Humanistic Mathematics Network Journal

Issue 10 Article 3

8-1-1994

Applied Mathematics Should Be Taught MixedGary I. BrownCollege of Saint Benedict

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Recommended CitationBrown, 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

Applied Mathematics Should be Taught Mixed

Gary J. BrownCol/ege of Saint Benedict

Saint Joseph, Minneso1l156374

I. INTRODUCTION

Many would argue that "the power of mathematics"is derived from the great variety of problems thatcan be modeled and so lved mathematically.Accord ing to The Power of Mathematics,Applications to Management and Social Science,by Whipkey, Whipkey and Conway, "the power ofmathematics is derived from two sources. First ,the same mathematical concept can be used to solvea multi tude of problems from diverse academicfields. For example, the algebra of matrices can beapplied to problems arising in mat hemat ics,physics , che mis try , economics. sociology,psychology and statistics" (Introduction, p 3).

The hidden message coming from such a derivationseems to be that (pure) mathem atics is this sel f­contai ned. autonomous abstract subjec t completelydevoid of any human value until it is "applied" to agiven problem or discipline. As a young student Ifirst felt this hidden message when encountering"modeling" diagrams such as the one in figure 1-:

One wa s supposed to tran sform a "real worldproblem" into a "mathematical model" and solvethe resulting problem mathematically. Then, onewas supposed to interpret the mathematicalsolutions in terms of the real world problem. Italways seemed that the "real world problem" wasco mplet ely separated from the " mathematicalproblem". As an aspiring mathematician, I hated todeal with the "real world problem" and onlywanted to deal with the "mathematical model."

The metaphor emanating from the term "applied"seems to reinforce this separation between (pure)mathematics and applied mathematics. Is ispossible to change metaph ors and reverse thi shidden message? Historically, the term "applied"was not used in the literature until about 170 yearsago ("applied" appeared in J. Gergonne's journalcalled •Annales de mathematiques pures etappliquees" that was first published in 1810).Prior to this, mathematics was divided into "puremathematics" and "mixed mathematics." Thepurpose of this paper is to outline historically themeaning of the term "mixed mathematics" and thento choo se aspects of its mean ing that can bemodified to presen t a different vision of how toteach applied math ematics. I will first argue thatthere is a dichotomy today between "pure" and"applied" mathematics . Then I will provide somehistorical analysis of mixed mat hemati cs inEngland in the early seventeenth century and inFran ce in the eighteenth century. This will befollowed by a brief examination of two nineteenthcentury practitioners of mixed mathemati cs,Gaspard Monge and William Whewell. Finally, Iwill provide some ideas for a possible new modelfor the teaching of applied mathematics that ispartially based upon a recent article in the Bulletinof the American Mathematical Society by ArthurJaffe and Frank Quinn. I an no, arguing that mixedmathematics is historically "better" than appliedmathematics, but that one can adapt some of themain ideas from mixed mathematics to our modemevolving view of applied mathematics. I hope that

Reol·vorlllproblem

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I!xpe~n\ ~y

(D.ta~ In"'",l\!\

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From Avner Friedman, current SIAM President,testifying to the House of Representatives on NSFfunding, and discussing the goa l s of themathematical sciences,

to lead the development and transfer ofapplications of mathematics to problems inscience, technology and industry.

All of these sources seem to reinforce the idea thatto do "applied mathematics" one must apply theoryA from "pure mathematics" to problem B from "thereal world." This separation did not seem to existprior to the French Revolution (It is difficult to findevidence of this separation in non-Europeancultures).

concerned with the study of physical,biological and sociological worlds... In arestricted sense, the term refers to the use ofmathematical principles as tools in the fields ofphysics, chemistry, engineering, biology. andsocial studies.

The term "mixed mathematics" occurred frequentlyin many so called "trees of knowledge" in theSeventeenth and Eighteenth centuries." One ofthe first prominent "trees of knowledge" comesfrom Francis Bacon's Of the Profictence andAdvancement of Learnings (1605). In Bacon'stree of knowledge (See Figure 2), philosophy wasdivided into three categories (Divine Theology,Natural, and Human). Natural Philosophy was

Ill. "MIXED MATHEMATICS IN EARLYSEVENTEENTH CENTURY ENGLAND

Figure 2

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Bacon's Tree of Knowledge (Human Learning)

Human

Divine or Natunl Theology

<Physics Cm.tr:I

<so;"""e M....physics < Pure Math 'I!lem.lic

Natunl Mix.d MathPerspective

Prud• • • •~EXperimental M\Uic•• P''''sphical.~ Astronomy

Me:ical osmogn.phyAIchi"'c!lmEngineeling

PHILoSoPHy

Before discussing "mixed mathematics:' it perhapsis necessary to demon strate the existence of adichotomy between " pure" and "applied"mathematics today. The first thing I did was toexamine how the terms "pure" and "applied" wereused in everyday English language. According toFunk and Wagnalls Dictionary,

Pure- Free from mixture or contact with thatwhich weakens or impairs or pollutes;considered apart from practical application,opposed to applied.Applied- To bring into contact withsomething; to devote or put to a particular use;as to apply steam to navigation or money topaymentor debts.

