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	Online Encyclopedia and Dictionary - Mathematics
		Nowadays mathematics is the investigation of axiomatically defined abstract structures using mathematical notation and symbolic logic.
		Mathematics is usually regarded as an important tool for science, even though the development of mathematics is not necessarily done with science in mind (See pure mathematics and applied mathematics.).
		Mathematics might accordingly be seen as an extension of spoken and written natural languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships.
	www.fact-archive.com /encyclopedia/Mathematics (2298 words)

	Abstraction (mathematics) - Wikipedia, the free encyclopedia
		Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications.
		Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures.
		Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract.
	en.wikipedia.org /wiki/Abstraction_(mathematics) (384 words)

	Mathematics - Crystalinks
		Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not utilize the scientific method.
		Mathematical logic is concerned with setting mathematics on a rigid axiomatic framework, and studying the results of such a framework.
		Discrete mathematics is the common name for the fields of mathematics most generally useful in theoretical computer science.
	www.crystalinks.com /math.html (3179 words)

	What is Applied Mathematics?
		Mathematics is the human endeavor which has come to provide definition and scope to these activities, in terms which employ the utmost precision of thought.
		The growth of mathematical roots, as mathematics, involves abstraction away from concern with particular objects towards an emphasis on relations among abstract objects, axiomatization, and establishment of the basic characteristics and facts of a subject by rigorous proof.
		Thus a distinction between applied and pure mathematics is one of the interest and purpose of the practitioner, it is not a fixed dichotomy or division of the subject areas of mathematics.
	kirk.math.twsu.edu /appliedmath.html (2558 words)

	Mathematics
		Mathematics 3 and 8 cover the basic calculus of functions of a single variable, as well as vector geometry and calculus of scalar-valued functions of several variables.
		Mathematics 17, "An Introduction to Mathematics Beyond Calculus", is a course designed for students interested in learning about some of the aspects of mathematics not usually encountered in the first years of mathematical studies.
		Prerequisite: Mathematics 8, or Mathematics 3 and 6.
	www.dartmouth.edu /~reg/courses/desc/math.html (7962 words)

	Simplification Using Abstraction « Abhijit Nadgouda @ iface
		Abstraction in Computer Science is very analogous to Abstraction in Philosophy and Abstraction in Mathematics.
		Abstraction can strip out details and provide a minimalist version of the idea which is easier to understand.
		Abstraction decouples the purpose from the means of achieving it.
	iface.wordpress.com /2006/04/29/simplification-using-abstraction (1140 words)

	UGResearch (Site not responding. Last check: 2007-10-13)
		It is the branch of mathematics that deals with the design and analysis of the techniques used to transmit secret information and therefore it employs a large amount of number theory and statistics.
		The mathematical model is based on Newton's Law of Cooling which states that the rate of change of temperature resulting from heat transfer across a single boundary is proportional to the difference in the temperatures on opposite sides of the boundary.
		The mathematical abstraction of concepts and relationships associated with a natural phenomenon allows us to both make predictions about its behavior as well as provide a logical structure with which to discover new insight.
	www.marywood.edu /math/ugresearch.htm (1845 words)

	Academic Advising for Math Majors
		Mathematics majors who have been highly successful (faculty recommendations and/or grades of 3.50 and above) in Calculus I and II as freshmen should be advised to elect MATH 340 Set Theory and Logic and MATH 375 Linear Algebra I during the fall semester of the sophomore year.
		A high cumulative college average, the mathematical maturity of the student and the student's enjoyment of the study of mathematics are three of the most important considerations of the mathematics faculty in making recommendations for a student to be admitted to the program.
		This level is characterized by a concern with a few very explicit mathematical objects, e.g., the integers, rational numbers, real numbers, the complex numbers, the Euclidean plane, etc. The next level of abstraction occurs when certain properties common to several different concrete mathematical objects are isolated and studied for their own sake.
	www.potsdam.edu /content.php?contentID=263D84494F24BEF9809630BA9FE6A3D2 (2970 words)

	Foundations of Mathematics (Site not responding. Last check: 2007-10-13)
		The study of the foundations of mathematics examines the origin or genesis of mathematical ideas and methods, as well as the structure and organization of the mathematical corpus, primarily with the goal of understanding, explaining, or justifying the perceived special qualities of mathematical knowledge.
		Although the study of mathematical foundations has always been a recognizable part of mathematics and philosophy, it is only since about 1900 that foundational study has emerged as a relatively independent discipline with its own methods, techniques and goals.
		Mathematics is traditionally conceived as the science of space (geometry) and of quantity (arithmetic).
	bahai-library.com /?file=hatcher_foundations_mathematics (14860 words)

