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89-497: From the time of the ancient Greeks , the philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus , mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli ) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with

178-631: A hyperplane at infinity for general dimensions , each consisting of points at infinity . In complex analysis the symbol ∞ {\displaystyle \infty } , called "infinity", denotes an unsigned infinite limit . The expression x → ∞ {\displaystyle x\rightarrow \infty } means that the magnitude  | x | {\displaystyle |x|} of  x {\displaystyle x} grows beyond any assigned value. A point labeled ∞ {\displaystyle \infty } can be added to

267-583: A finite and infinite amount of objects, atomic theory shows that objects are made from a specific number of atoms that form specific elements. Likewise, Zeno's arguments against motion have been challenged by modern mathematics and physics. Mathematicians and philosophers continued studying infinitesimals until they came to be better understood through calculus and limit theory . Ideas relating to Zeno's plurality arguments are similarly affected by set theory and transfinite numbers . Modern physics has yet to determine whether space and time can be represented on

356-514: A finite number of objects, there must be an infinite number of objects dividing them. For two objects to exist separately, according to Zeno, there must be a third thing dividing them, otherwise they would be parts of the same thing. This dividing thing would then itself need two dividing objects to separate it from the original objects. These new dividing objects would then need dividing objects, and so on. As with all other aspects of existence, Zeno argued that location and physical space are part of

445-405: A finite size, as there would always be a smaller part to take from it. Zeno also argued from the other direction: if objects do not have mass, then they cannot be combined to create something larger. In another argument, Zeno proposed that multiple objects cannot exist, because it would require an infinite number of objects to have a finite number of objects; he held that in order for there to be

534-790: A group. Almost half of the material in Euclid 's Elements is customarily attributed to the Pythagoreans, including the discovery of irrationals, attributed to Hippasus (c. 530–450 BC) and Theodorus (fl. 450 BC). The greatest mathematician associated with the group, however, may have been Archytas (c. 435-360 BC), who solved the problem of doubling the cube , identified the harmonic mean , and possibly contributed to optics and mechanics . Other mathematicians active in this period, not fully affiliated with any school, include Hippocrates of Chios (c. 470–410 BC), Theaetetus (c. 417–369 BC), and Eudoxus (c. 408–355 BC). Greek mathematics also drew

623-407: A kind of brotherhood. Pythagoreans supposedly believed that "all is number" and were keen in looking for mathematical relations between numbers and things. Pythagoras himself was given credit for many later discoveries, including the construction of the five regular solids . However, Aristotle refused to attribute anything specifically to Pythagoras and only discussed the work of the Pythagoreans as

712-410: A limit, infinity can be also used as a value in the extended real number system. Points labeled + ∞ {\displaystyle +\infty } and − ∞ {\displaystyle -\infty } can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us

801-489: A mathematical continuum or if it is made up of discrete units. Zeno's argument of Achilles and the tortoise can be addressed mathematically, as the distance is defined by a specific number. His argument of the flying arrow has been challenged by modern physics, which allows the smallest instants of time to still have a minuscule non-zero duration. Other mathematical ideas, such as internal set theory and nonstandard analysis , may also resolve Zeno's paradoxes. However, there

890-503: A mathematico-philosophic address given in 1930 with: Mathematics is the science of the infinite. The infinity symbol ∞ {\displaystyle \infty } (sometimes called the lemniscate ) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY ( ∞ ) and in LaTeX as \infty . It

979-542: A philosophical concept. The earliest recorded idea of infinity in Greece may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron , which means "unbounded", "indefinite", and perhaps can be translated as "infinite". Aristotle (350 BC) distinguished potential infinity from actual infinity , which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view,

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1068-500: A quantum system by observing it is usually called the Quantum Zeno effect as it is strongly reminiscent of Zeno's arrow paradox. In the field of verification and design of timed and hybrid systems , the system behavior is called Zeno if it includes an infinite number of discrete steps in a finite amount of time. Zeno's arguments against plurality have been challenged by modern atomic theory . Rather than plurality requiring both

1157-400: A satisfactory definition of a limit and a proof that, for 0 < x < 1 , a + a x + a x 2 + a x 3 + a x 4 + a x 5 + ⋯ = a 1 − x . {\displaystyle a+ax+ax^{2}+ax^{3}+ax^{4}+ax^{5}+\cdots ={\frac {a}{1-x}}.} Suppose that Achilles

