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Topic: Ideal mathematics

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	Ideal number - Wikipedia, the free encyclopedia
		In mathematics an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Kummer, and lead to Dedekind's definition of ideals for rings.
		An ideal in the ring of integers of an algebraic number field is principal if it consists of multiples of a single element of the ring, and nonprincipal otherwise.
		This means there is an element of the ring of integers of the class field, which is an ideal number, such that all multiples times elements of this ring of integers lying in the ring of integers of the original field define the nonprincipal ideal.
	en.wikipedia.org /wiki/Ideal_number (724 words)

	Constructive Mathematics
		To the Greeks mathematics was essentially geometry, and the quintessence of the subject was Euclid's Elements, where the deductive methods of the Eleatic school of philosophy, the logical procedures of Aristotle and Platonic idealism came together in the notion of an axiomatic system.
		It was hoped that by transforming the statements of mathematics into strings of meaningless symbols to be combined according to the rules of logic, whatever unavowed principles of reasoning had given rise to the paradoxes would be revealed.
		And when Brouwer attempted to rid mathematics of ideal notions while preserving certain of its long-established aspects, he introduced ideas that seemed to many mathematicians to be just as idealistic as those he was eliminating.
	digitalphysics.org /Publications/Cal79/html/cmath.htm (8036 words)

	Ideal theory -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-18)
		The ideal theory in question had been based on (additional info and facts about elimination theory) elimination theory, but in line with (German mathematician (1862-1943)) David Hilbert's taste moved away from (additional info and facts about algorithmic) algorithmic methods.
		The importance of the idea in general of a (A self-contained component (unit or item) that is used in combination with other components) module, more general than an ideal, probably led to the perception that ideal theory was too narrow a description.
		Northcott's 1953 Cambridge Tract Ideal Theory (reissued 2004 under the same title) was one of the final appearances of the name.
	www.absoluteastronomy.com /encyclopedia/i/id/ideal_theory.htm (232 words)

	Hilbert's Program
		It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent.
		Mathematics itself, however, operates with abstract concepts, e.g., quantifiers, sets, functions, and uses logical inference based on principles such as mathematical induction or the principle of the excluded middle.
		Reverse mathematics seeks to give a precise answer to this question by investigating which theorems of classical mathematics are provable in weak subsystems of analysis which are reducible to finitary mathematics (in the sense discussed in the preceding paragraph).
	plato.stanford.edu /entries/hilbert-program (7534 words)

	Untitled Document (Site not responding. Last check: 2007-10-18)
		Mathematics is so very familiar, present both in the natural world and in strictly human inventions.
		Leibniz sparked the trend when he famously published his desire to use mathematics in order to develop an all-encompassing universal language, which could be utilized in making all his vast interests more accessible.
		In Liebniz's ideal world, disagreements are always solvable: "And if someone would doubt my results, I should say to him: 'Let us calculate, Sir,' and thus by taking pen to ink, we should soon settle the question" (Preface, 15).
	www.contrib.andrew.cmu.edu /~lgreig/prose/mathLanguage.html (1870 words)

	C.S.U.S. Math Department - PAGE LABLE
		The key element in the design of our major is the mathematical maturity and intuition that our students develop through our program which will enable them to continue to be learners of mathematics, and to be models for their students as they approach mathematics.
		The mathematics that our students will be expected to teach or work with will change over a 40 year career, and the best method for students to respond to these changes is to be able to learn new areas of mathematics as needed.
		Professional mathematicians regard mathematical proof as the intrinsic essence of mathematics, and it is expected that undergraduates will arrive at an appreciation for the role of proof in mathematical discourse, as well as a grasp of the methods of proof that permeates all mathematical exposition.
	www.csus.edu /math/dept/part2.htm (2080 words)