	Springer Online Reference Works
		A mathematical idealization, related to a certain form of the concept of infinity in mathematics — the idea of a potential infinity.
		The acceptance of the abstraction of potential realizability logically leads to the principle of mathematical induction.
		Abstraction of potential realizability plays a special role in constructive mathematics, in which propositions concerning the existence of constructive objects that satisfy given conditions are regarded as propositions on the potential realizability of such objects.
	eom.springer.de /a/a010500.htm (158 words)

	NCTM: Standards: An Overview -- Principles and Standards for School Mathematics (Site not responding. Last check: 2007-10-13)
		Mathematics is a highly interconnected and cumulative subject.
		A well-articulated curriculum is necessary for teachers at each level to know what mathematics their students have already studied and will study in future grades.
		Teachers must know and understand deeply the mathematics they are teaching and understand and be committed to their students as learners of mathematics and as human beings.
	www.nctm.org /standards/principles.htm (1345 words)

	Pacific Union College \| Mathematics
		As a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth.
		Modern mathematics is a creative counterpoint of computation and deduction, rooted in data while unfolding in abstraction.
		Students majoring in mathematics will have experience with mathematical modes of thought which include modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols.
	www.puc.edu /PUC/academics/Academic_Departments/Math_Dept (323 words)

	Mathematics / 2006-07 PLU Catalog
		Mathematics is a many-faceted subject that is not only extremely useful in its application, but at the same time is fascinating and beautiful in the abstract.
		A placement test and background survey are used to help insure that students begin in mathematics courses that are appropriate to their preparation and abilities.
		MATH 203: History of Mathematics — MR, NS A study in the vast adventure of ideas that is mathematics from ancient cultures to the 20th century.
	www.plu.edu /print/catalog/degrees-courses/math.html (2123 words)

	Department of Mathematics
		The study of Mathematics is essential for all of the sciences, for Computer Science, for Economics, and for the social sciences.
		The Mathematics Award is presented to a graduating senior and a junior for excellence in Mathematics and service to the Department.
		The Sidney Goldwater Roth Prize in Mathematics is presented to the graduating senior who in the estimation of the Department shows the greatest Mathematical promise and who has been of greatest service to the Department and fellow students.
	www.nyu.edu /cas/dept/math.htm (578 words)

	MAD 3105 Discrete Mathematics II [FSU Math]
		There are two battles you fight at this level of abstraction in mathematics.
		The first is trying to understand the mathematics and how to use it in applications and in further theoretical development of the subject.
		You have successfully completed many mathematics courses and know that on a ``test'' you may not give or receive any help from a person or written material except as specifically designed acceptable.
	www.math.fsu.edu /~mott/discrete/mad3105sp03/MAD3105.html (628 words)

	Bookshelved Wiki: MathematicsItsContentMethodsAndMeaning
		The introduction explains mathematics as an abstraction of some of human thought, with two main ideas: numbers and three-dimensional objects.
		When farmers were very interested in partitioning their land, mathematicians would consciously develop the abstraction of straight lines...
		an abstraction which is impossible to realize in the real world, because things such as density and color are abstracted away (not to mention a couple of dimensions are missing, being 1-D).
	bookshelved.org /cgi-bin/wiki.pl?MathematicsItsContentMethodsAndMeaning (520 words)

	CIC Mathematics Department Microsite
		Recognizing that mathematics is the abstraction of logical relationships, the focus of teaching at CIC is to give each student the ability to use logic, together with factual knowledge, to solve problems they encounter.
		The mathematics staff in the CIC Secondary School is comprised of well qualified and experienced teachers who have also demonstrated a high level of concern for each student's academic and emotional well being.
		Projects and open-ended problems are embedded in the curriculum and encourage students to explore a variety of problem-solving strategies that transfer to situations beyond the mathematics classroom.
	www.cic-caracas.org /departments/mathematics (343 words)

	Mathematics and Art
		Appropriate mathematics is introduced to describe the window experiments.
		Using the new mathematics and an ordinary spreadsheet program, true perspective drawings are made with scatter plots and traditional or spreadsheet drawing tools.
		Phenomena observed in window experiments and spreadsheet drawings are validated as theorems on vanishing points, vanishing lines, rendering of regularly spaced objects, and various other drawing techniques valuable to the artist.
	www.indiana.edu /~iubmtc/Art/Art.html (277 words)

