718 SCHOOL SCIENCE AND MATHEMATICS

THE METHODS AND AIMS OF MATHEMATICAL SCIENCE.

BY LOTJIS C. KARPINSKI,University of Michigan, Ann Arbor, Mich.

Mathematics is the oldest recognized school discipline. Whi^this may not commend the subject, particularly to those whoare being disciplined thereby, yet the fact has profound historic alsignificance. If the names and the political methods of tenthousand nominal rulers of Egypt and Babylon and Greecehave been considered significant in the intellectual considerationof the history of mankind, how much more significant is a studywhich gives us some conception of ancient ways of thinking;how much more significant a study which gives us actual materialwith which the ancients occupied themselves.From the earliest times mathematics has been studied not

only for its practical value but also as a mental discipline.Among the Greeks Pytbogoras is given credit for consideringarithmetic from the philosophical point of view, aside fromany application. Fortunately for the practical the intellectualtreatment dominated the study of mathematics throughout itsperiod of most rapid development.For students in an American University mathematics should

be considered from the point of view of comprehending the worldin which we live. The world of today is the world of the scien-tist, the world of the engineer and the mechanic; it is the worldof the ehemist, the physicist, the biologist, and the physician-scientist; it is equally the world of the astronomer, of the artist,and of the thinker. Man is distinguished from other living crea-tures by his powers of reasoning; the world in which he lives isa world of thought and of people who think. Mathematics isan essential industry in any such conception of the world oftoday.

Consider for a moment the function of mathematics in thework of the engineer. All students, all who take any pride inbeing intelligent about our universe, should know somethingabout the problems and difficulties of engineering, the funda-mental and basic ideas upon which the great superstructure ofmodern engineering is erected. Society requires that we live inharmony; this demands an intellectual appreciation each of thethe other^s work.

^ lecture delivered in a course of lectures given to freshmen at the University of Michigan,for the purpose of acquainting the freshman with the fundamental notions of various depart-ments of knowledge.

AIMS OF MATHEMATICS 719

If you seek a monument to mathematics, look around.In every auditorium, in practically every large theatre, analgebraic equation, ^y^^^ ^d a related geometric curve, theparabola, are at the very foundation of the construction of thehall. In the Hill Auditorium at Ann Arbor, a hall seating morethan 5,000 people, the simple equation y2==70.02x, determinesthe form of the walls, determines the length of the stage, de-termines the height of the ceiling. Not any love for the mysticaldetermines this relationship but rather it is determined by thegeometrical properties of this curve combined with the physicalproperty of sound waves that they are transmitted in straightlines and reflected like rays of light.

In the building of a bridge, such as the recent tremendous1,000-foot arch in New York City, over Hell Gate, or such asthe Alexander III bridge in Paris, beauty is almost as funda-mental a consideration in the eye of the architect as safety andusuability. Here, too, it is found that this same parabola, acurve which can be cut from a circular cone, fulfills the mechani-cal requirements and at the same time offers satisfaction to theeye. Possibly you are so constructed that a great and beautifulbridge as a reflection of a simple algebraical equation and as areflection of a simple geometrical curve offers you no intellectualsatisfaction; however, that is a limitation similar to the lack ofappreciation of poetry or music or of other harmony and beautyin the universe. The bullet in its flight approximates the samecurve; the suspension bridge presents the same; the automobileheadlight is built upon the same model; so is the new roadwayover the Huron river, near Ann Arbor, and many another road-way.

Arithmetic and algebra, trigonometry and geometry, andanalytical geometry are all involved in the construction of sucha bridge, in the construction of such a hall. For the engineer itis evident, then, that the mathematics is a tool which he mustlearn to use even as he uses T-square and compass and triangle.

Quite as dependent upon mathematics as engineering is moderncommerce and industry. Of course the factory has the mathe-matical problems of the engineer; but the office and the adminis-tration has equally mathematical problems even more subtle,involving frequently the apparently uncertain human element.The binominal theorem, illustrated on next page, appears

at first sight as a perfectly harmless amusement for chil-dren, or for the academic youth, let us say. However, in

720 SCHOOL SCIENCE AND MATHEMATICS

(a+b^^a^+Sab+b2,(a+bV^a^+Wb+Sa^+b3,

(a+bV=^+4a3b+^a2b2+ . . ^.,

the hands of the economist this is a dangerous weapon, or, atleast, a powerful one. The price of a bond depends upon thistheorem; the actual cost of any instrument which depreciateswith use depends upon this mathematical formula. Yes, thesufficiency of a nickel for a street-car fare may hinge upon thebinomial theorem; the 2c or 3c a mile for a railroad fare dependsupon it; even the price of a sandwich or a pair of shoes involvesthe same formula. Replacement and insurance, life insuranceand health insurance and fire insurance are all founded andgrounded upon a mathematical substructure. If you have anysuspicion that your abilities lie in business administration, getacquainted with the binomial theorem. You will need it in yourbusiness.But these illustrations are drawn particularly from our material

