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Topic: Classical mathematics

	Classical mathematics - Wikipedia, the free encyclopedia
		Classical mathematics, as a term of art in mathematical logic, refers generally to mathematics constructed and proved on the basis of classical logic and ZFC set theory, i.
		Classical mathematics is sometimes attacked on philosophical grounds, due to constructivist and other objections to the logic, set theory, etc., chosen as its foundations.
		Defenders of classical mathematics argue that it is easier to work in, and is most fruitful; although they acknowledge non-classical mathematics has at time lead to fruitful results that classical mathematics could not (or could not so easily) attain, on the whole they argue it is the other way round.
	en.wikipedia.org /wiki/Classical_mathematics (239 words)

	Mathematics - Wikipedia, the free encyclopedia
		Historically, mathematics developed from counting, calculation, measurement, and the study of the shapes and motions of physical objects, through the use of abstraction and deductive reasoning.
		Nowadays, mathematics derives much inspiration from the natural sciences and it is not uncommon for new mathematics to be pioneered by physicists, although it may need to be recast into more rigorous language.
		Mathematics is inspiring to mathematicians because it has some intrinsic aesthetics or inner beauty, which is hard to explain.
	en.wikipedia.org /wiki/Mathematics (2627 words)

	[No title]
		They feel that constructivists who don't recognize classical mathematics as legitimate have serious philosophical delusions which cause them to work in very restricted areas of mathematics, but that their view of those areas is the same as that of the classical mathematician, once the definitions are understood correctly.
		Because constructive mathematics assumes less than classical mathematics, which assumes the law of excluded middle, any theorem in the former is a theorem in the latter---that is really the only sense in which I would assert that the classical universe is a model for constructive mathematics.
		But judging by the attempts of classical mathematicians to understand constructive mathematics solely in terms of recursive function theory---constructive real numbers as special kinds of real numbers---because they cannot shake their dependence on the law of excluded middle, I would say that it is much more difficult for them to understand what we are doing.
	www.math.fau.edu /richman/docs/Intrview.html (9565 words)

	E-Ren: Student Projects
		The growth of mathematics was not only retarded by war, but also by the injurious influence of traditional scholastic philosophy.
		Although the early Renaissance mathematics was basically an imitation of antiquity, characteristic of all learning of the period, the mathematicians eventually went beyond the Greek knowledge.
		The development of mathematics in Renaissance time was not only due to the ingenious of the mathematicians, but also it reflected and encouraged by the social and economic needs of Renaissance.
	www.idbsu.edu /courses/hy309/projects/math.html (3138 words)

	Classical Greek Mathematics (Site not responding. Last check: 2007-11-03)
		B.C., known as the classical period of Greek mathematics, mathematics was transformed from an ecclectic collection of practical techniques into a coherent structure of deductive knowledge.
		The Pythagorean's sought to found all of mathematics on number but were confounded by the discovery of incommensurable ratios in geometry.
		From very early in the classical period deduction is perceived as the primary method of arriving at mathematical truths.
	www.rbjones.com /rbjpub/maths/math005.htm (338 words)

	The Three Crises in Mathematics: Logicism,
		For the strict formalist "to do mathematics" is "to manipulate the meaningless symbols of a first order language according to explicit, syntactical rules." Hence, the strict formalist does not work with abstract entities, such as infinite series or cardinals, but only with their meaningless names which are the appropriate expressions in a first order language.
		However, although this kind of mathematics is often referred to as "foundations of mathematics," one cannot claim to be advancing the philosophy of mathematics just because one is working in one of these areas.
		Modern mathematical logic, set theory, and intuitionism with its modifications are nowadays technical branches of mathematics, just as algebra or analysis, and unless we return directly to the philosophy of mathematics, we cannot expect to find a firm foundation for our science.
	members.shaw.ca /kschindler/mathphil.htm (6210 words)

