Infinity is something which is boundless, endless, or larger than any natural number . It is often denoted by the infinity symbol ∞ {\displaystyle \infty } .
77-475: Rudolf von Bitter Rucker ( / ˈ r ʌ k ər / ; born March 22, 1946) is an American mathematician , computer scientist, science fiction author, and one of the founders of the cyberpunk literary movement. The author of both fiction and non-fiction, he is best known for the novels in the Ware Tetralogy , the first two of which ( Software and Wetware ) both won Philip K. Dick Awards . He edited
154-650: A BA in mathematics from Swarthmore College (1967) and MS (1969) and PhD (1973) degrees in mathematics from Rutgers University . Rucker taught mathematics at the State University of New York at Geneseo from 1972 to 1978. Although he was liked by his students and "published a book [ Geometry, Relativity and the Fourth Dimension ] and several papers," several colleagues took umbrage at his long hair and convivial relationships with English and philosophy professors amid looming budget shortfalls; as
231-475: A cerebral hemorrhage . Thinking he might not be around much longer, this prompted him to write Nested Scrolls , his autobiography. Rucker resided in Highland Park, New Jersey during his graduate studies at Rutgers University. The Ware Tetralogy Transreal Trilogy Transreal novels Other novels Collections Stories (by date of composition) Mathematician A mathematician
308-632: 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
385-467: 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, the Hellenistic Greeks had
462-468: 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 is a similar controversy concerning Euclid's parallel postulate , sometimes translated: If
539-570: A computer science professor at San José State University in 1986, from which he retired as professor emeritus in 2004. From 1988 to 1992 he was hired by John Walker of Autodesk as a programmer of cellular automata , which inspired his book The Hacker and the Ants . A mathematician with philosophical interests, he has written The Fourth Dimension and Infinity and the Mind . Princeton University Press published new editions of Infinity and
616-477: A financial economist might study the structural reasons why a company may have a certain share price , a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock ( see: Valuation of options ; Financial modeling ). According to the Dictionary of Occupational Titles occupations in mathematics include
693-410: 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
770-626: 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 a philosophical concept. The earliest recorded idea of infinity in Greece may be that of Anaximander (c. 610 – c. 546 BC)
847-400: A manner which will help ensure that the plans are maintained on a sound financial basis. As another example, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while
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#1732782990988924-464: A mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry , by deriving four corollaries to Thales's theorem . The number of known mathematicians grew when Pythagoras of Samos ( c. 582 – c. 507 BC ) established the Pythagorean school , whose doctrine it was that mathematics ruled the universe and whose motto
1001-515: 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
1078-441: 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., the cardinality of the line) is larger than the number of integers . In this usage, infinity
1155-627: A result, he failed to attain tenure in the "dysfunctional" department. Thanks to a grant from the Alexander von Humboldt Foundation , Rucker taught at the Ruprecht Karl University of Heidelberg from 1978 to 1980. He then taught at Randolph-Macon Women's College in Lynchburg, Virginia from 1980 to 1982, before trying his hand as a full-time author for four years. Inspired by an interview with Stephen Wolfram , Rucker became
1232-401: 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
1309-456: 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
1386-457: 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 the translation "the two straight lines, if produced indefinitely ...", thus avoiding
1463-416: Is Saucer Wisdom , a novel in which the main character is abducted by aliens . Rucker and his publisher marketed the book, tongue in cheek, as non-fiction. His earliest transreal novel, White Light , was written during his time at Heidelberg . This transreal novel is based on his experiences at SUNY Geneseo. Rucker often uses his novels to explore scientific or mathematical ideas; White Light examines
1540-420: Is mathematics that studies entirely abstract concepts . From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics , and at variance with the trend towards meeting the needs of navigation , astronomy , physics , economics , engineering , and other applications. Another insightful view put forth is that pure mathematics
1617-451: Is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on problems, often concrete but sometimes abstract. As professionals focused on problem solving, applied mathematicians look into the formulation, study, and use of mathematical models in science , engineering , business , and other areas of mathematical practice. Pure mathematics
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#17327829909881694-415: 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 the axioms of Zermelo–Fraenkel set theory , on which most of modern mathematics can be developed,
1771-730: 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
1848-515: 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
1925-400: Is not necessarily applied mathematics : it is possible to study abstract entities with respect to their intrinsic nature, and not be concerned with how they manifest in the real world. Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing
2002-436: 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
2079-433: Is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems . Mathematicians are concerned with numbers , data , quantity , structure , space , models , and change . One of the earliest known mathematicians was Thales of Miletus ( c. 624 – c. 546 BC ); he has been hailed as the first true mathematician and the first known individual to whom
2156-569: 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
2233-426: 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 the existence of Grothendieck universes , very large infinite sets, for solving
2310-458: 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
2387-676: The Schock Prize , and the Nevanlinna Prize . The American Mathematical Society , Association for Women in Mathematics , and other mathematical societies offer several prizes aimed at increasing the representation of women and minorities in the future of mathematics. Several well known mathematicians have written autobiographies in part to explain to a general audience what it is about mathematics that has made them want to devote their lives to its study. These provide some of
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2464-438: 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
2541-478: The graduate level . In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics; the students who pass are permitted to work on a doctoral dissertation . Mathematicians involved with solving problems with applications in real life are called applied mathematicians . Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of
2618-543: 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,
2695-586: The Italian and German universities, but as they already enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment , the same influences that inspired Humboldt. The Universities of Oxford and Cambridge emphasized the importance of research , arguably more authentically implementing Humboldt's idea of a university than even German universities, which were subject to state authority. Overall, science (including mathematics) became
