This dissertation has been 65—3869 microfilmed exactly as received
HUMBERD, Jesse David, 1921- RELIGIOUS BEARINGS IN THE DEVELOPMENT OF MATHEMATICS.
The Ohio State University, Ph.D., 1964 Education, history
University Microfilms, Inc., Ann Arbor, Michigan Copyright by
Jesse David Humberd
1965 RELIGIOUS BEARINGS IN THE DEVELOPMENT OP MATHEMATICS
DISSERTATION
Presented In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University
By Jesse David Humberd, B.S., B.A., M*A., B.D,
- 5 ;- - i s - # - j t
The Ohio State University 1964
Approved by
A d v ise r Department of Education ACKNOWLEDGMENTS
The author wishes to express his sincere thanks and appreciation to Dr, Harold P, Fawcett, teacher and adviser, for his inspiring personality in the classroom, and for his enthusiastic support in the preparation of this study. The author was first challenged by Dr, Fawcett in 1947, when in his first graduate course, Education 687, he was challenged to read more than twenty basic works in mathematics education. From that day to the present, he has known and valued greatly the ready enthusiasm and wise counsel of this master teacher.
It is with deep gratitude that the author recognizes the cooperation and encouragement of his wife, Laura, during the preparation of this study. Much of the inconvenience resulting from moving the family several times, for graduate work; and most of the extra effort at home caused by periods of the w riter’s absence, has been borne by her.
And to Lenora and Margaret, (or as they would have it, Lee and Meg), the author expresses thanks for their loyalty, during what must have seemed to be most of their lives. If they ever doubted that the study would ever be completed, at least they always hoped It would be.
ii VITA
December 21, 1921 Bom - Roann, Indiana
1943 B.S., Bryan University, Dayton, Tenn,
1942-1945 U. S . Navy
1947 B.A., Wittenberg College, Springfield, Ohio
1947 Teacher, Springfield, Ohio
1947-1948 Instructor, Wittenberg College 1948-1951 Teacher, Miami County Schools, Ohio
1950 M.A*, The Ohio State University
1954 B.D., C-race Theological Seminary Winona Lake, Indiana
1954-1958 Assistant Professor, Grace College Winona Lake, Indiana
1958-1959 Instructor, The Ohio State University School
1959-1964 Professor, Grace College Winona Lake, Indiana
FIELDS OF STUDY Major Field: Mathematics Education
Mathematics Education: Professors Harold P. Fawcett, Nathan Lazar, and John Kinsella
Science Education: Professors John S. Richardson and G. P. Cahoon
Higher Education: Professors Everett Kircher and Earl Anderson
iii CONTENTS
Chapter Page
I . INTRODUCTION ...... 1
Background for the study Need for the study Assumptions
PART I . RELIGIOUS BEARINGS IN THE DEVELOPMENT OP MATHEMATICS
I I . EARLIEST TIMES TO THALES (600 B .C .) . . 25
III. THE GREEK AGE 600 B.C. to A.D. 641 . . . 45
IV. THE MIDDLE A G E S ...... 83
V. THE SEVENTEENTH CENTURY...... 110
V I. THE MODERN A G E ...... 163
PART I I . RELIGIOUS BEARINGS IN THE CONTENT OP MATHEMATICS
V II. NUMBER...... 193
V I I I . MATHEMATICS— RELATIVE OR ABSOLUTE TRUTH? 218 IX. MATHEMATICS— INVENTION OR DISCOVERY? . . 236
PART I I I . RELIGIOUS BEARINGS IN THE TEACHING OP MATHEMATICS X. PRESENT PRACTICES IN THETEACHING OP MATHEMATICS...... 250
X I. SUGGESTED PRACTICES IN THE TEACHING OP MATHEMATICS...... 263
BIBLIOGRAPHY ...... 295
iv CHAPTER I
INTRODUCTION
Background for the study
The object of this study is to advance the thesis that religion has been an influential force in the history and development of mathematics. A review of the literature dealing with the progress of mathematics throughout the history of man disclosed a dearth of information relating to the role which religious influences have played in that development. The w riter’s responsibility for the teaching of mathematics in a Christian institution of higher learning,
and his concern for Christian education at all levels, led to this study. In a previous study, he was concerned with
the place of mathematics in the education of m inisterial
stu d en ts.A t that time evidence was found that mathematics
could contribute to the education of the minister in several
ways. Its study could provide certain tools essential to an educated man in his understanding of the quantitative aspects
of life. Mathematics could assist him in developing the
!j. D. Humberd, ”A Proposed Program in General Mathematics for the M inisterial Student” (unpublished Master's thesis, The Ohio State University, 1950).
1 ability to think clearly, and could contribute toward his acquaintance with the world of men and ideas, as well as with the world of nature and science. Subsequent experience has confirmed this evidence, and the writer has since that time completed theological training. He is now teaching in a Christian college in which all teachers are expected to be theologically trained, as well as academi cally prepared in one of the usual disciplines.
Religious bearings in mathematics. There is little provision in the usual college mathematics curriculum for presentation of much of the historical and cultural back ground information which could enrich the study of the facts and principles of mathematics. The demands of the usual courses are such that any dilution dare not be excessive.
Yet in the Christian school, there is an expressed desire and intent on the part of the faculty, and on the part of the constituency, to integrate all parts of the curriculum with the Christian philosophy under which the school o p e r a te s .
Different schools operate under different sets of aims and objectives. Accrediting agencies recognize the validity of this procedure by placing as the first criter ion in any self-study the clearly defined statement of the purposes of the institution. All additional criteria of evaluation relate to these clearly defined purposes. The purposes of a mathematics department In a
Christian school are similar to those in the mathematics department of any school. Almost everyone knows that mathematics serves very practical purposes in the market place, and in the office of the engineer. Most educated people would agree that mathematics carries the major burden of scientific reasoning, and permeates the major theories of physical science. Many are aware that mathematics has played a major role in the development of philosophy, has influenced the fine arts, and has provided models for logical thinking. Some have found in mathematics aesthetic values. In short, mathematics has been a major cultural force in civilization. But in all these aspects, mathematics is generally viewed as a secular science, devoid of religious overtones.
Such a gap in the literature may be of little moment to a student studying to pass a course, or to complete the requirements for a degree. Mathematics may become merely the science of calculating correct answers from certain available data, for the one studying for a career in science and engineering. Yet even such students recognize differences between good and poor teaching, and realize that a good mathematics teacher is one who teaches in such a way that "the whole is a great deal more than the sum of its parts." It is no secret that many educated people avoid the study of mathematics as an intellectual interest*
Unless forced by some external requirement to ’’take"
some mathematics, most people have been satisfied to get by with a minimum of knowledge secured by the end of the first or second year of high school* It has not even been necessary for future teachers of arithmetic in elementary schools to be acquainted with mathematics beyond the minimum required for entrance into college.
Many have returned to elementary classrooms after a
six- or seven-year period in which they had no formal contact with mathematics.
In recent years, the demand for more mathematical competency on the part of teachers at all levels has been widely made. A strong emphasis upon something called modern mathematics has aroused much comment, if not
consternation, among teachers in America. Millions of
dollars have been spent in a multitude of efforts to
improve the mathematics curriculum, and to provide special courses of Instruction in mathematics for
teachers to help them upgrade their understanding and facility. In some of these efforts, provision has been made for the inclusion of historical and cultural aspects
of the background of some of the mathematical content.
This has been included to increase interest, and to make more meaningful the operations and concepts presented.
The writer favors such efforts, and believes that in a •
Christian school, these historical and cultural contri butions should be joined with the religious bearings in the development of mathematics wherever practicable.
There are areas in the mathematics curriculum in which such integration finds natural expression. There are many areas in which such attempts would be artificial and strained. The cultural and religious bearings of some aspect of mathematics may appeal to one teacher, and another may not judge them worth inclusion. In the usual sequence of courses for mathematics majors, the
opportunities and time for such considerations will not be readily found. In Introduction to Mathematics, for non-majors, and in the study of the history of mathematics,
for both majors and others, the writer has been able to
take some first steps toward integration. These efforts have been seriously limited by the natural restrictions
of a small school, with infrequent offerings of such
courses, and the few mathematics students enrolled. Outlook for the future. The situation has changed. The writer is now faced with a new opportunity. The State
of Indiana has expanded the requirement in mathematics for
all elementary school teacher candidates. They must now
complete eight semester hours of mathematics in addition to the methods course. This new requirement expressly is not to be the usual first-year College Algebra, Trigonometry,
Analytic Geometry sequence, or similar content. It is to be a special course designed for the elementary teacher, and so created as to assist her in obtaining a better understanding of mathematics.
An examination of the dozen or more books which have appeared on the market to meet this demand has proved disappointing. Most of them are mere rearrangements
of the chapter contents of each other, or of other
elementary mathematics textbooks. Many of them do include topics which can be described as enrichment m aterials, particularly topics which relate to ideas being advocated
under the name of modem mathematics. But the examination
shows little or none of the integration of religion and mathematics.
The present study is not intended to provide an additional syllabus for a new course of study for these
elementary teachers in mathematics. It is foremost a
larger attempt to discover and make explicit those religious bearings of mathematics which will enrich any
program of mathematics in the Christian school, and by
extension, the work of any mathematics classroom. This
study was begun without any particular course or syllabus
In mind. However, the writer intends that the findings 7 of this study b© translated into classroom experience, and a later section will provide some direction toward the implementation of that intention.
The expectation that the search for the religious bearings in the development of mathematics might prove to be fruitful, was sharpened by one of the outstanding contemporary^ writers in this fields It would be possible to take any one of the fields that enjoyed close contact with mathematics in some period we have examined and trace the continuation and extension of that association right down to the present day. There is neither space nor time, however, to permit a comprehensive account of the relation of mathematics to art, science, philosophy, logic, and social sciences, religion, literature, and a dozen other major human activities and interests. It is hoped that enough has been said to support the thesis of this book, namely, that mathematics has played a predominant role in the formation of modern culture.2 It will be noted that one of the areas mentioned in this comment is religion. As a major human activity and interest, mathematics has been credited with providing man's best answers to fundamental questions about the nature of man and his universe. The writer agrees with this statement from the secular view, but he also agrees with the millions of every age who have believed that religion has provided man's best answers to the really fundamental questions about the nature of man and his
^Morris Kline, Mathematics in Western Culture (Hew York; Oxford University Press, 1953), p. 453. 5 Op. c i t . . p . ix . universe. This study has attempted to bring together these two areas, at least in a historical sense, and to combine some of their best answers.
Definition of the problem
The purpose of this investigation has been to determine, by an objective historical study of the development of mathematics, the role which religion has played in that development. In addition, that role has been examined in order to make explicit its relevance to the teaching of mathematics, and to provide some Illustrative teaching plans and procedures whereby the religious bearings of mathematics can be effectively presented in the classroom.
The need for the study
At the present time, there is little literature available in organized form which considers the extent to which religion has influenced the development of mathematics.
In a Christian, school, there is a recognized need to relate
the work of every aspect of the school activity to the basic purpose of the school, which is to present the
Christian view of life. The main goal Is to search out
God’s revelation to mankind, both in nature and in grace.
In many curricular areas, great strides have been made. Bibliographies, such as those in Teacher Education and Religion.4 outline hundreds of works which relate religion to the humanities, the social sciences, and the natural sciences. Mathematics is not among the
subjects included. Dr. Prank Gabelein, headmaster for
over forty years of a Christian school, and author of
several books on Biblical and educational subjects, examines the relations of school subjects with Christian truth, and concludes that mathematics is "The hardest
subject to integrate."5
Martin Hegland, after a period of half a century
in the study and reflection of Christian education, wrote
a book to demonstrate that "The whole process of education should be permeated with the Christian sp irit.”6 But the
word mathematics does not appear in his book. The Study and Research Committee of the Institute
of Christian Education of England and Wales were charged
with the problem of examining
The practical tasks and objectives of Christian teachers in secondary schools in terms of their school situations, and in the wider contemporary
4A. L. Sebaly (ed.), Teacher Educat ion and Religion (New York: The American Association of Colleges for Teacher Education, 1959), Bibliographies and notes, pages 245-285.
5Prank Gabelein, The Pattern of God13 Truth (New York Oxford University Press, 1954), p. 57.
6Martin Hegland, Christianity in Education (Minnea polis: Augsburg Publishing House, 1954), p. i. 10
social setting; and to consider how the Christian teacher can build up the attitude of pupils upon Christian foundations such as he himself accepts.7
This committee considered the approach to mathe matics and science together. Little reflection was given to mathematics itself, nearly every early reference being to "science and mathematics,11 after which the chapter was devoted to science alone.
One may find in the publications of some public school systems relating to the development of moral and spiritual values, some specific reference to the contri bution mathematics can make to these values: The
Development of Moral and Spiritual Ideals in the Public
Schools;8 Moral and Spiritual Values in Education.0 And in the handbooks and curriculum guides produced by those in the Christian school movement, one may find specific evidence of serious effort made to integrate religion and mathematics: Resource Units for Lutheran High Schools.10
Finding Truth in Arithmetic.11 But beyond a few such
7W. R. Niblett, Christian Education in a Secular Society (London: Oxford University Press, T§60), p. i. 8 Board of Education, City of New York, 1956, p. 11.
9Los Angeles City Schools, Publication No. 580, 1954, p. 95, 96.
10Board for Parish Education, The Lutheran Church, Missouri Synod (St. Louis, 1953). ^Commission on American Citizenship, The Catholic University of America (Boston: Ginn and Company, 1958). 11 examples, the great lack of ready reference to material
of integration presents a problem for the teacher of mathematics in a Christian school. Further examination of present practices in such schools, and some suggestions for integration of mathematics and religion will be found
in Chapters X and XI in this study. This lack of available references is the result
of a combination of factors: a belief that little rela tionship does exist between religion and mathematics, a
lack of concern over whatever relationships have been
found to exist, and the idea for many observers that
mathematics is completely secular in nature. Mathematics is thought to be so devoid of potential religious bearings,
that efforts to integrate academic and religious ideas
should be limited to matters of greater spiritual concern.
George R. LaNoue, after examining over one hun
dred textbooks used in science, mathematics, and languages,
to determine whether or not they were a part of the relig ious education in parochial schools, concluded that,
Sound educational principles demand that the subjects not be taught in a vacuum, and Catholic philosophy in particular states that religion must permeate every environment. Any attempt to create a strictly secular subject in a parochial school would result in poor, abstract education or religious h y p o c ris y .
1? George R. LaNoue, "The National Defense Educa tion Act and ‘Secular*. Subjects," Phi Delta Kappan, XLIII (June 1962), p. 386. 12
The devoted teacher in a Christian school who continues to teach no differently than he would teach in a secular school, must sooner or later sense a lack in his teaching# By accepting a position in such a school he has recognized a prior claim on his efforts.
Literature, history, mathematics, or whatever subject he teaches is no longer an end itself# It becomes a part of the means toward a larger end, that the student should
"grow in wisdom and stature, and in favor with God and man," The teacher will soon realize the lack of direction in the secular materials of his textbooks, and will seek
supplementary aids which can help in the integration of his subject matter with his religious convictions.
Survey of Present practices
The w riter^ correspondence with the leaders of the several Christian Day School groups across the country clearly indicates that they depend upon secular textbooks,
and "capable Christian teachers#"
Some people w ill insist that an objective teaching
of mathematics will be no different in either the secular
or Christian school. For example, David Lawrence said:
Certainly in the teaching of physics, or any other scientific subject, or in the teaching of mathematics, there is no likelihood of any religious doctrine being introduced.-^
^Column of February 14, 1962, Warsaw Times-Union. Warsaw, Indiana, 13
And at the present time, the National Defense
Education Act
Permits the U. S. Commissioner of Education to make loans to private schools to acquire science, mathematics, or foreign language equipment. We believe such loans are constitutional because the connection between loans for such purposes and the religious functions of a sectarian school seems to be nonexistent or minimal.14
This view is held even by some religious educators.
Aelred Graham, in summarizing his view of Catholic educa t i o n , s t a t e d , "When th e te a c h e r is d is c u s s in g any s u b je c t, except religion, it should, I think, be impossible to tell whether he is a Catholic or not."1^
Such a view contradicts the traditional educational philosophy of the Catholic church, and undermines the very purpose of a separate parochial school system. Assume that the church school is teaching religion only an hour or two each day, and for the remainder of the day does nothing different from the secular schools. It would be simpler, and much less expensive to send all children to public school for all the "secular" subjects and release
14G. R. LaNoue, op. c it., quoting from the "Memorandum on the Impact of the First Amendment to the Constitution upon Federal Aid to Education," submitted to Congress by the Department of Health, Education, and Welfare, p. 381.
■ ^ A e l r e d Graham, "Toward a Catholic Concept of Education," Harvard Educational Review, XXXI (Fall 1961), p . 408. 14 them to the churches for periods of religious education.
This plan is being worked out in some parts of the United
States at the present time.
However, the general objectives of Christian
schools, and the specific objectives of mathematics in
these schools, will continue to be explicit in their call
for some integration. At the very least, one must recog
nize that mathematics, as a product of man’s reason, is a
gift of God, designed to meet the needs of man in thinking in quantitative relationships.
Christian education is more than education ’’plus," --more than a department of religion added to a secular curriculum. It includes a spirit, an attitude and a method which make every subject in the curriculum come a l iv e w ith s p i r i t u a l m e a n in g s.1®
The apparent need for this study was confirmed in the mind of the writer by the results of a simple survey made of all the high schools associated with an inter
denominational group of Christian schools. All forty-nine
schools were conteacted by a simple questionnaire directed to the mathematics department.
Only twelve questionnaires were returned. The
survey was made early in the Spring, to avoid conflict with the busy closing season of the school year. It can
only be assumed that those who did not respond were
l^Martin Hegland, o£. cit.. p. 2. 15
uninterested, or that they had nothing to contribute.
All but two of the returns stated that there was a need
for supplementary materials for the teacher, to assist him
or her in making relationships more explicit between
mathematics and religion. The other two felt it should be
left completely up to the individual teacher.
The returns, while not a sufficient cross-section of mathematics teaching in Christian schools to provide
careful statistical treatment, came from eleven states,
from Massachusetts to California, and from Montana to Texas. The findings recorded in the following pages,
are to be compared with the statement earlier that the
Christian schools depend upon "capable Christian teachers," for integration of subject matter and religious objectives.
There were six questions asked. Each will be repeated here
and the responses examined briefly,
1. When you see the expression, "The Religious Bearings in the Development of Mathematics," what men, and which events in the history of m ath em atics come to your mind? Pour responses were blank, and five responded "none." The other answers referred to Biblical patriarchs
who built temples and arks under the direction of God, or
had reference to a vague knowledge of such men as Thales, Plato, Socrates, A ristotle, Pythagoras, Thomas Aquinas, and Saccheri. 16
2. How can reference to such men an# events contribute to a better understanding of mathematics in a Christian environment? Five responses were blank, and three said "don't know." Two stated that they would be glad to know more of the history of mathematics, and how reference to it could encourage the mathematically gifted student in the Christian school to continue in the field. The other two thought Christians should avoid the religious background of such men, for problems would be created which would lessen the effectiveness of the mathematics itself in the classroom.
3. How does mathematics in a Christian school suffer when "diluted" by reference to its religious heritage? Is this a price worth p ay in g ?
Six did not respond. One said all that was worth while could be done in two class periods. Others felt that such integration was possible and worthwhile in science, but not in mathematics. Three stated that they did not believe mathematics would suffer if it were related to its religious heritage.
4. Which books, or what literature, has been of greatest help to you In relating your mathematics teaching to Christian education? Six could think of none. One mentioned the ency clopedia, stating that he did not live in the city, and library facilities were meager. Four had had some contact 17 with supplementary materials prepared by Christian groups, and one took strong issue with the material which he had seen, because it was more religious than mathematical.
Two mentioned that their secular instruction in science had helped them see relationships.
5. What experiences do you recall in your own preparation in mathematics when religious influences were pointed out, or could have been?
Nine could think of none. One had heard of some interest in numbers in the Bible, and one had been directed to think of the use of mathematics in the build ing of cathedrals. Nearly everyone who responded to this question pointed out that most of their mathematics education had been in secular situations, and no relation ships were mentioned.
6. Assuming that the religious bearings of mathematics need to be nourished by Christian Schools, can this be best done by
a. Preparing special mathematics textbooks for Christian elementary and secondary sc h o o ls ? b. Providing supplementary materials for teachers to assist him or her in making relationships more explicit?
c. Leaving to the teacher the freedom to make whatever reference seems to him or her to have value in a given situation?
d. (your suggestion)
Only two of the responses indicated that special textbooks would be advisable, and both of these also 18 indicated that supplementary materials for the teacher should be made available. Pour additional responses felt that supplementary materials should be prepared. Pour thought that the whole matter should be left to the teacher, and two thought that it should be left to the teacher, but supplementary materials should be prepared which would help the teacher in making decisions.
Most of those responding to the questionnaire had had no experience in the history of mathematics. A large proportion of them had been drafted into the teaching of mathematics from other interests. One wanted no religious ly slanted problems with nuns and priests in them, and another thought problems with religious units would be a good way to keep Christian thoughts before the children.
One said it was best to "Get to the basis of
C hristianity, A WAY OP LIFE, and not burden our children down with non-essentials." There was one positive state ment to the effect that in-service workshops, or magazine articles on the subject, could help the Christian teacher in this work. A very definite response from one reader
i is quoted here:
I don't like your idea of making religion an issue with math.1 They don't contradict each other. The Bible is the guide to life. Math is the language for science. Opening a math class in prayer is sufficient unless one is teaching astronomy or logic. I don't mean to ignore religious issues, as they come up in class or as they are related to math, but I don't 19
think every lesson should be geared to provide a Biblical lesson. This is what we have Bible classes for.
The writer also corresponded with the education directors of the Christian schools, with the following as some of the response:
A capable Christian teacher should be able to supplement his teaching. . . so as to develop the proper perspective.
We w ill have to continue to depend upon the ability of our Christian teachers to add the perspective and the problems not now present in the various arithmetic series on the market.
The most important element in the Christian training of our young people is not the book so much as the teacher. Our schools teach all their subjects from the Christian viewpoint.
The more mature student is often taught by a better educated teacher, who is able to present a consistent Christian viewpoint.