	On Gödel's Philosophy of Mathematics, Chapter I
		Gödel argues forcefully that the concepts of classical mthematics are indeed understood and are "sufficiently clear for us to be able to recognize their soundness...."[1] Whether a mathematical concept is ever "completely given in mathematical intuition" or not does not seem to be a resolvable issue.
		Mathematics has a propensity for employing physical or "thing" language, and this does have considerable heuristic value because the metaphors chosen are usually clever and appropriate.
		Mathematics for Gödel is boundless, having its beginning in the rudiments of logic, extending up to classical analysis, the higher axioms of infinity, and beyond to bolder, richer but as yet undiscovered theories.
	www.friesian.com /goedel/chap-1.htm (4807 words)

	Constructive Mathematics
		It is on these interpretations of disjunction and existence that mathematicians have built the grand, and apparently impregnable, edifice of classical mathematics which serves a foundation for the physical, the social, and (increasingly) the biological sciences.
		Mathematics arises when the subject of two-ness, which results from the passage of time, is abstracted from all special occurrences.
		However, the comparison with classical mathematics should not be made superficially: in order to understand that there is no real contradiction here, we must appreciate that the meaning of such terms as “function” and even “real number” in intuitionistic mathematics is quite different from that in the classical setting.
	plato.stanford.edu /entries/mathematics-constructive (6375 words)

	Inconsistent Mathematics
		Inconsistent mathematics is the study of the mathematical theories that result when classical mathematical axioms are asserted within the framework of a (non-classical) logic which can tolerate the presence of a contradiction without turning every sentence into a theorem.
		In addition, mathematics has a metalanguage; that is, names for mathematical statements and other parts of syntax, self-reference, proof and truth.
		Gödel's contribution to the philosophy of mathematics was to show that the first three of these can be rigorously expressed in arithmetical theories, albeit in theories which are either inconsistent or incomplete.
	plato.stanford.edu /entries/mathematics-inconsistent (2006 words)

	[No title]
		On the contrary, she justifies the objective existence of mathematical concepts on the intimate relation between mathematics and science: Since physics is about real things and mathematics is indispensable for physics, then mathematics is also about real things.
		While Plato considered mathematical knowledge to be a priori certain and necessary, the Quine/Putnam approach leads to no such conclusions: If mathematics is objective because it is embedded in scientific theory, it can hardly be considered a priori; and there is likewise little support for certainty or necessity.
		Mathematical truth\endheading \ext {\it I believe there are exactly $15, 747, 724, 136, 275, 002, 577,\mathbreak 605, 653, 961, 181, 555, 468, 044, 717, 914, 527\!, 116\!, 709\!,\mathbreak 366\!, 231\!, 425\!, 076\!, 185\!, 631\!, 031\!, 296$ protons in the universe, and the same number of electrons}.
	www.ams.org /journals/bull/pre-1996-data/199501/199501019.tex.html (3819 words)

	The Math Forum: Talk of the Nation, Science Friday
		Mathematics is more than just number counting is what she is saying.
		There are all sorts of ways that mathematics creeps into the various disciplines and subjects and if you have not seen it, you are going to have a devil of a time.
		William Dunham is professor of mathematics at Muhlenberg College in Allentown, PA and author of The Mathematical Universe from Wiley.
	www.mathforum.com /social/articles/npr-tape.html (5554 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		On the one hand, whether in mathematics as elsewhere, on pain of circularity or infinite regress knowledge of the meaning of an expression cannot in general consist in an ability to state that meaning, since we would need to know already the meanings of the expressions in which the meanings are stated.
		Certainly it must be conceded that the classical conception of truth implies that there are truths that cannot be proved on the basis of currently accepted axioms: whichever of CH or its negation is true is a well-known example.
		If we replace classical logic in empirical science by a logic in which the meaning of the particles is given by elimination rules or falsification conditions, then all statements will be potentially falsifiable in the sense of its being in principle possible to recognize them as false if they are false.
	www.princeton.edu /~jburgess/TennantReview.doc (6208 words)