Secondly, I looked at many general source bookssuch as encyclopedias. Accordin g to the WorldBook Encyclopedia,

The work of mathematicians may be dividedinto pure mathematics and applied mathemasics.Pure mathematics seeks to advancemathematical knowledge for its own sake ratherthan for any immediate practical use...appliedmathematics seeks to develop mathematicaltechnique for use in science and other fields.

Also from the Mathematics Dictionary edited byJames and James, I found the following:

applied mathematics: A branch of mathematics

II. THE EXISTENCE OF A DICHOTOMYBElWEEN "PURE" AND "APPLIED"

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the changing of metaphors will result in reversingthis hidden message of applied mathematics.

divided into Science and Prudence; Science wasdivided into Physics and Metaphysics. Finallymathematics was classified under Metaphysics (SeeFigure 2).

According to Bacon , ma thematics belonged tometaphysics because its aim was to inquire aboutfixed and constam causes not indefinite causes. Hesaid "all other form s are the most abstracted andseparated from matter and therefore most proper toMetaphysics."

After placing mathematics in the category ofmetaphysics Bacon subdivided mathematics into"pure" and "mixed". He said "to pure mathematicsbelong tho se sc iences which handle Quantityentirely severed from matter and from the Axiomsof Natural Philosophy. These are Geometry andArithmetic...Mixed Mathematics has for its subject

According to d'Alembert, "quantity,the object of mathematics, could beconsidered either alone andindependent of real and abstractthings from which one gainedknowledge of it"

some axioms and parts of natural philosophy, andconsider quantity determined as it is auxiliary andincident unto them. For many parts of nature canneither be inverted with sufficient subtlety nordemonstrated with sufficient perspicuity withoutthe aid and intervening of mathematics."

Bacon added to the Ancient Greek's list of the"composite sciences" the branches of architecture(important in the 16th Century Renaissance),Engineering and co smography (science thatde scribes the universe including astronomy,geography and geology). The original "compositesciences" consisted of astronomy, harmonics andoptics.

The term "mixed mathematics" was not only usedby Bacon but appeared in ' he Oxford EnglishDictionary 1648 in conjunction with John Wilkin's"Math Magick". Here 1quote from the dictionary:"The Mathematics are being Pureor Mixed."

The reference to John Wilkins pertains to the

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former Puritan clergyman and popularizer ofBacon's approach to science. Wilkins was one ofthe founder> of the Royal Soc iety (1660), 'he firstscientific organization whose purpose was thepromotion and advancement of science and itsapplications to the improvement of society. Thebook "Math Magick" refers to the longer title "MathMagick: The Wonders that may be Performed byMechanical Geometry. " Mechanical Geometry wasdescribed as "one of the most easy , useful and yetmost neglected parts of mathematics. It is a liberalart like astronomy and music:' Topics included adiscussion of pulleys, cranes, bow s and catapultsused in levers and wedges; the possibilities ofvarious kinds of machines, such as submarines andcarriages propelled by sails; and " secret andspeedier ways of attacking forts by approaches andgalleries:' (inventions in fortification )

The purpose of "Math Magick" was to familiarizethe average person with the basic and long­accepted principles of mechanics. Wilkins beginswith a defense of mechanics as a liberal art that wasmore like astronomy and music than the so called"illiberal sciences" which involved some physicalactivity such as manufacturing or trade. The basicsubject of mechanics was the relationship betweenweight and power. Weight was no longer to beconsidered a "natural quality, whereby condensedbodies do of them selves tend downwards," but"an affection which might be measured" Quantity,the subject of seventeenth century mathematics,could therefore pertain to qualities of physicalobjects such as weight and power. If mechanicswere to become a mathematical science then onecould not separate the physical from the theoretical.The scientist must use the proper mixture oftheoretical reasoning, direct observation andexperimentation.

During the seventeenth century there were otheradvocates of a careful handling of the mixedmathematical sciences such as John Wallis andIsaac Barrow in England, Marin Mersenne and thefamous mathematics textbook writer ChristopherClavius on the Continent. However, any study ofmixed mathematics must include the famousphilosophe and geometer of the Enlightenment,Jean d'Alemben (1717-1783).