interests. The astronomer handles problems which are moreremote so far as our bread and butter needs are concerned. Theastronomer will tell you that you are even now moving upon anellipse, the path of the earth about the sun being an ellipse,a^/a2^-^2/^^ 1, with the sunnot at the center ofthe path but some1,500,000 miles off center at the focus of the ellipse. So theother planets move in elliptical orbits about the sun. Probablythe power of mathematics has never been shown more vividlythan by two astronomers working independently who calculateda planet into existence. Astronomers had noted certain irregula-rities in the motion of one of the planets. By a study of theseperturbations or disturbances, Adams, an Englishman, andLeverrier, a Frenchman, computed that these disturbances mustbe caused by an unknown planet whose orbit they were able todetermine. Then when this planet again came within the rangeof the telescopes Adams instructed certain astronomers to turntheir telescopes to a certain spot in the heavens where the newplanet actually was found�Neptune.The beauty of mathematics is found, for the mathematically

inclined, at least, in the contemplation, for example, of series ofnumbers:

1,3,5,7,9,11,13, : . .2n-l.1,4, 9, 16, 25, 36,. 49, . . . n2.

or in form:

AIMS OF MATHEMATICS 721

The question which the mathematician asks when confrontedwith such a series is:

"Axe there any more at home like you?57 The discovery of thepentagon, the five sided regular polygon as a possible con-struction with ruler and compass, marked the beginning ofa new period in mathematics, really logical geometry. Follow-ing this the cosmic solids were constructed.The five cosmic solids are the tetrahedron, cube, octahedron,

dodecahedron, and icosahedron. There are only five of theseregular solids possible in our universe of three dimensions.The mathematician builds up a system of geometry, each

proposition more or less following upon the preceding. This isan intellectual problem, primarily; it is occasioned, however, bythe world in which a thinking being finds himself.

Just as geometrical forms present themselves to reflectivebeings so also a first degree equation presents itself. To theEgyptian as a;+^/7==19, the number which when its seventhis added gives 19. Then pure quadratics and cubic equationsfollowed much later. The solution of the quadratic came bygeometrical methods quite early to the Greeks; the analyticalsolution, was obtained much later by the Arabs and Hindus.Here the astonishing fact is found that these two diveser

routes of number and form lead to a consideration of funda-mentally related problems. The problem solvable by quad-ratics corresponds to a geometric problem solvable with rulerand compass. The Greeks solved geometrically; the Hindusand the earlier Egyptians solved numerically. The Arabs com-bined the two. But this wonderful connection between geometryand algebra was not fully exploited until Descartes in theseventeenth century, 1637, invented or rather discovered theanalytical geometry. Here the straight line is represented by alinear or first-degree equation; the circle by a quadratic equa-tion of simple type; o^+^^lOO; the parabola, ellipse, andhyperbola by other simple quadratics.Another step in this direction’ of discovering fundamental

unity was made with the discovery at the end of the eighteenthcentury of the analytical representation of complex numbers, bya Dane, Caspar Wessel. The connection then was shownbetween these geometrical figures, regular polygons inscriba-ble in a circle, and the equations of type

a;5=i; ^4=^Now a mechanical mathematical genuis like the great electrician

722 ] SCHOOLJSCIENCE ANDfMATHEMATICS

Steinmetz finds these imaginary numbers very convenient in hisbusiness furthering the development of electricity, scientific andpractical. Similarly mathematical considerations of a theoreti-cal type led up to the present wireless telegraph and telephone.The imaginary may suggest to some of you the fourth dimen-

sion which is frequently given as an illustration of mathematicalnonsense. Of course much nonsense has been written in thistopic, but not by mathematicians. Spiritualists have had thenerve to locate the spirits in the fourth dimensions, and domes-ticated them to rap on tables, tip over chairs, or perform otheruseful tasks. For the mathematician, of course, the fourthdimension exists no more and no less than the fifth or the first,or than the third, to which the world of sense corresponds. Aline is real to the mathematician, but real only in thought; thisline on the blackboard is solid, actually solid; you can measureit; you can weight it. A plane is also a solid as soon as you cantouch, taste, or handle it; it is a plane only in thought.

In conclusion, I wish to discuss the method of work of mathe-maticians because it is the method of work, of creative thoughtin every field of human endeavor, whether of music or art or

literature or mathematics. None of you students here may everdevelop into creative mathematicians; there would be no occasionfor surprise if that were true; however, Michigan may wellexpect that a number of you may develop into creative workersin literature, in engineering, in medicine, in intellectual workof one kind or another. All of you who’ profit by a college educa-tion should be compelled somewhere and somehow to thinkdeeply and consecutively, consistently, in some field of humanendeavor.An inquiry was made some years ago into the method of work

of mathematicians. Such men as Poincare responded. Poin-care mentioned that the idea for his first real contribution tomathematicis came to him as he put his foot on the step of anomnibus while engaged in his period of two weeks^ militaryservice. Again and again men’have reported that creative ideascame to them suddenly in the night, or at some time whenapparently not actively working on the particular subject.However, no man has ever reported that any idea came tohim in a field in which he had not previously and -even re-

cently been working. No mathematician has reported that theideas came fully elaborated. Work, work, work; that has beenand ever will be the absolutely essential condition for crea-

tive thinking along any line which is worthwhile for human be-

ngs.