	Constructive Mathematics
		Much more of classical mathematics was preserved than had been thought possible, and no classically false theorems resulted, as had been the case in other constructive schools such as intuitionism and Russian constructivism.
		The main thrust of constructive mathematics was in the direction of analysis, although several mathematicians, including Kronecker and van der Waerden, made important contributions to constructive algebra.
		It is important to realize that constructive algebra is algebra; in fact it is a generalization of classical algebra in that we do not assume the law of excluded middle, just as group theory is a generalization of abelian group theory in that the commutative law is not assumed.
	www.math.fau.edu /Richman/html/construc.htm (1293 words)

	Citations: Elements of Mathematics - Bourbaki (ResearchIndex)
		Godel [11] But formalized mathematics cannot in practice be written down in full, and therefore we must have confidence in what might be called the common sense of the mathematician.
		Introduction A formal mathematical proof is a finite sequence of formulas, each element of which is either an axiom or the result of applying one of a fixed set of mechanical rules to previous formulas in the sequence.
		The main results of the present paper are devoted to dcpo s (D; admitting a directed set P of projections with range in the set K(D) of compact elements of (D; We call these dcpo s P domains and give....
	citeseer.ist.psu.edu /context/2841/0 (2626 words)

	>The Origins of Greek Mathematics
		In actual fact, our direct knowledge of Greek mathematics is less reliable than that of the older Egyptian and Babylonian mathematics, because none of the original manuscripts are extant.
		This school had amongst its chief pursuits the use of mathematics to understand the function of the universe.
		During the classical period, mathematics and philosophy were favored.
	www.math.tamu.edu /~don.allen/history/greekorg/greekorg.html (1554 words)

	Challenging the role of mathematics in engineering education
		Engineering educators are concerned about the ability of today's students to handle the, largely classical, mathematics which traditionally underlies the teaching of engineering.
		At the same time engineering as practiced by graduates is becoming increasingly less dependent on a knowledge of classical mathematics, as the tools used by professional engineers become embodied in computer packages.
		What educational research has been done on this type of mathematics in engineering and what ongoing initiatives and developments are relevant.
	www.chemeng.ed.ac.uk /people/jack/Maths/project.html (566 words)

	Computational Mathematics
		Mathematical models arise in a wide variety of fields, including business, economics, engineering, finance, medicine, and science.
		The application of computer methods to simulate such models was traditionally called "scientific computation," though the practice has spread far beyond its roots in science to encompass problems arising in all areas of society.
		There is a significant demand for people educated in the field of computational mathematics; that is, those who are able to deploy effectively a wide range of mathematical and computational techniques in areas of application.
	www.math.uwaterloo.ca /navigation/Prospective/programs/cm.shtml (327 words)

	Reverse Mathematics - Articles and Information (Site not responding. Last check: 2007-11-03)
		Reverse mathematics is the branch of mathematics concerned with what are the minimal axioms needed to prove the particular theorem.
		It turns out that over a weak base theory, many mathematical statements are equivalent to the particular additional axiom needed to prove them.
		The primary difference between doing classical mathematics in set theory (ZFC) and doing it in second order arithmetic is that in second order arithmetic one deals with codes for sets rather than sets themselves (except sets of integers).
	www.breakpt.org /article/Reverse_Mathematics (269 words)

	20th Century Schools in the Philosophy of Mathematics (Site not responding. Last check: 2007-11-03)
		Most closely associated with Brouwer, this school is characterised by a wholesale rejection of the methods characteristic of classical set theory, particularly of the treatment of infinity pioneered by Cantor.
		It is often characterised as the view that logic and mathematics are mere formal games and have a legitimacy independent of the semantic content of these formalisms, provided only that we can be reassured of the consistency of the formal systems.
		Hilbert's "programme" for resolving the paradoxes was to seek a "finitary" consistency proof for the whole of classical mathematics.
	www.rbjones.com /rbjpub/philos/maths/faq027.htm (448 words)

	Mathematics - Wikipedia, the free encyclopedia
		An online encyclopedia of mathematics, focusing on classical mathematics.
		An online math encyclopedia under construction, focusing on modern mathematics.
		This page was last modified 11:43, 13 August 2005.
	www.wikipedia.org /wiki/Mathematics (2627 words)