IV. JEAN D'ALEMBERT'S VIEWS OF MIXEDMATIIEMATICS

O'Alembert was a key advocate of extending the

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kind of thinking involved in geometry to decisionmaking in society. As a philosophe he consideredhimself as a literary man who exercised publicresponsibility and whose function was to criticize,analyze, and redefine intellectual norms andinstitutional practice. He became one of the leadersamong a group of philosophes that calledthemselves rationalists.

D'Alernbert thought that rational thinking involvedthe processes of analysis and methodical doubt.Analysis was the process of starting with an ideaand breaking it into simpler ideas until one reachessimple ideas (atoms), Simple ideas are self-evidenttruths based upon sensory receptions. One wouldknow from experience that so and so is a simpleidea. Then, once the simple idea (first principles)was found by analysi s, one tried to construct adeductive chain of ideas that built back to theoriginal idea. This was the process of synthesisfound in geometry.

Notice that this fonn of analysis is similar to thatfonn used in algebra. Consider examples I and 2below:

E3ample I: Solve 23+3=2Assume there is a solution x.

analysis --. 2x=-13=-1/2

synthesis -+ x=·In.23=-123+3=2

Example 2: Solve 232+3=2Assume there is a solution x.

analysis -+ 2x2=·1Impossible

synthesis --+ Hence one of the elements in thechain is false

In Example I, one assumes the equation is true andfinds "simpler" statements that follow from theassu mption. Eventually, one will arrive at a"simplest" statement that cannot be broken downany further . Then one tries to build a chain ofstatements from this "simplest" statement to theoriginal sta tement. Such a procedure willconstitute a proof that there is a solution to2x+3=2. In proofs involving mixed mathematics(which might include example 2 as a trivial

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example), one must factor in experience whenanalyzing each step in the chain. This is apparentin example 2 where an assumption of 2x2+3=2having a real solution will lead to a positive realnumber equalling a negative real number. Thechain has been severed and one must re-examinethe problem.

D' Alembert was convinced that man hadknowledge on only a few links that joined thatgigantic chain of each discipline together. Hence,the philosophe had to look at arguments with ahealthy dosage of methodical doubt for errors inany chain. The philosophe especially had to searchfor abuses of theory as might occur in the secondexample above . Furthermore. the philosopheshould use methodical doubt then " indicating thepaths which have deviated from the truth, so that itfacilitates the search for the path which conducts toit." This chain always had to factor in commonexperience (the use of the sense perceptions), i.e,one should never separate the pure from lheapplied.

Now d'Alembert was involved in the 1750s withDiderot in the Encyclopedia project. They wantedto classify all of knowledge and d 'Alembert wasespecially interested in classifying mathematics.After all, there were many advances in mathematics

For Bacon "quantity" was"determined by fixed or constantcauses" while for d'Alembert wasapplicable to objects that wereindependent of real things (pure) orcould be applicable to Objectsinvolving physical things (mixed).

since the last time someone had tried to do such aproject. We find d'Alernbert's classification ofmathematics in his famous Discours Prellmlnalre.(See d'Alembert' s tree of knowledge) .

According to d'Alernbert, "quantity, the object ofmathematics, could be considered either alone andindependent of real and abstract things from whichone gained knowledge of it, or it could beconsidered in their efforts and investigatedaccord ing to real or supposed causes; thisreflection leads to the division of mathematics into

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hysico-Mathematics

Figure 3:d'Alembert's Tree of Knowledge

An of Conjecturing (Analysis of Games of Chance)

Optics. proper

Plic~DiOPtriCS' perspective

Caroptrics

Acoustics

< StaticsMechanic

Dynamics

~coSmOgraPhY

eometric Astronomy ChronologyGneminics

Numerical Arithemetic

Arithmeti< <ElementaryAlgebra <Differential Calculus

InfinitesimalIntegral Calculus

<Elementary {Military Architecture Tactics)Geometry

Transcendental (Theory of Curves)

Mathematics

ScienceofMan

Particular Physic ~AnatomY

PhysiologyZoolog Medicine

Veterinary MedicineHorsemanshipHunting

Physical AstronomyMeteorologyCosmology

BotanyMineralogyChemistry

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pure mathematics, mixed mathematics, andphysico-mathematics. Abstract quantity, the objectof mathematics, is either numerable or extended.Numerable abstract quantity has become the objectof arithmetic and extended abstract quantity that ofgeometry.

Notice d 'Alemben's idea of "quantity" differedfrom Bacon 's view. For Bacon "quantity" was"determined by fixed or constant causes" while ford'Alemben it was applicable to objects that wereindependent of real things (pure) or could beapplicable to objects involving physical things(mixed).

