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SPOM5 Jinni Nationa D00052MT 225012 10 175 690 SE 029 6713 AOT101 Schaaf, Villias L., Ed. TITLE Reprint Series: What is Contemporary Mathematics. RS-3. IMSTITOTIOM Stanford Univ., Calif. School Mathematics Study Group. SPOM5 Jinni Nationa. :Lance Foundation, Washington, D.C. P08 DATE 66 lorE 41p.: For related documents, see SE 029 676-690 EDRS PRICE EF01/PCO2 Plus Postage. DESCRIPTORS Curriculum: *Enrichment: *Instruction: *Logic: *Mathematical Applications: Mathematics Education: *Modern Mathematics: Secondary Education: *Secondary School Mathematics: Supplementary Reading Materials IDENTIFIERS *School Mathematics Study Group ABSTRACT This is one in a series of SHSG supplementary and enrichment pamphlets for high school students. This seriesmakes gvailatle espcsitory articles which appeared ina variety of thematical periodicals. Topics covered includg: (1) thenature of hematics:(2) mathematical inutility and the advance of science: had (3) logic. (59) *********************************************************************** Reproductions supplied by EDRS are the best thatcan be made from the original document. *********************************************************************** U S DEPARTMENTOF NEAL TN "PERMISSION TO REPRODUCE THIS EDUCATION II INELFAIIIE MATERIAL HAS BEEN GRANTED BY NATIONAL INSTITUTE Of EDUCATION 'tt {,L ttM 444 FPI 644 PRO I C 4 PA' °, A W 4 ,.4 I MONA HI pi 1,4 :ttlA ,(N r)W Al t",,04". Y,4 " nk DPIPoOP4S 'OA PP (P4,PIC Y LiEppf ,4 t11 1 4. N441,4)4441 PYST.tt,TE TO THE EDUCATIONAL RESOURCES A! ")41 O% .0,441 INFORMATION CENTER (ERIC)." 1 C 1,6 by The Beard al Trustees al lb. laland &milordAuthor UsiverilitY All rights swerved Natal in the Maud %ass J Anowica Financial support for the School MathematicsStudy Group has been provided by the Notional Science Foundation. Mathematics is such a vast and rapidly expanding field of study that there are inevitably many important and fascinating aspects of the subject which do not find a place in the curriculum simply because of iack of time, even though they are well within the grasp of secondary school students. Some classes and many individual students, however, may find time to pursue mathematical topics of special interesto them. The School Mathematics Study Group is preparing pamphlets designed to make material for such study readily .accessible. Some of the pamphlets deal with material found in the regular curric- ulum but in a more extended manner or from a novel point of view. Others deal with topics not usually found at all in the standard curriculum. This particular series of pamphlets, the Reprint Series, makes available ex- pository articles which appeared in a variety of mathematical periodicals. Even if the periodicals were available to all schools, there is convenience in having articles on one topic collected and reprinted as is done here. This series was prepared for the Panel on Supplementary Publications by Professor William L Schaaf. His judgment, background, bibliographic skills, and editorial efficiency were major factors in the design and successful completion of the pamphlets. Panel on Supplementary Publications R. D. Anderson (1942-66) Louisiana State University, Baton Rouge M. Philbrick Bridges (1962-64)Roxbury Latin School, Westwood, Mass. Jean M. Calloway (1962-64) Kalamazoo College, Kalamazoo, Michigan Ronald J. Clark (1962-66) St. Paul's School, Concord, N. H. Roy Dubisch (1962-64) University of Washington, Seattle W. Engene Ferguson (1964-67) Newton High School, Newtonville, Mass. Thomas J. Hill (1962-65) Montclair State College, Upper Montclair, N. J. L. Edwin Hirschi (196548) University of Utah, Salt Lake City Karl S. Kalman (1962-65) School District of Philadelphia Isabelle P Rucker (1965-68) State Board of Education, Richmond, Va. Augusta Schurrer (1962-65) State College of Iowa, Cedar Falls Merrill E. Shanks (1965-68) Purdue University, Lafayette, Indiana Henry W. Syer (1962-66) Kent School, Kent, Conn. Frank L. Wolf (1964-67) Carleton College, Northfield, Minn. John E. Yarnelle (1964-67) Hanover College, Hanover, Indiana PREFACE Mathematics is a magnificent cultural heritage. Although its origins are obscure, it is presumably one of the oldest concerns of man, reaching back into time by some six thou.sand years. Next to the invention of language itself, the creation of mathematics is without doubt the most subtle, powerful, and significant achievement of the human-mind. Just what is mathematics? This is more easily asked than answered, at least in any brief sort of way. Mathematics has been described as the "science of self-evident truths"; it has been called the "science which draws necessary conclusions"; it has been defined as the "universal art apodeictic." Bertrand Russell's celebrated characterization of mathematics as the "subject in which we do not know what we are talking about or whether what we say is true" is illuminating if it is not misinterpreted. The late Edward Kasner, in his whim- sical way, called mathematics the science "that uses easy words for hard ideas." A sophisticated observer might say