	Lars Birkedal / Realizability Bibliography
		Realizability and Shanin's algorithm for the constructive deciphering of mathematical sentences.
		Constructive assertions in an extension of classical mathematics.
		Mathematical independencies in the pure calculus of constructions.
	www.itu.dk /people/birkedal/realizability/index.html (2374 words)

	Ancient and Classical Mathematics
		We will be concerned with both the development of certain mathematical ideas and the roles mathematics has played in the historical cultures and societies we study.
		That is, we will be interested in the mathematicians who created the mathematics, the teachers who communicated the mathematics, the students who learned the mathematics and the people who paid for it all.
		Various short mathematical and writing assignments will be given during the semester; these will add up to about 30% of the grade.
	it.stlawu.edu /~dmelvill/323 (997 words)

	Mathematics Chair Created At SLU With Gift From Texas Couple (Site not responding. Last check: 2007-11-03)
		He joined the faculty at St. Lawrence in 1991, and has published widely in mathematics, mathematics history and philosophy.
		Specific research fields of Melville's include Lie algebras, Mesopotamian mathematics and ancient and classical mathematics, and many of his most recent publications are on the topic of quantum deformations.
		Melville is the Frank P. Piskor Faculty Lecturer for 2005, and will give a campus lecture in the spring on the topic "Teaching and Learning Mathematics in Mesopotamia." Martha Root Peterson, a native of Colton, is a 1962 graduate of St. Lawrence, with a degree in mathematics.
	www.stlawu.edu /news/petersonchair.html (477 words)

	History of Mathematics: Greece (Site not responding. Last check: 2007-11-03)
		Fowler, D. The mathematics of Plato's academy: a new reconstruction.
		Classical mathematics: a concise history of the classical era in mathematics.
		The birth of mathematics in the age of Plato.
	aleph0.clarku.edu /~djoyce/mathhist/greece.html (268 words)

	On Gödel's Philosophy of Mathematics, Abstract (Site not responding. Last check: 2007-11-03)
		An attempt is made to explicate and analyze Kurt Gödel's philosophy of mathematics with emphasis on his defense of classical mathematics, and his rejection of intuitionism, and the vicious circle principle.
		Gödel's belief in the real existence of mathematical objects is examined.
		It is argued that one need not accept Gödel's pronounced realism in order to assent to his methodology of mathematics.
	www.friesian.com /goedel/abstract.htm (72 words)

	Find in a Library
		Classical mathematics : a concise history of the classical era in mathematics
		To find a library, type in a postal code, state, province, or country.
		WorldCat is provided by OCLC Online Computer Library Center, Inc. on behalf of its member libraries.
	worldcatlibraries.org /wcpa/ow/baa4bcbd5b3f2a70a19afeb4da09e526.html (41 words)

	Mathematics in Aristotle (Key Texts) by Thomas Heath : Book (Site not responding. Last check: 2007-11-03)
		Mathematics in Aristotle (Key Texts) by Thomas Heath : Book
		Book Subject -- Aesthetics, General, History and Surveys - Ancient and Classical, Mathematics, Philosophy, Ancient Greece, Ancient Western philosophy to c 500, Philosophy of mathematics, Philosophy of science
		Data retrieved from amazon.com Sunday August 14 2005 at 1:19 AM Mathematics in Aristotle (Key Texts)
	www.crimsonbird.com /cgi-bin/a.cgi?j=1855065649 (101 words)

	Amazon.ca: Books: Introduction to Classical Mathematics I, from the Quadratic Reciprocity Law to the Uniformation Theo (Site not responding. Last check: 2007-11-03)
		Amazon.ca: Books: Introduction to Classical Mathematics I, from the Quadratic Reciprocity Law to the Uniformation Theo
		Introduction to Classical Mathematics I, from the Quadratic Reciprocity Law to the Uniformation Theo
		Top of Page : Introduction to Classical Mathematics I, from the Quadratic Reciprocity Law to the Uniformation Theo
	www.amazon.ca /exec/obidos/ASIN/0792312317 (142 words)

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