Looking at the tree diagram, we see " science ofman" divided into "mathematics" and "particularphysics". "Mathematics" was divided into "pure"and "mixed". "Pure" was divided into arithmeticand geometry. Arithmetic divided into numericalarithmetic and algebra (higher order arithmetic).Algebra divided into "elementary" and"infinitesimal". Infinitesimal involved differentialsand hence calculus was thought of as manipulation

If mathematical thinking was to beextended to all disciplines,d'Alembert argued that one mustguard against mistaking an empty"non-meaningful" mathematicalderivation for a genuine law ofnature by incorporating someexperimental data into hisreasoning.

of differentials and integrals. Mixed mathematicswas divided into mechanics, astronomy, optics,acoustics, pheumatics and art of conjecturing(probability theory).

There are many question raised by this treediagram.

(1) Why is "An of Conjecturing" classified as"Mixed Mathematics'[

(2) Why isn't Calculus classified as "MixedMathematics'[

(3) What is the difference between "MixedMathematics," "Physico-Mathematics," and"Particular Physics"?

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According to d'Alembert, the difference betweenphysics and mathematics was the following:"'Particular physics ' is properly only a systematiccollection of experiments and observations. On theother hand the physico-mathematical sciences, byapplying the mathematical calculations toexperiments, sometimes deduce from a simple andunique observation a large number of inferencesthat remain close to geometrical truths by vinue oftheir certitude." Examples of physico-mathematicswould be "a single experiment on the reflection oflight produces all of Catoprrics, or the science ofmirrors; ... a single experiment on the accelerationof falling bodies opens up the laws of descentdown inclined planes and the laws of movementsof pendulums; Rameau reduced all of harmony to asmall number of simple and fundamental chords,of which the others are only combinations orinversions."

Addressing the question of why the Art ofConjecturing was classified as "mixedmathematics," d'Alembert argued that suchproblems could involve physical situations. Heclassified the Art of Conjecturing into threebranches. The first involved the analysis ofprobabilities that might occur in games of chancewhere the occurrences follow certain rules. Thesecond branch was the detennination of probabilitywhen the data were not based on mathematicalrules but drawn from past experience. The thirdbranch dealt with those sciences in which it wasrare or impossible to provide demonstrations.These were further divided into speculative(physics and history) and practical (medicine andjurisprudence). Hence, the an of conjecturing inmixed mathematics involved situations whereenough physical information was known to fill inthe deductive chain of ideas.

The geometer, according to d'Alemben, had todistinguish between the physical laws andmathematical laws of probability. His mostfamous example was called the St. PetersburgParadox. It can be stated as follows: Two playersA and B playa fair coin toss game. If the cointurns up heads on the first toss, B gives A oneducat, if heads does not tum up until the secondtoss, B pays A two ducats, if heads does not tumup until the third toss, then B pays A four ducats,and so on, until the case if heads does not occuruntil the nth toss, A wins 2R- 1ducats from B. The

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question is how much should A pay B to play thegame.

A PURELY MATIlEMATICAL APPROACH TOSOLVING THE ST. PETERSBURG PARADOX:

Let Ei = event that the first (i-1) tosses of a coin aretails and the ith toss is a head.

Pi =probability that Ei occurs.di = expected payoff for player A if Ei

occurs.Then Pi=1/Zi, di=Zi.l and

-

= -

Hence, B should charge no less than infinity ducatsfor A to play this game, which is absurd.

D'Alemben argued that the probability terms, thePi, inflated the summed expectation. Hence, the Pineeded to be adjusted. He argued that it might be"metaphysically" possible for a fair coin to turn uptails 1000 or even n times in succession, butexperience dismissed such outcomes as physicallyimp ossible. In fact, if such a run of successivetail s occurred, then most observers would positsome underlying cause, such as an asymmetriccoin. Not only would all heads or tails sequencesnever occur, but that some mixed sequences would

D'Alembert's rational mecbanicswas widely accepted in the 18thCentury primarily because it hadthe authority of Newton behind it.Further, it seemed to strip awaymotive causes that Newton had usedand was based upon observed facts.

be repeated two or three times. By including themainly "metaphysical" possibilities of a unifonnsequence on an equal footing with more physicallyplausible ones, d'Alemben claimed the paradoxwas based on false premises. He argued that a

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reasonable man can make judgments that disagreewith the co nventional mathematical the ory,whenever his experience showed that the world didnot act the way the theory predicted. He said"would it not be astonishing if these formulas bywhich one proposes to calculate under certainty didnot parti cipate in the uncenainty themselves. Ifmathematical thinking was to be extended to alldisciplines, d'Alemben argued that one must guardagainst mistaking an empty "non-meaningful"mathematical derivation for a genuine law of natureby incorporating some experimental data into hisreasoning.