2772-545: The Mind in 1995 and in 2005, both with new prefaces; the first edition is cited with fair frequency in academic literature. As his "own alternative to cyberpunk," Rucker developed a writing style he terms transrealism . Transrealism, as outlined in his 1983 essay The Transrealist Manifesto , is science fiction based on the author's own life and immediate perceptions, mixed with fantastic elements that symbolize psychological change. Many of Rucker's novels and short stories apply these ideas. One example of Rucker's transreal works
2849-517: 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
2926-823: The Seashell, and the Soul: What Gnarly Computation Taught Me About Ultimate Reality, the Meaning Of Life, and How To Be Happy summarizes the various philosophies he's believed over the years and ends with the tentative conclusion that we might profitably view the world as made of computations, with the final remark, "perhaps this universe is perfect." Rucker was the roommate of Kenneth Turan during his freshman year at Swarthmore College. In 1967, Rucker married Sylvia Bogsch Rucker (1943–2023). Together they have three children. On July 1, 2008, Rucker suffered
3003-404: The best glimpses into what it means to be a mathematician. The following list contains some works that are not autobiographies, but rather essays on mathematics and mathematicians with strong autobiographical elements. Infinity 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
3080-456: 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
3157-523: The concept of infinity , while the Ware Tetralogy (written from 1982 through 2000) is in part an explanation of the use of natural selection to develop software (a subject also developed in his The Hacker and the Ants , written in 1994). His novels also put forward a mystical philosophy that Rucker has summarized in an essay titled, with only a bit of irony, "The Central Teachings of Mysticism" (included in Seek! , 1999). His non-fiction book, The Lifebox,
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3234-409: 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
3311-500: The focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. The German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of
3388-1060: The following. There is no Nobel Prize in mathematics, though sometimes mathematicians have won the Nobel Prize in a different field, such as economics or physics. Prominent prizes in mathematics include the Abel Prize , the Chern Medal , the Fields Medal , the Gauss Prize , the Nemmers Prize , the Balzan Prize , the Crafoord Prize , the Shaw Prize , the Steele Prize , the Wolf Prize ,
3465-409: 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
3542-633: The imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, and localized constructs, applied mathematicians work regularly in the study and formulation of mathematical models . Mathematicians and applied mathematicians are considered to be two of the STEM (science, technology, engineering, and mathematics) careers. The discipline of applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry; thus, "applied mathematics"
3619-484: 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,
3696-464: 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
3773-424: 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 the foundation of calculus , it remained unclear whether infinity could be considered as
3850-580: The kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of "freedom of scientific research, teaching and study." Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education , and then proceed to specialize in topics of their own choice at
3927-474: The king of Prussia , Fredrick William III , to build a university in Berlin based on Friedrich Schleiermacher 's liberal ideas; the goal was to demonstrate the process of the discovery of knowledge and to teach students to "take account of fundamental laws of science in all their thinking." Thus, seminars and laboratories started to evolve. British universities of this period adopted some approaches familiar to
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#17327829909884004-640: 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
4081-485: 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
4158-575: 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
4235-402: 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
4312-531: The probability and likely cost of the occurrence of an event such as death, sickness, injury, disability, or loss of property. Actuaries also address financial questions, including those involving the level of pension contributions required to produce a certain retirement income and the way in which a company should invest resources to maximize its return on investments in light of potential risk. Using their broad knowledge, actuaries help design and price insurance policies, pension plans, and other financial strategies in
4389-484: The real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research. Many professional mathematicians also engage in the teaching of mathematics. Duties may include: Many careers in mathematics outside of universities involve consulting. For instance, actuaries assemble and analyze data to estimate
4466-921: 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
4543-577: The science fiction webzine Flurb until its closure in 2014. Rucker was born and raised in Louisville, Kentucky , son of Embry Cobb Rucker Sr (October 1, 1914 - August 1, 1994), who ran a small furniture-manufacture company and later became an Episcopal priest and community activist, and Marianne (née von Bitter). The Rucker family were of Huguenot descent. Through his mother, he is a great-great-great-grandson of Georg Wilhelm Friedrich Hegel . Rucker attended St. Xavier High School before earning
4620-403: 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
4697-403: The seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle , and at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics . Moving into the 19th century, the objective of universities all across Europe evolved from teaching the "regurgitation of knowledge" to "encourag[ing] productive thinking." In 1810, Alexander von Humboldt convinced
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#17327829909884774-421: 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
4851-441: 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
4928-478: 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
5005-522: 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
5082-653: 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
5159-411: 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
5236-943: Was Al-Khawarizmi . A notable feature of many scholars working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics , maths and astronomy of Ibn al-Haytham . The Renaissance brought an increased emphasis on mathematics and science to Europe. During this period of transition from a mainly feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli (founder of accounting ); Niccolò Fontana Tartaglia (notable engineer and bookkeeper); Gerolamo Cardano (earliest founder of probability and binomial expansion); Robert Recorde (physician) and François Viète (lawyer). As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in
5313-680: 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
5390-697: Was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria ( c. AD 350 – 415). She succeeded her father as librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria punished her, presuming she
5467-448: 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
5544-430: 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
5621-447: 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
5698-585: Was involved, by stripping her naked and scraping off her skin with clamshells (some say roofing tiles). Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based primarily on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages
5775-431: Was ongoing throughout the reign of certain caliphs, and it turned out that certain scholars became experts in the works they translated, and in turn received further support for continuing to develop certain sciences. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support
5852-399: 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
5929-556: 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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