Mathematics
Mathematical concepts and theorems need not correspond to anything in the physical world. Insofar as a correspondence does exist, while mathematicians and physicists may select axioms and postulates that seem reasonable and intuitive, it is not necessary for the basic assumptions within an axiomatic system to be true in an empirical or physical sense. Thus, while many axiom systems are derived from our perceptions and experiments, they are not dependent on them.
For example, we could say that the physical concept of two apples may be accurately modeled by the natural number 2. On the other hand, we could also say that the natural numbers are not an accurate model because there is no standard "unit" apple and no two apples are exactly alike. The modeling idea is further complicated by the possibility of fractional or partial apples. So while it may be instructive to visualize the axiomatic definition of the natural numbers as collections of apples, the definition itself is not dependent upon nor derived from any actual physical entities.
Nevertheless, mathematics remains extremely useful for solving real-world problems. This fact led physicist Eugene Wigner to write an article titled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".
See also
Notes
- ^ No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art depends on the artist's imagination (see Euclid).
- ^ Peirce, p.97
- ^ Steen, L.A. (April 29, 1988). The Science of Patterns. Science, 240: 611–616. and summarized at Association for Supervision and Curriculum Development.
- ^ Devlin, Keith, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996, ISBN 9780716750475
- ^ Jourdain
- ^ Eves
- ^ Peterson
- ^ The Oxford Dictionary of English Etymology, Oxford English Dictionary
- ^ Sevryuk
- ^ Earliest Uses of Various Mathematical Symbols (Contains many further references)
- ^ See false proof for simple examples of what can go wrong in a formal proof. The history of the Four Color Theorem contains examples of false proofs accepted by other mathematicians.
- ^ Ivars Peterson, The Mathematical Tourist, Freeman, 1988, ISBN 0-7167-1953-3. p. 4 "A few complain that the computer program can't be verified properly," (in reference to the Haken-Apple proof of the Four Color Theorem).
- ^ Patrick Suppes, Axiomatic Set Theory, Dover, 1972, ISBN 0-486-61630-4. p. 1, "Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects."
- ^ Waltershausen
- ^ Einstein, p. 28. The quote is Einstein's answer to the question: "how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" He, too, is concerned with The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
- ^ Popper 1995, p. 56
- ^ Ziman
- ^ "The Fields Medal is now indisputably the best known and most influential award in mathematics." Monastyrsky
- ^ Riehm
- ^ Clay Mathematics Institute P=NP
References
- Benson, Donald C., The Moment of Proof: Mathematical Epiphanies, Oxford University Press, USA; New Ed edition (December 14, 2000). ISBN 0-19-513919-4.
- Boyer, Carl B., A History of Mathematics, Wiley; 2 edition (March 6, 1991). ISBN 0-471-54397-7. — A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Courant, R. and H. Robbins, What Is Mathematics? : An Elementary Approach to Ideas and Methods, Oxford University Press, USA; 2 edition (July 18, 1996). ISBN 0-19-510519-2.
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7.— A gentle introduction to the world of mathematics.
- Einstein, Albert (1923). "Sidelights on Relativity (Geometry and Experience)". P. Dutton., Co.
- Eves, Howard, An Introduction to the History of Mathematics, Sixth Edition, Saunders, 1990, ISBN 0-03-029558-0.
- Gullberg, Jan, Mathematics—From the Birth of Numbers. W. W. Norton & Company; 1st edition (October 1997). ISBN 0-393-04002-X. — An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. — A translated and expanded version of a Soviet mathematics encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM, and online [1].
- Jourdain, Philip E. B., The Nature of Mathematics, in The World of Mathematics, James R. Newman, editor, Dover, 2003, ISBN 0-486-43268-8.
- Kline, Morris, Mathematical Thought from Ancient to Modern Times, Oxford University Press, USA; Paperback edition (March 1, 1990). ISBN 0-19-506135-7.
- Monastyrsky, Michael (2001). "Some Trends in Modern Mathematics and the Fields Medal". Canadian Mathematical Society. Retrieved on 2006-07-28.
- Oxford English Dictionary, second edition, ed. John Simpson and Edmund Weiner, Clarendon Press, 1989, ISBN 0-19-861186-2.
- The Oxford Dictionary of English Etymology, 1983 reprint. ISBN 0-19-861112-9.
- Pappas, Theoni, The Joy Of Mathematics, Wide World Publishing; Revised edition (June 1989). ISBN 0-933174-65-9.
- Peirce, Benjamin. "Linear Associative Algebra". American Journal of Mathematics (Vol. 4, No. 1/4. (1881). JSTOR.
- Peterson, Ivars, Mathematical Tourist, New and Updated Snapshots of Modern Mathematics, Owl Books, 2001, ISBN 0-8050-7159-8.
- Paulos, John Allen (1996). A Mathematician Reads the Newspaper. Anchor. ISBN 0-385-48254-X.
- Popper, Karl R. (1995). "On knowledge", In Search of a Better World: Lectures and Essays from Thirty Years. Routledge. ISBN 0-415-13548-6.
- Riehm, Carl (August 2002). "The Early History of the Fields Medal". Notices of the AMS 49 (7): 778-782. AMS.
- Sevryuk, Mikhail B. (January 2006). "Book Reviews" (PDF). Bulletin of the American Mathematical Society 43 (1): 101-109.
- Waltershausen, Wolfgang Sartorius von (1856, repr. 1965). Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend. ISBN 3-253-01702-8.
- Ziman, J.M., F.R.S. (1968). "Public Knowledge:An essay concerning the social dimension of science".
External links
| Find more about Mathematics on Wikipedia's sister projects: | |
|---|---|
 |
Dictionary definitions |
 |
Textbooks |
 |
Quotations |
 |
Source texts |
 |
Images and media |
 |
News stories |
 |
Learning resources |
At Wikiversity you can learn more and teach others about Mathematics at:
- Online Encyclopaedia of Mathematics [2] from Springer. Graduate-level reference work with over 8,000 entries, illuminating nearly 50,000 notions in mathematics.
- Some mathematics applets, at MIT
- Rusin, Dave: The Mathematical Atlas. A guided tour through the various branches of modern mathematics. (Can also be found here.)
- Stefanov, Alexandre: Textbooks in Mathematics. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: MathWorld: World of Mathematics. An online encyclopedia of mathematics.
- Polyanin, Andrei: EqWorld: The World of Mathematical Equations. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- Planet Math. An online mathematics encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- Metamath. A site and a language, that formalize mathematics from its foundations.
- Mathematician Biographies. The MacTutor History of Mathematics archive Extensive history and quotes from all famous mathematicians.
- Cain, George: Online Mathematics Textbooks available free online.
- Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas. In The Dictionary of the History of Ideas.
- Nrich, a prize-winning site for students from age five from Cambridge University
- 'FreeScience Library->Mathematics ' The mathematics section of FreeScience library
- Open Problem Garden, a wiki of open problems in mathematics
- Applications of High School Algebra
|
|||||
For more information review our copyright contact and privacy policy.