Now d'Alemben' s rational mechanics epitomizedwhat he meant by mixed mathematics. Rationalmechanics in the 18th century was viewed as asystem of propositions linked by the concepts ofmatter, force and motion to the fundarnentallawsof abstract dynamics, usually Newton's laws ofmotion being mentioned. By applying these lawsproperly, all ob servable phenomena weresupposedly reducible to the basic underlyingconcepts of matter, motion and force. Butd'Alemben restricted his rational mechanics to thestudy of observed effects. Since mass and forcecould not be observed (he called them "motivecauses") they were thrown out and his mechanicswas based on his three laws of motion, involvinginertia, compounding of motion, and equilibriumrespectively. These laws of motion involvedequations of impenetrable matter and impact. Allphysical motion, d'Alemben believed, could bedescribed as some combination of these threeconditions.

D'Alembert's rational mechanics was widelyaccepted in the 18th Century primarily because ithad the authority of Newton behind it. Further, itseemed to strip away motive causes that Newtonhad used and was based upon observed facts . Allother discipline s in the mixed mathematics realmco uld look to the methodology of rationalmechanics as a proper way of doing things.

Because o f [heir importance to the military,engineerin g and the advancement of technology,subjects such as navigation, warfare and hydraulicsbecame important are as co study in mixedmathemat ics. In navigation, the problem oflongitude {i.e. the problem of determining a ship'sposition at sea) involved Euler's theory of themoon with Tobias Mayer' s revisions of the valuesfor the positions of the stars.

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In warfare, the problem of project ile moti oninvolved the so called quadra tic theory of balli sticsin combination with expe rimental data on thevelocity of projectiles shot from a cannon.

In hydraulics. the problem of building waterworksand ships involved the theory of architecture andhydrodynamics with experimental results. In all ofthese cases, mixed mathematics was involvedbecause all quantities used magnitudes subsistingin material bodies and interwoven everywhere withphysical considerations. All subjects modeled theirapproach to mixed mathematics after that of rationalmechanics.

V. SOME PROPONENTS OF TIlE TEACHINGOF MIXED MATHEMATICS DURING THENINETEENTH CENTURY

(a) Gaspard Monge

French geometers such as Laplace and Condorcetcontinued their advocation for mixed mathematicsin the spirit of Jean d 'Alembert. Laplace reiteratedd'A1emben's idea that mixed mathematics involvedthe search for first principles and the building of adeductive chain when he said "natural phenomenaare mathematical results of a small number ofinvariable laws: ' However, the major advocate forthe teaching of mixed mathematics in the spirit ofd'Alemben was Gaspard Monge, known today asthe Father of Descriptive Geometry.

Monge was a major advocate of using physicaldrawings in doing geometry and an advocate ofincorporating mathematics and science in aneducation intended to be broadly based. He was amajor advocate of integra ting science andmathematics into a humanistic educati on. Mongedid not just teach mathematics, he molded futureFrench citizens. His descriptive geometry lentitself beautifully to both the theoretical and practicalsides of the cu rriculum because it consisted of apurely rational theory which could be translatedinto concrete graphic relations . This involvedlearning the essential skills of mechanical drawingby passing constantly fro m the abstract to theconcrete and back again. Monge expressed thisidea of mixing mathematics best in the preface ofhis famous Geometrie descripti ve: "The secondobject of descriptive geometry is to deduce fromthe exact de scription of bodies all whichnecessarily follows from their forms and respectivepositions. In thi s sense it is a means of

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investigating truth; it perpetually offers examples ofpassing from the known to the unknown; and sinceit is always applied to objects with the mosteleme ntary shapes, it is nece ssary to introduce itinto the plan of national education and make use ofthis geometry of the representation anddetermination of the e lements of machines bywhich man, controlling the forces of nature,reserves for himself, so to speak, no other labor inhis work but that of his intelligence."

In contrast to Monge 's position of integratingdescriptive geometry into a humanistic eduction,we have Cauchy's po sition that mathematicsshould be treated educationally as a separate,specialized discipline. According to Cauchy, "l etus then admit that there are truths other than thosein algebra, realities other than those of sensibleobjects. Let us ardently pursue mathematicswithout trying to extend it beyond its domain."Different fields are different, Cauchy asserted: theyrest on different fonns of evidence, use differentkinds of arguments, and generate different kinds of

According to Cauchy, "let us thenadmit that there are truths otherthan those in algebra, realities otherthan those of sensible objects. Letus ardently pursue mathematicswithout trying to extend it beyondits domain."

knowledge. Cauchy ' s mathematics was an austereand rigidly circumscribed subject for whi ch heoffered no justification outside of its own integrity.