The above summary at least indicates that there is some respect in which integration between mathematics and religion will occur. The results of the questionnaire seem to indicate that if such integration is to take place, the teachers will need to be provided with some direction.
Supplementary m aterials are needed which w ill help them to understand more clearly how this work can be done.
Christian education embraces the work of teachers such as that described by George P. Thomas:
If a professor is a Christian theist, he has a perspective from which he can interpret the facts and deal with the issues in his field more adequately. 20
If Christian theism is true, it asserts the ultimate Truth about reality as a whole. Therefore, it provides the key for the interpretation of all the truths in special fields of knowledge which deal with particular aspects of reality. God is not only the absolute Light; he is the Source of the imperfect light which illuminates every field of reality we e x p lo re .
Definition of terms
Religious bearings. Any relationship between religious ideas and mathematical ideas.
These relationships developed in some cases because of the church-related education of some of the men, and their consequent theological orientation. In other cases, they developed from the natural response of the intelligence of man to understand the God of the Word and the God of the world as the same God. In man’s search for the proper language for his understanding of nature, he has found the language of mathematics to be the best, at least up to the present time.
Christian school. One in which the teaching of every subject matter area reflects its religious bearings.
Specifically, this includes parochial schools with explicitly stated objectives for religious purposes. It also includes public schools, and institutions at any level on the education ladder, which desire to enrich all
■^George F. Thomas, Religious Perspectives in College Teaching (New Haven; The Edward W. Hazen Founda tion, 1951), p. 12. 21 subject matter areas by considering the impact which religion has made and can make to them. Integration. The employment of man's intellectual resources in the search for truth wherever it may lead him, with the constant acknowledgment in both purpose and practice that mind and reasoning power are gifts of God to man, given that man might have dominion over and subdue the creation into which God has placed him.
Integration should not mean the dilution of the mathematical content so that lessons become moral lessons rather than mathematical ones. It does not mean school- as-usual with occasional or frequent religious exercises, or required chapel services, although these may be included. Integration is not spiritually designed pseudo numerology, nor tacked-on moral emphases.
Assumptions on which the study is based 1. Mathematics as a major human interest and acti vity will be one of the subject matter areas taught in
Christian schools, as well as in other schools.
2. There are relationships between religious and mathematical concepts which can naturally arise from a study of mathematical content, as well as fran. the historical development of mathematics. These can enrich the mathematics program without harmful dilution of the mathematics content or competence. 22
3. The relationships between mathematical and religious concepts w ill be developed independently of
the usual textbooks, 4. The religious bearings of mathematics can be
nourished through well planned use of relevant supple mentary m aterials• 5. Teachers in Christian schools will find this
study useful in providing one source of such materials.
Procedures followed in the study This study is a historical and critical research
of the religious bearings in the development, the content,
and the teaching of mathematics. It has been made from
a study of books and research which relate to the history
and the teaching of mathematics.Personal experience, consultation, and correspondence over the years color
some of the discussion. Some detailed attention is
given to a few of the major areas of concern relating to the integration of mathematics and religion, areas
which have been the object of critical research.
General plan of the study This introduction began with the background for
the study and defined its purpose. It will conclude with
a consideration of related literature. Following the Introduction, the study consists of
three major parts, and a bibliography. 23
Part I consists of five chapters representing summaries of the findings of historical study from library sources dealing with the history and development of mathematics, from the earliest days to the present.
Part II is concerned with the religious bearings in the content of mathematics. Prom the material content of mathematics itself have been selected some relation ships between religious and mathematical thought. These topics include the religious use of number, the question of the origin of mathematical content by means of inven tion or discovery, and the relation between absolute and relative truth.
Part III represents an attempt to indicate the religious bearings in the teaching of mathematics• This part of the study examines the stated practices and ob
jectives of those schools which at present are doing
something in integration of religion and mathematics. Some suggestions and ideas are included which may suggest ways in which a teacher can adapt the material of this
study to his own use. The study closes with a bibliography of some of
the books and periodicals which have been found helpful. PART I
RELIGIOUS BEARINGS IN THE DEVELOPMENT
OF MATHEMATICS
24 CHAPTER II
EARLIEST TIMES TO THALES (600 B.C.)
The beginning. When one seeks to find a beginning in a chronological study, he is forced to make a decision on somewhat arbitrary grounds as to what will determine that beginning. It is the purpose of this investigation to follow the relationships which have existed between mathematics and religion, and it is important that the earliest practicable date be found.
Until recently, many of the books of the history of philosophy and mathematics naturally began with the
Greeks, and specifically with the philosopher Thales,
Now, through the patient efforts of Otto Neugebauer and
P. Thureau-Dangin for the Egyptians and Babylonians, and
Joseph Needham for the Chinese, the contributions of earlier civilizations have been given more recognition.
Prom Biblical accounts and archaelogical discoveries, it is apparent that certain arithmetic operations and geometric designs were in wide use before the time of
Abraham and Hammurabi. The Egyptian calendar was introduced about 4000 B.C., and the great pyramid constructed about
2900 B.C.
25 26
However, although the methods involved in the arithmetic and geometry of these early peoples show great ingenuity and imagination, there seems to have been little or no attempt by any of these early scholars to isolate and describe basic principles which could be the beginnings of a system of mathematical thought. This was left for the Greeks to develop, and later civilizations to improve. Pew things seem more certain at the present time than the fact that the earliest recorded evidence of mathematical, reasoning, as some pure mathematicians now use the term, is the contribution of Greek speaking communities of the Mediterranean about 600 B. C. Pew th in g s are a ls o more c e r ta in than thuu they themselves did not exaggerate their self- confessed and considerable indebtedness to the temple cultures of Egypt and Mesopotamia, that the priestly custodians of the calendar jealously guarded the secrets of how they arrived at the discoveries for which their records disclose no proof and that they had some good reasons for confidence in the reliability of at least some of them. What level of mathematical skill the earliest settled communities which have left remains elsewhere may have attained is a question to which we have as yet no answer; but it is unlikely that their achievements were comparable to those of Egypt and Mesopotamia. We can say with assurance that Egypt and Mesopotamia emerge first in the history of the written record; and that the record of their progress covers more than two m illen n ia B. C.-*- Some day the amazing Greek miracle may turn out to be the reflection of the ancient Babylonian miracle. A still further study may perhaps reveal
^Lancelot Hogben, Mathematics in the Making (Garden City: Doubleday and Co., 1960), p. 50. 27
that the Babylonian miracle, too is no Babylonian miracle. But that it is a part and parcel of this wonderful wonder, which is man, meditating and wondering at the marvelous glory of the Creator and His Creation.2 Priests or scribes? What class of people were responsible for the first mathematics? Most commentators ascribe quite naturally the early efforts in mathematics to the priests, and some religious significance is given to its use. Aristotle perhaps was the first to state simply, "Thus the mathematical sciences originated in the neighborhood of Egypt, because there the priestly class was allowed leisure."^ Van der Waerden, after noting this remark of Aristotle, goes on to state that in his opinion, there was no well organized estate of priests at this early time, and the problems are problems of the surveyor and the erector of granaries. Therefore he concludes that the earliest mathematics, including that of the Rhind Papyrus, was intended for use in a school for scribes.4
Neugebauer believes that the early association of the Chaldaeans of the Bible with astrologers and magicians
is reflected today in the wide-spread idea that most of
2Solomon Gandz, "Studies in Babylonian Mathematics," Osiris, VIII (1948), p. 40.
^Van der Waerden, Science Awakening (New York: Oxford University Press, l'96l), p. lfj.
4Ib id ., p. 17 28 the Babylonian documents are concerned with religion, magic, or number mysticism. But he states, In fact, however, the overwhelming majority of cuneiform texts concern economic items. Tens of thousands of such documents were unearthed, and especially for the earliest period of writing, the economic records are almost the only class of existing documents and the number signs are among those signs which one can read with certainty even for periods when the interpretation of the other signs is very problematic. Kline seems to take issue with this conclusion, and adopts more nearly the idea of Aristotle,
It is a mistake to believe that mathematics in Egypt was confined just to the solution of practical problems. This belief is as false for those times as it is for our own. Instead, we find upon closer investigation, that the exact expression of man's thoughts and emotions, whether artistic, religious, scientific, or philosophical, involved then, as today, some aspect of mathematics. In Babylonia and Egypt, the associations of mathematics with painting, architecture, religion, and the investi gations of nature was no less intimate and vital than its use in commerce, agriculture, and construction.®
The rules of calculation in the early days were
difficult and hidden from the ordinary person. It is
rather disconcerting to see the claim of the Rhind
Papyrus, that it gives "complete and thorough study of
all things, insight Into all that exists, knowledge
5 Otto Neugebauer, The Exact Sciences in Antiquity (Providence; Brown University Press, 1957), p7 l8.
®Morris Kline, Mathematics in Western Culture (New York; Oxford University Press, 1953), p. 17*. 89 of all secrets,"7 only to find merely some arithm etic calculations. Those who could manipulate such, secrets were held in awe* A man skilled in the art was regarded, as endowed with almost supernatural powers • This may explain why arithmetic from time immemorial was so assidiously cultivated by the priesthood.®
It is generally accepted by historians of mathema tics that the secrecy maintained by the p rie sts was a hindrance to the long range development of mathematics. T. L. Heath, after giving five reasons for the great Greek period of mathematics and science, adds,
Last, but not less important, the Greeks possessed the advantage over the Egyptians and th e Babylonians of having no priesthood which could monopolize learn ing as a preserve of its own, with the inevitable result of sterilising it by keeping it bound up with religious dogmas and prescribed and narrow routine. They had no Bible or Koran to interfere with the free use of their reasoning faculties. Their religion was just adequate enough to satisfy natural sentiment, but left free scope for the exercise of all their Intellectual powers even when this tended to uproot and supplant religious dogmas.9 Bell agrees with Heath that the Greek mathematicians performed a great service in shattering the tradition of secrecy fostered by self-perpetuating priesthoods. He
7 Van der Waerden, o j d . c i t ., p. 16.
O °Tobias Dantzig, Number, the Language o f Science (New York: The Macmillan Co., 1933), p. 25. 9T. L. Heath, "Greek Mathematics and Science," Mathematics Gazette, XXXII (July 1948), p. 122. 30 further points out that a later attempt to continue the same se c re t monopoly on a rith m e tic was h a lte d , and the
Greeks were able to advance their form of civilization with greater certainty.
The attempt of Pythagoras to carry on the secretive tradition of Babylonia and Egypt was quickly dissipated and enlightenment was put within the grasp of any unsanctified vulgarian _ with the will and intelligence tog rasp for it.
Positive contributions. However, the situation was not all bad. The mathematics of Babylonia and Egypt was basically empirical, and useful in the market place. It was also used for contemplation of the heavens. These priests and scribes were concerned with astronomy and calendar questions, with the construction of tombs, temples, and other religious buildings, as well as with practical problems of land measurement, surveying and primitive engineering. Through the priests was the continuity from one generation to the next of the achieve ments in the arts and sciences. Jastrow states:
Prom the earliest to the latest period the priests continued to act as the teachers of the people. With the art of writing in the hands of the priests, the secrets of the gods could be unlocked by them only. . . much of the Babylonian literature has thus come down to us in the form of school editions, and this applies also to mathematical tablets, chronological and geographic lists and medical prescriptions for which
lOE. T. Bell, The Development of Mathematics (New York: McGraw-Hill Book Co.^J In c.’, 1945), p. 36. 31
there followed instruction in the temple service. The sciences which were evolved out of this cult, such as astronomy and mathematics in connection with astrology, and medicine and botany as an outcome of the incantation rituals were likewise in the hands of the priests
To priests we owe most of the technical and historical documents that have come down to us. They developed and maintained libraries before the middle of the third millenium B. C,, contain ing 20,000 to 30,000 tablets Standards of weight and measure were kept in the temple. Thus the Bible speaks of the ”shekel of the sanctuary,” or the standard shekel, and Amos accuses the priests of using short weights in their measures. The concentration of learning, and the keeping of standards in the temple at that time is not surprising. Some later generation might well discover standard units of the twentieth century locked up in the vaults of the
Bureau of Standards in Washington D. C., or in Paris.
The geom etric p a tte rn s on e a rly p o tte ry may originally have had religious or magic meanings, but much of this has been lost, and the esthetic appeal gradually has become dominant. It is certain that religious ceremonies were deeply permeated with magic, and this magical element was incorporated into existing conceptions of number and form, as well as in the music,
•^Morris Jastrow, The Civilization of Babylonia and Assyria (Philadelphia: 77”B7 Lippincott Co.^ I9l5), p. 275.
12Ibid., p. 46. 32 sculpture, and drawing of the day. There were magical numbers such as four and seven, and magical figures such as the Pentalpha and the Swastika. The numerology of medieval times, and of the present day, is a carryover from the magical rites dating back to earliest times.
Was it the magical qualities of mathematics which dominated throughout the centuries? Or were the social and economic roots most important? Perhaps the surest answer is th a t both ex isted side by s id e . At le a s t one must recognize that both the religious, or magical, and the socio-oconomic aspects of early life were under the direction of the priests. The static character of the Orient imparted a fundamental sanctity to its institutions which facilitated the identification of religion with the state apparatus. The bureaucracy often shared this religious character of the state; in many oriental countries priests were the administrators of the domain. Since the cultivation of science was the task of the bureaucracy, we find in many-- but not all--oriental countries that the priests were the outstanding carriers of scientific knowledge One of the chief influences which the priests had
over the people was their responsibility for keeping the
calendar, and maintaining the festivals in track with
the seasons. When rebellions or famines occurred, it was often concluded that something was wrong with the
-*-3Dirk J. Struik, A Concise History of Mathematics (New York: Dover Pub1ications, IncT, I948"J7 p. 15. 33 calendar, and the mathematicians were asked to re c o n s tru c t i t . The calendar year was worked out differently in different parts of the world. For some time the Babylonians had a calendar year of 360 days with a correction of a five day month* The Mayan priesthood used a calendar with eighteen months of twenty days each, and a five day month added. For
some time, the year was begun approximately at the time of the vernal equinox, and the time of the winter months
simply was not counted at all. The year went from seed time to harvest, with a cold gap in the middle. At the most primitive level, however, the custodians of the
Egyptian calendar seem to have settled the length of the
year as 365'|- days as early as 4000 B. C.
It is no secret that in many places the priests monopolized all learning, including mathematics. Whether
they should be scorned for thus secreting valuable knowledge
for their own ends, or praised for thus preserving what
knowledge they collected, is somewhat a matter of choice
for the reader. Certain it is that knowledge has usually
led to power, and by restricting the spread of knowledge,
man has ever hoped to reduce the likelihood that others
would be able to challenge that power. In addition,
ignorance leads to fear, and people who are afraid turn to leaders who will promise to guide and reassure them. 34
In this manner, the priests reinforced their position, and were able to maintain their control over the people*
On the other hand, during all of this time, the difficulties of preserving knowledge were enormous* Fables and true accounts of battles and other important events could be handed down from one generation to another by word of mouth. Mnemonic devices helped to insure the accuracy of this type of tradition to a degree. But anyone who has participated in a game of "gossip,11 knows the weakness of complete dependence upon this method.
In these early days there was no easy method of writing, no material oh which to record, and no means of mass
communication such as is common today. Perhaps, if there
had been no group which assiduously and secretly main
tained these records, the account which has come down to
the present time would never have been kept, and the
march of civilization would have been slowed down.
The calen d ar year was always lo sin g i t s r e la tio n
to the seasons for the masses of people, because they
were not initiated into its secrets. Even a well-prepared
calendar would soon confuse, because there was always
that odd quarter of a day to account for. Prediction of
a religious holiday, or the flooding of the Nile, even a
few days in advance required an accurate knowledge of the
motions of the heavenly bodies, and of mathenatics which 35 was possessed only by the priests* Kline is sure that
the priests were careful to keep this knowledge a secret
in order to secure power over the uninformed masses.
In fact, it is believed that the Egyptian priests knew the solar year, that Is, the year of the seasons to be 365^ days in length, but deliberately withheld this knowledge from the people. Knowing also when the flood was due, the priests could pretend to bring it about with their rites while making the poor farmers pay for the performance.1^
Kline continues to protest the work of the priests,
and hastens to generalize concerning the effect which
such periods of religious influence have had on the advance
of civilization. He credits the rapid advances of the
few hundred years of Greek ascendency, and the forward
scientific advance of the past few hundred years with the absence of such a dominant priestly class.
The theocracies of Babylonia and Egypt compare very unfavorably with civilization in which there was no dominant priestly class. We shall see that the few hundred years during which the Greek flourished and the last few hundred years of our modern era produced infinitely more knowledge and progress than the millenniums of the two ancient civilizations.15
Again, it must be said that both good and bad
effects resulted from the monopoly of the priests.
Religious mysticism through its wonder about life, death,
and the changing aspects of nature, fastened onto mathema
tics and resulted in astrology. But through wonder about
14Iv*orris Kline, o£. cit., p. 19. 15Loc. cit. 36 the heavens, and the orderliness of the patterns of nature, there came astronomy, and the correct science which has survived.
Children of Israel. The student of the Bible is aware of the pagan priesthoods and mysticism of the early days, and will wonder as well about the contribution of the Israelites during these early days. In many respects they were, as many Christians are today, "not of this world.*' To the average secularly-oriented man, education is a preparation for existence within some kingdom or country ruled by his fellow man. To Christians, education and knowledge are only t emporary, and the eternal and important effort of man is his preparation for existence within the kingdom of God in heaven, under the rule of
God, fo re v e r.
Israel as a nation operated under such a religious philosophy. However, it is both possible and very neces sary for some Christians in this present age to concern themselves with the science and thought of the secular world, and so it was in the times of the children of
Israel. The Bible tells of Joseph, placed second only to the Pharaoh of Egypt, administering the production of food, and the feeding of multitudes of people during the seven years of plenty, and the seven years of famine.
The mathematical problems of building granaries, and of 37 record keeping, and of bargaining for the exchange of properties for food, would tax the mind of all but a very few people. Later, the Bible records the work of
Moses in organizing the children of Israel, and leading them out of the land of Egypt. This was a position of some responsibility which required some education and executive ability. Stiil later, during the Babylonian captivity,
Daniel and his friends were chosen to occupy positions of authority in the king’s court. Jeremiah earlier had prophesied the captivity would last for seventy years, and Daniel knew enough about the calendar, that he recognized that the seventy year period was coming to an end. Solomon certainly employed men who toad a great knowledge of mathematics. The problems of gathering materials from the entire world for the building of the temple of God as described in the Bible, and for the building of Solomon's own palace, required understanding
of mathematics for their solution. The matters of
logistics alone would be staggering.
But the nation of Israel was a nation called of
God, to represent him in areas other than science and mathematics. Their particular work on earth was to give
to the world the written revelation of God, and to provide
the channel for the Messiah. As a nation, they were to be 38 an example of God’s providential care upon a people who would serve Kim, and Cod’s wrath upon a poopIn who would fursrd:o or turn against Jin.
Che simple (Iorsc3:L) model of the universe displays no scientific curiosity, but it does demonstrate that principle of God-maintained order in the world, which underlies the whole of Old Testament thought and is so magnificently affirmed in Psalm 104. This conception of order sprang from the Hebrew religious awareness and, despite the unscientific way in which it is expounded, the scientist of the 20th century might profitably meditate on its far-reaching implications. °
Hardly any references to mathematical and astro nomical subjects are found in the Bible, not even in a way which would indirectly enable us to arrive at some conclusion about the knowledge of the Jews after their return from the Babylonic exile. The nearest motive for such material would be the building of the Tabernacle in Exodus and of the Solomonic Temple (first book of K^ngs); yet the reference in the latter (7:23) which has been often quoted as determing the number pi as 3, can be taken in many d if fe re n t a s p e c ts, and is n e ith e r negatively nor positively conclusive. The celestial bodies and their courses are frequently mentioned, in particular, some fixed stars in Job 9:9. The material contained in the Mishna and in the Talmud (of Jerusalem and Babylon) is abundant. However, nov/here do these passages deal with purely mathema tical problems, but rather with application of mathematical and astronomical conceptions to Halahic problems. '
Although the historian may claim that the Jewish
nation has not produced a large number of geniuses in
the highest sense in the fields of mathematics and other
*^E. w. Keaton, Everyday L jfe in Old Testam ent Days (Hew York: Chas. Scribner’s Son3, 1956"]~i P* 187. ■^Abraham A. P raenkel, ’’Jewish Mathematics and A stro nomy," Scripts Mathematics XXV (1960), p. 47.. 39 sciences in the distant past, he must admit that there has been a multitude of prominent Jewish mathematicians and scientists, perhaps a greater number than corresponds to the comparative number of Jews in modern civilization.
What has been said of the nation of Israel of Old
Testament days, can well be said for many Christians of today. The possibility of alternate explanations of nature, or differing philosophies of life, do not excite the attention of many. The very atmosphere negates the idea of a Christian university which is advocated and hoped for by a relatively small group. And so it was with Israel:
Even when, it is cautiously qualified, the term "science" is so blatantly wide of the mark In any account of Hebrew thought that it may be used without danger as a convenient way of grouping together a nuigLbqr of subjects which now properly fall into the scientific category. And not only now. Already in Old Testament time, the Babylonians were calculating the volumes of pyramids, making astonishing studies in algebra, solving problems of surveying, compiling astronomical tables, listing and roughly classifying plants, animals, birds, fish, and stones, and establishing formulas for metal alloys. This capacity as observers merits the description of their achievement under such headings as mathematics, botany, zoology, and mineralogy. Unless we count Solomon (I Kings 4:53), the Israelites have nothing to offer which is even vaguely comparable. As the great Oriental scholar VVm. Robertson Smith observed a long time ago, the Hebrew "values nature only in so far as it moves and affects him, or is capable of being moved and affected by him. . • to him nature is what he feels 40
as he beholds it." Such imperious subjectivity is not the foundation on which the natural sciences are b u ilt.
China. Another of the early civilizations which needs to be considered in this study is China. Recent e f f o rts of Joseph Needham have c l a r i f i e d much of the understanding about the development of science, mathema tics and civilization in general in that great nation.
