Mathematics

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Mathematics is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies these concepts.^[1] Abbreviated forms of the name include math (in American English) and maths (in British English).

Main conceptual divisions[]

Quantity[]

Quantity is a fundamental concept related to counting distinct objects and measuring magnitudes that occur along a continuum (examples of the latter idea include length of time, physical length, mass, area, volume, and so forth). This dual nature of quantity is captured in the everyday concepts of "how many" versus "how much", and by the more technical terms discrete versus continuous.

Areas of mathematics primarily focused on quantity include:

Structure[]

Structure refers primarily to logical structure, as opposed to physical structure (see also the section on Space below).

Areas of mathematics primarily focused on notions of structure include:

Space[]

Space refers to both physical and conceptual notions of space, not simply "outer space" (for which see Astronomy and Cosmology).

Areas of mathematics primarily focused on notions of space include:

Change[]

Change...

Areas of mathematics primarily focused on change include:

Calculus
Differential equations
Dynamical systems

History[]

Main article: History of mathematics

Prehistoric[]

The origins of mathematical thought lie in the concepts of number, magnitude, and form.^[2] Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. The idea of the "number" concept evolving gradually over time is supported by the existence of languages which preserve the distinction between "one", "two", and "many", but not of numbers larger than two.^[2]

Long before the earliest written records, there are drawings that indicate some knowledge of elementary mathematics and of time measurement based on the stars. For example, paleontologists have discovered in a cave in South Africa, ochre rocks about 70,000 years old, adorned with scratched geometric patterns.^[3]

Also prehistoric artifacts discovered in Africa, dated between 35,000 and 20,000 years old,^[4] suggest early attempts to quantify time.^[5] The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be as much as 20,000 years old. One common interpretation is that the bone is the earliest known demonstration^[4] of sequences of prime numbers and of Ancient Egyptian multiplication.

Predynastic Egypt of the 5th millennium BC pictorially represented geometric spatial designs.^[6]

Ancient[]

The Babylonian mathematical tablet *Plimpton 322*, dated to 1800 BC

In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years.^[7]^[8] Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.^[9] The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC.^[10] Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time.^[11]

Hellenistic mathematics emerged in the late 4th century BC, representing a synthesis of Greek, Egyptian and Babylonian mathematics. Circa 300 BC, Egyptian Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof.^[12] His book, Elements, was highly influential.^[13] Another influential mathematician of antiquity was Archimedes (c. 287 – c. 212 BC) of Syracuse (Sicily).^[14] He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola.^[15] Another achievement of Hellenistic mathematics was conic sections, developed by Apollonius of Perga in Asia Minor (modern Turkey) during the 3rd century BC.^[16]

Medieval[]

The Indian-Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and then Islamic mathematics, from where they were transmitted to the Western world via Islamic mathematics.^[17] Other notable developments of Indian mathematics include the development of the sine function, and later an early form of infinite series.^[18]^[19]

During the Golden Age of Islam, especially during the 9th and 10th centuries, Islamic mathematics saw many important innovations. Some of the achievements of Muslim mathematicians during this period include the development of algebra and algorithms (see Muhammad ibn Mūsā al-Khwārizmī), the development of spherical trigonometry,^[20] the addition of the decimal point notation to the Arabic numerals,^[21] the discovery of all the modern trigonometric functions, al-Kindi's introduction of cryptanalysis and frequency analysis, the development of analytic geometry by Ibn al-Haytham, the beginning of algebraic geometry by Omar Khayyam, the first refutations of Euclidean geometry and the parallel postulate by Nasīr al-Dīn al-Tūsī, the first attempt at a non-Euclidean geometry by Sadr al-Din, the development of an algebraic notation by al-Qalasādī,^[22] and many other advances in algebra, arithmetic, calculus, cryptography, geometry, number theory and trigonometry.