As time advanced during the nineteenth century,Cau chy's view tended to win out over Monge'sview. Public assertions that scientific knowledgeis irrelevant to humanistic education can be foundas early as 1802. The final years of GaspardMonge epitomized thi s change from a holi stic toseparatist view of mathematics. Monge tau ghtgeometry at the Ecole Poly technique until therestoration in 1816 at which time the school wasclosed for several months. What is more, he wassummarily removed from his position in theInstitute and another appointed to take his place.He died miserably soon thereafter.

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If Cauchy's separatist view of mathematics beganto prevail in French mathemat ics in the earl ynineteenth century, then one must also concludethat d'Alemben's view of mixed mathematics mustbe waning during this period. If mathematics is tobe done separately from other disciplines. howcould it be mixed with these disciplines? On theother hand. it makes perfectly good sense to talkabout how a body of mathematics could be appliedto some discipline. Since the mathematics hadbeen already established. one could simply " lift"the needed mathematics from its total embodimentand "insen" it where it could be applied in thatdiscipline.

(b) William Whewell

As Cauchy's approach to mathematics waswinning out over Monge's approach on theEuropean Continent, the changes in mathematics inEngland were far more gradual. The English had amore "exemplary" view of mathematics whichtended to grow stronger, both institutionally andphilosophically, well into the middle of the 19thCentury. The ir view was based upon Newton 'sapproach to calculus using flux ions and to theunified Enlightenment view of science. By 1800.Newton 's approach to the calculu s was still beingtaught at inst itutions like Cambridge. Thisapproach was more rigorous than the AncientGreeks ' perspective of geometry but more difficultto learn and apply to problems than the calculus ofCauc hy. There was an early 19th centuryawakening of English interest in Con tinentalMathematics led by a shon-lived organization ofCambridge students known as the Analytic Society(1812-18 13). This group led by Charles Babbage.George Peacock and John Herschel advocated themore analytical methods of calculus using Lebnitznotati on. Although this group died out. many ofits aspirations had effects on how mathematics wastaugh t at Cambridge. By the 1820s, theCambridge mathematics exam. the Tripos wasradically modified to include some of these analytictechniques. However, the most powerful theme atCambridge still remained the connectedness anduniversaiity of knowledge. The person thatepitomized this approach was William Whewell.

In 1800 the mathema tics curriculum at Cambridgeincluded arithmetic, algebra, trigonometry.geometry, fluxions, mechanics, hydrostatics,optics and astronomy. The foc us was fromNewton's Principia. When Whewell came along

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there was pressure from the Analytic Society toch ange the curriculum tow ards Cont inentalmathematics. whewelt forged a compromise. Hewanted the students to be grounded in phys icalrealities- in pulley machines, and forces ratherthan math ematical symbols. In his textbookElementary Treatise on Mechanics (1819) he placeda considerable portion of mechanics prior to anydiscussion of diffe rential calculus. Today. weargue that calculu s should be taught before and atworst concurrently with mechanics. Whewellthought the best way to achieve rigorous result swas through geometric-physical reasoning and toconfirm the results thus achieved by "more direct"reasoning . It was his mission to preven t theestablishment of the study of abstract analysis as adiscipline independent of, and as prestigious, asmixed mathematics. He accomplished this missionwith his many textbooks that were primarily usedduring the first half of the 19th century and hisinfluence in the writing of the Tripos exams. Thisextended even as late as 1848 when many changeswere being considered with regard to the Triposexam. By 1854 (James Clerk Maxwell's year) theTripos exam consisted of a ratio of three appliedmathematics problems to every two puremathematical problems; problems cal1ing forsynthet ic -geometric solutions, includingNewtonian ones. made up more than 40% of theexamination. Obviously the emphasis was still on

Whewell thought the best way toachieve rigorous results wasthrough geometric-physicalreason ing and to confirm the resultsthus achieved by " more direct"reasoning.

mixed mathematics, i.e, geometry over algebra.intuitive rather than abstract rigor, on detail morethan generization, extensiveness more thanintensiveness, on problem solving rather thanmathematical processes. and on Newton rather thanLagrange.

Even in the 2nd half of the 19th century mixedmathematics did not go away . There were battlesbetween Arthur Cayley supporting ContinentalMathematics and Henry Airy supponing mixedmathematics. Only after another generation of

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mathematicians led by Bertrand Russell and G.H.Hardy do we clearly see a decline in mixedmathematics.