But the conclusions about the mathematics of China at an early age are very uncertain still. The oriental mind has provided instances of societies which may have had no centralized political organization, but which have achieved a large measure of cohesion through some common forms of religious ritual. In such cases it often happens that a number of groups will agree in granting a unique ritual status to the chief of a particular group without in any way accepting his secular executive authority. Here again, as in the case of the Egyptians and Babylonians, the authority rests in the one whose responsibility it is to maintain a calendar. In China, this was the emperor who had the task of regulating the cycle of months and seasons for the benefit of his subjects. William Edward Soothill, professor of Chinese language and literature at Oxford from
1920 to 1935 gives a d e s c rip tio n of th is re s p o n s ib ility
•**8E. W. Heaton, o£. c it., p. 185.
v 41 which was attached to the privileged position of the
Chinese emperor: The Chinese emperor had the task of regulating the cycle of months and seasons for the benefit of mankind. He was essentially a "corn king," with the duty of assuring a good harvest through the magical control of natural forces. The loess and alluvial lands of the Yellow River Basin form one of the most fertile agricultural areas of the world, but they are subject to disastrous famines from two opposite kinds of catastrophe—drought due to the failure of the spring rains, and floods due to excessive rain fall in summer. The great variability of rainfall is, indeed, one of the most important characteristics of North China. . • and to an exceptional degree it focused the attention of the Chinese in the formative period of their cultural evolution on the cycle of the seasons, and the relation of the times-periods of the months and y ea r; a t th e same tim e i t encouraged a notion of the cosmos as an order of things normally beneficial in operation, but liable to catastrophic breakdowns, as an equilibrium of forces always in danger of becoming unbalanced. For preserving the harmony of these forces, how ever, mankind had an effective agency in religious ritual. No amount of toil in the fields would avail the farmer if the spring rains were lacking, but he was not entirely helpless; he had a representative and mediator with the powers of the sky, a wise man who was the Son of heaven and endowed with the power of the rain-bringing dragon. By performing the proper ceremonies at the proper times, according to the calendar which it was his business to formulate, the emperor promoted the harmony of the productive forces of nature and assisted them in their work.1 Dr. S o o th ill a lso quotes from M. G-ranet in h is La Civilization Chinoise in support of this argument: "le
souverain est essentiellment l ’auteur d’un calendrier
exact et bienfaisant. . . Quant a lui, il regne, sans
W. E. Soothill, The Hall of Light, A Study of Early Chinese Kingship (London: Lutterworth Press, 1951), p . x v ii. 42 penser a gouverner. II s’emploie a creep, ou plutot a pn secreter, l'ordre." Although the prestige of the old priestly king ship varied during the centuries, and sank to a very low ebb during the period of the warring states, Dr.
S o o th ill goes on to say,
Until the Revolution of 1911, the Chinese monarchy never lost its sacerdotal character; for Confucian orthodoxy, the emperor was a person of cosmic significance, the Unique Man.1'2!
Summary. When one writes about the developments of science and mathematics of other ages, it is difficult and very likely impossible to follow a middle course between two extremes which are equally erroneous. Hogben expresses the difficulty in this way: On the one hand, one may underrate its achievements ' through misunderstanding of what were its needs and what its difficulties. On the other, it is a common fault to discern anticipation of later discoveries of which people of earlier times could have had no pre-vision, unless gifted with superhuman subtlety or second sight.
An additional difficulty is always at hand. The
subjective inclinations of the investigator are apparent in the assumptions, hidden or expressed, which guide his
thinking. For example, Van der Waerden makes a re lig io u s
inference when he quotes Aristotle as saying "Thus the mathematical sciences originated in the neighborhood of
2°Loc . cit. *^Loc. cit. 22Hogben, ojo. c i t ., p. 53. 43
Egypt, because there the priestly class was allowed leisure."^3 On the other hand, Neugebauer states "A much more sophisticated attitude is represented by
Aristotle, who considers the existence of a 'leisure
class,1 to use a modern term, a necessary condition for
scientific work."24 This latter quotation eliminates
the reference to the religious bearings of the early
work in mathematics#
Thus, the suggestions of what the priestly
mathematicians did or didn't do, and the reasons for their actions, cannot be described on purely objective
evidence. But the objective evidence available to the
historian continues to increase, and the future student
may find more definitive answers to his questions. The
present knowledge of the mathematical attainment of
these early civilizations is capably summed up by
C a rro ll Newsoms
Our knowledge of the development of arithmetic among the various groups who successively dominated ancient Babylonia has been obtained from a study of approxi mately three hundred tablets out of a total of about half a million that have been made available for scientific study as the result of a series of archaeo logical excavations. • . . Most of our knowledge of e a rly Egyptian mathematics comes from a study of the so-called Moscow papyrus and the Rhind papyrus. The former work with a commentary was published in 1930; the latter work was similarly published in 1927. . . . In China, it must be noted, the arithmetical and geometric disciplines were well developed by the
^3Van der Waerden, ojd . cit., p. 15. ^Neugebauer, o£. cit., p. 151. 44
second century A. D.; their status at that time was undoubtedly a consequence of many centuries, and possibly several millennia, of activity on the part of sch o lars
Prom the vantage point of the twentieth century, one might be tempted to say that the chief effect of the religious bearings of mathematics during this earliest of times was to retard the progress of civilization, through the secrecy of the priests. However, it is only fair to point out once more, that the lack of the knowledge of printing, paper-making, and efficient means of communication were also factors in the concen tration of learning among a few. Furthermore, the very ac tio n s which seem to have re ta rd e d le a rn in g , made i t possible for what learning did exist to be passed on from one generation to another.
^C arroll V. Newsom, Mathematical Discourses (Englewood Cliffs: Prentice-IIall, Inc., 1964), pT 10 ff• CHAPTER III
THE GREEK AGE 600 B.C. to A.D. 641
The beginning of this period is not difficult to
select, although there was no specific termination date given f o r the e a r l ie r p erio d . One of the most commonly
accepted dates in the history of mathematics, philosophy, or science, is that of Thales. Born in Miletus about
640 B.C., he lived for nearly a century, and traveled
widely. He financed his travels through business ingen
uity, and his practical understanding of the mathematics
of his day. He is especially noted in the history of
mathematics for laying the foundation of deductive
geometry, and for being a teacher of Pythagoras.
Thales visited Egypt, and was acquainted with
the astronomy and mathematics of the Orient, so that he
was a b le to p re d ic t the e c lip se of the sun in 585 B.C.
He appreciated his debt to the priests of Egypt and
advised his pupil Pythagoras to visit them also. Nothing
more is known of Thales which concerns the study of
religious influences in mathematics. But more will be
said of his pupil Pythagoras a little later. 46
A natural date for the close of the period of
Greek influence is not so readily agreed upon. There is sufficient reason to claim that the age of Greek influence has never ended, and has continued until the present.
Major publications have recently indicated a revival of interest in Greek culture and civilization in America in recent times. However, for the purpose of this study, it is worthwhile to find a date which marks a transition from the active Greek tradition to a different age.
In a study involving religious influences in history, it is only naturally of importance that one find a chronological break with the birth of Christ. And this date is significant in the history of man’s use of mathe matics. For it is from this date that man now records all the events of history. Everytime one uses the calendar, uses a piece of money, looks at the daily newspaper or any periodical, or puts the date on a check, he is witnessing to the fact that nearly two thousand years ago an extra ordinary event took place. No aspect of man’s existence has been unaffected by that event.
But the Greek tradition persisted into the new era, through the Roman period, until the light was temporarily dimmed during the period calxed the Dark
Ages. David Eugene Smith closes the period around A. D.
500 v/ith Boethius, the last of the great Romans, Sarton, 47 who begins his consideration of Greek culture with Homer in the ninth or eighth centuries B.C., and of Greek science with Thales and Pythagoras in the sixth century
B.C., ends the age with the closing of the Academy of
Athens by Justinian in A.D. 529. Most other historians, including Bell, Ball, Struik, and Eves, extend the close of the period to A.D. 641, when the city of Alexandria was burned by the Arabs.
It may be misleading to attempt to classify all the mathematics of this long period of more than a thousand years by some simple title as ’'Greek Mathematics."
Neugebauer, who has spent much of his life exanining the ancient records and analyzing the mathematics of earlier cultures which preceded the Greek age, cautions concerning this oversimplification:
We are fairly well acquainted with three mathema- ticians--Euclid, Archimedes, and Apollonius—who represent one consistent tradition. We know only one astronomer, Ptolemy. And we are familiar with about equally many minor figures who more or less follow their great masters. Thus what is called "Greek" mathematics consists of the fragments of writings of about 10 or 20 persons scattered over a period of hundreds of years,^
Any attempt to abstract a common type from such a small amount of material, and then to establish some great principle v/ith which to connect mathematics with
^Otto Neugebauer, The Exact Sciences in Antiquity (Providence: Brown University Press, 1957), p. 190. 48 other areas of life, is perhaps a hasty and dangerous generalization. But the consensus of historians is that there did indeed exist a fundamental difference between the utilitarian mathematics of the earlier ages, and that of the Greek mathematicians. The Greeks sought knowledge
of what was universal and eternal rather than immediate
and passing. They changed the character of mathematics from
an empirical science into a deductive system of thought.
They changed the approach to content from that of manipu
lation of concrete objects, to the thinking of abstract
ideas. Where the Egyptians thought of a wheel, the Greek
mind saw a c i r c le , where the Egyptian saw a s tre tc h e d
rope, the Greek imagined a straight line. What was lost
in ease of thinking, v/as gained in generality.
The Greeks preferred the abstract concept because it was to them permanent, ideal, and perfect, whereas physical objects are shortlived, imperfect, and corruptible. The physical world was unimportant except in so far as it suggested an ideal one; man was more im portant than men.
For over two thousand years, Mesopotamia had
maintained a great civilization, and had contributed to
science, law, art, mathematics, medicine, literature, and
religion. But with the Persian conquest of 639 B.C.,
and the subsequent invasions of Parthians, Greeks, and
Romans, l i t t l e rem ained of her e a r l i e r g lo ry . F o rtu n a tely
^Morris Kline, Mathematics in Western Culture (Oxford University Press’, 1953), P» 49 for the continuation of the development of astronomy and mathematics, the great kings, Cyrus and Darius had been tolerant of captured peoples, and interferred with neither their cultures nor their religions. The Bible records the careful treatment which the Israelite^ received from these leaders, in the first chapter of the book of Ezra. Such care on the part of the captors of Babylon must be credited v/ith preserv in g much of the le arn in g of the Bab7/lonians until such time as the Greeks had assimilated it, and as
Plato observed, "Whatever we Greeks receive v/e improve and perfect." Had a barbarous and destructive horde overthrown
Babylon, the efforts of the succeeding generations to advance the cause of civilization would have been much more d i f f i c u l t .
This mixture of Hebrews, Assyrians, Phoenicians, and Greeks formed an amalgamation of culture which resulted in a new type of civil organization, the city states of
Greece. The new social order created a new type of man.
The merchant trader v/as in touch with the world, and at the same time knew more independence than man had ever enjoyed. This independence v/as the result of a struggle, and this recognition of struggle was embedded in the thinking of the West. The static outlook of the Orient did not accompany the other contributions which were carried to the Western civilization. Geographical 50 d isc o v e rie s were being made which, would not be surpassed until the sixteenth century by Western Europe. This new
Greek man Recognized no absolute monarch or power supposedly vested in a static deity. Moreover, he could enjoy a certain amount of leisure, the result of wealth and of slave labor. He could philosophize about this world of his. The absence of any well established religion led many inhabitants of these coastal towns into mysticism, but also stimulated its opposite, the growth of rationalism and the scientific outlook.5
The great contribution of the Greeks was the recognition of the power of human reason. By applying reasoning to mathematics they completely altered the nature of the subject. Instead of the ordinary empirical uses of mathematics having a predominate role in the education of the intellectual elite, a systematic rational study of the whole subject was undertaken. The Greeks found, perhaps to their amazement, that when they had laid out the rules for reasoning about ideal lines and circles and angles, they had produced knowledge that experience alone would never have imagined. Success in this direction naturally led to experimentation in other directions.
Other civilizations before and after them looked upon nature as capricious, terrifying and arbitrary. They sought magical formulas and religious ritual to appease the mysterious and feared forces. But the Greeks
°Dirlc Struik, A Concise History of Mathematics (New York: Dover Publications*]! 1948), p. 41, 51
confident that the secrets of nature could be discovered,
and that mathematics was the means through which these
discoveries could be made. They Dared to affirm that nature was rationally and indeed mathematically designed, and that man’s reason chiefly through the aid of mathematics, would fathom that design. The Greek mind rejected tradi tional doctrines, supernatural causes, superstitions, dogma, authority, and other such trammels on thought and undertook to throw the light of reason on the processes of nature.^ Religion of the Greeks. Having succeeded with the
• processes of nature and mathematics, the Greeks proceeded
to apply this same type of reasoning to other areas of
life, including religion. In so doing, they were initiat
ing a cycle which would be repeated over and over again
throughout the history of man:
Our spiritual life is also influenced by science and technology. In a measure but rarely fully under- istood. The unprecendented growth of natural science In the seventeenth century was followed ineluctably by the rationalism of the eighteenth by the deification of reason, and the decline of religion; an analogous development had taken place earlier, in Greek times. In a similar manner, the triumphs of technology in the nineteenth century were followed in the twentieth by the deification of technology. Unfortunately, man seems to be overly inclined to deify whatever is powerful and successful.
But what of their religion? It was not a definite set of doctrines formulated in a set creed, and supported
^Morris Kline, Mathematics, A Cultural Approach (Reading, Massachusetts: Addis on-Wesley Publishing Company, In c ., 1962), p. 16.
^B. L. Van der Waerden, Science Awakening (Oxford University Press, 1961), p. 3. by an organization separate and distinct from the state.
It v/as rather a mixture of legends handed down orally by one generation to the next, and in constant revision in the minds and memories of poets and priests. Such priests as there were were mainly public officials, appointed to perform certain religious rites.
The r e lig io n included re c o g n itio n of many gods, but no real v/orship. Every power of nature was presumed to be a spiritual being, and nature became a company of spirits. However, these gods, although incomprehensible and often hostile, were brought down to man’s size. If they became angry, they could be mollified, and they could also be played one against another. Not only was external nature so deified, but man’s own passions, love and hate, war, wisdom, and even his appreciation of the fine arts were all given spiritual forms.
The gods were in one sense beings in human forms, perhaps superior to man, spending much time on earth involved v/ith t'he affairs of man. They were in fact the ancestors of the race, and thus the real founders of society. The whole social structure of Greece was thus permeated v/ith the spirit of religion. But the religion was merely the spiritual side of their political life. It v/as ritualistic, not conducive to true worship. It was elaborately designed for man to feel at home in this world. 53
This religion was not what Christians think of as
religion, which concerns the relation of the soul to God,
the sense of sin, and the need for repentance and grace. i It was a mechanical religion. There was no anticipating
what a god might do, and the art of divination and interpretation of signs was very important. One performed
sacrifices to g ain favor, and neglected them at the risk
of punishment. Albert Newman, one of the great church historians of the nineteenth century, summarizes their
religion in this statement:
Their religion was a polytheistic personification of the powers of nature resting on a semi-pantheistic conception of the world. Their gods and goddesses were the embodiments no less of the baser passions of the human soul than of the nobler qualities, and the moral ideals of the people were low. The idea of sin as an offense against a holy God and as involving guilt was almost wholly absent. Sin was conceived of rather as ignorance, as a failure to understand one's true relations. There is no adequate recogni tion of the personality of God or the personality and responsibility of man,®
This religious outlook was shared by the several
city-states, and provided some unity, but it was not
sufficient•to overcome the fatal defect of the Greek civilization, which was the failure of the independent
city-states to join for mutual protection against the
outside world.
®Albert Henry Nev/man, A Mas ual of Church H istory, Vol. 1: Ancient and Medieval Church History, to A.D. 1517 (Philadelphia: The American Baptist Publication Society, 1899), p. 20-21, 54
Greek philosophy. A Greek who was not too inquisitive might accept all the religious ritual, and move through the calendar of special feasts and fasts, charmed with the beauty of the forms, and enjoying the familiar legends with out ever questioning whether it all be true. But once he ra ise d the question, and broke the s p e ll, he became a philosopher attempting to work out answers on his own. This was the spirit of the Greek age. When the Apostle Paul visited Athens, he was encountered by the philosophers who brought him to Mars1 Hill and wanted to know what he meant by what he said .
Thou b rin g est c e rta in strange th in g s to our ears; we would know th erefo re what these things mean. (For all the Athenians and strangers which were there spent their time in nothing else, but either to tell, or to hear some new thing.
The Greeks who questioned sought a materialistic basis for things as they were, and more particularly, for the origin of things. In the period between Thales and
Plato, one finds in succession such men as Anaximander,
Pythagoras, Anaximenes, Leucippus, and Democritus. These men sought some principle of representation of matter which could provide the unity they desired. All things were in te rp re te d at one time or another as having th e ir
origin in water, air, fire, chaos, and indivisibles or atoms. These philosophers were alike in ignoring the
vActs 17:20, 21. 55 supernatural and proceeding on the basis of physical causation. Some were therefore persecuted because of their atheism, and their teaching in opposition to the current gods.
Pythagoras. One of these great thinkers v/as
Pythagoras, a student of Thales. He based his philosophy upon the. postulate that number is the source of the various qualities of matter. This led him to dwell upon the mystic properties of number and number relationships.
Although historical knowledge of Pythagoras is very lim ited, and no record of his own has been preserved, early writers vie with each other in the invention of stories relating to his travels and his teachings. Aristotle went so far as to place Pythagoras in a class by him self when he classified living beings v/ith reasoning powers in three categories: God, man, and Pythagoras.
Pythagoras is said to have left his home town of Samos when he was not yet tv/enty, and traveled to Babylon,
India, and Egypt, This was during the sixth century B.C., a time in history when Buddha was introducing his doctrine in India, and Confucius and Lao-Tze were laying the foundation for their philosophic cults in China. Whether
Pythagoras came in touch with them or not, at least the
age v/as one which was ripe for great movements of the mind and spirit. 56
Pythagoras returned to Greece after thirty years of wandering, and established a near-religion based completely upon number. This was a brotherhood of students patterned perhaps after those which he had seen in India among the Buddhists, and similar in turn to the later monastic orders in Europe, Pythagoras introduced rites of abstinence and purity and meditation. These were meant to cleanse the soul and to provide an escape from the common world. His students thought him to be the personi fication of the highest wisdom, and super-human. Only after a period of testing and rigorous selection was an initiate permitted to hear the voice of the master from behind a curtain. Only after additional years of living in accordance with strict regulations, and being sworn to secrecy was he permitted to see the master himself. There were many such mystic cults, and mystery groups at the time, but what made the brotherhood of Pythagoras different from any other was the insistence that everything could be explained in terms of number.
Y/hat distinguished the Pythagoreans from all others, is the road along which they believe the elevation of the soul and the union with God to take place, namely by means of mathematics. Mathematics formed a part of their religion. Their doctrine proclaims that God has ordered the universe by means of numbers. God is unity, the world is plurality and it consists of contrasting elements. It is harmony which restores unity to the contrasting parts and .which moulds them into a cosmos. Harmony is divine, it consists of numerical ratios. Whosoever acquires full understanding of this 57
number-harmony, he becomes himself divine and im m o rta l,9 Apparently a full history of Greek geometry and astronomy, covering the period to 375 B.C., and including a thorough story of Pythagoras was w ritten by Euderaus, a student of Aristotle, Unfortunately this has been lost, and only a brief account of this history remains through a record written in A.D, 400 by Proclus in his commentary on Euclid, Eves9 states that this commentary of Proclus constitutes the most reliable source of information for the early period of Greek mathematics.
The Pythagorean influence has been felt in all ages since his own. Prom that time on, throughout the records of mathematics history, one reads of Pythagorean doctrines, Pythagorean influences, Pythagorean mathema ticians, Neo-Pythagoreans, and often simply Pythagoreans, For most people the contact is limited to the Pythagorean theorem encountered in geometry,
Pythagoras held a strange mixture of fact and fantasy. He taught that the planets were divine, anima ted, rational beings which were enabled by their own reason to describe eternal circles. His doctrine of the harmony of the celestial spheres was based on the assumption
9B. L, Van der Waerden, oj>, c it., p. 93. a ^Howard Eves, An Introduction to the History of Mathematics (Rinehart and Company, Inc., New York, 1953), p , 53 • 58 that they were separated from each other by intervals corresponding to the relative length of strings combined to produce musical harmony. The soul he regarded as a harmony, chained temporarily to the body as a punishment.
Much stress was placed on a series of contrasts or antitheses, and many of these ideas appeared later in the beliefs of the Gnostics, with which the early
Christians had to contend.
The position which Pythagoras held in the minds of the intellectual world in succeeding generations can be illustrated by a statement from Dantzig:
Among other things, he was credited with being the originator of the heliocentric hypothesis, in spite of the indisputable evidence that this hypo thesis had been first propounded by Aristarchus, a contemporary of Archimedes. The belief that Pythagoras had taught that the earth revolved around the sun persisted even after the contributions of Copernicus had been made public. In fact, in 1635, when Galileo was tried for his heresies, the immortal document of the Inquisition, in listing his errors, called the heliocentric hypothesis a Pythagorean doctrine. This brilliant subterfuge spared the Holy Office the embarrassment of indicting Copernicus, an ordained priest of the Church. 0
Pythagoras had built a beautiful philosophical structure on the belief that everything could be attributed to number. The discovery which humiliated him and spoiled his theories was that the common whole numbers were not
10g>obias D antzig, The Bequest of the Greeks (New York: Charles Scribner’s Sons, 1955), p. 31. sufficient for even the most rudimentary consideration of mathematics found beneath his feet every day. He had attempted to free mathematics from practical application.
His followers had pursued mathematics as a kind of religious contemplation, as a way to approach eternal tr u t h .
He had preached like an inspired prophet that all nature, the entire universe in fact, physical, metaphysical, mental, moral, mathematical—every thing-- is built on the discrete pattern of the integers 1, 2, 3, . . . and is interpretable in terms of these Clod-given bricks alone; God he declared, indeed, _is "number," and by that he meant common whole number. • . One obstinate mathematical discrepancy demolished Pythagoras' discrete philoso phy, mathematics, and metaphysics. . . This was what knocked his theory flat: it is impossible to find two whole numbers such that the square of one of them is equal to twice the square of the other.
