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Applied Mathematics Should Be Taught Mixed Gary I Humanistic Mathematics Network Journal Issue 10 Article 3 8-1-1994 Applied Mathematics Should Be Taught Mixed Gary I. Brown College of Saint Benedict Follow this and additional works at: http://scholarship.claremont.edu/hmnj Part of the Applied Mathematics Commons, Educational Methods Commons, and the Science and Mathematics Education Commons Recommended Citation Brown, Gary I. (1994) "Applied Mathematics Should Be Taught Mixed," Humanistic Mathematics Network Journal: Iss. 10, Article 3. Available at: http://scholarship.claremont.edu/hmnj/vol1/iss10/3 This Article is brought to you for free and open access by the Journals at Claremont at Scholarship @ Claremont. It has been accepted for inclusion in Humanistic Mathematics Network Journal by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Applied Mathematics Should be Taught Mixed Gary J. Brown Col/ege of Saint Benedict Saint Joseph, Minneso1l156374 I. INTRODUCTION The metaphor emanating from the term "applied" seems to reinforce this separation between (pure) Many would argue that "the power of mathematics" mathematics and applied mathematics. Is is is derived from the great variety of problems that possible to change metaph ors and reverse this can be modeled and solved mathematically. hidden message? Historically, the term "applied" Accord ing to The Power of Mathematics, was not used in the literature until about 170 years Applications to Management and Social Science, ago ("applied" appeared in J. Gergonne's journal by Whipkey, Whipkey and Conway, "the power of called •Annales de mathematiques pures et mathematics is derived from two sources. First, appliquees" that was first published in 1810). the same mathematical concept can be used to solve Prior to this, mathematics was divided into "pure a multitude of problems from diverse academic mathematics" and "mixed mathematics." The fields. For example, the algebra of matrices can be purpose of this paper is to outline historically the applied to problems arising in mathematics, meaning of the term "mixed mathematics" and then physics, che mis try , economics. sociology, to choo se aspects of its mean ing that can be psychology and statistics" (Introduction, p 3). modified to present a different vision of how to teach applied math ematics. I will first argue that The hidden message coming from such a derivation there is a dichotomy today between "pure" and seems to be that (pure) mathematics is this self­ "applied" mathematics. Then I will provide some contai ned. autonomous abstract subject completely historical analysis of mixed mathemati cs in devoid of any human value until it is "applied" to a England in the early seventeenth century and in given problem or discipline. As a young student I Fran ce in the eighteenth century. This will be first felt this hidden message when encountering followed by a brief examination of two nineteenth "modeling" diagrams such as the one in figure 1-: century practitioners of mixed mathemati cs, Gaspard Monge and William Whewell. Finally, I One was supposed to tran sform a "real world will provide some ideas for a possible new model problem" into a "mathematical model" and solve for the teaching of applied mathematics that is the resulting problem mathematically. Then, one partially based upon a recent article in the Bulletin was supposed to interpret the mathematical of the American Mathematical Society by Arthur solutions in terms of the real world problem. It Jaffe and Frank Quinn. I an no, arguing that mixed always seemed that the "real world problem" was mathematics is historically "better" than applied co mpletely separated from the " mathematical mathematics, but that one can adapt some of the problem". As an aspiring mathematician, I hated to main ideas from mixed mathematics to our modem deal with the "real world problem" and only evolving view of applied mathematics. I hope that wanted to deal with the "mathematical model." Reol·vorlll Abstrac\ problem De ·w I!xpe~ n\ ~y Re ults ~ D.ta In"'",l\!