1246-451: A series of paradoxes that used reductio ad absurdum arguments, or arguments that disprove an idea by showing how it leads to illogical conclusions. Furthermore, Zeno's philosophy makes use of infinitesimals , or quantities that are infinitely small while still being greater than zero. Criticism of Zeno's ideas may accuse him with using rhetorical tricks and sophistry rather than cogent arguments. Critics point to how Zeno describes

1335-618: A similar topology . If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough. The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation . To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe. Greek mathematics Greek mathematics refers to mathematics texts and ideas stemming from

1424-456: A standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid ) that the whole cannot be the same size as the part. (However, see Galileo's paradox where Galileo concludes that positive integers cannot be compared to the subset of positive square integers since both are infinite sets.) An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity

1513-463: Is a similar controversy concerning Euclid's parallel postulate , sometimes translated: If a straight line falling across two [other] straight lines makes internal angles on the same side [of itself whose sum is] less than two right angles, then the two [other] straight lines, being produced to infinity, meet on that side [of the original straight line] that the [sum of the internal angles] is less than two right angles. Other translators, however, prefer

1602-787: Is an important difference between Greek mathematics and those of preceding civilizations. Greek mathēmatikē ("mathematics") derives from the Ancient Greek : μάθημα , romanized :  máthēma , Attic Greek : [má.tʰɛː.ma] Koinē Greek : [ˈma.θi.ma] , from the verb manthanein , "to learn". Strictly speaking, a máthēma could be any branch of learning, or anything learnt; however, since antiquity certain mathēmata (mainly arithmetic, geometry, astronomy, and harmonics) were granted special status. The origins of Greek mathematics are not well documented. The earliest advanced civilizations in Greece and Europe were

1691-593: Is another one of the main sources of present day knowledge about Zeno. Zeno is one of three major philosophers in the Eleatic school, along with Parmenides and Melissus of Samos . This school of philosophy was a form of monism , following Parmenides' belief that all of reality is one single indivisible object. Both Zeno and Melissus engaged in philosophy to support the ideas of Parmenides. While Melissus sought to build on them, Zeno instead argued against opposing ideas. Such arguments would have been constructed to challenge

1780-729: Is called Dedekind infinite . The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size". Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers . Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite sequences which are maps from

1869-529: Is generally agreed that he was one of the Seven Wise Men of Greece . According to Proclus , he traveled to Babylon from where he learned mathematics and other subjects, coming up with the proof of what is now called Thales' Theorem . An equally enigmatic figure is Pythagoras of Samos (c. 580–500 BC), who supposedly visited Egypt and Babylon, and ultimately settled in Croton , Magna Graecia , where he started

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1958-425: Is known for certain, except that he was from Elea and that he was a student of Parmenides . Zeno is portrayed in the dialogue Parmenides by Plato , which takes place when Zeno is about 40 years old. In Parmenides , Zeno is described as having once been a zealous defender of his instructor Parmenides; this younger Zeno wished to prove that belief in the physical world as it appears is more absurd than belief in

2047-510: Is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, c = ℵ 1 = ℶ 1 {\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}} . This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory , even assuming the Axiom of Choice . Cardinal arithmetic can be used to show not only that

2136-462: Is only known to have written one book, most likely in the 460s BC. This book is told of in Parmenides , when the character of Zeno describes it as something that he wrote in his youth. According to Plato's account, the book was stolen and published without Zeno's permission. Zeno's paradoxes were recorded by Aristotle in his book Physics . Simplicius of Cilicia , who lived in the 6th century AD,

2225-436: Is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01 . Achilles does overtake the tortoise; it takes him The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable , innumerable, and infinite. Each of these

2314-407: Is still debated in the present day, and no solution to his paradoxes has been agreed upon by philosophers. His paradoxes have influenced philosophy and mathematics, both in ancient and modern times. Many of his ideas have been challenged by modern developments in physics and mathematics, such as atomic theory , mathematical limits , and set theory . Zeno was born c. 490 BC. Little about his life

2403-569: Is still used). In particular, in modern mathematics, lines are infinite sets . The vector spaces that occur in classical geometry have always a finite dimension , generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in functional analysis where function spaces are generally vector spaces of infinite dimension. In topology, some constructions can generate topological spaces of infinite dimension. In particular, this

2492-453: Is the case of iterated loop spaces . The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters and can have infinite or finite areas. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake . Leopold Kronecker was skeptical of