The diagonal of any one of the many square t i l e s on the floor of the temple was incommensurable with a side of the same square tile. Pythagoras was confounded to find such an "irrational" idea invading his closed system of thought. Unlike so many of his successors,
Pythagoras, after trying unsuccessfully to keep this discovery a secret, accepted the consequences, and proceoded to erect an even greater ediface.
The diagonal of a u n it square is equal to the square root of two. This diagonal can be constructed
H e . T. B e ll, Ivlen of Iviathematics (ilew York; Simon and Schuster, 1937), p. 21, 60 exactly geometrically, but cannot be measured in any finite number of steps. The square root of two can "be calculated to any required finite number of decimal places by the processes one learns in school, but the decimal never repeats, nor does it ever terminate.
In this discovery Pythagoras found the taproot of modern mathematical analysis. Issues were raised by this simple problem which are not yet disposed, of in a manner satisfactory to all mathematicians. These concern the mat hems::: ioal concepts of the infinite (the unending, the uncountable), lim its, and continuity, concepts which are at the roots of modern analysis. Time after time, the paradoxes and sophisms which crept into mathematics with these apparently indispensable concepts have been regarded as finally eliminated, only to reappear* a generation or two later, changed, but yet the same,12
Today, the approximations of science, and the solution of practical problems to a reasonable degree of accuracy are sufficient for most people. Modern science knows no exact measurement, nor are there perfect in stru ments constructed. The theoretical ideals of pure mathema tics, the perfect models of deductive geometry, like the ideals of the Greek philosophers, are goals which lie beyond apparent reality, but which form the basis Tor the wonders of modern civilization. It is not necessary, for most people, to be shown by deductive reasoning what they are willing to accept as true.
The mathematician’s insistence on finding a deductive proof even when the truth of the conclusion
1SE. T. Bell, op. cit., p. 22. 61
is undoubted has led to the sarcastic comment that reason is the slow and tortuous method by which those who do not know the t r u t h d isco v er it* ^
Plato* In Plato (427-347), Greek philosophy made its nearest approach to Christianity• Church historians have recognized his influence on the intellectual world for centuries. No Greek writer exerted so much influence on the Jewish thought of the last centuries before Christ, or on early and later Christian thought. Eusebius, the father of church history, said that Plato alone of all the Greeks reached the vestibule of truth and stood upon the threshhold.14
Coolidge quotes at some length from Jowett’s translation of the Republic, after which he concludes:
It is perfectly clear from all this that Plato had a very exalted idea of the importance of mathematical study, finding three different reasons for it. First, it Is evident that mathematics has great practical utility. . . Secondly, he believed that mathematics affords valuable mental training, for he asserts that anyone who has studied geometry is infinitely quicker of comprehension than one who has not. , • And lastly, he found the supreme justification of mathematics in this, that it leads to absolute truth, as it exists in th e mind of God. The search f o r th is is the h ig h est object of man’s endeavor,
Plato was not a productive mathematical scholar. But his influence upon other mathematicians has been very
l3lvlorr5s Kline, Mat he mat ics and the Phys leal World (New York: Thomas Y. Crowell Company, 1959’}, p. 18.
l^ A lb ert Nev/man, _op. c i t ., p. 23.
-LSjulian l . Coolidge, The Mat he mat ics of Great Amateurs (New York: Dover Publ'i c a t ion s , I n c ., 1963), p. 4. great. He had the highest opinion of the value of the subject. But the spiritual height of his views, together v/ith his fundamental lack of mathenatical knowledge, led to confused statements in his writings. For example, in the Republic, he speaks of a geometrical number which he calls "The Lord of better and worse births," and many writers have tried to interpret this Platonic number.
Smith^ suggests one theory, that the number 12,960,000, or sixty to the fourth power, was the number. This number played an important part in the mysticism of the
Hindus and Babylonians, and either Pythagoras or Plato might have picked it up in the course of their travels.
Van der Waerden considers the attention of Plato on the problem of the duplication of the cube. This for
Plato v/as the central problem in solid geometry. Van der
Waerden quotes from the Eplnomls, a posthumous work of P lato :
However, what is divine and marvelous for those who understand it, and reflect upon it, is this that through the pov/er, which is constantly whirling about the duplication and through its opposite according to the different proportions, the pattern and type of all nature receives its mark.-^
Van der Waerden continues to discuss the problem of duplication, which includes the operation of doubling numbers, areas, volumes, and ratios, and finally observes
16David E. Smith, History of Mathematics, Vol. I (New York: Dover Publications, Inc.^ 1951), pT 89.
■^Van der Waerden, o£. cit., p. 156. 63
There is in this entire world of ideas an unmistakably mystical element. It is not a sober natural science, which is speaking here, but a mystical surrender to the divine Creator of the disciple who has been initiated into the secrets of numbers and of harmony. Because the human spirit, which has the divine spark within itself, is mystically united with the divine spirit, it penetrates into the marvelous plan on which nature has been constructed. In this mystical region, everything flows together; one does no longer distinguish between musical intervals and the corresponding numerical ratios, nor between geometric and material bodies. The human soul communicates with the divine soul; thinking is no longer human, i t has become d iv in e . A ll of th is is characteristically Pythagorean.18 Allman, quoting from Plutarch, refers to this same problem of duplication. Given two figures, to construct a third which shall be equal to one of the two, and similar to the other. Diogenianus asks the teacher what this problem has to do with the forms of the constituent elements of bodies such as air, earth, fire, and water, which have been described,as'the regular solids. The answer is a mixture of philosophy and mathematics which gives at the same time the lofty thoughts, and mathematical paucity of this great mind;
You will easily know, if you call to mind the division in the Timaeus, which divided into three the things first existing, from which the Universe had its birth; the first of which three we call God (Theos, the arranger), a name most justly deserved; the second we call matter, and the third ideal form. • . God was minded, then, to leave nothing, so far as it could be accomplished, •undefined by limits, if it was capable of being
1 8Loc. c it. 64
defined by limits; but (rather) to adorn nature with proportion, measurement, and number: making some one thing (that is, the universe) out of the material taken all together; something that would be like the ideal form and as big as the matter. So having given himself this problem, when the two were there, he made, and makes, and for ever maintains, a third, viz, the universe which, is equal to the matter, and like the model. ^-9
The regular solids were of cosmic meaning and had theological value to the neo-Platonists. Plato built the world of triangles, using the isosceles right triangle, and the 30-60-90- degree triangle, these being half of the square and half of the equilateral triangle respectively.
With four equilateral triangles he made a tetrahedron, and associated it with fire, with eight equilateral triangles he made a regular octahedron which he assigned to air. Twenty equilateral triangles produced a regular icosahedron which v/as assigned to water. Six squares made a cube, and this hard resistant shape was assigned to earth. The fifth regu lar combination, the dodecahedron, made of twelve hexagons was "used by God in the delineation of the Earth."
Plate may have traveled to Egypt, It is said that he visited there partly for purposes of trade, and chiefly that he might acquire knowledge. Whether directly from
Egypt, or through the Pythagoreans of his own land, he came to appreciate geometry, and Is said to have placed above the entrance to his school of philosophy (the Academy)
•^George J. Allman, Greek Geometry from Thales to Euclid (London: Longmans, Green 1: Co., 18397, P* 30. 65 the words "Let no one ignorant of geometry enter my doors."
He is also reported to have spoken of God as "The Great
Geometer."
The pure mathematician claims no truth at all for the axioms and the theorems of the various geometric systems which he investigates. His only claim is that the theorems logically follow from the axioms. The physicist, who applies a geometric system to physical reality, expects only to establish a very modest practical kind of truth to the system. He is aware that v/ith other suitable substitutions, he might find other systems which would be equally valid, and which he could verify through experi mentation. This view of geometry v/as foreign to Plato, and he scorned those who would thus use geometry for practical purposes. For him, such use v/as the "mere corruption and annihilation of the good of geometry, which was thus turning its back on the unembodied objects of pure intell igence, to recur to sensation, and ask help from matter.
For Plato, the Euclidean geometry v/as not an abstract system, in the modern sense, but a doctrine which is either true or false, the notions of truth and falsehood being understood in an absolute sense. The concepts of Euclidean geometry are for Plato, not variables, but certain properties and relationships intimately related to our sense perception of space. Geometry possesses absolute and exact truth. Although popular geometry, which deals v/ith the world of the senses, contains at most an inferior, approximate
L. Coolidge, o£. c i t ., p. 12. 66
truth, perfect instances of the Euclidean Ideas exist in the ideal realm. 1 To speak of Plato's view of Euclidean geometry is of course an anachronism, for Euclid lived nearly a cen tury later. However, the term carries sufficient meaning for the reader to understand the purpose of the comment.
Euclid. To speak of Euclid at all, from a relig ious point of view, is at best a guess. Smith states that of the life of Euclid nothing definite is known. He was the most successful textbook writer the world has ever known, and his Elements have appeared in over a thousand editions since the invention of printing. No other book, except the Bible, has been distributed so widely, nor transmitted so carefully. It is not so much with his work, but with the effects of it, that this study is co n c e rn e d .
Euclid lived long enough after Thales, Pythagoras, and Plato, for their work in deductive geometry to have
undergone some study. But no one had collected the Ideas
of these men, and organized them into a system of know ledge. This was to be Euclid's contribution to civiliza
tion, and this was to be the only time in history when one
man could successfully Include in his works all the essen
tia l parts of all the mathematical knowledge of his time.
^Anders Wedberg, Plato's Philosophy of Mathematics (Stockholm: Almqvist & Wiksell, 1955), p. 46. 67
He may not have contributed much original content of his own, but he created a pattern of thought, which though challenged, withstood all the challengers for nearly two thousands of years. This pattern continues to be taught in the classrooms of schools in the twentieth century, virtually unchanged. Kline recognises two pre eminent values in Euclidean geometry: The first is the evidence it supplies of the power of human reason to derive new knowledge by deductive reaso n in g . The thousand and more theorems all follow from ten simple axioms. . • The second supreme value of Euclidean geometry is the evidence it proffers of the rational and indeed mathematical design of n a tu re .
It v/as Euclidean geometry which opened man's eyes to the possibility of tracing the design in nature, and understanding it t tiro ugh the means of mathe matics. For two thousand years after Euclid’s time, it had been believed that his plane and solid geometry was more than a body of consistent propositions. It was thought not only that the system was logically sound, but that it v/as an exact account of physical space. Kant, in the eighteenth century, went so far as to declare that
Euclidean geometry is the only description which human beings could give of space, because they are incapable of thinking of space in other than Euclidean terms. But it
op Morris Kline, Mathematics and the Physical World (Hew York: Thomas Y, Crowell Company, 1959) , p . 89. 6 8 is not at all certain that Euclidean geometry is an exact description of physical space. Other geometries, equally valid, have been created. In the past one hundred years, what has amounted to a revolution has taken place in philosophical thought. Scientific thought, and the study of the nature of mathematics itself, has resulted in some different conclusions. Mathematical thought is no longer confined within the restrictions of Euclidean geometry.
Kline paints a dramatic picture of the effect of the discovery of non-Euclidean geometry in the nineteenth century:
The creation of non-Euclidean geometry cut a devasting swath through the realm of truth. Like religion in ancient societies mathematics occupied a revered and unchallenged position in Western thought. In the temple of mathematics reposed all truth, and Euclid was its high priest. But the cult, Its high priest, and all its attendants were stripped of divine sanction by the work of the unholy three: Bolyai, Lobatchevsky, and Riemann. . » In depriving mathematics of its status as a collection of truths, the creation of non-Euclidean geometries robbed man of his most respected truths and perhaps even of the hope of ever attaining certainty about anything.
But this does not make of the deductive geometry of Euclid a barren wasteland, or a false system of thought. It Is still taught In most of the schools of the -world today, and provides the basis for the engineering marvels of man. Man continues to build bridges and houses by the use of circles and parallel lines. To the contemplative
O ~ ^Mathematics in Western Culture, p. 429. 69 philosopher there is yet a source of inspiration in the thought pattern so well illustrated by the geometric m ethods.
Ever since the days of Plato one of the most impressive, persistent, and fruitful of philosophical conceptions has been that of a world of ideas transcending the world of sense. That powerful con ception, because of its sheer ideality, has been specially congenial to the temper and habit of mathe matical minds. Perhaps it was mathematical in origin. At all events, shortly after its rise, it was both greatly clarified and greatly reinforced by Euclid's Elements. For this soundest of all the works that have come down to us f rorti antiquity deals with points, lines, planes, triangles, circles, spheres, cubes, cylinders, cones, and the like; but all such terms denote pure ideas and not concrete things in the world of sight and touch. Herein men have found a deep source of religious consolation, for, because such ideas are wholly unaffected by temporal vicissitudes, one feels in contemplating them that one is literally in touch with things eternal. Thereby too, theology ^as been greatly stimulated, advanced, and fortified. *
Schools. There were several famous schools estab
lished during the years of the Greek period. Pythagoras
settled at Crotona on the southeastern coast of Italy,
P lato e sta b lish e d his Academy a t Athens, and the g re a te s t
mathematical school was the one at Alexandria. This
city, founded by Alexander the Great about 520 B. C.,
was to last for centuries, until its destruction by the
Arabs in A. D. 641. With this school are associated
such names as Eucxid, Archimedes, Apollonius, Erastothenes j
24 Cassius J. Keyser, Mathematics and the Quest ion of Cosmic Mind (New York: Scripta Mat hematic a, 1935)’," "p. 54. 70
Ptolemy, Hero, Diophantus, and a host of others. Not a trace of the famous school, its library, or its museum has been preserved to the present day, but there is no question but that great contributions to civilization were developed there.
There were religious influences in the school at
Alexandria. The Jews shared in the great literary acti vity. It was here that the Septuagint, the Greek version of the Old Testament, was produced. Influential Jews were scattered throughout the civilized world, and their com.iercial importance generally recognized. The record in the Bible^S 0f the many p laces from which the Jews had come to Jerusalem for the day of Pentecost, indicates how widespread was their settlement.
The passing of hundreds of years has always resulted in cultural and political changes. But much of man's learning has been kept in trust by scholars of each generation, and handed down to succeeding generations.
Mathematicians have appeared in every age since the time of Euclid, to continue his ideas, to discuss them, and to pass them on. It is not possible to include all such mathematicians in this study. However, mention should be made of Nicomachus, who was born not far from Jerusalem about the end of the first century. He wrote detailed
25Acts 2:9, 10. commentaries on a rith m e tic , which, were d estin ed to be used for centuries; for his Greek arithmetical treatise, translated into Latin by Boethius, continued in active use well into the 17th century, and was used by European church schools almost exclusively for the subject of arithmetic in the 10th, 11th, and the 12th centuries. Two distince types of speculative arithmetic were current, the Boethian, and a mystical arithmetic involving contemplation of the numbers appearing in the Bible.
By the second century after Christ, the world was very different from what it had been at the estab lishment of the school at Alexandria. The world was
Roman politically speaking, although the intellectual ideas and the languages of the sciences were still predominately Greek. Christianity had been taught for some time, and had prospered, but its influence had been almost unfelt in the political and scientific w orlds.
Ptolemy. One of the great scientists of history was the early astronomer, Ptolemy. lie left no biography
It is not certain exactly where or when he lived and died, but it must have been during the second century
A. D. Astrology was the scientific passion of his day.
Men who were scientifically minded, but still loyal to pagan traditions were seeking to replace the ancient
2GLouis C. K arpinski, The H istory of A rithm etic (Chicago: Rand Mclially & Co., 1925),” p. 62. 72 mythology with something scientific, and they settled
on astrology. Stemming from Greek astronomy and Chaldean astrology, it was a compromise between the popular religion and monotheism; the concept of sidereal immortality which it fostered reconciled astronomy with religion; it was a kind of scientific pantheism indorsed by men of science as well as by philosophers, especially by neo-Platonists and Stoics.27
Astrology claimed to be both science and religion
at the same time, a mixture which has been the source of
conflict incapable of being resolved satisfactorily in
the best of climates.
Ptolemy is known for two great works, the Tetra-
biblios and the Almagest. The Tetrabiblios was a compila
tion of Chaldean, Egyptian, and Greek folklore. It was so
complete and well organized that it has remained as a
standard work of astrology to the present day •
The Almagest was called the Mathematical Synthesis
by Ptolemy. The Arabs called it the Almagest, which means
the Very Greatest. This book has been translated into
many languages. It is noted in every geography and science
classroom as the basis for the geo-centric theory. It is
almost unbelievable that it was not translated into English
until 1952, when Catasby Taliaferro did so for inclusion in
the Great Books of the Western World.
27George Sarton, Ancient Science and Modern Civilization (Lincoln: University of Nebraska Press, 1954), p. 59. 73
The Almagest was a mathematical explanation of the then-known facts of astronomy. It included the astronomical knowledge available about A.D. 150, and describes the geo centric system. This is the explanation of the solar system
centered around the earth.
Ptolemy had rejected the ideas of Aristarchus, who anticipated Copernicus, and had accepted those ideas
of Hipparchus, because the latter agreed betteu with the
observations which Ptolemy had. Better agreement with
the hello-centric ideas of Aristarchus and Copernicus
was not possible until fourteen hundred years later,
when Kepler used the observations of Tycho Brahe, and replaced the circular trajectories of the planets with
elliptical ones. It would not have been sufficient to
place the sun in the center, and continue to imagine that
the planets revolved about In circles.
Geocentricism is a theory with which all of the appearances then observed were consistent. In this sense, we might, if we so chose, call geocentricism ’’true" for Ptolemy. This ’’truth" is susceptible of "confirmation," whenever we succeed in demonstrating appearances by means of the th e o ry . .'.e can become certain of the theory not - as a physical fact, but as a consistent interpretation of the appearances, at le a s t of the appearances known to Ptolemy. Indeed, even today, one might get a consistent interpretation of the appearances, however complex, on a geocentric basis; but it is no longer worth the effort.28
2&Joseph Sikora, "Geocentricism in the Syntax is Hathematica," The New Scholasticism (Washington, D. C.: Catholic University of America, XXXII, January 1958), p. 72. 74
The difficulties with the Ptolemaic theories were evident almost from the beginning, and they continued to increase throughout the middle ages. Astronomers, whether
Jewish, Christian, or Muslim, attempted to reconcile observations with Ptolemy, and found disagreements between facts and theory. Additional epicycles had to be chosen to fit the theory to the facts.
Ilot only was the p ic tu re d is to r te d by having placed the earth in the center, and by having the other planets revolve in circular orbits rather than elliptical ones, but the Almagest also held up the advance of science by the use of sexagesimal fractions. Even after decimal numbers were well established, decimal fractions were discouraged, and it was not until 1585 that the superior ity of decimal fractions over common fractions was clearly explained by Simon Stevin. This struggle is not yet over, and evidences of i t may be found in the homework of the fifth grade children in any American school.
This study is concerned with Ptolemy, as with
Euclid, not so much for his mathematical content, but for the effect which his work had upon the religious world for such a long period of time.
Ptolemy told the public what it wanted to believe. The public believed it. The evidence was before their eyes. They could see the sun climb up in the morning and the stars sweep across the sky at night. To them this evidence was final. Besides it was what they wanted to believe. It tickled their vanity. This 75
world of th e ir s was f a r and away the g r e a te s t and most important thing in the universe, and to that thought they clung. Christianity swept in. Its teachings meshed perfectly with the Ptolemaic theory. Man and his home, the world, were the results of direct, special creation at the hands and mind of the Ruler of the Universe. It was all very simple and clear. Man was exalted by his own interpretation of nature.29
Education and the church. The church flourished
on this idea for fourteen centuries, but the advance of
science and learning was slowed. One might ask why most of the men of sciece were pagans after three_or four centuries
of missionary efforts? It was partially because the schools
were pagan, restricted to neo-Platonism, and to various forms of mysticism. Science was largely a study of
astrological delusion and superstition which had nothing
to offer the Christian.
This was much too learned and too objective for the common man and woman who craved a liv in g f a i t h and a religion which was personal, emotional, and colorful. Those cravings were satisfied in varying degrees by a number of oriental religions, of which Christianity was for a long time the least conspi- cious. The development of Christianity, early and late, is one of the mysteries of the world; it is the sacred mystery- in the highest sense. The events which guided the Church and caused its final triumph in the face of innumerable calamities are so incredible, or call them miraculous, that Christian spologists have used them as clinching proofs of the truth and super iority of their faith.
^9Crove Wilson, The Human Side of Science (New York: Cosmopolitan Book C orporation, 19291', P« S3*
•^George Sarton, op. cit., p. 96. 76
After the Empire had become nominally Christian, about 315, the pagan schools continued to function at
Athens and Alexandria. The Christians did organize schools
of their own, and it is said that in the school of
Constantinople the mathematicians were probably Christians.
In the second century, Origen, one of the church leaders
of the day, and head of the catechistic school of Alexan
dria at the age of eighteen Put his pupils through a severe scientific training, especially in geometry, and led them by th is means to a study of the m ysteries of the f a i t h .
Origen was the creator of a theology which followed
the ideas of Plato, and rested consciously upon contempor
ary science. He was the most learned man in the early
church, and wrote widely. He attempted to reconcile the
Christian faith, and science, and from the view of Christian
ity, originated many of the ideas which caused the doctrinal
controversies of the following centuries. The faithful of the first Christian congregations were concerned primarily with the propagation of the Gospel, with the spreading of the revelation of their faith, with the freedom of worship, and with the moral rebirth of man. This mission made them enemies of '’pagan” learning. Tertullian (160-222) regarded philosophy as the real fountainhead of all heresy, and combatted worldly knowledge as folly before God.3-"
3-**An Outline of Christianity, Vol. II, ’’The Builders of the Church” (Hew York: Bethlehem Publishers, Inc., 1926), p. 58.
Joseph E. Hofmann, The History of Mathematics (New York: Philosophical Library, 1957), p. 46. \ 77 This attitude was partially due to the interpreta tion of the passages of the gospels which made the early Christians expect the soon end of the world. When they
found that the existing world, with all its weaknesses
and strange properties, must be conquered for the Gospel,
and that they were to live in it, the attitude of some of
them changed.