\ Pl\!dic\ ') ('-_""/) Results C Figure I HMNJournal #10 I the changing of metaphors will result in reversing concerned with the study of physical, this hidden message of applied mathematics. biological and sociological worlds... In a restricted sense, the term refers to the use of II. THE EXISTENCE OF A DICHOTOMY mathematical principles as tools in the fields of BElWEEN "PURE" AND "APPLIED" physics, chemistry, engineering, biology. and social studies. Before discussing "mixed mathematics:' it perhaps is necessary to demon strate the existence of a From Avner Friedman, current SIAM President, dichotomy between " pure" and "applied" testifying to the House of Representatives on NSF mathematics today. The first thing I did was to funding, and discussing the goa l s of the examine how the terms "pure" and "applied" were mathematicalsciences, used in everyday English language. According to to lead the development and transfer of Funk and Wagnalls Dictionary, applications of mathematics to problems in Pure- Free from mixture or contact with that science, technology and industry. which weakens or impairs or pollutes; considered apart from practical application, All of these sources seem to reinforce the idea that opposedto applied. to do "applied mathematics" one must apply theory Applied- To bring into contact with A from "pure mathematics" to problem B from "the something; to devote or put to a particular use; real world." This separation did not seem to exist as to apply steam to navigation or money to prior to the French Revolution (It is difficult to find paymentor debts. evidence of this separation in non-European cultures). Secondly, I looked at many general source books such as encyclopedias. Accordin g to the World Ill. "MIXED MATHEMATICS IN EARLY Book Encyclopedia, SEVENTEENTH CENTURY ENGLAND The work of mathematicians may be divided into pure mathematics and applied mathemasics. The term "mixed mathematics" occurred frequently Pure mathematics seeks to advance in many so called "trees of knowledge" in the mathematical knowledge for its own sake rather Seventeenth and Eighteenth centuries." One of than for any immediate practical use...applied the first prominent "trees of knowledge" comes mathematics seeks to develop mathematical from Francis Bacon's Of the Profictence and technique for use in science and other fields. Advancement of Learnings (1605). In Bacon's Also from the Mathematics Dictionary edited by tree of knowledge (See Figure 2), philosophy was James and James, I found the following: divided into three categories (Divine Theology, applied mathematics: A branch of mathematics Natural, and Human). Natural Philosophy was Bacon's Tree of Knowledge (Human Learning) Divine or Natunl Theology P H Physics Cm.tr:I I L so;"""e M....physics < Pure Math 'I!lem.lic o < S Natunl Mix.d Math o Perspective P Prud•••• ~EXperimental M\Uic H < •• P'.~'''sphical Astronomy y Me:ical osmogn.phy Human AIchi"'c!lm Engineeling Figure 2 2 HMNJournal #/0 divided into Science and Prudence; Science was former Puritan clergyman and popularizer of divided into Physics and Metaphysics. Finally Bacon's approach to science. Wilkins was one of mathematics was classified under Metaphysics (See the founder> of the Royal Society (1660), 'he first Figure 2). scientific organization whose purpose was the promotion and advancement of science and its According to Bacon , ma thematics belonged to applications to the improvement of society. The metaphysics because its aim was to inquire about book "Math Magick" refers to the longer title "Math fixed and constam causes not indefinite causes. He Magick: The Wonders that may be Performed by said "all other forms are the most abstracted and Mechanical Geometry." Mechanical Geometry was separated from matter and therefore most proper to described as "one of the most easy, useful and yet Metaphysics." most neglected parts of mathematics. It is a liberal art like astronomy and music:' Topics included a After placing mathematics in the category of discussion of pulleys, cranes, bow s and catapults metaphysics Bacon subdivided mathematics into used in levers and wedges; the possibilities of "pure" and "mixed". He said "to pure mathematics various kinds of machines, such as submarines and belong tho se sciences which handle Quantity carriages propelled by sails; and " secret and entirely severed from matter and from the Axioms speedier ways of attacking forts by approaches and of Natural Philosophy. These are Geometry and galleries:'(inventions in fortification ) Arithmetic...Mixed Mathematics has for its subject The purpose of"Math Magick" was to familiarize the average person with the basic and long­ According to d'Alembert, "quantity, accepted principles of mechanics. Wilkins begins the object of mathematics, could be with a defense of mechanics as a liberal art that was more like astronomy and music than the so called considered either alone and "illiberal sciences" which involved some physical independent of real and abstract activity such as manufacturing or trade. The basic things from which one gained subject of mechanics was the relationship between knowledge of it" weight and power. Weight was no longer to be considered a "natural quality, whereby condensed bodies do of them selves tend downwards," but some axioms and parts of natural philosophy, and "an affection which might be measured" Quantity, consider quantity determined as it is auxiliary and the subject of seventeenth century
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