	Essence as a Platonic ideal
		Platonic analysis aims to understand the physical world in terms of the ideals that capture the real essence that is dimly reflected in physical existence.
		Mathematical captures the mathematical properties of the ideal circle as an abstract structure.
		Mathematics avoids making implicit assumptions by limiting itself to one primitive entity, the empty set, with no structure of its own.
	www.mtnmath.com /whatth/node7.html (464 words)

	Bard Mathematics \| Courses
		There is no particular mathematical prerequisite for this course, though algebraic computations and deductive proofs of some of the main results are required.
		Prerequisites: Precalculus mathematics (Mathematics 110 or the equivalent).
		This course, which provides an overview of one of the oldest and most beautiful areas of mathematics, is ideal for any student who wants a taste of mathematics outside the calculus sequence.
	math.bard.edu /courses (1473 words)

	POM Contributed Paper Sessions (Site not responding. Last check: 2007-10-18)
		Earlier literature in mathematics had established that Godel's Proof could never support either conclusion: the proof deals with mathematical statements which could be proved (true) as a result of being based on arithmetic, mistakenly presumed to be (because of arithmetic's elementary nature) at the foundation of Science.
		Evidently, there is a tension between the way mathematics is and the level the learner is trying to reduce it to --to facilitate his or her accommodation of the concept.
		An important aspect of the nature of mathematics is the question of what a mathematical object is. A key component of this question is whether mathematical objects are thoughts inside people's minds or are entities external to human beings.
	www.wooster.edu /pom_sigmaa/PhilOfMathCPS.html (4976 words)

	Template -- Teachers @ Work - Mark Teadwell
		The emphasis here is on using these two mathematicians to show the mathematics is all about investigation of patterns and how those patterns can be described and predicted using the language of mathematics.
		These mathematics enrichment activities cover a wide range of contexts and themes and provide students with an interactive multimedia environment where they can practice basic mathematical principles as well as interact and learn the relationship between mathematical concepts as they are presented.
		Mathematical applets are small computer programs which run automatically over the Internet providing a visual sequence of events which students can engage with an experience what happens as different variables are changed.
	teachers.work.co.nz /www_Mch_2005.htm (4100 words)

	Ideal Cut Diamonds (Site not responding. Last check: 2007-10-18)
		Comparing ideal cut diamonds beside a poorly cut diamond and the differences are amazing.
		In 1919, Marcel Tolkowsky empirically calculated the ideal proportions of a round diamond as part of his Ph.D. thesis in Mathematics.
		It should be obvious that finding a ideal cut diamond to Tolkowsky's ideal diamond specifications is an expensive undertaking.
	www.canadadiamonds.com /ideal-cut-diamonds.htm (323 words)

	Reading of former days (Site not responding. Last check: 2007-10-18)
		mathematics!) because it is very likely, based on current cognitive science, that mathematics is done in our brains using structures which evolved for other purposes, e.g.
		In particular, that mathematics is similar to the most common use of human language, namely, gossip.
		One of their conclusions is that the actual form of mathematics is determined by our brains and what they are capable of doing - in other words, if there is a sort of Platonic ideal mathematics, there is no way that we could know about it.
	webpage.pace.edu /sshaver/oldreading.html (595 words)

	Biblioteca Digital del INEAM - La Educación - (119) III - 1994
		The philosophy that mathematics is dynamic is modeled in these courses by students exploring operations and properties of number, conjecturing and testing hypotheses with multibase blocks, fraction bars, geoboards, calculators, graphs, and the like.
		A year-long upper division sequence in some area of mathematics should be studied in depth with the purpose of having students (1) think deeply about mathematical ideas through exploration, conjectures, and justification of these hypotheses by way of discussion and proof, and (2) develop intellectual curiosity and self-directed learning.
		In sum, these recommendations call for an increase in the mathematics content studied by prospective teachers of mathematics at all levels, as well as for fundamental change in the nature of prospective teachers collegiate mathematics course experiences and the ways in which those courses are taught.
	www.educoas.org /portal/bdigital/contenido/laeduca/laeduca_119/articulo5/call.aspx (1074 words)