Many notable Islamic mathematicians from this period were Persian, such as Al-Khwarizmi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.^[23] Arabic mathematical texts were translated to Latin during the Middle Ages and made available in Europe.^[24]

Modern[]

During the early modern period, mathematics began to develop at an accelerating pace in Western Europe, with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation, the introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and the proof of numerous theorems.^[25]

Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics.^[26]

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."^[27]

References[]

Citations[]

↑ Wikipedia:Mathematics
↑ ^2.0 ^2.1 (Boyer 1991, "Origins" p. 3)
↑ Henahan, Sean (2002). "Art Prehistory". Science Updates. The National Health Museum. Retrieved 2006-05-06.
↑ ^4.0 ^4.1 Williams, Scott W. (2005). "An Old Mathematical Object". MATHEMATICIANS OF THE AFRICAN DIASPORA. SUNY Buffalo mathematics department. Retrieved 2006-05-06.
↑ Mathematics in (central) Africa before colonization
↑ Thom, Alexander, and Archie Thom, 1988, "The metrology and geometry of Megalithic Man", pp 132-151 in C.L.N. Ruggles, ed., Records in Stone: Papers in memory of Alexander Thom. Cambridge Univ. Press. ISBN 0-521-33381-4.
↑ See, for example, Wilder, Raymond L.. Evolution of Mathematical Concepts; an Elementary Study. passim.
↑ Zaslavsky, Claudia (1999). Africa Counts: Number and Pattern in African Culture.. Chicago Review Press. ISBN 978-1-61374-115-3. OCLC 843204342.
↑ Kline 1990, Chapter 1.
↑ Mesopotamia pg 10. Retrieved June 1, 2024
↑ Boyer 1991, "Mesopotamia" pp. 24–27.
↑ Mueller, I. (1969). "Euclid's Elements and the Axiomatic Method". The British Journal for the Philosophy of Science 20 (4): 289–309. doi:10.1093/bjps/20.4.289. ISSN 0007-0882. JSTOR 686258.
↑ Boyer 1991, "Euclid of Alexandria" p. 119.
↑ Boyer 1991, "Archimedes of Syracuse" p. 120.
↑ Boyer 1991, "Archimedes of Syracuse" p. 130.
↑ Boyer 1991, "Apollonius of Perga" p. 145.
↑ Ore, Øystein (1988). Number Theory and Its History. Courier Corporation. pp. 19–24. ISBN 978-0-486-65620-5. https://books.google.com/books?id=Sl_6BPp7S0AC&pg=IA19.
↑ Singh, A. N. (January 1936). "On the Use of Series in Hindu Mathematics". Osiris 1: 606–628. doi:10.1086/368443. JSTOR 301627.
↑ Kolachana, A.; Mahesh, K.; Ramasubramanian, K. (2019). "Use of series in India". Studies in Indian Mathematics and Astronomy. Sources and Studies in the History of Mathematics and Physical Sciences. Singapore: Springer. pp. 438–461. doi:10.1007/978-981-13-7326-8_20. ISBN 978-981-13-7325-1.
↑ Syed, M. H. (2005). Islam and Science. Anmol Publications PVT. LTD.. pp. 71. ISBN 8-1261-1345-6.
↑ Saliba, George (1994). A history of Arabic astronomy: planetary theories during the golden age of Islam. New York University Press. ISBN 978-0-8147-7962-0. OCLC 28723059.
↑ O'Connor, John J.; Robertson, Edmund F., "Abu'l Hasan ibn Ali al Qalasadi", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Qalasadi.html .
↑ Faruqi, Yasmeen M. (2006). "Contributions of Islamic scholars to the scientific enterprise". International Education Journal (Shannon Research Press) 7 (4): 391–399. https://eric.ed.gov/?id=EJ854295.
↑ Lorch, Richard (June 2001). "Greek-Arabic-Latin: The Transmission of Mathematical Texts in the Middle Ages". Science in Context (Cambridge University Press) 14 (1–2): 313–331. doi:10.1017/S0269889701000114. https://epub.ub.uni-muenchen.de/15929/1/greek-arabic-latin.pdf.
↑ Kent, Benjamin (2022). History of Science. 2. Bibliotex Digital Library. ISBN 978-1-984668-67-7. http://rguir.inflibnet.ac.in/bitstream/123456789/16963/1/9781984668677.pdf.
↑ Archibald, Raymond Clare (January 1949). "History of Mathematics After the Sixteenth Century". The American Mathematical Monthly. Part 2: Outline of the History of Mathematics 56 (1): 35–56. doi:10.2307/2304570. JSTOR 2304570.
↑ Sevryuk 2006, pp. 101–109.