VI. 1liE DECLINE OF MIXED MA1liEMATICSIN 1liE NINETEENTII CENTURY

(a) A Consequence of the French Revolution­Changing Views of Probability Theory

As mentioned earlier, the rise of military andengineering schools brought an increase in thestudy of mixed mathematics. Furthermore, theimportance of mathematics as a method ofsearching for the truth was important to someEnlightenment philosophes. All of these trendscollided with the tremendous upheavals of theFrench Revolution.

The French Revolution seemed to create adiscontinuity and a rethinking of some of the trendsof the 18th century. Philosophes of the l Srhcentury stressed the imponance of enlightening theindividual. Society would ultimately prosper onlyif its citizens were enlightened individuals. Thismeant the individual had to master rational thinkingwhich was based upon the reasoning of geometry(synthesis) and to a lesser extent algebra (analysis).Mathematically. this mean that psychologicalarguments could be mixed with mathematicaltechnique in solving problems such as the St.Petersburg paradox and the Inoculation Problem

The events of the French Revolution seemed toshatter this belief that "good sense" was monolithicand a constant for a selected few. Passions seemedto prevail over reason . and what was "good sense"and who practiced it was no longer so clear. Thechaos and many political shifts from the Revolutionin 1789 to the restoration of the Bourbons in 1814shook the confidence out of the remainingphilosophes, some of whom suggested that "goodsense" is more intuitive than rational. Mixedmathematics involved the proper combination oftheory with experience. but by the time of therestoration of the monarchy. these ideas seemed tosome philosophers diametrically opposed.

While the philosophers of the Enlightenmentthought the way to improve society was byconcentrating on enlightening the individual(individual ...... society). the social scientists of the19th century thought one can improve the lot of theindividual by improving society (society......

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individual). This change of worldly viewsespecially affected how probability theory wasused. The philosophes used it by considering therational individual while the social scientist used itin statistics to say something about the "averageperson with properly A." For example. Queteletused the frequentist interpretation of probabilitytheory and the normal distribution to find the mean

The events of the French Revolutionseemed to shatter this belief that"good sense" was monolithic and aconstant for a selected few. ·Passions seemed to prevail overreason, and what was "good sense"and who practiced it was no longerso clear.

in the case of statistical regularities of data. andthus associated a number to some social trait of theaverage man. The average man could be assigned"a penchant for crime" equal to the number ofcriminal acts committed divided by the totalpopulation. In this way a set of discrete acts bydistinct individuals was transformed into acontinuous magnitude. "the penchant:' which wasan attribute of the average man. It was theresponsibility of the social scientist to apply themathematical theory to the socialcham; the normaldistribution was applied to all kinds of social (andlater biological) phenomena.

(b) Some Mathematical Discoveries of the 19thCentuIy

If the rise of statistical methods affected howscholars viewed the role of mathematics in the "realworld," then mathematical events occurring later inthe 19th century had a funher influence on thischanging view of mathematics. Here I amreferring to (1) the discovery of non-Euclideangeometry, (2) the discovery of non-commutativerings (i.e. Hamilton's quaternions), (3) thediscovery of nowhere differentiable buteverywhere continuous functions on the real line.(4) the changing relationship between geometryand mechanics.

The discovery of non-Euclidean geometry shiftedthe foundations of mathematics towards arithmetic

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and away from geometry. No longer was there anexample of absolute, infallible knowledge.Furthermore , the sensory percept ions could nolonger be trusted. This was especially re-iteratedwith the discovery of non-intuitive mathematicalentities like the quaternions and nowheredifferentiable, everywhere continuous functions.The truth of such propositions must rely on moreformalistic arguments and less on experience.

With regard to mechanics. recall d'Alemben usedhis rational mechanics as a model for the othermixed mathematical sciences. His rationalmechanics was kind of an extension of geometrywith three axioms of motion added. However,with the discovery of non-Euclidean geometry andthe rise of experimental physics in the 19th centurysome physicists thought that mechanics was less abranch of geometry and more one of analysis(calc ulus).

VII . THE CHANGING MEANING OFAPPLIED MATHEMATICS TODAY

In a recent article appearing in the July, 1993 issueof the Bulletin of the American MathematicalSociety, Anhur Jaffee and Frank Quinn discuss"Theoretical Mathematics: Toward a CulturalSynthesis of Mathematics and TheoreticalPhysics."

According to Jaffe and Quinn, mathematics shouldbe divided into "theoretical" and "rigorous"mathematics. They argue that applied mathematicsis generated in two stages: First intuitive insightsare developed, conjectures are made, andspeculative outlines of justifications are suggested.This they call "theoretical" mathematics. Then theconjectures and speculations are corrected and aremade reliable by proving them. This they call"rigorous" mathematics. Later in their article. theysay "pure" was used in the past instead of" theoretical" but the term "pure" is "no longerconunon."