2581-924: The Antikythera mechanism , the accurate measurement of the circumference of the Earth by Eratosthenes (276–194 BC), and the mathematical and mechanical works of Heron (c. 10–70 AD). Several centers of learning appeared during the Hellenistic period, of which the most important one was the Mouseion in Alexandria , Egypt , which attracted scholars from across the Hellenistic world (mostly Greek, but also Egyptian , Jewish , Persian , among others). Although few in number, Hellenistic mathematicians actively communicated with each other; publication consisted of passing and copying someone's work among colleagues. Later mathematicians in

2670-646: The Archaic through the Hellenistic and Roman periods, mostly from the 5th century BC to the 6th century AD, around the shores of the Mediterranean . Greek mathematicians lived in cities spread over the entire region, from Anatolia to Italy and North Africa , but were united by Greek culture and the Greek language . The development of mathematics as a theoretical discipline and the use of deductive reasoning in proofs

2759-511: The Dedekind cut , developed by Richard Dedekind , who acknowledged Eudoxus as inspiration. Euclid , who presumably wrote on optics, astronomy, and harmonics, collected many previous mathematical results and theorems in the Elements , a canon of geometry and elementary number theory for many centuries. Menelaus , a later geometer and astronomer, wrote a standard work on spherical geometry in

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2848-459: The Hellenistic Greeks had a "horror of the infinite" which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving the infinitude of the prime numbers , Euclid "was the first to overcome the horror of the infinite". There

2937-495: The Minoan and later Mycenaean civilizations, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs , they left behind no mathematical documents. Though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on

3026-412: The cardinality of the line) is larger than the number of integers . In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object. The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among

3115-526: The cosmos together rather than physical or mechanical forces. Aristotle (c. 384–322 BC), the founder of the Peripatetic school , often used mathematics to illustrate many of his theories, as when he used geometry in his theory of the rainbow and the theory of proportions in his analysis of motion. Much of the knowledge about ancient Greek mathematics in this period is thanks to records referenced by Aristotle in his own works. The Hellenistic era began in

3204-436: The extended real numbers . We can also treat + ∞ {\displaystyle +\infty } and − ∞ {\displaystyle -\infty } as the same, leading to the one-point compactification of the real numbers, which is the real projective line . Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and

3293-533: The Cantorian transfinites . For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986) . A different form of "infinity" is the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor . In this system,

3382-552: The Circle ), and a proof that the area enclosed by a parabola and a straight line is 4/3 times the area of a triangle with equal base and height ( Quadrature of the Parabola ). Archimedes also showed that the number of grains of sand filling the universe was not uncountable, devising his own counting scheme based on the myriad , which denoted 10,000 ( The Sand-Reckoner ). The most characteristic product of Greek mathematics may be

3471-508: The Eleatic idea of a single entity of existence . By the time that Parmenides takes place, Zeno is shown to have matured and to be more content to overlook challenges to his instructor's Eleatic philosophy. Plato also has Socrates hint at a previous romantic or sexual relationship between Parmenides and Zeno. It is unknown how accurate the depiction in Parmenides is to reality, but it is agreed that it bears at least some truth. Zeno died c. 430 BC. According to Diogenes Laertius , Zeno

3560-926: The Eleatic school, as his arguments built on the ideas of Parmenides, though his paradoxes were also of interest to Ancient Greek mathematicians . Zeno is regarded as the first philosopher who dealt with attestable accounts of mathematical infinity . Zeno was succeeded by the Greek Atomists , who argued against the infinite division of objects by proposing an eventual stopping point: the atom. Though Epicurus does not name Zeno directly, he attempts to refute some of Zeno's arguments. Zeno appeared in Plato's dialogue Parmenides , and his paradoxes are mentioned in Phaedo . Aristotle also wrote about Zeno's paradoxes. Plato looked down on Zeno's approach of making arguments through contradictions. He believed that even Zeno himself did not take

3649-414: The Hellenistic and early Roman periods , and much of the work represented by authors such as Euclid (fl. 300 BC), Archimedes (c. 287–212 BC), Apollonius (c. 240–190 BC), Hipparchus (c. 190–120 BC), and Ptolemy (c. 100–170 AD) was of a very advanced level and rarely mastered outside a small circle. Examples of applied mathematics around this time include the construction of analogue computers like

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3738-512: The Riemann sphere taking the value of ∞ {\displaystyle \infty } at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations (see Möbius transformation § Overview ). The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In