Along with Origen, they found a new aim, which was
to build scientific foundations under the faith. Science
was recognized as having a measure of truth, if it were
still imperfect in comparison with the revelations of their faith. Origen,
Like Plato, regarded geometry as the methodical p ro to ty p e. The same view was advocated in t h e e a s t by Eusebius of Caesarea (265-339) and Gregory of Nyssa (335-595), and in the West by Marius Victorinus in the fourth century, who was the capable translator of numerous neo-Platonistic writings, and of a few treatises of Aristotle in logic, and above all by his great pupil, Augustine (354-450), bishop of Hippo. They were all filled with a conviction of the profound significance of matheiiiatical insight and were devoted to math.matleal studies, even if they were not active themselves as discoverers, commentators, or educators in this domain.^
In the second century, also, Justin Martyr, after
searching for truth in the several philosophies of his
day, was asked, when he ap p lied to a Pythagorean,
If he had studied music, astronomy, and geometry, not as we might expect for any practical purpose but because these studies "wean the soul from sensible
®®Loc. c i t . 78
objects and surrender it fit for that which appertains t o th e m i n d . 1*3 4
After this rebuff, he went to a learned Platonlst and studied Platonic philosophy for a time. Shortly after this, he met an aged Christian, and was converted to
Christianity. Many of the Christians had been embittered by persecution at the hands of the Romans, and had turned against all pagan learning. As soon as Christianity was legalized, these Christians attempted to forbid all pagan learning. As the Church grew stronger, it condemned and attacked such secular activity. Some of the Church fathers ridiculed astronomy and physical science, and claimed the
Bible as the source of all knowledge, including scientific knowledge. St. Cyril, who became bishop of Alexandria in
412, decided to put an end to all pagan and Jewish learn ing. It was during his rule that Hypatia, one of the earliest known women mathematicians, was murdered. She had been carefully taught by her father, and was recog nized as a scholar. She attempted to revive Platonism, and thought reason sufficient for the solution of all problems. She was seized by a Christian crowd, and taken to a church, where it was hoped she would recant. The mob got out of control, and she was murdered.
^ O u tlin e o f C h r i s t i a n i t y (New Y ork: B ethlehem Publishers, Inc., 1926), p. 45. 7 9 In 529 A.D. the'Eastern Roman emperor Justinian closed all Greek schools of philosophy at Athens, includ ing the Academy, The destruction of what remained at
Alexandria, both Christian and pagan, was completed by the
Mohammedans in 641 A.D. They argued that if the books contained anything which was contrary to the teaching of
Mohammed, the books were wrong. If they were in agreement with his teaching, they were superf luoiis. Y/ith such specious argument, the dusk of twilight settled upon learning in the schools of Greece.
The persecutions which the Christians practiced against pagan learning may not be justifiable, but from their point of view, they were being faithful to the purposes for which they were upon the earth. The fight for the truth, against darkness, has led into some very strange directions throughout history. The record of the Crusades, the Inquisition, the witch-hunts, and the many persecutions and counter-persecutions occupies a large part of the history- of mankind. Christianity has been a minority struggle against the world since its beginning.
There arose in the next few centuries intense theological debate and activity within the church, because of great controversy over heresies. The various kinds of
Christians held more enmity against each other than they did against the infidel. Refuges from the various perse cutions against the Arians, hestorians, and the Eutychians, 80 carried Greek science to the East, and helped in no small measure to prepare intellectual weapons outside the
Christian world, weapons which would in some cases, soon be used against it. Intolerance and persecution are self-defeating.
A re c e n t example is th a t of H itle r d riv in g from Germany the very scientists who later developed the weapons which wouM have made Germany n early in v in c ib le . The hunger for knowledge and the search for truth cannot be eradicated by imperial decree. Refugees carry the wisdom and know ledge to another place, and mankind continues to go on. Strangely, the Christian persecutions can be credited in no small measure with the preservation of learning through the ever approaching dark ages.
Greek scholars were driven out of the Greek world and helped to develop Arabic science. Later the Arabic writing was translated into Latin, into Hebrew, and into our own vernaculars. The treasure of Greek science, most of it at least, came to us through that immense detour. We should be greatful not only to the inventors, but also to all the men through whose courage and obstinacy the ancient treasure finally reached us and helped to make us what we are.^5
Summary. There were fo u r g re a t influences which
summarize for this age the religious bearings in the
development of mathematics:
First, the number mysticism of Pythagoras. This was elaborated upon, and provided a system of mystical
^ E ^George Sarton, op. cit., p. 1 1 1 . 81 numerological investigation which continued to run parallel with the development of scientific mathematics to the present century. That this number symbolism did not corrupt the work of the early church is a testimony to the marked simplicity of the original Christian faith,
Christianity v/as interested in a personality, rather than a principle, and it was only after the writings of Origen and Augustine that the medieval number philosophy recieved
its religious blessing. Its detailed complexity does have
some fascination, but cannot be elaborated upon here.
Second, the perfection of Euclid’s thought pattern and content. The method provided the Inspiration for the more geometrico of Descartes and Spinoza, and for Leibniz'
expectation that he could find a perfect presentation of
the Gospel on logical grounds. The "absolute truth" of
Euclid became the model for all truth, and the structure
of the Declaration of Independence is a good indication of its Influence Into modern times.
Third, the Ptolemaic interpretation of the solar
system. This appealed to the religious mind, since it placed man at the center of God’s interest and intent.
It became so complex that when it was explained to
Alphonso X in 1252, he exclaimed, "Had God consulted him at the creation, the universe would have been on a better
and simpler plan.1" The hold which this theory of the 82 solar system had on the Church can be indicated by the persecution which accompanied attempts to replace it by the Copernican theory. The book of Copernicus was not removed from the Index of prohibited books of the Catholic church "until 1822. Fourth, the violent persecutions of the early centuries. These persecutions of Christians, Jews, and pagans, and the consequent scattering of the educated and intellectual scholars throughout the world had the effect of conserving the knowledge which had been accumu la te d . A ll was not lo s t in th e clo sin g of the Academy and the burning of the school at Alexandria. The book of the Acts in the Bible records that "Therefore they that were scattered abroad went everywhere preaching the word,"^° The same observation can be applied to the secular knowledge of the day.
^ A c ts 8:4 CHAPTER IV
THE MIDDLE AGES
This period included approximately one thousand years. It was a period of great events, and of transi tion. The first half of this time was so lacking in what is called progress that it has been referred to as the Dark Ages. During these centuries, the Mohammedans rose to power, and dominated much of the civilized world, only to fall back into relative obscurity. It was the age of the Crusades, the Renaissance, and the Reformation.
This period of time saw the Invention of printing and the development of paper-making, the rise of the universities, and the opportunity for civilisation to advance once more after a long recess.
For the development of mathematics, one can say that little was accomplished in the West during the time covered by the Dark Ages, but much that had been -previously developed was preserved and transmitted. Basically unchanged, it passed through the difficulties of transla tion from one language to another, and on to another. As for the religious bearings in the development of mathema tics during this period, one must admit that religious
S3 84 influences helped to slow down the advance of learning at the beginning of the period. But there were religious influences which helped in the conservation of the mater ial and method, and religious ideas played a large part in helping to bring about the revival of learning after the Dark Ages were over.
After the capture of Alexandria by the Arabs in
641, some of the teachers went to Constantinople, As a result of the collapse of the Vi/'estern Empire, the West
lost the knowledge of the Greek language. The principal writings of Greek philosophy and mathematics had never been translated into Latin. But in the Eastern Empire,
the scientific tradition survived. Greek science was
combined with other fundamentals of knowledge in Persia,
India, and Mesopotamia, and led to the development of a
study of computation by figures, and the development of
trigonometry.
Very little else of any great importance was
added to the Greek works, nor are the names of any of
these Byzantine scholars of great importance in the
history of mathematics. Constantinople was captured by
the Turks in 1453, and any evidence of a Greek school of
mathematics there disappeared. This time many of the
scholars took refuge in Italy, and the books they brought back to Europe came as a revelation of new knowledge. 35
These Greek reproductions were joined with the transla tions which were then being made from the Arabic language of the work of Euclid, Ptolemy, and other earlier writers.
There is some evidence in the fifth century of contact between China and the rest of the world. The
Buddhist missionaries and pilgrims from India led to the translation of an arithmetic and some of the astronomical works of the Brahmans, which stimulated the activity of Chinese scholars in these fields.
Since religion was closely related to astronomy, and astronomy to mathematics, the influence of this interchange of religious thought must have been stimulating to the science of China,
Mathematical activity of a creative type continued in the East, In 662, Severus Sebokht, a Syrian Bishop, writing in a monastery on the Euphrates, referred to the
"Science of the Hindus. . . and of the easy method of their calculations, and of their computation, which surpasses words. I mean that made with nine symbols,"2
He stated further,
If those who believe because they speak Greek, they have reached the limits of science, should know these things, they would be convince^ that there are also others who know something.0
^David E. Smith, History of Mathematics (New York: Dover Publications, Inc."^ 1951), p." 143.
2E. R. Turner, "The Hindu-Arabic Numerals," Popular Science Monthly, LXXXI (December 1912), p. 609.
^Joseph Needham, Science and C iv iliz a tio n in China (Cambridge: University Press, 1959), p. 149. 86
Although the discovery of the alphabetic principle in ancient Y/est Asia gave convenience to the writing of language all over the world, there had been an unfortunate application of that principle to numerical notation. The temptation to use all the letters of the alphabet and not stop with the first nine ]sd to the strangulation of mathematical computation. One result was the elaborate numerological symbolism which accom panied the Greek and Hebrew languages for hundreds of years. Strangely, it was in a non-alphabetic culture that the earliest form of a decimal place-value system was developed. Y/ithout this system, which is almost universally adopted among the nations of the world, the unified modern world of science would have been impossible.
The Bishop mentioned only nine signs, indicating that he must have been in contact with those who used empty spaces in computing, and did not yet use the zero symbol. Among the Christian peoples of Europe there was no widespread use of the ten symbols as they were developed by the Hindus, until the Christians found them in Spain.
The earliest date of their introduction from Spain is thought to be that of Gerbert, who as Sylvester II was
Pope from 999 to 1003. He brought back the Hindu-Arabic numerals from the Saracens among whom he had studied. 87
For several centuries in the West, the only places where studying took place were the Benedictine monasteries. The mathematics of modern Europe can be traced back to the teaching given in these monasteries, and later in the cathedral schools. This teaching was largely based on the mathematics derived from Roman sources, and this in turn had been based on the teaching received from the Greeks.
During the latter half of the eighth century Charles the Great commanded schools in cathedrals and monasteries. Probably we shall be correct in saying that such mathematics as were taught in one of these schools did not go beyond the geometry of Boethius, the use of the abacus and multiplication table, and possibly the arithmetic of Boethius; while except there or in a Benedictine cloister it was hardly possible to get opportunity for study."
The mathematics taught was limited to the method
of keeping accounts, and a knowledge of the rule by which the dato of Easter could be determined. The monk had
renounced the world, and there was no real reason why
he should learn more of science than was required for
the services of his church and monastery.
Benedict had become disappointed with his life,
and had become a herm it. A fter th ree years, he re a liz e d
that the negative approach to existence, with its ascetic
suppression of productive powers, was not logical for
him. He began to attract followers, and he gathered them
~W. W. R. Ball, A Primer of the History of hathema- tics (London: Macmillan and Co., Ltd., 1906), p. 97. 88
Into groups of twelve. Ills rules were eight hours of work, eight hours of study, and eight hours of sleep per day. Prom each of these periods, one-half hour was allowed for silent prayer and adoration. He gathered a g re a t lib r a r y , and founded the m onastery a t Monte Cassino in 529. By the thirteenth century, there were about 1300
Benedictine abbeys.
St. Augustine had said that at least one mathema tician was necessary for each monastery for the express purpose of calculating the church year, and maintaining the calendar. But in the early yearsof the Benedictine order there was little time for leisure, and no mathemati cian of any importance appears in the history. The first such person is Venerable Bede (673-735). Bede wrote on the calendar and the calculation necessary for its use.
He was the first scholar whose mathematics appeared to be i n t e l l e c t u a l l y p repared. Some h is to ria n s th in k of him less as a monk than a mathematician. But his mathematics needed only include the simple operations of addition, subtraction, multiplication, and division. He gave a description of finger notation, and was interested in ancient number theory, but his greatest contribution was his compiling of the works of older authors.
In the year in which Bede died, Alcuin (735-804) was born. He spent some time In the court of Charlemagne 89 as a teacher, and together with Charlemagne attempted an ambitious project of education of the common people.
All of the liberal arts were to be taught there, but in such a way as that each should bear reference to religion, for this was regarded as the final end of all learning. . . In short, the thought of both of the king and of th e sc h o lar who labored w ith him was to refer all things to religion, nothing being considered as truly useful which did not bear some relation to th a t en d .5 Alcuin also made a collection of mathematical puzzles and riddles, many of which are used today. His greatest student was Rabanus, who was both a theologian and a m athem atician.
The outstanding mathematician of the Benedictine orJ.er was Cferbert. He went to Spain to complete his education in mathematics and science. He has been called
"The first mind of his time, its greatest teacher, and most universal scholar. . . He aroused interest in mathematics because of his own interest in the subject.
He developed an abacus in which he used counters with the
Hindu-Arabic symbols written on them. The ninth century was the period when Greek mathematical works were translated into Arabic. The tw e lfth century is known fo r the tr a n s la tio n from Arabic into Latin. Much of this work was done by the Benedictine
5Augusta T. Drane, Christian Schools and Scholars (New York: G. E. Stechert & Co., 1910), p. 120.
5IIenry Taylor, The Mediaeval Mind (Cambridge: Harvard University Press, 1914), pT 224. 90 monks, among whom was Adelard of Bath, who is perhaps the best known translator of England.
He studied at Toledo, Tours, and Laon, and com pleted his science education by travel in Spain, Italy, Horth Africa, Asia Minor, Egypt, and possibly Arabia, where he may have become acquainted with Arabian scholarship, Cn his travels he collected numerous mathematics manuscripts. He was one of the first to translate Euclid into Latin, perhaps from Arabic, although since he knew Greek, he might have worked with the original.''' The original contributions of these monks to the development of mathematics were few. But they made a real contribution by preserving and spreading; knowledge. They kept alive an interest in mathematics by copying manuscripts, by careful treatment of the manuscripts, and by their teaching. Their contribution was not limited to mathematics, but included the translation of literary works as w ell.
Cassiodorus, a man who had served in the government of Emperor Theodoric, joined the monastery a t about seventy years of age, and lived to be more than ninety, lie thought the monks should -work more than eight hours a day, at least until t ’ey had copied the great works of literature so that every monastery would have a library equal to that at Monte Cassino. He traveled from monas tery to monastery encouraging the translators and writers,
^Sister Joanne Huggle, "Benedictine Contributors to Mathematics, from the 6th to the loth Centuries," American Benedictine Review IV (Spring 1953), p. 44. 91
Cassiodorus is also the one who set the bounds on the liberal arts at the number seven, because of the seven pillars of wisdom mentioned in the Bible: ’’Wisdom hath builded her house, she hath hewn out her seven pillars."® These seven areas provided the education for a free man. They were divided into two groups: the quadrivium or Pythagorean group (arithm etic, geometry, spherics, and music); and the trivium (grammar, dialectics, and rhetoric.) According to Ball® students entered the medieval school when quite young, and spent four years in the study of the trivium* Many went no farther. The title of bachelor of arts was conferred upon completion of this course. Those who went on, studied the quadrivium for an additional three years, and received the master of arts degree, which was simply a license to teach.
The cathedrals, which were constructed by the monks, indicate a depth of mathematical skill and an understanding of geometric principles. The construction of towers, belfries, and arches not only exhibit a high degree of engineering knowledge and skill, but mathematics is shown to be the servant of the arts.
®Proverbs 9:1
9\V. W. R. Ball, A Short Account of the History of Mathematics (Dover Publications, Inc., I§60), p. 14l. 92
The Order of St. Benedict had almost made a monopoly of the exact sciences. It was there where were formed the able architects and ecclesiastical engineers who erected so many magnificent edifices throughout Europe. The cathedral and monastic schools of the eleventh and twelfth centuries became the forerunners of the medieval universities.
In 622, Mohammed fled to Medina, and returned to
Mecca triu m p h an tly . Then began an e n tir e ly new p erio d in the history of the culture of Europe. The cultures of the Byzantine and Persian empires were not destroyed, but absorbed. In 635', the Caliph Omar moved the seat of his government to Damascus, and made plans to conserve the knowledge which he found there.
Damascus was a ce n te r of G-raeco-Roman and of Semitic culture. The law gave religious freedom to Jews and to Christians because they believed in one God. The pagans had to be converted. Arabic, the sacred language, became the world language of the Moslem empire. Tolerance, wise government, concern for safety and welfare, the nurture of the arts and the sciences, these are the characteristics of the Arab dominion. In 145 after Hejira (766 A.D.), the Caliph Al-Mansur erects the fabled city of Baghdad not far from the ruins of Selucia and of Babylon. Princely stipends attract Jewish, Syrian, and Persian scholars and artists.^--*- When theArab conquerors settled In cities, they became su b ject to d iseases unknown to them in the d e s e rt.
■^Paul La Croix, Science and Literature in the Mid dle Ages (London: Bickers & Sons, 1878), p^ 82.
■^B. L. Van der Waerden, Science Awakening (Oxford University Press, 1961), p. 57. 93
They contacted Greek doctors. The study of medicine had been largely restricted to Greeks and Jews, and the Arabs
likely contacted Greek learning through these doctors, and encouraged them to settle in Damascus and Bagdad.
The Arabs demanded the Greek works on medicine,
and became interested in the other sciences. Missions
were sent to Constantinople and to India for the purpose
of bringing back whatever manuscripts could be found. By
the ninth century translations of Euclid, Apollonius,
Ptolemy, and others had been made, and some of these
translations are the oldest copies in existence today
of these works. The Arabs, though quick to appreciate
the work of others, did not attempt systematically to
develop the subject of mathematics beyond what they found.
The Moors had e sta b lish e d Moslem c u ltu re and r u le
in Spain in the eighth century, and by the tenth century
had attained a high degree of civilisation in that place. The moving of the center of this Moslem culture from
Bagdad to the institutions of higher learning in Spain,
had great consequences for the development of the intel lectually dormant Europe. It was now geogrjp hically easy
for the results of Hindu and Chinese mathematics to make
their way into Western Christendom. There was some
attempt made to keep this knowledge from the Christians in 94
Europe, but during the twelfth century copies of the books used in Spain were taken to Europe, An English monk,
Adelhard of Bath, under the disguise of a Mohammedan student, attended some lectures at Cordova about 1120, and obtained a copy of Euclid’s Elements, This copy, trans lated into Latin, was the foundation of all editions known in Europe u n til 1553, when the Greek te x t was recovered.
The work of translating the Arabic texts into
Latin, was undertaken by monks, Jewish physicians and merchants who studied in the Moorish universities in
Spain, and by the thirteenth century, Latin translations of the works of Euclid, Ptolemy, and other Greek authors were as widely available as could be possible before printing from movable type began. By the end of the fourteenth century, the first five books of Euclid were a part of the curriculum at many of the European univer- s i t ie s •
Another of the scholars who helped introduce
Moorish learning into Europe was Abraham Ben Ezra, who lived in the twelfth century. He was a distinguished
Jewish rabbi, and a doctor. He wrote an arithmetic in vh ich he explained the system of numeration with nine
symbols and zero, and explained the fundamental processes
of arithmetic. He was also interested in astrology,
especially in finding the number of ways in which planets 95 could come into conjunction, as this had an important bearing upon astrological predictions.
The astrologer believed the destiny of countries, nations, and individuals to be indicated in the heavens by the various positions of the planets. Two planets meeting at the same place form a conjunction and exert a special significance on the development of events, the significance varying with the indivi dual stars. A conjunction of three planets has a greater influence than two, etc. The most dreaded conjunction is of course the one of all seven planets referred to in an early Hindu tradition, and was expected to return in 26,000 years.12
The Jewish people during the Middle Ages, not
only supplied the doctors and mathematicians, and as such conserved the intellectual writings of their own, and of
other generations. They also continued the number symbo lism of the earlier ages, as used by the Babylonians, and
strengthened and elaborated by Pythagoras. The Talmud
contains a great many numerical references, many of which are attempts to explain the details of the Scripture, or to clarify some part of the ceremonial life of the people. One example of the use of mathematics in the
Talmud is this discussion of the length of a ladder necessary to scale a wall ten handbreadths in height.
The first answer is the length of the diagonal of a square
ten handbreadths on each side. Of course a shorter ladder
could be used to scale the wall, as it did not need to
•^Jekuthiel Ginsberg, "Rabbi Ben Ezra on Permuta tions and Combinations," Mathematics Teacher XV (October 1922), p. 350. 96 be at a forty-five degree angle with the wall, nor did it need to reach the top of the wall.
Rabbi Judah citing Samuel ruled: A wall ten handbreadths high requires a ladder of fourteen handbreadths in length to render it permissible for use. R. Joseph ruled: Even a ladder of thirteen handbreadths and a fraction is sufficient. Abaye ruled: even one of eleven handbreadths and a fraction suffices. R. Huna, son of R. Joshua ruled: Even one of seven handbreadths and a fraction suffices.13
There are many such illustrations which could be cited which concern mathematics in the Talmud.
It was in fact, impossible to understand certain parts of the Talmud as the students in great contentional yeshibas did, without a considerable knowledge of mathematical principles, and it is instructive that in the seventeenth century, we find appended to legal decisions of a German Rabbi, a list of propositions of Euclid needed for the elucidation of the Law.** Other scholars, both Jewish and Christian, contri buted to the translation and spread of knowledge. Emperor
Frederick II helped to spread the knowledge of Arab mathematics throughout western Europe. As the Jewish physicians were tolerated in Spain because of their medical skill and scientific knowledge, Frederick engaged a staff of learned Jews to obtain copies of the Arabic works and Greek manuscripts, and translate them.
■^Rev. Dr. Israel W. Slotki, The Talmud: Erubin (London: The Soncino Press, 1938), p. "543.
14Israel Abrahams, Jewish Life in the Middle Ages (Philadelphia: The Jewish Publication Society of America, 1911), p. 368. 97
I t was by th is means th a t w estern Europe was able once more to assist in the development of science and mathe m atics • By the end of the thirteenth century translations of the works of Euclid, Archimedes, Apollonius, Ptolemy, and of several Arab authors were obtainable from this source. Prom this time then we may say that the development of science in Europe was independent of the aid of the Arab schools.