that mathematics is what a mathema- tician does. But what does a mathematician do? Or we might ask: How does he go about doing whatever it is that he does? Why does he do these things? Who really cares what a mathematician does? You will find some answers to these questions in the essays that follow. But we should like to give you a bit of perspective first. You must realize that mathematics has had a long history and that it is a far cry from pre-Babylonian mathematics to the mathematics of today. Further- more, during this long span there were many "ups and downs." There were barren gaps and intervals of stagnation, as well as brilliant break-throughs and lucid periods of creativity. In this connection, there are several ways of regard- ing the development of mathematics. The conventional interpretation suggests that this development has been an evolutionary process in which successive generations of mathematicians built upon the contributions of their predecessors without tearing down what had gone before (Hankel ). A less conventional viewpoint, suggested by Spengler in his Decline of the W est. holds that there never was a "continuous growth" of a single discipline called mather,. :ics; what seems to be the development of a single broad subject actually is a p;..ral- ity of independent mathematicts ). Each new and independent mathematic follows another, not by building upon its predecessor, but by repudiating it and reflecting a subsequent alien culture. Perhaps a more realistic approach than either of these is that suggested by E. T. Bell, namely, a series of rather distinct epochs, each of which grew by expansion or extension out of the "residue" of the previous epoch. Professor Bell has indicated seven such major epochs, together with newer concepts or more powerful tools which survived from the earlier period and heralded the beginning of the next epoch. Be that as it may, we are here concerned with the nature of mathematics as we know it today. Stared very briefly and by no means completely, contem- porary mathematics is characterized by its sweeping generalizations; its pure Ill abstractions; its succinct symbolism; its precisicm of language; its ideals of rigorous thinking; and above all, its prime concern with patterns of ideas, with the structure of forms, and with the qualities of relationships. Theessence of mathematics is that, unlike the physical sciencep, it is wholly "man-made", being essentially independent of the external world. In thissense, it is an art atuot a science. What is described in these essays is the mathematics of today. What of the mathematics of tomorrow? It has been suggested by Bell thatwe may find it necessary to give up the idea of "one all-inclusive kind of mathematics." We may have to abandon the concept of the continuous and embrace the discrete in mathematics. What such mathematics will be like, nobody can foretell: "the mathematics of the twenty-first century may be so different from that of the twentieth century that it will scarcely be recognized as mathematics." William L. Schaaf 6 iv Contents THE NATI/RE OF MATHEMATICS MATHEMATICAL INUTILITY AND THE ADVANCE OF SCIENCE . 13 ARE LOGIC AND MATHEMATICS IDENTICAL? BIBLIOGRAPHY . 35 FOR FURTHER READING AND STUDY . 36 V ACKNOWLEDGMENTS The SCHOOL MATHEMATICS STUDY GROUP MI= this opportunity to express its gratitude to the authors of thesearticles for their generosity in allow- ing their material to be reproduced in this manner: Mina Rees, who, atthe time that her paper was first published, was dean of graduate studies at the City University of New York; Carl Boyer, who, when his article first appeared, was a member of the Department of Mathematicsof Brooklyn College of the City University of New York; and Leon Henkin, who, at the time that his paper was written, was associated withthe Mathematics Department of the University of California at Berkeley and the Association for Symbolic Logic. The SCHOOL MATHEMATICS STUDY GROUP is also pleased to express its sincere appreciation to the several editors and publishers who have been kind enough to allow these articles to be reprinted, namely: THE MATHEMATICS TEACHER MINA REES, "The Nature of Mathematics",* vol. 55 (October 1962 ), pp. 434-140. SCIENCE CARL B. BOYER, "Mathematical Inutility and the Advance of Science", vol. 130 ( July 3, 1959), pp. 22-25. LEON HENKIN, "Are Mathematics and Logic Identical?", vol. 138 (Nov. 16, 1962 ), pp. 788-794. Dean Ree s paper also appeared hi Scielffs, vol. 138 (Oct. 5, 1962), pp. 9.12. VII 0 Li FOREWORD If one may be allowed a slight oversimplification, modernmathematics is characerized largely by its concern for abtract structures, whilemathematics prior to the nineteenth century leaned somewhat more heavily onintuition. Yet, as pointed out in the present essay,the interplay of intuitive consklerations and abstract formulations has proved exceedingly fruitfuL In this discussion of the nature of mathematics, the authorappropriately comments upon the relation of mathematics toscience (a matter which is again discussed in the second essay).
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