	Constructive Mathematics (Site not responding. Last check: 2007-10-18)
		Constructive mathematics is pertinent to the PRL project because all of our logics have had a constructive core.
		Its modern expression was brought into focus by Cantor and Hilbert, who referred to it as "ideal" mathematics.
		Brouwer's response to the crisis was to say that the problems arose from the nature of "ideal" or "existential" mathematics and did not arise in the computational tradition.
	cs.cornell.edu /Info/Projects/NuPrl/Intro/ConstrMath/constrmath.html (917 words)

	Annihilator - TheBestLinks.com - Ideal, Mathematics, TheBestLinks.com:Find or fix a stub, Ring (mathematics), ... (Site not responding. Last check: 2007-10-18)
		Annihilators are a concept that occurs in ring theory, a branch of mathematics.
		Annihilators are always one-sided ideals of their ring: If a and b both annihilate S, then for each s in S, (a+b)s=as+bs=0, and for any c in R, (ca)s=c(as)=c0=0.
		The annihilator of M is even a two-sided ideal: (ac)s=a(cs)=0, since cs is another element of M.
	www.thebestlinks.com /Annihilator.html (194 words)

	Supplements-Others (Site not responding. Last check: 2007-10-18)
		This series is suitable for the Exploring Math Series, and is ideal for classroom use.
		A Glossary defining all the mathematical terms used in the book in a precise way, so the book is self-contained.
		The prerequisite knowledge is no more than what is already taught in Additional Mathematics, but the problems given here are of a higher level of difficulty and require 'mathematic maturity' from the student to solve them.
	www.singaporemath.com /supplement_secmath.htm (1032 words)

	Template -- Teachers @ Work - Mark Teadwell
		Mathematics is about using and practising the language of problem solving.
		About Mathematics provides a wide collection of links to mathematics web sites as well as locally produced material which is presented on the site.
		Although presented as a scientific study, teaching epidemiology is more of an integrated study and includes mathematics, social studies, history and physical education.
	teachers.work.co.nz /www_June_2004.htm (4349 words)

	archive/cnp_test_2000
		The construction is based on the existence of a canonical isomorphism between the skein algebra of the torus and the subalgebra of the noncommutative torus generated by noncommutative cosines.
		The elements of this ideal are in some sense "orthogonal" to the vector whose entries are the colored Kauffman brackets of the knot.
		The generalized version of the A-polynomial, the noncommutative A-ideal, is a finitely generated ideal of polynomials in the quantum plane.
	www.math.buffalo.edu /archive/cnp_test_2000.html (8098 words)

	Supp_Pri Math (Site not responding. Last check: 2007-10-18)
		It is written to complement the Primary Mathematics U.S. Edition textbooks and workbooks and follows the same sequence of topics.
		throughout our many years of teaching mathematics to pupils of varying abilities and of different levels, we have identified common mistakes and errors, which are highlighted throughout the book.
		is ideal for enrichment and challenge for 6th grade and higher.
	www.singaporemath.com /supplement_primath.htm (2844 words)

	Graduate Program - General Info (Site not responding. Last check: 2007-10-18)
		Mathematics is ideal training for producing such generalists who can understand and solve a variety of problems arising in many different technical areas.
		This is a specialized degree designed to provide secondary school mathematics teachers with more depth in mathematics combined with a core of relevant professional courses in education.
		Finally, students who plan to pursue a Ph.D. in mathematics should include a sequence in analysis and a sequence in algebra in their plans of study.
	www.unomaha.edu /~wwwmath/graduate/info.html (425 words)

	ideal - OneLook Dictionary Search
		Ideal, ideal : UltraLingua English Dictionary [home, info]
		Phrases that include ideal: ideal gas, ego ideal, beau ideal, ideal gas law, ideal solid, more...
		Words similar to ideal: apotheosis, edenic, idealistic, idealless, nonesuch, nonpareil, nonsuch, paragon, saint, utopian, best, dream, exemplar, model, paradigm, perfect, standard, unparalleled, more...
	www.onelook.com /?w=ideal&ls=a (396 words)