Jaffe and Quinn argue that mathematicians haveeven better experimental access to mathematicalreality than the laboratory sciences have to physicalreality. They say. «this is the point of modeling: aphysical phenomena is approximated by amathematical model; then the model is studiedprecisely because it is more accessible." Today.the same person is likely to be doing "theoreticalmathematics" and "rigorous mathematics" in

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solving a problem. This is especially true in theareas of string theory, confo rmal field theory andropological quamumfield theory.

Finally, they argue that the bifurcation ofmathematics into theoretical and rigorouscommun ities has part ially begun but has beeninhibited by consequences of improper speculation.They argue that "speculative mathematics"ought tobe publishable but with the stipulatio n that it can beacknowledged as speculative.

VlIl. CONG.USION

I wish to argue that applied mathematics should betaught " mixed" allow ing for "speculativemathematics." Solving mathematical problems hasalways involved factoring in "experience" even ifthat experience could be non-intuitive. To say thatsome mathematics was applied to some physicalproblem is a distortion and misleading. This viewtends to trivialize the modeling process.

"Mixing mathematics" involves breaking down agiven problem into simpler pans until one arrives at"first principles." One is supposed to create achain of truths starting from these first principlesand logically arrive at a solution to the problem. Itseems reasonable in today's changing mathematical

Why should acceptable mathematicsalways be in some finished formfree from dead ends andspeculations? Why shouldacceptable mathematics be freefrom the motivations that lead tothe theorems?

world, that this chain could include speculativestatements , conjectures, refutations. dataaccumulation, experimental strategies and evensome incorrect conclusions as long as this chain isconstructed in a logical progression. The studentcould be required in some son of notebook toinclude any analytical or synthetic way of thinkinginvolved in the solving of a problem. This mayinclude the stripping down to "first principles" thathave no obvious or apparent relevance to theoriginal problem. It could include the use of"common experience" to make conjectures or

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refutations. It would be the opposite of the lean,economical and formalized kind of writtenmathematics emphasized today. Why shouldacceptable mathematics always beinsomefinishedform free from dead ends and speculations? Whyshould acceptable mathematics be free from themotivations that lead to the theorems? Why does ithave to appear from the finished product thatmathematics hasbeen applied to a problem when itwas really mixed? Why do we continue toreinforce this distorted view of appliedmathematics? Mathematics should be taught mixedand clearly advocated to students asbeing mixed.

References

D'Alembert, Jean. (1963). Preliminary Discourseto the Encyclopedia ofDiderot translated byR. Schwab. New York: Bobbs-Merri!Company.

Bacon, Francis . (1858) The Works of FrancisBacon edited by J. Spedding, R. Ellis andO. Heath. London.

Becher, Harvey. (1980). William Whewell andCambridge Mathematics. HistoricalStudies in the Physical Sciences. XI, 1-48.

Bittinger, Marvin. (1992). Calculus. 5th Edition.Reading. Massachusetts: Addison Wesley.

Brown , Gary (1991). The Evolution of the Tenn"Mixed Mathematics." Journal of theHistory of Ideas, 52, 81·102.

Brown, Gary . The Changing Meaning of AppliedMathematics. To appear.

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Jaffe , Arthur & Quinn, Frank. (1993 ).Theoretical Mathematics: Toward aCultural Synthesis of Mathematics andTheoretical Physics. Bulletin of theAmerican Mathematical Society, 29, 1. 1­14.

Monge, Gaspard (1847). Geometrie descrip tive.suivie d' une theorle des ombres et de laperspective, extroue des papiersdel' aUleurpar M. Brisson. Paris. 7th edition.

Whipkey, K.L., Whipkey, M.N., & Conway,G.W. (editors) (1981). The Power ofMathematics , Applications to Managemenland Social Sciences. New York: JohnWiley & Sons.

Wilkins, John (1802) . The Mathematical andPhilosophical Works of the Righl ReverendJohn Wi/kins. London: C. Wh ittinghamfor Vernor and Hood.

Footnotes

• This approach to modeling appears in Bittinger(1992), page 76. It would be an interesting studyto examine to what extent the Bittinger diagramrepresents the presentation of modeling inelementary mathematics textbooks over the lastfifty years . I am conjecturing that it would typlifyother such diagrams.

.. For a more thorough, scholarly explanation,including complete citations and footnotes, of thehistorical evolution of the term "mixedmathematics" see Brown (1991) and Brown (toappear).

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