3827-858: The Roman era include Diophantus (c. 214–298 AD), who wrote on polygonal numbers and a work in pre-modern algebra ( Arithmetica ), Pappus of Alexandria (c. 290–350 AD), who compiled many important results in the Collection , Theon of Alexandria (c. 335–405 AD) and his daughter Hypatia (c. 370–415 AD), who edited Ptolemy's Almagest and other works, and Eutocius of Ascalon ( c. 480–540 AD), who wrote commentaries on treatises by Archimedes and Apollonius. Although none of these mathematicians, save perhaps Diophantus, had notable original works, they are distinguished for their commentaries and expositions. These commentaries have preserved valuable extracts from works which have perished, or historical allusions which, in

3916-544: The absence of original documents, are precious because of their rarity. Most of the mathematical texts written in Greek survived through the copying of manuscripts over the centuries. While some fragments dating from antiquity have been found above all in Egypt , as a rule they do not add anything significant to our knowledge of Greek mathematics preserved in the manuscript tradition. Greek mathematics constitutes an important period in

4005-406: The actual phenomena of happenings and experience with the way that they are described and perceived. The exact wording of these arguments has been lost, but descriptions of them survive through Aristotle in his Physics . Aristotle identified four paradoxes of motion as the most important. Each paradox has multiple names that it is known by. Zeno's greatest influence was within the thought of

4094-429: The arguments seriously. Aristotle disagreed, believing them to be worthy of consideration. He challenged Zeno's dichotomy paradox through his conception of infinity, arguing that there are two infinities: an actual infinity that takes place at once and a potential infinity that is spread over time. He contended that Zeno attempted to prove actual infinities using potential infinities. He also challenged Zeno's paradox of

4183-532: The attention of philosophers during the Classical period . Plato (c. 428–348 BC), the founder of the Platonic Academy , mentions mathematics in several of his dialogues. While not considered a mathematician, Plato seems to have been influenced by Pythagorean ideas about number and believed that the elements of matter could be broken down into geometric solids. He also believed that geometrical proportions bound

4272-597: The attributes of different ideas as absolutes when they may be contextual. He may be accused of comparing similarities between concepts, such as attributes that physical space shared with physical objects, and then assuming that they be identical in other ways. Zeno rejected the idea of plurality , or that more than one thing can exist. According to Proclus , Zeno had forty arguments against plurality. In one argument, Zeno proposed that multiple objects cannot exist, because this would require everything to be finite and infinite simultaneously. He used this logic to challenge

4361-442: The axioms of Zermelo–Fraenkel set theory , on which most of modern mathematics can be developed, is the axiom of infinity , which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on

4450-450: The complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold , or Riemann surface , called the extended complex plane or the Riemann sphere . Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in

4539-524: The existence of Grothendieck universes , very large infinite sets, for solving a long-standing problem that is stated in terms of elementary arithmetic . In physics and cosmology , whether the universe is spatially infinite or not , is an open question. Ancient cultures had various ideas about the nature of infinity. The ancient Indians and the Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as

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4628-416: The existence of indivisible atoms. Though the first part of this argument is lost, its main idea is recorded by Simplicius. According to him, Zeno began the argument with the idea that nothing can have size because "each of the many is self-identical and one". Zeno argued that if objects have mass, then they can be divided. The divisions would in turn be divisible, and so on, meaning that no object could have

4717-407: The existence of multiple objects, and his arguments against motion. Those against plurality suggest that for anything to exist, it must be divisible infinitely, meaning it would necessarily have both infinite mass and no mass simultaneously. Those against motion invoke the idea that distance must be divisible infinitely, meaning infinite steps would be required to cross any distance. Zeno's philosophy

4806-408: The first transfinite cardinal is aleph-null ( ℵ 0 ), the cardinality of the set of natural numbers . This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege , Richard Dedekind and others—using the idea of collections or sets. Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as

4895-431: The foundation of calculus , it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers , showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e.,

4984-419: The history of mathematics : fundamental in respect of geometry and for the idea of formal proof . Greek mathematicians also contributed to number theory , mathematical astronomy , combinatorics , mathematical physics , and, at times, approached ideas close to the integral calculus . Eudoxus of Cnidus developed a theory of proportion that bears resemblance to the modern theory of real numbers using

5073-487: The ideas of pluralism , particularly those of the Pythagoreans . Zeno was the first philosopher to use argumentative rather than descriptive language in his philosophy. Previous philosophers had explained their worldview, but Zeno was the first one to create explicit arguments that were meant to be used for debate. Aristotle described Zeno as the "inventor of dialectic ". To disprove opposing views about reality, he wrote