One interesting sidelight to this general march
of mathematics progress through translation was the contribution of Dionysius Exiguus to the calendar. In
A.D. 525, at the req u est of Pope S t. John I, he prepared
a calendar which marked the birth of Christ as the begin
ning of the new era. He discarded the Alexandrian era of
Diocletian, reckoned from A.D. 284, on the ground that he
"did not wish to perpetuate the name of the great persecu
tor, but rather to number the years from the Incarnation
of Our Lord Jesus Christ, Dionysius did not compute as accurately as he might have, and he reckoned the birth
of of Jesus to be in the year 754 of the founding of Rome
(A.U.C.), although early Christians had placed it in the
year 750. The Gospels state that Christiwas born under
Herod the Great, and the nativity is usually chosen as
occurring about 4 B.C., according to Dionysius* account.
^5W. W. R. Ball, A Primer of the History of Mathematics (London: Macmillan and Co., Ltd., 1906), p. 47.
■^"Chronology," Encyclopedia Britannica, Vol. 5, p. 528. 98
This calendar was adopted in Rome in the sixth century, but the Pope did not adopt it until the tenth century, and other parts of Europe delayed adoption until the eleventh century. For a short time in France in the eleventh century there was an attempt to establish a calendar based on the Era of Passion. This would have dated events from thirty-three years after the Incar n a tio n , but th is movement was s h o rt-liv e d .
Europe itself had contributed nothing during the
early days of the dark ages. The Germanic tribes of
central Europe and the Gauls of Western Europe were
actually barbarians with a very primitive civilization.
When the Roman Empire had collapsed, the Church had
taken on the task of civilizing and converting the bar
barians. Since the Church had just been successful in its fight against the pagan Greek learning, and the
barbarians were not educated, mathematics and science
:'.fere practically unknown in Europe until about the beginning of the tv.'elfth* century.
Bede had provided a course of study for priests
in which arithmetic was given a proper place. Alcuin in the eighth century together with Charles the Great des ignated
Arithmetic as one of the subjects to be taught to children in the schools attached to religious foundations, and in such a way some instruction in 99
elementary arithmetic was given to children in many parts of Europe from the ninth to the fifteenth c e n tu r ie s . In spite of this beginning, practical arithmetic from the tv/elfth to the sixteenth centuries was commonly taught by laymen o u tsid e the sch o o ls, in a way similar to modern instruction in music and dancing,
There were indications of the type of thinking
which would develop during the seventeenth century,
Anselm of Canterbury and Peter Abelard at the beginning
of the twelfth century developed a clear technique which
* became the forerunner of the Scholastic method of inquiry,
Alanus ab Insulis attempted to present an axiomatical ly
constructed picture of theology on the basis of mathemati
cal deductive methods, anticipating Descartes and Spinoza,
Hugo of St. Victor believed that unshakable truths can
be won only by the application of reason, and that exper ience is subject fundamentally to the possibility of
delusion. Roger Bacon, on the other hand, believed that
there is but one ultimate test for knowledge, and that is
experience. He recognised only one way of organizing such
knowledge into a science, namely by showing Its conformity
to the laws of mathematics.
But these were just the first movements, and a
far cry from the combined voices which would be heard a
few centuries later. Education was still confined to the
-^Louis C. Karpinski, The History of Lathematics (Chicago: Hand L'clially & Co., 1925), pT 170. 100 church, and given over almost completely to the under standing of the Bible, as fostered, expounded, and dictated by the church fathers. The church continued to fight against the reading of Greek and Latin master pieces which were full of mythology and pagan teaching. Philosophy and literature and science were of small impor tance 'compared to the salvation of the soul. There was no reason to study natural phenomena ■until the nature of God, and the relation of the human soul to God was understood.
All intellectual interests were absorbed into theological questions. A teleological viewpoint was generally held, nature was placed here for man, and the earth was at the center of the universe because man was the most important creation of God. Teachings not directly ecclesiastical were hereticd. and suppressed.
The little mathematics which was kept alive was for the purpose of keeping an accurate calendar, and for maintaining the church fasts and festivals. This required only a little arithmetic, geometry, and astronomy, and was carried out by the monks because they had the most educa tion. In some places they had accepted Plato’s dictum that the study of mathematics trained the mind for philoso phy, by replacing the v/ord philosophy with the word theology.
But the study of mat hem t ics was not fo r the purpose of developing or learning mathematics. It was to help develop 101 the mind so that it oould reason more efficiently about th eology. By the twelfth century, Europeans were trading freely with the Arabs. The acquisition of culture had not been a part of the program of the Crusaders. But, there can be no doubt that the Crusades did help spread culture throughout Europe. For three centuries the
Christian powers had attempted to impose their culture upon the hoslems. But the net result had been that the culture, and the learning of the Arabs had slowly and surely penetrated into Europe. The learning from the Arabs was largely responsible for the revival of European learning which was to start in Italy a few years later.
The Crusades were a mixture of religious fervor and terror. Yet they did accomplish some far reaching, though unexpressed, goals. They broke up feudalism, and led people to be bound by the common law, and they brought new sciences into Europe. It may be that the
Crusades saved France, Germany, the Scandanavian countries, and B r ita in from Mohammedanism. For w ithout the constant attention and force needed in the Holy Land, the Mohammedans might have overrun Western Europe.
One of the problems which was faced by the leaders of the Crusades was that of an army of occupation. Pierre
Dubois had a Scheme of Education prepared in 1309 which 102 included a plan to educate certain specially selected young people to fit them to hold the Holy Land and to convert the oriental peoples. These young people were to be educated in Latin, Greek, Arabic, and other languages, logic, Bible, medicine, surgery, and Moreover in mathemutical sciences on account of their many utilities, especially touched upon in the little book Super Utilitatibus made by brother Roger Bacon of the order of Minorities. It will be advisable to instruct some disciples of this found a tio n , as they s h a ll appear to show in te llig e n c e , skill and speed therein, but rather dwelling on those matters which may be of service in t aking and keeping the Holy Land. It is especially desirable that every catholic should know written figures, the situation and places of the elements, their magnitudes and shapes; the thickness of the celestial orbs, their magnitude; the velocity, motion, and influences of sun, moon, and other s ta r s : and how sm all the e a rth is compared to them, and how great with respect to man: so that admiration of these may swell the praise of thoir Creator, and that, repelling the lust for things worldly, man may not grow proud because of all these Inferior things, which are as nothing in the universe that contains them all, and should be regarded as nothing. The scheme goes on to suggest that some be taught .
to make instruments such as burning glasses and other
instruments useful for war, which can be made through the application of mathematics and natural science.
Roger Bacon, to whose book the above writer had
referred, was a believer in experimentation. He applied
mathematics and science to the construction of practical
l s Lynn Thorndike, U niversity Records and L ife in The Middle Ages (New York: Columbia University Press, 1944), p. 140. 103 instruments. He went so far as to state a wish that he could destroy the works of Aristotle, for he recognized that if science continued to build upon the ignorance or incomplete knowledge of an earlier age, it would not meet the needs of a new age.
The difficulty forseen by Bacon did materialize. The church frowned upon mathematics as a ''black art” and went so far as to exclude its study at the University of Paris in the thirteenth century. '.Yhen the works of Aristotle began to return to public attention, by the end of the thriteenth century, its impact was great. Aristotle was recovered through Arabic translations and through the discovery of the Greek originals. The church not only accepted Aristotle, but adopted his views as the only authority in matters of science.
Because his views were accepted by the church, they had to be reconciled with the existing doctrines. A group of philosophers, known as the Scholastics, of whom Thomas
Aquinas is the most noted, took on the task. Aristotle had stressed the us of reason, the Church stressed faith.
He used deductive reasoning to account for all the pheno mena the average man was likely to encounter, and answered all questions he was likely to raise about the universe and man's place and purpose in it.
Great courageous spirits like Abelard and St. Thomas Aquinas dared to introduce into Catholicism the concepts of Aristotelean logic, and thus founded 104
scholastic philosophy. But when the Church took the sciences under her wing, she demanded that the forms in which they moved by su b je cte d to the same uncon ditioned faith in authority as were her own laws. And so it happened that scholasticism, far from free ing- the human spirit, enchained it for many centuries to come, until the very possibility of free scientific research came to be doubted. At last, however, here too daylight broke, and mankind, reasaured, determined to take advantage of its gifts and to create a know ledge of nature based on independent thought. The dawn of this day in history is known as the Renaissance, or the Revival of Learning.-^-®
The Renaissance is generally understood to have begun in Italy, The activity against the Papacy had kept things in a fluid condition, and encouraged intellectual individuality. Because of its geographical position, Italy had become the center of trade, and the wealthy Italian merchants brought back manuscripts and employed copiers to make copies available.
The discovery of America, and the opening of a new route to China around Africa, broadened horizons, and the Europeans met Mohammedans, Chinese, and American
Indians. Protestants opposed Catholics, often just because it was the thing to do. Many who could not under stand the conflict became confused, began to question all religious teaching, and turned to the study of the physical, world. The discovery of printing led to cheap and plenti ful books, and took from the wealthy and the ecclesiastical
l^Tobias Dantzig, tqanber, the Language of Science (New York: The Macmillan Co., 1933), p. 84, 105 scholar the final monopoly on learning. Europe was revolting against t b e scholastic tyranny of thought and the established a u t h o rity of the church. The renaissance v/as not an unmixed, "blessing: The R enaissance was pagan to the core. . . filled Italy w ith violent, remorseless, haughty tyrants, who •fcb.±r*sted for power and glory, and whose geniality v/as only attained by contempt for all lav/s o f morality. It was brilliant with creative power*, "but also moral and corrupt.^0
It was the a g e which produced a Caesare Borgia,
a iviachiavelli, and. popes who brought murder, licentious
ness, aggrandizement;, and luxurious living to the papal
throne. But it a ls o was the age of Erasmus, Luther, and
Lelanchthon. These were part of a humanistic movement
against the re stric tio n s of scholastic philosophy and
ecclesiastical aut h o rity .
The new im p u lse to study nature, and to apply
reason and experim entation to that study, led to the
study of m athem atics. Scholars had the Greek writers
to reinforce the id e a that mathematics provided the
language and sym bols for the study of physical space.
The use of reason w as a challenge in itself. There
seemed to be o nly o n e sure source of tru th , the processes
and concepts of m athem atics. The scientists accepted the
idea that through, mathem atics nature was revealed by God.
^Clyde L. Manschreck, A History of Christlanity (Englewood C l i f f s : P re n tic e -H a ll, Inc., 19S4), p. 2, 106
Stifel, who had been a monk, and who had accepted the doctrines of Luther, unfortunately believed that he had discovered the true way of interpreting the Biblical prophecies. He announced that the world would come to an end in 1553. Many peasants, because of his scientific reputation, and ecclesiastical connections, accepted his assurance and were ruined. He sought safety in prison,
where he continued his mathematical studies. He made some
contributions such as the use of the symbols 1A, 1AA, and
1AAA for the unknown quantity, its square, and its cube,
which shewed at a glance the relation between them.
Ramon Lull, in the fourteenth century, antici pating Leibniz' all inclusive method, believed that he
could develop an all-embracing scientific system of knowledge. He devised what he believed to be an infalli
ble and universal logical method of proving t..at the
mysteries of faith are not contrary to reason. It v/as
his aim to teach the Christian theology so logically that the Moslems could not fail to see the truth. His work,
called the ars magna, used letters to represent certain ideas, and was a mechanical plan which would be a general
axiomatic system for all of science. He felt it could
be used also as a final proof of the truths of Christianity. His works had a decisive influence upon Leibniz three hundred years later. 107
Summary. The Dark Ages had seen the birth pangs of a new age. Rome had conquered parts of Europe and had imposed some of its customs and laws on the peoples.
When the Empire failed, the Church was the only powerful organization remaining in Europe. In order to convert the heathen the Church had to introduce and support schools, and supply leaders. Everything was barren, and the bar barians had to become acquainted with writing, with government institutions, with law, ethics, and the
Christian religion. All these matters occupied the attention and energy of the Christian leaders. The onward development of mathematics ceased for a time. The progress that was made was contributed by the
Hindus and Arabs. But even the ideas of place value, base ten, negative numbers, and the idea of zero were not really absorbed into the whole body of mathematics and learning until the seventeenth century. There were too many people who didn’t yet know simple principles.of calcu lation, and the Involved number mysticism of the middle ages was hard to displace. Host of the contact which mathematics had with scholars during the period was in the Church. The new spirit of inquiry, congenial to the Protestant belief in individual responsibility, and the Catholic emphasis on the reasonableness of a universe designed by God, together were to form a new program for science. Science 108 was about to discover the mathematical relationships that underlie and explain all natural phenomena, and thus reveal the grandeur and glory of God's handiwork.
Whether one can call the mathematics translators of the middle ages mathematicians or not, is a matter of choice for the observer. Today Otto lieugebauer and
Joseph Needham are following in the tradition of those men, and expending almost endless effort in an attempt to recover the learning of another age. Certainly their contributions to civilisation have been great. They have added to the storehouse of knowledge. The world may regret the loss of progress during the dark ages, but the lamp of learning had not gone out. The faithful Christian men in the monasteries and universities preserved and trans mitted the ancient learning so that the scholars of a new age might make great strides in the advancement of learning.
The religious bearings of mathematics during the middle ages were of great importance. The Church provided both the men, and the places for the preservation of learn ing. At the close of the Dark Ages, when man ags.in began to stir, and to think for himself, it v/as Jewish doctors and scholars who contributed to the advance of mathematics.
Serious study v/as concentrated later in the universities, and these too were theologically oriented, as they grew out of the cathedral or church schools. 109
Until printing was invented, and it became possible for comparatively obscure v/riters to be known, manuscripts available for study v/ere centered in the church controlled institutions. There was little need for secular learning in mathematics and science until the expansion of travel and trade made such knowledge necessary.
V/hat v/as said of the priests of Egypt and Babylon might be said of the Church of the middle ages. The mono poly on learning had a double effect. It hid from the people the principles and facts of mathematics without which a civilization cannot advance. But, It preserved for posterity, the principles and facts of mathematics
which might have been lost if left to the whims of a barbaric and pagan people.
Whether the secrecy and monopoly of learning of the church was the cause or the result of the lack
of learning of the middle ages is a question which
cannot be solved here. What is recognized here is that, whatever v/as worth preserving, translating, and transmit
ting from earlier generations to the renaissance, was maintained and passed 011 by religious men, and because of religious ideals, hathematics throughout the middle ages was very dependent upon religion. CHAPTER V
THE SEVENTEENTH CENTURY
Background of the period
After the destruction of the school at A lex an d ria, and the consequent close of the Greek age, G reek m athe matics, and indeed all mathematics, lay dorm ant f 0 1 * about a thousand years. That long period, v a r io u s ly referred to as the dark or middle ages, was d e s c r ib e d in the preceding chapter. It was significant i n -fclie development of mathematics more for the c o n s e rv a tio n of earlier works rather than the creation of new m a t hiemati- cal ideas. It was a time when people with, e d u c a tio n and a sense of scholarship were busy, but were o c c u p ie d with copying books rather than studying nature. I t w s_s a time for contemplation, rather than a time for the se e lc in g of ways to co n tro l n atu re. The middle ages provided the assurance tti£L-fc; because someone had previously observed, c a l c u l a t e d , and concluded, the total of knowledge available to t h e human mind had already been revealed. Once the p u rp o s e of a plant, a planet, or an organism had been d e te rm in e d , and its suitability for man’s use found out, the s u b j e ct was
110 I l l closed to further study. Usually the church made the final decision as to the importance or truth of any m a tte r . A storehouse of knowledge and information is a good thing to have. And the view that what has been written has been worthwhile and useful, is basic to modern science. The multitude of tables of information found in handbooks of science and mathematics are of great importance, and a source of much time saving information. One does not wish to redetermine all the boiling points, or every logarithm, or the many other values which can be easily obtained from reliable sources. Only by analyzing first what has been done, and then arranging and re-arranging one’s own thoughts, can duplication be avoided, and progress be made most efficiently.
But the arranging and re-arranging of information found in tables of value, and the transmitting of pre vious conclusions do not in themselves advance the cause of science. New observations and new experiences must be followed by more experiments and new formulations of theories. And these in turn must be tested and used to suggest new hypotheses or new problems to be solved.
A century of transition. The beginning of the seventeenth century marked a transition point in many 112 ways. It was not just a revolution in science and mathematics, though the work of this study will be concerned primarily with mathematics and its relation to the religious atmosphere in which it developed. All of the great men involved in this movement, with the exception of Copernicus who died in 1543, were active in some part of the seventeenth century. And almost everyone of them was orthodox, or at least an active religious man. It was an age when a man was expected to stand up and be counted religiously.
The reformation had put an end to the religious and intellectual solidarity of the nations, and the contest between Rome and Protestants absorbed the mental energy of Europe, During the second half of the 16th century science was therefore very little cultivated, and though astronomy and astrology attracted a fair number of students, still theology was thought of first and last. And theology had come to mean th e most l i t e r a l ac cep tan ce o f ev ery word of Scripture; to the Protestants of necessity, since they denied the authority of Popes and Councils, and to the Roman Catholics from a desire to define their doctrines more narrowly and to prove how unjustified had been the revolt against the Church of Rome, There was an end of all talk of Christian Renaissance and of all hope of reconciling faith and reason; a new spirit had arisen which claimed absolute control for Church authority,1
But the mathematics developed during the seven teenth century was considerable: Fermat began the modern theory of numbers, Descartes invented analytic geometry, Cavalieri followed by Newton and Leibniz established the
^Mark Graubard, Astrology and Alchemy, Two Fossil Sciences (New York: Philosophical Library, 1953), p, 218, 113 calculus, and Pascal and Desargues extended geometry
Into the new directions of projective geometry. Napier invented logarithms, Kepler and Galileo expanded the horizons of astronomy, and the slide rule, telescope, and calculating machines were invented.
This was also an age of literature. Shakespeare,
Milton, Corneille, Moliere, and Bunyan are but a few of the familiar names. It was an age of religion: the King James1 Version of the Bible was produced in 1611, the
Pilgrims landed in Massachusetts in 1620, and the Shorter
Catechism was written in 1647, "What is the chief end of man? Man’s chief end is to glorify God and to enjoy him forever." The fervor of feeling between Catholic and
Protestant was at its height during the first part of the cent lory.
The mathematics of the age was an international activity. The Polish Copernicus was followed by the
Italian Galileo and the German Kepler, and they in turn were followed by the English Napier and the French Fermat and Descartes. These were succeeded by the great English mathematician Newton, and his German contemporary Leibniz.
Much of th e advance o f s c i e n t i f i c le a r n in g was th e p ro d u ct
of the organized activity of the learned societies and
academ ies. 114
The Academy in Naples, the first of these scientific organizations, was established in 1560, but was soon closed by the Pope on suspicion of black arts. Another lasted for a short time in Rome after
1603, and still another briefly in Spain after 1657,
The two which had the greatest effect on later scientific advancement were the Royal Society of London, organized in 1662, and the French Academy in 1666,
In this study of the religious bearings in the development of mathematics, it is significant to note that there was very close relationship between these men and the church, and in nearly every instance, the motives for their work must be explained in reference to this relationship.
David Eugene Smith,^ in his monumental History of
Mathematics, listed hundreds of events, and dozens of names of men in a chronological table in the back of the book.
In the Greek age he classified seven names as mathematical names of greatest importance; Thales, Pythagoras, Plato, m A ristotle, Euclid, Archimedes, and Diophantus, No additional name is joined to this rank until during the seventeenth century when six more names are included:
Napier, Descartes, Fermat, Pascal, Newton, and Leibniz.
In the following century, Smith designates only five more
^David Eugene Smith, History of Mathematics (New York: Dover Publications, Inc., 1951). 1X5 names in the category of greatest importance: Euler,
Lagrange, Gauss, Laplace, and Legendre. He does not attempt to select any during the more modern period.
E. T, Bell,3 in his Men of Mathematics (The Lives and
Achievements of the Great Mathematicians from Zeno to
Poincaire), included no mathematician between Archimedes in the second century, and Descartes in the seventeenth century. The long night was now over, and the new day of mathematical development had begun.
The status of mathematics at the start of the age At the close of the discussion of the Greek age, there were mentioned four ideas which summarized the relationship which religion and mathematics shared: Pythagorean number mysticism, Euclidean geometry, the Ptolemaic geocentric theory of the solar system, and religious persecution which affected the spread of learning throughout the world. It is of some interest to examine these four aspects of mathematics and religion at the beginning of the seventeenth century.
Pythagorean number mysticism had been expanded beyond comprehension during the middle ages through religious writings. For hundreds of years the numeral
system had been based on alphabetic symbols and a word
could mean either a number of a word. For example, if the letters from a to i in the English alphabet were to 116 represent the numbers from one to nine respectively; j to r to represent 10 to 90, and s to z to represent
100 to 800, the three letters toe might mean either
200 plus 60 plus 5 or 265, or it might actually mean a part of the foot. Such speculation had been applied in the greatest detail to much of the Biblical writings, and many volumes were written in efforts to interpret the messages suggested by such numerological observation.
The rules had multiplied over the years, until it was possible for a word to have many values, and the values
to have a variety of interpretations.
The same numeral procedures were applied to
areas other than the Bible. Francos Siszi, a Florentine,
in a book entitled Pianoi Astronomica, in 1611 declared:
There are seven windows given to animals in the domicile of the head through which the air is admitted to the tabernacle of the body to enlighten, to warm, and to nourish. What are these parts of the micro-cosmos? Two nostrils, two eyes, two ears, and a mouth. So in the heavens as in a microcosraos there are two favorable stars, two unpropitious, two luminaries, and Mercury, undecided and indifferent. From this, and many other sim ilarities in nature, such as the seven metals, etc., which it were tedious to enumerate, we gather that the number of the planets is necessarily seven. Moreover, these satellites of Jupiter are invisible to the naked eye, and therefore, do not e x i s t . 4
This preoccupation with numbers has persisted to
the present day. A remnant from the astrological ideas
of ancient Babylon, and the Pythagoreanism of Greece,
4Mark Graubard, o£. crt., p. 218. 117 these two currents of thought entered Into Christianity by way of the Gnostics. They had been accepted by many of the church fathers, including Augustine, and the authority of these church fathers, including the most learned scholars, had attached itself to the ideas•
The middle ages had elaborated upon the earlier views, and passed them on to the Renaissance.