Sources[]

Bouleau, Nicolas (1999). Philosophie des mathématiques et de la modélisation: Du chercheur à l'ingénieur. L'Harmattan. ISBN 978-2-7384-8125-2.
Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). New York: Wiley. ISBN 978-0-471-54397-8. https://archive.org/details/historyofmathema00boye/page/n3/mode/2up.
Eves, Howard (1990). An Introduction to the History of Mathematics (6th ed.). Saunders. ISBN 978-0-03-029558-4.
Kleiner, Israel (2007). A History of Abstract Algebra. Springer Science & Business Media. doi:10.1007/978-0-8176-4685-1. ISBN 978-0-8176-4684-4. LCCN 2007932362. OCLC 76935733. https://books.google.com/books?id=RTLRBK-wj6wC.
Kline, Morris (1990). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press. ISBN 978-0-19-506135-2. https://archive.org/details/mathematicalthou00klin.
Monastyrsky, Michael (2001). "Some Trends in Modern Mathematics and the Fields Medal". CMS – Notes – de la SMC (Canadian Mathematical Society) 33 (2–3). http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf.
Oakley, Barbara (2014). A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra). New York: Penguin Random House. ISBN 978-0-399-16524-5. https://archive.org/details/isbn_9780399165245. "A Mind for Numbers."
Peirce, Benjamin (1881). Peirce, Charles Sanders. ed. "Linear associative algebra". American Journal of Mathematics 4 (1–4): 97–229. doi:10.2307/2369153. JSTOR 2369153. Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C. S. Peirce, of the 1872 lithograph ed. Google Eprint and as an extract, D. Van Nostrand, 1882, Google Eprint. https://books.google.com/books?id=De0GAAAAYAAJ&pg=PA1&q=Peirce+Benjamin+Linear+Associative+Algebra. .
Peterson, Ivars (1988). The Mathematical Tourist: Snapshots of Modern Mathematics. W. H. Freeman and Company. ISBN 0-7167-1953-3. LCCN 87033078. OCLC 17202382.
Popper, Karl R. (1995). "On knowledge". In Search of a Better World: Lectures and Essays from Thirty Years. New York: Routledge. Bibcode 1992sbwl.book.....P. ISBN 978-0-415-13548-1. https://archive.org/details/insearchofbetter00karl.
Riehm, Carl (August 2002). "The Early History of the Fields Medal". Notices of the AMS 49 (7): 778–782. https://www.ams.org/notices/200207/comm-riehm.pdf.
Sevryuk, Mikhail B. (January 2006). "Book Reviews". Bulletin of the American Mathematical Society 43 (1): 101–109. doi:10.1090/S0273-0979-05-01069-4. https://www.ams.org/bull/2006-43-01/S0273-0979-05-01069-4/S0273-0979-05-01069-4.pdf.
Whittle, Peter (1994). "Almost home". In Kelly, F.P.. Probability, statistics and optimisation: A Tribute to Peter Whittle (previously "A realised path: The Cambridge Statistical Laboratory up to 1993 (revised 2002)" ed.). Chichester: John Wiley. pp. 1–28. ISBN 978-0-471-94829-2. http://www.statslab.cam.ac.uk/History/2history.html#6._1966--72:_The_Churchill_Chair.