actual infinity; for example, instead of saying that there are an infinity of primes, Euclid prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers. The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders: In this work, two basic types of infinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asaṃkhyāta ('countless, innumerable') and ananta ('endless, unlimited'), between rigidly bounded and loosely bounded infinities. European mathematicians started using infinite numbers and expressions in a systematic fashion in the 17th century. In 1655 John Wallis first used the notation ∞ {displaystyle infty } for such a number in his De sectionibus conicis and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of 1 ∞ . {displaystyle { frac {1}{infty }}.} But in Arithmetica infinitorum (1655 also) he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending '&c.' For example, '1, 6, 12, 18, 24, &c.'