	Drake Economics
		This program, offered in cooperation with the Department of Mathematics, is ideal for students planning graduate study in economics (PhD or MA programs).
		Such students choose a major advisor in either the Department of Mathematics or the CBPA, depending on their college of enrollment.
		This program, offered in cooperation with the Department of Mathematics, is ideal for students planning graduate study in economics (PhD or MA programs) or business (PhD programs or quantitatively-oriented MBA programs).
	www.drake.edu /cbpa/econ/majors.html (478 words)

	Klaus F. Jørgensen
		Specifically one here tries to secure some more ideal parts of mathematics relative to parts of mathematics that are epistemically unproblematic.
		These tools can be seen as reducing some ideal mathematics to some unproblematic parts of mathematics, both with respect to consistency and computational content.
		They also provide a frame work for the evalutation of mathematical principles which at first sight look non-constructive but on a closer view in a specific framework nevertheless are constructive, as for instance Markov's principle, extensionality and restricted forms of independence-of-premise.
	www.akira.ruc.dk /~frovin/work.htm (280 words)

	Student calculator is ideal as a child learning tool for mathematics formula use in the school.
		Student calculator is ideal as a child learning tool for mathematics formula use in the school.
		Developed by the Dovada research team, this student calculator is ideal for mathematics formula use in the school, home, office or any engineering or research areas, anywhere mathematical or physics information is regularly used or required.
		These departments fall only under the category of physics and mathematics and similar educational departments and the licensed copies are given in return for supplying some software analytical appraisal.
	www.dovada.com.au /calculator.htm (398 words)

	Ideal child learning tool, mathematics formula calculator with built in metrics conversion calculator
		The student calculator is ideal in teaching child mathematics and boasts many inbuilt features, which are not limited to, but include,
		Developed by the Dovada research team, the student calculator is ideal for mathematics formula use in the school, home, office or research areas, anywhere mathematical or physics information is used or required.
		These departments fall only under the category of physics and mathematics educational departments and the licensed copies are given in return for software analytical appraisal.
	www.dovada.com /calculator.htm (370 words)

	01.06.12: Multi-Sensory Manipulatives in Mathematics: Linking the Abstract to the Concrete
		Mathematics is an element present in virtually every career (especially if you are looking to make money).
		The focus of this unit is the use of multi-sensory manipulatives in the teaching/learning of mathematics.
		I have also observed that many of the veteran mathematics teachers are not using manipulatives (this is not due to lack of supplies because the New Haven district Mathematics Department is very generous).
	www.yale.edu /ynhti/curriculum/units/2001/6/01.06.12.x.html (4204 words)

	Andrew D. Irvine :: UBC Department of Philosophy (Site not responding. Last check: 2007-10-18)
		It argues that Frege's concerns can best be understood as questioning Hilbert's implicit importing of content into ideal mathematics.
		Finitary mathematics, unlike ideal mathematics, is claimed to involve genuine propositions.
		Frege's objections are thus primarily directed against ideal mathematics.
	www.philosophy.ubc.ca /irvine/student/vogt.htm (100 words)

	Bibliography
		Mathematical Proceedings of the Cambridge Philosophical society 100: 31-41.
		Keywords: ideals in algebraic number theory, in algebraic geometry, modern algebra, ideals in analysis and topology.
		Nineteenth-Century Mathematics in the Mirror of its Literature: A Quantitative Approach.
	www.rzuser.uni-heidelberg.de /~proquet2/bib.html (5203 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		] A principal ideal of a ring given by a single element that has properties analogous to those of the prime numbers.
		In classical physics and in special relativity, the principle that the laws of physics take the same mathematical form in all inertial reference frames.
		In general relativity, the principle that the laws of physics take the same mathematical form in all conceivable curvilinear coordinate systems.
	www.accessscience.com /Dictionary/P/P43/DictP43.html (2683 words)

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