Scientists of the first order often turned to numerology; nor is it a long way from this to the revived Pythagoreanism of the 15th and 16th centuries in which Copernicus, Galileo, and Kepler were so deeply immersed. The structure of the universe gradually revealed by these astronomers was founded squarely upon mathematics, then coming to the fore front throughout the universities and academies of Europe. Galileo wrote on one occasion, "Philosophy is written in that vast book which stands forever open before our eyes, I mean the universe; but It cannot be read until we have learnt the language and become fam iliar with the character in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures without which it 3s humanly impossible to comprehend a single word."5 Kepler, In the first year of the seventeenth century, Tycho Brahe took on an assistant in his observatory. This assistant, Kepler, was to advance the science of astronomy throughout his life, and to have an influence on all study of the solar system.
He was a believer in astrology and number mysticism.
His search for patterns throughout Brahe's records of the motions of the heavenly bodies, was motivated first
C. A. Patrides, "The Numerological Approach to Cosmic Order During the English Renaissance," Isis, XL (1958), p. 394. 118 by his understanding of the view that the planets were related to the Platonic regular polygons, and could be
explained by means of a numerical pattern. He published
a tract in 1596 under the title The Cosmic Mystery, which
reflected the spirit of the period so well that both
Galileo and Brahe responded with flattering comments, and
Dantzig says it was this tract which so impressed Brahe
that he invited young Kepler to become his assistant.
Dantzig quotes from the tract:
Before the universe was created, there were no numbers except the Trinity, which is God himself. . . There remain six bodies, the sphere and the five regular polyhedra. To the sphere corresponds the outer heaven. For the universe is twofold: dynamic and static. The static is the image of God-Essence, while the dynamic is but the image of God-Creator, and is therefore of a lower order. In its very nature, the round corresponds to God and the flat to his creation. Indeed the sphere is threefold: surface, centre, volume; so is the static world: firmament, sun, ether; and so is God: Son, Father, Spirit. On the other hand, the dynamic world is represented by the flat-faced solids. Of these there are five; when viewed as boundaries, however, these five determine six distinct things; hence the six planets that revolve about the sun. This is also the reason why there are but six planets. . • Thus was I led to assign the cube to Saturn, the Tetrahedron to Jupiter, the Dodecahedron to Mars, the Icosahedron to Venus, and the Octahedron to Mercury.®
Since number was the first model of things in the
mind of the Creator, it was expected that explanations
of material phenomena not only could, but of necessity
Tobias Dantzig, Bequest of the Greeks (New York; Charles Scribner’s SonsV 1^35), p. “55. ------119 must be found In harmonious mathematical relationships discoverable in these phenomena.
Hence to the Renaissance neo-Platonist, the simplest and most harmonious mathematical system capable of representing the facts would, by virtue of these very qualities, be closest to reality and consequently the most acceptable system.” The geometric pattern of Spinoza. Euclidean geometry had remained practically unchanged during the middle ages.
It had been carefully translated from Greek Into Arabic, from Arabic into Latin, and eventually into the French, English, and German of the Renaissance. But it was still wholly Euclid. The authority which the perfection of this system gave to geometry, and to mathematical methods in general, was the basis for the general belief that only through mathematics could the problems of nature be solved. Spinoza wanted to think out more clearly than had ever been done before, the ultimate reasons for life.
Therefore, he chose as the model for his thinking, the geometric pettern of Euclid.
Deeply trained in Jewish scholasticism, Cartesian philosophy, and Cartesian science, it occurred to him that the rigorous form of argument used by Euclid to establish geometric truth might be extended to the demonstration of philosophical truth. His first published book was a geometrical interpretation of Cartesian philosophy. For some fifteen years (1661- 1675), most of his energy was spent in the elaboration of a treatise of ethics which he wanted to be as exact
^Francis R. Johnson, Astronomical Thought in Renaissance England (Baltimore, 19371),p . §6. 120
as the geometric Elements. This immense effort was not entirely successful, but as far as I know, no other philosopher has gone beyond Spinoza in the direction indicated by him.8
Seventeenth century science stressed the
importance of mathematics, and preferred a priori reasoning to the method of empirical observation. The accepted accuracy of mathematics and the certainty of its conclusions provided the ideal method for solving any problem. If anything could be proved mathematically, it was believed
and commanded general acceptance. It is easy to understand
why Spinoza, following Descartes’ example, adopted the method of geometric demonstration for proving philosophical
truth. However, almost every critic or commentator of
Spinoza has recognized that the method of Geometry is not adequate for the treatment of philosophical truth.
But the critics, besides showing the faulty of nature of his Spinoza’s deductions, argue further that this is so, because philosophy, by its very nature, is not amenable to geometrical treatment. • • the sciences are departmental, abstract, and make many presuppositions. Philosophy is concerned with the real, with the whole, and tries, as far as possible, to dispense with presuppositions altogether. Their difference being so fundamental, it would be a wonder if a method which deals with the categories of space, number, and quantity were equally applicable to a sphere which is also concerned with matters of s p i r i t . 9
8George Sarton, "Spinoza,11 Isis X (1928), p. 12.
9Senat Kumar Sen, "The Method of Spinoza," The Philosophical Quarterly. XXXI (July 1958), p. 33. 121
Galileo. The Ptolemaic view of the solar system was the accepted view of the church throughout the seventeenth century, even though Copernicus* view had been published earlier in the year 1543. Many had accepted the new view as a simpler one, and therefore more certain. Galileo
seemingly had always believed in Copernicus* view that the earth and the planets revolved around the sun, but was afraid to publish it on account of the ridicule and perse
cution it would have aroused. However, he became certain that the discovery of the satellites of Jupiter made the truth of the Copernican theory almost unassailable, and felt that he must announce his findings.
But Galileo ran into difficulties. He couldn’t get his opponents to look at Jupiter through his telescope,
and the orthodox party resented his teaching. On February
24, 1616, the Inquisition declared that to imagine the sun
to be the center of the solar system was absurd, heretical, and contrary to Holy Scripture. For many years thereafter, it was necessary for the Jesuits who taught the Copernican
theory to treat it as an hypothesis from which, though false, certain results would follow.
This subterfuge had already been advanced at the
time of the publishing of Copernicus’ work in 1543. The
one to whom Copernicus had given his book to be printed
had passed the responsibility on to a Lutheran minister,
Andrew Osiander, who wrote a second preface to the book. 122
This preface indicated to the reader that the theory was merely an artificial device which served to give future positions of the earth, and had nothing to do with any Galileo attempted the same excuse when he published his Dialogue Concerning the Two Chief Systems of the World in 1632. However, it didn’t work, and he was attacked on the grounds that he had been commanded in 1616 not to teach the Copernican theory. He replied that he had been told only that he was not to hold or defend the theory. In spite of his age, and the cold winter, he was called to Rome when he was seventy. He had to face the Inquisition the following summer, recant, and promise to leave the teaching of theology to the church. Galileo insisted that the Bible allowed different views from those which the church held. I am inclined to think that Holy Scripture is intended to convince men of those truths which are necessary for their salvation and which being far above man’s understanding cannot be made credible by any learning or by any other means than revelation. Bub th a t th e same God who has endowed us w ith s e n s e s , reason, and understanding does not permit us to use them and desires to acquaint us in another way with such knowledge as we are in a position to acquire for ourselves by means of those faculties—that, it seems to me I am not bound to believe, especially concerning those sciences about which the Holy Scriptures contain only small fragments and varying explanations; and this la precis Illy the case with astronomy, of which there is so little that the planets are not all enumerated, only the sun and the moon, and once or twice Venus under the name of Lucifer.^-O ■^Mark Graubard, ojd. c i t . . p . 220. 123 This trend of thought was sure to get Galileo into trouble. So strongly were his views expressed, and so thinly were his intents disguised, that the pressures of th e common f a i t h and common se n se com bined to b r in g him to trial. His heresy is evident: 1. Many chapters of Holy Scripture cannot be taken literally. 2. In discussing natural phenomena the Bible is not authoritative. 3. Biblical commentators err in matters of s c ie n c e . 4. Scripture Is authentic only in matters of f a i t h . 5. Many passages in the Bible require a new Interpretation.11 The religious view of learning. Supreme faith in the Bible, and veneration of the accumulated writings of the ancients, meant to the Church that education was to be restricted to transmission of learning, and that no provision be made for additional experimentation or original thinking. In the schools, the study of philosophy and mathematics was made to sound profound, but Didactic lectures were given, and in addition to attending these, students defended their theses at public disputation. The dissertations were never supposed to bring in new subject matter, they were only to give the student a chance of combining, and re-combining, of practicing and exercising, what he had acquired from A ristotle or other authorities.12 1 ;lL o c . c i t . (Condensed and paraphrased). 12J. P. Pulton, "Robert Boyle and His Influence on Thought in the 17th Century," Isis. XVIII (1932), p. 81. 1 2 4 Transmission of learning is a valid mode of instruction. Certainly it is necessary for each generation to pass on to the next, the accumulated knowledge and experience which it has gained. If this were not done, there would be a great waste of time and effort, and no generation would be able to progress very far beyond the one which preceded it. When there is a r body of information^ to be handed down with very little change, transmission is an efficient mode of instruction. The Naval and M ilitary Academies have often selected promising young officers from graduating classes to be instructors, to pass on to the next class the material which has been developed and given to them. Much teaching in theological seminaries is of this nature. Critical monographs on Bible texts are written as a partial ful fillment of the requirements for the B. D. degree, and they are essentially are a combining and a synthesis of what has been written in various commentaries, together with the student*s view. The fourth characteristic of the Greek age to be brought to attention is the persecution of different religious groups by one another. It- is a matter of record that the seventeenth century was not an easy time in which to live. The conflicts between Galileo and the church, could be elaborated upon, and expanded to include other 125 men such as Arnauld and Pascal, Descartes destroyed an early copy of his writings when he heard of the difficulty which Galileo had with his presentation of the Copernican th e o ry . The religious persecutions of the seventeenth century were largely a result of the Catholic and Protes tant groups sparring for control of countries and for the minds of men. Anyone who was prominent and learned enough to be included in the history of the age, was no doubt involved in some way in the controversies of the church, A casual comment in a history book that de Moivre, nA Hugenot who settled in London after the revocation of the Edict of Nantes,’* may have behind it a very real story. It is not difficult to understand the intensity of the religious fervor of the age. The motives of the church leaders of that day, are the same motives to which church and missionary leaders give allegiance today. And even the secular world enforces laws and patterns of behavior based on authority and common consent. Criminals are punished who disobey the current statement of the law. So much more should they be punished who disobey the law of God, And as the removal of a criminal may provide peace for those who remain, so, the persecution and destruction of an enemy of the faith 126 may allow for the more certain propagation of the truth to those who remain* The church leaders in both camps were sure they had a monopoly on absolute truth, and they had to persuade all others to their beliefs, or fail in their mission. The beings and principles which are the concern of religion are beings and principles eternal to us and independent of us, eternal in the heavens and surrounded with an aura of absolute truth. This absolute truth is thought of as intrinsically know- able, by revelation if not by more mundane methods. . . Given the view that there exists an absolute truth, and given furthermore'the conviction that one has found absolute truth, the intolerance of the inquisition or the brain-washings of the Communists become logically inevitable.13 Trends in the seventeenth century Robert Merton has made a detailed and comprehensive study of the seventeenth century in England. He has noted a shift in emphasis from religion and theological interest to a marked attention to literature, ethics and art, in the earlier days of the Renaissance, followed by the special interest which for the past three centuries has centered in science and technology. The famous Savilian and Lucasian professorships of mathematics at Oxford and Cambridge were established during the seventeenth century, as were chairs of Astronomy, Chemistry, Botany, and Natural Philosophy. ^Percy W. Bridgman, "Quo Vadis,1* Science and Ideas (Englewood C liffs: Prentice-Hall, Inc., '1964'), p. 2‘7¥I 127 Merton found that toward the end of the century, the vocational choices of the educated people had changed from the religious professions to those with more u tili tarian goals. He examined the Dictionary of National Biography, and recorded the numbers of outstanding people whose contributions could be accredited to various occupa tions, He arranged these names in five-year periods, and further analyzed the information in twenty-year periods. At the close of each twenty-year period, beginning with 1601, and continuing for five more periods to the year 1701, he recorded the nu3-;b©r making their greatest contri bution in religious endeavor during each of the six periods. These were as follows: 102, 114, 80, 52, 35, and 28, This represents a rapid and a definite change, Merton asserts: Paradoxically, but inevitably, then, this religious ethic, based on rigid theological foundations, furthered the development of the very scientific disciplines which later seem to confute orthodox theology.14 Merton insists that the figures given above do not necessarily mean a decrease in religious devotion to truth. It represents rather a trend from the monastic view of the relative worthlessness of matter, to the study of science as the study of God's works. In a later 14Robert K. Merton, "Science, Technology, and Society in Seventeenth Century England," O siris. IV (1938), p . 458. study Merton-^ notes that the Protestant view was for the most part favorable to science, while the Catholic influ ence restricted the advance of science by suppressing its t study. In the establishment of the Royal Society in 1662, forty-two of the sixty-eight concerning whom information about their religious orientation is available were clearly Puritan, in spite of the fact that the Puritans constituted a relatively small minority in the English population. It is interesting to note also, that the group who founded the Royal Society specifically excluded discussions of theology in their deliberations. The trends noted, by Merton in the Seventeenth century in England are trends which the religious world has noted whenever Christianity has taken education and accompanying skills to some new missionary area. Christianity taught thousands of Chinese to read, and the missionaries had to leave. In Africa, the missionary expends great energy preparing a small group to read and help spread the Gospel, only to find that once the skill is learned, the m aterialistic rewards of a government job become more attractive than the work of the Church. And in a Christian college, the Education and Business departments are always subject to the criticism of those who know of some young person who had entered school to ■^Robert K. Merton, "Puritanism, Pietism, and Science, Science and Ideas (Englewood Cliffs: Prentice Hall, Inc., 1964) , “pi S32-269. 129 prepare for the Church work, only to b ecome a teacher, or to enter the business world* The work of the early leaders of the Renaissance had succeeded only in delaying the scientific revolution. With the invention of printing, and better methods of paper making, the spread of learning was greatly facili tated. There was a rush to get the older materials widely circulated, and the older ideas became more firmly fixed than ever* The great teachers and scientists of the early renaissance were soon followed, however, by those who were free and eager to explore and experiment on their own. There were many areas of knowledge to be investi gated at once. Man was reminded again that advances in culture and civilization are made along the entire front together, and one field does not advance far beyond all others. In fact, any new discovery in one area usually generates interest and new discovery in a host of other related areas. Long-distance voyages were now undertaken. These required a first rate scientific, and secular study of astronomy. In this search, there was a need for better methods, and better tables of values in mathematics. Logarithms were invented, and the usefulness of the life of an astronomer was doubled. The compass, slide rule, 130 telescope, and calculating machine were invented and placed in use* Great academies and societies through which most of the new ideas would flow were established* The time had come to c ease respecting only the knowledge which had filtered down from some previous generation and authority, and to make provision for the percolation of new ideas to meet new challenges. The profound social and economic effects of these explorations of general information are common Which of the events or ideas were chiefly causes, and which were effects, of the advancement are questions which have occupied the attention of countless s c h o la r s . The acceptance of the spherical shape of the earth, and the consequent difficulty, though a necessity, to project the roundedness of a sphere upon a plane, led to the study of projections, and eventually projective geometry. Mercator issued his famous projection of the globe in 1568. Although scorned by geographers, it has been the most valuable projection for nautical navigation. The question of finding one’s location at sea, was a matter of knowing his latitude and longitude. The latitude problem had been generally solved much earlier, but the longitude problem remained to be solved, until new mathema tical and practical minds could invent and perfect an 131 accurate clock. This was not successfully accomplished until the eighteenth century. The system of grids on the globe was to be followed by a similar system in analytic geometry. These cartesian grids were to have a greater effect on the world than many mechanical inventions, or explorations of the surface of the globe. Religious bearings in mathematics The original search, to find man's place in the creation of God, continued to be the primary search of astronomer, mathematician, and philosopher. For centuries the Ptolemaic idea had been challenged informally by those who advocated either a moving earth, a heliocentric system, or both. The original theory of Ptolemy had been modified by the addition of epicycles, until it had lost its classical appeal of simplicity. There was a need for a new theory, which would match all the obser vations. The sky was there, and the objects available for observation for whomever wished to look. The problem was to find out the laws of motion. The belief that the planets were rational beings, acting on their own volition, sounded no more plausible than the Greek mythology from which it had come, and was no longer satisfactory to empirically minded scientists and mathematicians. Galileo believed that "nature's great book is written in mathematical symbols,11 and that it was man's right and destiny to read that book. His investigations led him to develop the laws which describe the motion of falling bodies to the earth. He made the amazing dis covery that motion is just as natural a state as rest, and contrived experiments to prove his discovery. His science was not purely inductive in the sense of deriving generalizations only from experimentation, but was the verification of assumptions resulting from observation and thought, Aristotle had said heavier objects fell faster than light ones, and no one had dared or cared to admit for some two thousand years that he knew it wasn't so, Galileo dared. Whether he dropped balls from the tower of Pisa or not is of no great concern. 1964 was the four hundredth anniversary of his birth, and a great deal was said of Galileo, including the statement that he did not drop the balls, the tower was not yet leaning, and he didn't need such dramatic proof for what he knew to be true. Development of mathematical symbols. Mathematics is a symbolic language. It could not advance without the development of symbolism suited for the new advances. The Greeks had rejected the symbolism of the Babylonians, and had forced mathematics into a rigorous mold from 133 which it could not break into freedom until new symbolism was invented which would provide the means toward systematization and generality. The development of symbols to represent algebraic quantities and operations, together with the use of the Hindu-Arabic numerals and zero, made it possible for number theory to shake off mysticism, and advance by leaps and bounds. "Most of the operational and other symbols now in use in elementary arithmetic and algebra date from the sixteenth and seventeenth cen tu ry * The signs for addition and subtraction can be found in commercial arithmetic works as early as 1489, but their use as operational symbols dates from the beginning of the seventeenth century. The symbol for equality was first suggested by Robert Recorde in 1557. The symbol for m ustiplication was introduced in 1631 by Oughtred, a clergyman, whose Clavis Mathematicae, after Rapier's Descriptio in 1614, Was the most influential mathematical publication in Great Britain in the first half of the seventeenth century. It was one of the few books that contributed to laying the foundations of the mathematical know ledge of Newton as he was starting on his career. Oughtred exceptionally emphasized the use of mathema tical symbols, and althoagh he introduced more than one hundred of them, only three have come down to modern times. These are: cross-sign for multiplica tion, the four dot sign of proportion, and our s^Lgn 16A. Wolf, A History of Science. Technology and Philosophy in the 16th and T7th Centuries (New York: Harper & Brothers, 1§59), p. 162. 134 for similarity* About 1622 Oughtred invented the slide rule, and his priority in the invention is unassailable* Oughtred was by profession a minister of the Gospel, but his avocation was the teaching of mathematics, and the writing of mathematics books. Among.,his pupils were John W allis, and Christopher W ren. ' The use of a dot as a symbol was due to Leibniz. No mathematician was more taken up with symbolism than he was, and he hoped to perfect mathematics to the point that he would have an instrument which would serve to exalt the intellect as much as the telescope had served to perfect the vision of the astronomer. Perhaps no mathematician has seen more clearly than Leibniz the importance of good notation in mathematics. . . . No other mathematician has advanced as many symbols which have retained their place to the present time as has Leibniz. . • Among Leibhniz1 symbols which at the present time enjoy, universal or well nigh universal recognition and wide adoption are his dx, dy, his sign of integration, his colon for division, his dot for multiplication, his geometric signs for sim ilarity, and congruence, his use of the Recordian sign of equality in writing proportion, and his double suffix notation for determinants *^8 The symbol for division was first printed in 1659 in a work by a Swiss mathematician, J. H. Rahn. The symbols for "greater than" and "less than" were developed early in the seventeenth century by Harriot. The methods of indicating powers of algebraic quantities ■^R. C. Archibald, Outline of the History of Mathematics (Oberlin: Mathematical Association o f A m erica, 1949J, p. 32. ■^Florian Cajori, "Leibnitz, the Master Builder of the Mathematical Notations," Isis VII (1925), p. 413, 417. 135 were finally developed into the modem index notation (for positive integral powers) by Descartes in 1637. This notation was extended to represent roots and reciprocals of powers by Wallis and Newton, who showed how fractional and negative indices could be used for this purpose. Simon Steven suggested in 1585 a notation for decimal fractions, joining to each figure an index showing the place a digit occupied to the right of units place. Vieta at the beginning of the s eventeenth century proposed the comma prefixed to the decimal quantity, and Napier in 1617 used a point. The geometric symbols used in plane geometry, although ancient in origin, were revived and used considerably by Herigone and Oughtred in the seventeenth century. The first use of the Greek symbol £i for the ratio of the circle to the diameter was by William Jones in New Introduction to the Mat hematics, 1706, p. 243.^ Computation, which had been bound to the abacus for hundreds of years, was greatly facilitated by the introduction of these many symbols, and the methods of using them. Decimals began to appear, logarithms were invented, and instruments such as Napier’s bones, Pascal’s n q D. E. Smith, A Source Book in Mathematics. (New Y ork: Dover P ublicatio n s"^ 1^5*57, p . 345. 136 calculating machine, and Oughtred’s sl±<±Q xule, increased the ease with which problems were att^ clc^ . and solved, Kepler, What Galileo had done f o r 5 motions near the surface of the earth, Kepler was to do !*<=: the motions of the heavens. His religious understandinp: -was th a t of medieval theology, that God had placed m o.