5162-477: The infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound". Achilles races a tortoise, giving the latter a head start. Etc. Apparently, Achilles never overtakes the tortoise, since however many steps he completes,

5251-462: The integers is countably infinite . If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable . Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes. One of Cantor's most important results

5340-1135: The lack of original manuscripts, the dates for some Greek mathematicians are more certain than the dates of surviving Babylonian or Egyptian sources because a number of overlapping chronologies exist, though many dates remain uncertain. Netz (2011) has counted 144 ancient authors in the mathematical or exact sciences, from whom only 29 works are extant in Greek: Aristarchus , Autolycus , Philo of Byzantium , Biton , Apollonius , Archimedes , Euclid , Theodosius , Hypsicles , Athenaeus , Geminus , Heron , Apollodorus , Theon of Smyrna , Cleomedes , Nicomachus , Ptolemy , Gaudentius , Anatolius , Aristides Quintilian , Porphyry , Diophantus , Alypius , Damianus , Pappus , Serenus , Theon of Alexandria , Anthemius , and Eutocius . The following works are extant only in Arabic translations: Zeno of Elea Zeno of Elea ( / ˈ z iː n oʊ  ...   ˈ ɛ l i ə / ; Ancient Greek : Ζήνων ὁ Ἐλεᾱ́της ; c.  490  – c.  430 BC )

5429-742: The late 4th century BC, following Alexander the Great's conquest of the Eastern Mediterranean , Egypt , Mesopotamia , the Iranian plateau , Central Asia , and parts of India , leading to the spread of the Greek language and culture across these regions. Greek became the lingua franca of scholarship throughout the Hellenistic world, and the mathematics of the Classical period merged with Egyptian and Babylonian mathematics to give rise to Hellenistic mathematics. Greek mathematics reached its acme during

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5518-638: The notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism , an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism . In physics , approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them. The first published proposal that

5607-484: The number of points in a real number line is equal to the number of points in any segment of that line , but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space. The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval ( − ⁠ π / 2 ⁠ , ⁠ π / 2 ⁠ ) and R . The second result

5696-565: The order of 1 ∞ . {\displaystyle {\tfrac {1}{\infty }}.} But in Arithmetica infinitorum (1656), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "1, 6, 12, 18, 24, &c." In 1699, Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas . Hermann Weyl opened

5785-402: The positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of

5874-480: The present. Zeno's philosophy shows a contrast between what one knows logically and what one observes with the senses with the goal of proving that the world is an illusion; this practice was later adopted by the modern philosophic schools of thought, empiricism and post-structuralism . Bertrand Russell praised Zeno's paradoxes, crediting them for allowing the work of mathematician Karl Weierstrass . Scientific phenomena have been named after Zeno. The hindrance of

5963-914: The same properties in accordance with the Law of continuity . In real analysis , the symbol ∞ {\displaystyle \infty } , called "infinity", is used to denote an unbounded limit . The notation x → ∞ {\displaystyle x\rightarrow \infty } means that  x {\displaystyle x} increases without bound, and x → − ∞ {\displaystyle x\to -\infty } means that  x {\displaystyle x} decreases without bound. For example, if f ( t ) ≥ 0 {\displaystyle f(t)\geq 0} for every  t {\displaystyle t} , then Infinity can also be used to describe infinite series , as follows: In addition to defining

6052-403: The second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical systems , including smooth infinitesimal analysis and nonstandard analysis . In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field ; there is no equivalence between them as with

6141-421: The signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero , namely z / 0 = ∞ {\displaystyle z/0=\infty } for any nonzero complex number   z {\displaystyle z} . In this context, it is often useful to consider meromorphic functions as maps into

6230-450: The single object that exists as reality. Zeno believed that for all things that exist, they must exist in a certain point in physical space. For a point in space to exist, it must exist in another point in space. This space must in turn exist in another point in space, and so on. Zeno was likely the first philosopher to directly propose that being is incorporeal rather than taking up physical space. Zeno's arguments against motion contrast

6319-440: The square. Until the end of the 19th century, infinity was rarely discussed in geometry , except in the context of processes that could be continued without any limit. For example, a line was what is now called a line segment , with the proviso that one can extend it as far as one wants; but extending it infinitely was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points but

6408-544: The stadium, observing that it is fallacious to assume a stationary object and an object in motion require the same amount of time to pass. The paradox of Achilles and the tortoise may have influenced Aristotle's belief that actual infinity cannot exist, as this non-existence presents a solution to Zeno's arguments. Zeno's paradoxes are still debated, and they remain one of the archetypal examples of arguments to challenge commonly held perceptions. The paradoxes saw renewed attention in 19th century philosophy that has persisted to

