Usopeo

Makala hii kuhusu "Usopeo" inatumia neno au maneno ambayo si ya kawaida; matumizi yake ni jaribio kwa jambo linalotajwa. Wasomaji wanaombwa kuchangia mawazo yao. Kamusi za Kiswahili ama zinatofautiana kuhusu matumizi ya neno hili au hazieleweki au hazina neno kwa jambo linalojadiliwa.

Usopeo (kwa Kiingereza: Infinity) ni hali ya kutokuwa na mipaka, kikomo wala mwisho. Huwakilishwa kwa alama ya ∞, inayoitwa alama ya usopeo. Tangu zamani za Wagiriki wa kale, katika falsafa hali hiyo imekuwa mada ya mijadala mingi.

Katika karne ya 17, kwa kuanzishwa kwa alama ya usopeo na mfumo wa kalkulasi, wataalamu wa hisabati walianza kufanyia kazi mfululizo usio na mwisho. Wakati wataalamu wa hisabati wakihangaika na misingi ya kalkulasi, haikubaki wazi ikiwa usopeo ungeweza kuchukuliwa kama namba au ukubwa.

Mwishoni mwa karne ya 19, Georg Cantor alipanua somo la hisabati la usopeo kwa kusoma seti zisizo na mwisho, akionyesha kuwa zinaweza kuwa za saizi mbalimbali. Katika matumizi hayo, usopeo ni dhana ya kihisabati, na vitu vya kihisabati visivyo na mwisho vinaweza kusomwa na kutumiwa kama kitu kingine chochote cha kihisabati.

Dhana ya kihisabati ya usopeo inaboresha na kupanua dhana ya kale ya falsafa. Miongoni mwa kanuni za nadharia seti ya Zermelo–Fraenkel ni kanuni ya usopeo (axiom of infinity), ambayo inahakikisha uwepo wa seti zisizo na mwisho. Dhana ya kihisabati ya usopeo na utumiaji wa seti zisizo na mwisho hutumiwa sana katika hisabati, hata katika maeneo kama kumbu (combinatorics). Kwa mfano, uthibitisho wa Wiles wa nadharia ya mwisho ya Fermat unategemea uwepo wa ulimwengu wa Grothendieck, ambazo ni seti kubwa sana za usopeo.

Marejeo

• Cajori, Florian (1993) [1928 & 1929], A History of Mathematical Notations (Two Volumes Bound as One), Dover, ISBN 978-0-486-67766-8
• Gemignani, Michael C. (1990), Elementary Topology (tol. la 2nd), Dover, ISBN 978-0-486-66522-1
• Keisler, H. Jerome (1986), Elementary Calculus: An Approach Using Infinitesimals (tol. la 2nd)
• Maddox, Randall B. (2002), Mathematical Thinking and Writing: A Transition to Abstract Mathematics, Academic Press, ISBN 978-0-12-464976-7
• Kline, Morris (1972), Mathematical Thought from Ancient to Modern Times, New York: Oxford University Press, ku. 1197–1198, ISBN 978-0-19-506135-2
• Russell, Bertrand (1996) [1903], The Principles of Mathematics, New York: Norton, ISBN 978-0-393-31404-5, OCLC 247299160
• Sagan, Hans (1994), Space-Filling Curves, Springer, ISBN 978-1-4612-0871-6
• Swokowski, Earl W. (1983), Calculus with Analytic Geometry (tol. la Alternate), Prindle, Weber & Schmidt, ISBN 978-0-87150-341-1
• Taylor, Angus E. (1955), Advanced Calculus, Blaisdell Publishing Company
• Wallace, David Foster (2004), Everything and More: A Compact History of Infinity, Norton, W.W. & Company, Inc., ISBN 978-0-393-32629-1

Vyanzo

• Aczel, Amir D. (2001). The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity. New York: Pocket Books. ISBN 978-0-7434-2299-4.
• D.P. Agrawal (2000). Ancient Jaina Mathematics: an Introduction, Infinity Foundation.
• Bell, J.L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.
• Cohen, Paul (1963), "The Independence of the Continuum Hypothesis", Proceedings of the National Academy of Sciences of the United States of America, 50 (6): 1143–1148, Bibcode:1963PNAS...50.1143C, doi:10.1073/pnas.50.6.1143, PMC 221287, PMID 16578557.
• Jain, L.C. (1982). Exact Sciences from Jaina Sources.
• Jain, L.C. (1973). "Set theory in the Jaina school of mathematics", Indian Journal of History of Science.
• Joseph, George G. (2000). The Crest of the Peacock: Non-European Roots of Mathematics (tol. la 2nd). Penguin Books. ISBN 978-0-14-027778-4.
• H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
• Eli Maor (1991). To Infinity and Beyond. Princeton University Press. ISBN 978-0-691-02511-7.
• O'Connor, John J. and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor' Archived 2006-09-16 at the Wayback Machine, MacTutor History of Mathematics archive.
• O'Connor, John J. and Edmund F. Robertson (2000). 'Jaina mathematics' Archived 2008-12-20 at the Wayback Machine, MacTutor History of Mathematics archive.
• Pearce, Ian. (2002). 'Jainism', MacTutor History of Mathematics archive.
• Rucker, Rudy (1995). Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton University Press. ISBN 978-0-691-00172-2.
• Singh, Navjyoti (1988). "Jaina Theory of Actual Infinity and Transfinite Numbers". Journal of the Asiatic Society. 30.

Viungo vya nje

• Infinity on In Our Time at the BBC
• A Crash Course in the Mathematics of Infinite Sets Archived 2010-02-27 at the Wayback Machine, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1–59. The stand-alone appendix to Infinite Reflections, below. A concise introduction to Cantor's mathematics of infinite sets.
• Infinite Reflections Archived 2009-11-05 at the Wayback Machine, by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1–59.
• Grime, James. "Infinity is bigger than you think". Numberphile. Brady Haran. Ilihifadhiwa kwenye nyaraka kutoka chanzo mnamo 2017-10-22. Iliwekwa mnamo 2013-04-06.
• Hotel Infinity
• John J. O'Connor and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor' Archived 2006-09-16 at the Wayback Machine, MacTutor History of Mathematics archive.
• John J. O'Connor and Edmund F. Robertson (2000). 'Jaina mathematics' Archived 2008-12-20 at the Wayback Machine, MacTutor History of Mathematics archive.
• Ian Pearce (2002). 'Jainism', MacTutor History of Mathematics archive.
• The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity
• Dictionary of the Infinite (compilation of articles about infinity in physics, mathematics, and philosophy)

Makala hii kuhusu mambo ya hisabati bado ni mbegu. Je, unajua kitu kuhusu Usopeo kama historia yake au mahusiano yake na mada nyingine? Labda unaona habari katika Wikipedia ya Kiingereza au lugha nyingine zinazofaa kutafsiriwa? Basi unaweza kuisaidia Wikipedia kwa kuihariri na kuongeza habari.