	Shepherd College Mathematics (Site not responding. Last check: 2007-10-13)
		At Shepherd University, you will explore mathematical realms never dreamed of in elementary algebra.
		Mathematical methods extend to all endeavors that require analysis, reason, logic, and abstraction.
		Shepherd University mathematics students pursue careers in accounting, actuarial science, advanced studies, business, civil service, computer science, design, engineering, education, financial analysis and research.
	www.shepherd.edu /mathweb/mathematics.html (105 words)

	MATH228 (Site not responding. Last check: 2007-10-13)
		An introduction to discrete/combinatorial mathematics, and to abstraction, theorems and proofs.
		This is a mathematics course, not a computer science (or any other kind of science) course.
		The emphasis in the title of the textbook should be on the word MATHEMATICS, not on the clause after the comma.
	www.wesleyan.edu /wesmaps/course9899/math228f.htm (159 words)

	Augsburg - Math
		For Professor Coult, the beauty and challenge of mathematics comes not only from the discipline itself, but from the myriad of ways in which mathematics is used to understand the world around us.
		As an applied mathematician, he is motivated by this connection between the abstraction of mathematics and the real world.
		"This type of research, delving into difficult mathematics, but also connecting to computational science and physics, is what draws me to the field," he said.
	www.augsburg.edu /mathematics/coult.html (287 words)

	RUMEC RUMEC-related Papers and Preprints
		Reflective abstraction and mathematics education: The genetic decomposition of induction and compactness, by Ed Dubinsky and P. Lewin.
		The constructive aspects of reflective abstraction in advanced mathematics, by Ed Dubinsky.
		One theoretical perspective in undergraduate mathematics education research, by B.
	www.cs.gsu.edu /~rumec/Papers/index.htm (843 words)

	UH: MATHEMATICS AND REALITY - 19.2.2002
		Mathematical models are an abstraction of the system they represent.
		Abstraction allows for the simplification of the system because it is not necessary or even desirable for it to be exact or replicate the exact mechanisms.
		explained in simplified mathematical terms so that their essential qualities are characterized in a concrete manner.
	www.hi.is /~joner/eaps/ltwh655.htm (1040 words)

	Abstraction in Mathematics (cont'd) (Site not responding. Last check: 2007-10-13)
		"The propositions of mathematics have, therefore, the same unquestionable certainty which is typical of such propositions as "All bachelors are unmarried".
		Mathematical abstraction results in the elimination of content.
		Abstraction in computer science consists in the enlargement of content.
	www.d.umn.edu /~tcolburn/cs2511/lectures/abstraction/text9.htm (58 words)

	Abstraction in mathematics #2 [Tesugen]
		Regarding abstraction, Olof said (if I understood correctly) that the procedure is (always?) to move to more and more abstract levels and then back up; as if it’s about finding the “root abstraction”, backing up from there and questioning each abstraction on the way.
		This is easy to relate to programming, where you also often try to find common abstractions, but obviously not anyway near the meticulous search for abstractions Olof explained.
		You’re only interested in an abstraction if it means an opportunity to reuse code between two more concrete classes, or to achieve flexibility of some kind – such as being able to deal with only “people” in a system that contains employees, managers, customers etc.
	tesugen.com /archives/02/09/mathabstraction2 (298 words)

	abstraction - OneLook Dictionary Search
		ABSTRACTION : 1911 edition of the Encyclopedia Britannica [home, info]
		Phrases that include abstraction: hardware abstraction layer, lambda abstraction, data abstraction, beta abstraction, abstraction creation, more...
		Words similar to abstraction: abstract, abstractedness, abstractional, abstractive, extraction, generalization, more...
	www.onelook.com /?w=abstraction (323 words)

	Abstraction in mathematics [Tesugen]
		I had lunch with my friend Olof, who’s into applied math, and I had asked him if there are any parallels in mathematics to abstraction in code (software).
		You do do things in mathematics that resemble abstraction.
		For example, Olof said that you use approximation to simplify the scenario (forgive me if these terms aren’t at all applicable) and that it’s often a question of finding the best approximation, in order to produce as exact results as possible, when things are so complex that exactness is impossible.
	tesugen.com /archives/02/09/mathabstraction (351 words)

	Deparment of Mathematics - Iona College
		Faculty Programs Undergraduate Courses Graduate Courses Course Schedules Other Mathematics Sites Contact Us
		These programs offered by the department are designed to assist students develop their command of mathematical tools and of mathematical modes of thinking including intuition, analysis of argument, modeling, abstraction, generalization, proof, and making connections both within mathematics and between mathematics and other disciplines.
		The offerings of the Department of Mathematics have as their goals to enable students to enjoy liberal learning, life-long learning, and productive citizenship while at the same time preparing them for graduate and/or professional education and careers in education, business and industry.
	www.iona.edu /artsscience/departments/math (99 words)

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