6497-563: The style of the Elements , the Spherics , arguably considered the first treatise in non-Euclidean geometry . Archimedes made use of a technique dependent on a form of proof by contradiction to reach answers to problems with an arbitrary degree of accuracy, while specifying the limits within which the answers lay. Known as the method of exhaustion , Archimedes employed it in several of his works, including an approximation to π ( Measurement of

6586-411: The theory of conic sections , which was largely developed in the Hellenistic period , starting with the work of Menaechmus and perfected primarily under Apollonius in his work Conics . The methods employed in these works made no explicit use of algebra , nor trigonometry , the latter appearing around the time of Hipparchus . Ancient Greek mathematics was not limited to theoretical works but

6675-473: The tortoise remains ahead of him. Zeno was not attempting to make a point about infinity. As a member of the Eleatics school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument. Finally, in 1821, Augustin-Louis Cauchy provided both

6764-494: The translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period. Zeno of Elea ( c.  495 – c.  430 BC) did not advance any views concerning

6853-470: The universe have infinite volume? Does space " go on forever "? This is still an open question of cosmology . The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature, one will eventually return to the exact spot one started from. The universe, at least in principle, might have

6942-690: The universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds : "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds." Cosmologists have long sought to discover whether infinity exists in our physical universe : Are there an infinite number of stars? Does

7031-411: The use of set theory for the foundation of mathematics , points and lines were viewed as distinct entities, and a point could be located on a line . With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as the set of its points , and one says that a point belongs to a line instead of is located on a line (however, the latter phrase

7120-404: The younger Greek tradition. Unlike the flourishing of Greek literature in the span of 800 to 600 BC, not much is known about Greek mathematics in this early period—nearly all of the information was passed down through later authors, beginning in the mid-4th century BC. Greek mathematics allegedly began with Thales of Miletus (c. 624–548 BC). Very little is known about his life, although it

7209-675: Was projective geometry , where points at infinity are added to the Euclidean space for modeling the perspective effect that shows parallel lines intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane , two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not be distinguished in projective geometry. Before

7298-764: Was a pre-Socratic Greek philosopher from Elea , in Southern Italy ( Magna Graecia ). He was a student of Parmenides and one of the Eleatics . Zeno defended his instructor's belief in monism , the idea that only one single entity exists that makes up all of reality. He rejected the existence of space , time , and motion . To disprove these concepts, he developed a series of paradoxes to demonstrate why they are impossible. Though his original writings are lost, subsequent descriptions by Plato , Aristotle , Diogenes Laertius , and Simplicius of Cilicia have allowed study of his ideas. Zeno's arguments are divided into two different types: his arguments against plurality , or

7387-448: Was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the locus of a point that satisfies some property" (singular), where modern mathematicians would generally say "the set of the points that have the property" (plural). One of the rare exceptions of a mathematical concept involving actual infinity

7476-455: Was also used in other activities, such as business transactions and in land mensuration, as evidenced by extant texts where computational procedures and practical considerations took more of a central role. Although the earliest Greek mathematical texts that have been found were written after the Hellenistic period, most are considered to be copies of works written during and before the Hellenistic period. The two major sources are Despite

7565-428: Was further subdivided into three orders: In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. 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

7654-443: Was introduced in 1655 by John Wallis , and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology . Gottfried Leibniz , one of the co-inventors of infinitesimal calculus , speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying

7743-563: Was killed while he was engaged in a plot to overthrow the tyrant Nearchus . This account tells that he was captured, and that he was killed after he refused to give the names of his co-conspirators. Before his death, Zeno is said to have asked to whisper the names into Nearchus's ear, only to bite the ear when Nearchus approached, holding on until he was killed. The writings of Zeno have been lost; no fragments of his original thoughts exist. Instead, modern understanding of Zeno's philosophy comes through recording by subsequent philosophers. Zeno

7832-399: Was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves , curved lines that twist and turn enough to fill the whole of any square, or cube , or hypercube , or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in

7921-555: Was that the cardinality of the continuum c {\displaystyle \mathbf {c} } is greater than that of the natural numbers ℵ 0 {\displaystyle {\aleph _{0}}} ; that is, there are more real numbers R than natural numbers N . Namely, Cantor showed that c = 2 ℵ 0 > ℵ 0 {\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph _{0}}} . The continuum hypothesis states that there

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