~xr . at the center of the universe, and had created the sun, m o ►xi, and stars for him; and that therefore they must go a ro ia a : .d. the e a rth . If one had asked Kepler why he studied t h e ] avens, and nature, when God was not yet fully known, he would have answered that he studied nature that God. .ight be known. He was also convinced with Pythagoras th ; ^ the universe is a systematic, harmonious structure whos© ctions are explainable by mathematical law. The pu.Tz> ished theory of Copernicus gave direction to his thought; He was not studying the heavens to learn astronomy, •till less to improve navigation procedures, although had been asked to help in reforming the calendar which s then in a confused state. When Kepler announced his third 1 he did i t with the announcement of a system of heavenly Larmonies, a •'music of the spheres," enjoyed by th e s i x which had a soul provided for this purpose. He comt>:X_ .ed a quantita- tive mathematical conception of things w l h an intense concern for religious matters. His o rig 3 _ Lai vocational 137 interest was in the field of theology, and from a letter he wrote to a friend, it is known that he was restless until he discovered that he could celebrate God in his astronomical work. He and Copernicus were looking, not only for mathematical explanations (of the universe, but for the simplest and most harmonious explanation. The Ptolemaic system lost out to the Copernican, not because it couldntt be explained, but because the Copernican system was simpler, and thus superior, although it had no greater measure of certainty. The support of the theory of Copernicus was bound to cause controversy. It seemed to run counter to those Biblical passages which assumed the centrality of the earth, and the movement of the sun. And the identity of the purpose and place of the earth in the universe was now in qiestion. The closely knit compact universe was no m ore. Only men possessed of unshakable convictions in regard to the importance of mathematics in the design of the universe v/ould have dared to uphold the new theory against the powerful opposition it was sure to encounter. . . The fight to establish the helio centric theory weakened the stranglehold of ecclesias tic ism on the minds of man. The mathematical argument proved more compelling than the theological one, and the battle for the freedom to think, speak, and write was finally won.20 20 Morris Kline, Mathematics in Western Culture (New York: Oxford University Press, 1953), p. 120, 125. 138 The structure of the planetary system was such that Rheticus, who was Copernicus* assistant could say "The planets show again and again all the phenomena which God d e s ir e d to b e seen from th e e a r t h . " Actually, however, the initial conditions of our planetary system were chosen in such a way that all the satellites of the sun--and our own satellite as well—behave with great modesty. Their orbits can be closely approximated by circles such that the simplest model of a circular motion with constant speed leads immediately to very reasonable results for the description of the solar and lunar phenomena. On the other hand, the deviations from the trivial circular orbits are just great enough to be observed and to challenge an explanation, but small enough such that again comparatively simple modifications of the trivial solution give satisfactory results. This statement, made in the 1950’s by a scientist, reminds the writer of a book written in the latter part of the nineteenth century by Rev. Thomas K ill. Rev. Kill at different times was president of Antioch College, and of Harvard University. He writes nearly the same idea as the quotation above, but directs his attention toward the religious bearings in these motions of the universe: In the beginning the Creative Spirit embodied in the material universe, those laws and forms of motion which were best adapted to the instruction and development of the created intellect. The circle and the ellipse are among the simplest of figures, defined by the simplest laws. Accordingly, the Creator has strewn examples of the circular form around us on every side; and by the pictured 210tto Neugebauer, The Exact Sciences in Antiquity (Providence: Brown University Press, 195T), p. 15S. .i 139 alphabet of the heavens, called our attention to the consideration of elliptical orbits# When in the course of ages, some of the comparatively easy problems of astronomy had been successfully solved, problems of more difficulty were gradually brought into view; and phenomena which were not obvious, not pictured alphabet, but the fine print of creation, led men into the hidden knowledge of optics, electricity, chemistry, and other forms of molecular physics. The course of history and of scientific progress has been precisely what it might have been had1 God designed to educate man; to reason with them, and teach them the sciences; for there has been a constant presentation of simpler truth, whereby man has been led to the acknowledgment of those less obvious, and this is essentially r e a s o n in g . ^2 These two quotations represent the differences which are between an objective evaluation of man's search for trubh, and a view which seeks religious bearings in all parts of that search. Kepler would have appreciated Rev. H ill's interpretation of man's search for better ways of explaining nature, by seeing the process as one of the discovery of eternal laws as they are revealed to man. The examination of the laws of the heavens and other laws of nature was going on in various parts of the world during the seventeenth century. Most scientists of that day were religious men, and saw in the passiveness of nature, reflected by Kepler and Newton in their laws of motion, the need of a God to move it, or to link it with the human w ill. The physical scientists have not on the whole been so subject to the anti-religious ^Thomas H ill, Geometry and Faith (Boston; Lee and Shepard, 1882), p. 25. 140 criticism that the biological scientists have received because of their singleness of thought concerning the evolutionary hypothesis. The age of reason. At Worms in 1521, Luther had appealed to right reason, and to Scripture, giving historical precedence to the use of reason in testing faith, and in interpreting Scripture. After Newton published his Principia in 1687, and had explained the scientific method as a convincing logical basis for furthering investigation, it became increasingly harder for the church to insist upon its absolute authority. Newton’s ideas were common knowledge, and Newtonianism for Ladies was widely circulated. These men, such as K epler and Nev/ton, were not a n t i- r e lig io u s . They had not forseen that their findings would work toward destruc tion of ecclesiastical authority. But once they published their findings, based on mathematical understandings, they felt that they must resist unwarranted attacks upon them. The universe became a realm in which the laws of cause and effect reigned, and there was no need for the action of an arbitrary divinity. Men questioned all forms of authority, and gloried in their newly found powers of reasoning, throwing all caution to the wind. Scientific achievements were happening often enough to assure anyone that the laws worked, and there was no compulsion to doubt 141 reason as a universal guide. The revelations of the religious leaders were questioned as no more sacred or reliable than the discoveries of the divinely-given universal reason of mankind. Man became more interested in the God of Nature than in the God of Scripture, and natural reason as the discoverer of that God of Nature acted as the regulative force in the affairs of man, result ing in a generalized ethical theism, a tolerant broad ening of religious forms, a scepticism with regard to supernatural claims, and a confidence in human ability to solve all mysteries. Free will was extolled, predestination and depravity decried. Former distrust of man's reason and human culture, as seen in the traditional emphasis on original sin gave way to dependence on reason and the glorification of man's culture as the flower of that reliance. The Bible and institutional ecclesiasticism seemed anachronistic.23 When reason was enthroned, and man became his own authority, he did not stop with the church, but threatened government and the divine right of kings. Reaction soon set in from many directions. Phil osophers tried to show the futility of reliance on reason, which leads only to scepticism and self-destruction. The pietists and evangelical groups continued to demon strate that fervent, emotional religion had power to change man's lives, and that arid intellectualism is not a final nor sufficient solution for any of the real problems of man. Finally, in the following century, in 2«z Clyde L. Manschreck, A History of Christianity. (Englewood Cliffs: Prentice-Iiall, Inc., 1964), p. 219. 142 the French Revolution, man experienced or witnessed with horror the destructive anarchy of rampant reason without a u th o rity . It was thought at the time that the application of honest and disinterested intelligence to the conduct of public affairs would suffice to bring men out of a certain darkness into the clear light of a better world in which reason alone would assure Improvement in the condition of man and benevolence in his treatment of his fellows. This expectation was refuted, created more problems than it settle#, and led to the disillu sionment and even the destruction of those who served it. ^ Philosophy combined with science supplied the background for the spirit of the age. The scientist and mathematician, sensing the certainty of their conclusions, began to apply mathematical methods to more subtle matters. Rene Descartes, interested only in what was clear and certain, set himself to doubt everything until he could find some basis or point of departure which could not be doubted. To his own satisfaction he found such a basis in his own thinking existence. Cogito ergo sum. From this basis, step by step, with mathematical clarity, he sought to prove the existence of God, and to show that matter, which had its source in God, had been given both extension and mechanical motion by God. O 4. George V. Taylor, "Prospectus for a Christian Consideration of the French Revolution," Christian Scholar, XL (March 1967), p. 23. 143 The premises of Descartes, that all conceptions must be doubted until adequately demonstrated, and that adequate proof must have the certainty of mathematics, seemed a safe program to follow. The Protestant Revolu tion, which had led to man's insistence upon the right of the individual to think his own thoughts, even of heavenly matters, provided a fertile field for such thought, Man felt that the method of free inquiry was a safe way of salvation, and if honestly followed, would lead into participation in God's promises. William Chillingworth, an apostate from the Church of England who later renounced Rome and returned to the Anglican fold, is sometimes called the father of the rationalists in religion. In 1638, he published The Religion of Protestants, A Safe Way of Salvation, in which he said: For my part, I am certain that God hath given us our reason to discern between truth and falsehood, and he that makes not this use of it but believes th in g s he knows not why, I sa y , i t is by chance th a t he believes the truth and not by choice; and that I cannot but fear, that God will not accept this sacrifice of fools,25 John Locke, later in the century, strengthened the trend to rationalism. He believed that religion, although above reason or perhaps beyond experience could not be contradictory to experience, and he sought to show the reasonableness of Christianity. The combination of 25 Clyde L, Manschreck, o jd , c i t ., p. 225 144 religious fervor, mathematical ingenuity, and the most careful attention to logic and order was a combination which should successfully explain matters of faith, even as the rules of mathematics had been so successful in explaining the rules of nature. This mathematical method had been employed by Descartes. At the request of friends, Descartes reluctantly agreed to display his metaphysics in synthetic geo metric forms, the more geometrico. He gave 10 definition?, 7 postulates, 16 axioms, four proposi tions, and one corollary. The more geometrico was not important in Descartes* synthesis of mathematics and philosophy; the analysis, similar to mathematical analysis was important. There were three ultimate realities in Descartes system of philosophy: mind, matter, and God. He considered that the attribute of matter was geometric extension. Mind or thinking substance could not be mathematical, since it was the antithesis of geometric extension. Descartes thought that mathematical truths were established by God and they were immutable only because he was unchangeable. For him the objects of mathematics could be indefinite but not infinite, only God was infinite. He gave three proofs of the existence of God and declared that one of these was parallel to the proof that the three angles of a triangle were equal to two right angles. Another of his proofs can be put in the form of the reductio ad absurdum, or indirect proof, of mathematics• Pascal. Another example of a brilliant mathematical mind embroiled in the religious questions of the day was Pascal. He went from religion to mathematics, and back to religion several times. On each occasion, he attri buted his decision to some revelation or feeling that it ^6Richard Herbert, Some Educational Implications of Descartes Synthesis of Mathematics and Philosophy (Abstract of Contribution-to Education #£78) (Nashville: George Peabody College for Teachers, 1940), p. 4. 145 was GodTs will. In his brief lifetime of less than f o r ty y e a rs , he was ab le to Achieve a 7-fold immortality, as a mathematician, physicist, as an inventor, as a philosopher, as a theologian, as the creator-in-chief of his nation Is greatest prose, and as a religious devotee,^? However, most of his life he devoted himself single-mindedly to religious considerations. He had accepted a triple thought pattern which could accept evidence of the senses in science, the dictates of reason in mathematics, and faith as the basis for his religious belief. He planned a great apology for the Christian religion, and wrote down scattered notes from time to time, but his health which had never been good, failed, and the apology never was even clearly outlined. History reports that his thoughts were collected, and pasted together to fit pages more or less at random, and were published as the Pensees, Only occasionally do mathematical ideas appear in this work, but enough appears that they have been cited as a first attempt at a philosophy of mathematics. His contributions to mathematics included developing with Fermat the theory of probability. He developed a doctrine of ’’Probable opinions” under the name of Montalte in his replies to the Jesuits, The Jesuits had developed 27 C, J, Keyser, ”A Mathematical Prodigy,” Scripta Mathematics, V (April 1938), p, 9, 146 a doctrine of probabilism, which permitted them to absolve any course of action as perfectly acceptable to Christ if any authority could be cited in support of it* In answer to this doctrine, Pascal said, MI am not content with the probable, I seek the sure*"^ He felt that God should reveal himself more clearly if he existed* But he reasoned on the grounds of his own laws of proba bility that even if the probability that God exists and that the Christian faith is true be small, the reward for belief is an eternity of bliss. On the other hand, if it is false, the loss caused by believing the Christian d o c trine is so slight, that it is far better to wager on the ex isten ce of God, Pascal is - remembered particularly for the theorem which bears his name. It states that the three points determined by producing the opposite sides of a hexagon inscribed in a conic are collinear, a most productive theorem from which has been deduced over four hundred corollaries* He worked out the problem of the cycloid, wrote extensively on the triangular arrangement of the coefficients of the powers of a binomial so that this arrangement is since known as Pascal’s triangle. And his work with indivisibles anticipated calculus. The names of Newton and Leibniz are most common in the ^Emile Cailliet, Pascal, The Emergence of Genius (New York: Harper & Brothers, 1961), p. 245* 147 history of calculus, but many mathematicians of earlier times had made beginnings, and Pascal m s such an one* When Leibniz in 1673 visited Huygens, the latter urged him to study the works of Pascal, turning over to him the treatises of 1658-59. Leibniz later reported that it was during his reading of the Traite (Traite des sinus du quart de cercle) that a light suddenly burst upon him, and that he then realized that the determination of tangents to a curve depended upon the ratio of the difference in the ordinates and abscissas as these became infinitely small, and that quadratures depended upon the sums of the ordinates of infinitely thin rectangles,28 Leibniz. Leibniz is chiefly remembered in the history of mathematics for his contributions to the calculus. He also has been credited with the invention of many of the symbols used in modern mathematics, which fact has been noted earlier. He may also be noted for the religious bearings on mathematics contained in his writings. He was fascinated with the possibilities of applying the principles of mathematical logic to religious or missionary persuasion, and found in the binary number system and the complex or imaginary number some outlook for a mystic bond between reason and imagination. Concerning the imaginary number he said, The Divine Spirit found a sublime outlet in that wonder of analysis, that portent of the ideal Carl B. Boyer, "Pascal: The Man and the Mathematician," Scripts Mathematics, XXVI (December 1963), p. 301. 1 4 8 world, that amphibian between being and not-being, which we call the imaginary root of negative unity,29 His discovery of the binary system, according to which number may be represented by a definite succession of the two ciphers zero and one only, led him to identify God with the one or unity, and the void with the zero, God created all things out of nothing, just as all integral numbers may be formed by the digits zero and one in the binary system, Laplace reports in his Oeuvres that the idea was so pleasing to Leibniz that He communicated the idea to the Jesuit Grimaldi, President of the Mathematical Board of China, with the hope that this emblem of the creation might convert to Christianity the reigning emperor who was particularly attached to the sciences.30 He also noted the series 1 + x + x*+ x3+xV • • • When x is equal to negative one, the sum of the series is l/l-x or ■§•. But if written with x equal to negative one, and expressed in the form (1-1)■►(1-1)+(1-1). . . the sum is zero, Leibniz refers to another Italian Jesuit Grandi, who saw in this example the proof of creation of the universe out of nothing, as revealed in the Scriptures. If the series had the sum of \ and y et was equal to zero, the divine power had truly created something out of nothing. 29 Tobias Dantzig, Number, the Language of Science (New York: Macmillan Co., 1933), p. 33. ®^Quoted in Robert E. Moritz, On Mathematics and Mathematicians (New York: Dover Publications, 1958), p. 163, 149 The climax of the work of Leibniz was to be a perfected, complete presentation of the faith, mathema tically presented# First he carefully set forth his qualifications for the work: It will be thought that the author has merely transcribed and problematized, and is probably a superficial mind little versed in the mathematical sciences and consequently hardly capable of true demonstration. In view of these considerations I have tried to disabuse everyone by pushing myself ahead a little further than is common in mathematics where I believe I have made discoveries which have already received the general approval of the greatest men of the day, and which will appear with brilliance whenever I choose# This was the true reason for ray long stay In France--to perfect myself in this field and to establish my reputation, for when I went there I was not yet much of a geometrician, which I needed to be in order to set up my demonstrations in a rigor ous way# So I want first to publish my discoveries in analysis, geometry, and mechanics, and I venture to say that these will not be inferior to those which Galileo and Descartes have given us# Men will be able to judge from them whether I know how to discover and to demonstrate# I did not study the mathematical sciences for themselves, therefore, but in order someday to use them in establishing my credit and furthering piety.31 Leibniz planned to set forth a demonstration which would be incontestable, and as certain as anything which had ever been proved by mathematical calculation. The method would prove to be of more worth to the mind of man than the telescope or microscope has proved to the eye of man# The demonstration was to be so complete and simple ^G ottfried Wilhelm Leibnez, Philos ophical Papers and Letters, Tr. and ed. by Leroy E. toomker (Chicago: University of Chicago Press, 1956), p# 400-401. 150 that any fallacies which result will be easily detected errors in calculation, Leibniz continues his letter to John Frederick, Duke of Brunswick-Hanover, in the Fall of 1679: This will become the great method of discovering truths, establishing them, and teaching them irresistably when they are established. Nothing could be proposed that would be more important for the Congregation for the Propagation of the Faith, F or, when th is language is once e s ta b lis h e d among missionaries, it will spread at once around the world. It can be learned in several days by using it, and will be of the greatest convenience in general Intercourse. And, wherever it is received, there will be no difficulty in establishing the true religion which is always the most reasonable, and in a word, everything which I shall develop in my work on Catholic Demonstrations. It will be as impossible to resist its sound reasoning as it is to argue against arithmetic. You can judge what advantageous changes will follow everywhere in piety and morals and, in short, in increasing the perfection of mankind. Before one remarks with amusement upon such thoughts, consider the twentieth century scientist who holds that the scientific method provides the only way to knowledge, and makes almost the same promises as Leibniz: A religion which seeks to become global in character cannot base itself upon supernatural revelation but instead must found itself upon a kind of truth and method of ascertaining it that can be found and accepted by all men everywhere— science and its method alone has given us this kind of truth; therefore, a global religion ought to found itself upon science and its method.33 3 2 L oc. c i t . 33G-race Edith Cairns, "Liberal Theology for a Global World," Journal of Bible and Religion, XVIII (April 1950), p. 106. 151 Napier. John Napier, the theologically oriented Scot, failed to realize the full importance of his inven tion of logarithms. The chief business of his life was to show that the pope was anti-Christ, but his favorite amusement was the study of mathematics and science. In 1588, the pope helped outfit a large fleet of Spanish ships, the famous Armada, against England. Queen Eliza beth appealed to the nationalism of her subjects, and they outfitted two hundred small, swift ships with guns heavier than the Spanish ones, and defeated the Armada. The invasion of the Spanish Armada in 1588 le d Napier, as an ardent Protestant, to take a consider able part in church politics. In January 1594, he published his first work "A Plaine Discovery of the Whole Revelation of St. John.” This book is regarded as of considerable importance.in the history of Scottish theological literature, as it contained a method of interpretation much in advance of the age.34 Both Napier and Newton felt that their mathematical works were of lesser value than their theological writings. Pascal felt that mathematics was something to be taken up or laid aside at the will of God. Mankind can consider themselves fortunate indeed, even if these greatest of contributions were only sidelines. One can only conjec ture what might have been accomplished if these men had considered their mathematical responsibilities as most im portant. °^E. W. Hobson, John Napier and the Invention of Logarithms. 1614 (London: Cambridge Univ. Press, 19lT7. 152 It la impossible to mention all of the contributions made to mathematics by religious men during the seventeenth century. Smith gives some religious connections of about thirty men during this period who made contributions sufficiently important to be mentioned in the history of mathematics. But one short example will be cited, and then a consideration of one of the most important men, Newton, Wallis, John Wallis, was for 54 years the Savilian professor of geometry at Oxford, He was also a doctor of divinity, chaplain to the king, Charles II, and one of the founders of the Royal Society. The recognition of his contributions to the development of mathematics suffers chiefly because of the greatness of his contempor ary, Newton, His work included the method of indivisibles, work on infinite series, the length of an element of a curve, and he made the first serious attempt in England to write the history of mathematics. He was the first to record the imaginary number graphically. He wrote a fanciful description of the three dimensions of a cubical body, extended infinitely to speak of the Spirit of the infinite God. Tho we should allow , th a t a Cube cannot be infinite (because a Body, and therefore a finite Creature): yet a Spirit may; such as is the Infinite God. And therefore no Inconsistence, that there be Three Personalities (each infinite, and all equal), 153 and yet but One Infinite God, essentially the same with those Three Persons. I add Further, that such Infinite Cube, can therefore be but One and those Three Dimensions can be but Three, (not more nor fewer). For if Infinite as to its Length (Eastward and Westward), and ab to its Breadth (North and South), and as to its Heighth (Upward and Downward), it will take up all imaginable space possible, and leave no room either for more Cubes or more Dimensions. And if this infinite Cube were (and shall be) Eternally so, its Dimensions also must be Infinite and Co-Eternal . ^ This was a part of a twenty page pamphlet entitled "The Doctrine of the Blessed Trinity Briefly Explained," enclosed in a letter to a friend, and dated London 1690. Newton. Galileo had solved the problem of the motions of objects near the surface of the earth. Kepler had explained satisfactorily the motions of the heavenly bodies. But there remained yet the necessity of combining in one great theory, the idea of motion. This was to be the contribution of Newton. Lagrange remarked later that Newton was the greatest genius that ever lived, and the most fortunate, since it is only once that a system of the universe can be established. The twentieth century theory of Relativity of Einstein may invalidate that statement of Laplace. Newton’s theory of gravitation has been called "indisputably and incomparably the greatest scientific discovery ever made." It was a fitting climax to the *^R. C. Archibald, "Wallis on the Trinity," American Mathematical Monthly, XLIII (January 1936), pp. 35-37. 154 heliocentric theory firmly established by Copernicus, and Kepler's laws explaining the elliptic orbits of the planets. Pope expressed the general feeling of the age when he sa id : Nature and Nature1s laws lay hid in night God said, "Let Newton be I" and all was light. The outstanding fact that colors every other belief in this age of the Newtonian world is the overwhelming success of the mathematical interpretation of nature. Galileo found that he could explain and predict motion by applying the language of mathematics to the book of nature. Descartes had generalized from this method and had pictured the universe as a great machine. Kepler had applied the rules to the heavenly bodies, and had discovered the lav/s he was seeking. Torricelli, Galileo, Pascal, Boyle, and Roemer, all had experimented and had found results which are yet taught in physical science classrooms. The great formulator of the seventeenth century science was born in 1642, the very year of Galileo's death. Although he did not publish his Principia until 1687, he had laid the foundation for most of his work before he was twenty-five. He had glanced at Euclid, and found it "trivial." He had had a poor start in school. But by his twentieth year, he was in Cambridge. During the years 1665 and 1666 the plague closed the University, and Newton was sen t home. I t was during th ese years th a t he thought 155 out the principles of gravitation, the fluxions or differential calculus, and did his work with Ught and the spectrum. Upon returning to Cambridge, he was recognized almost at once as a brilliant mathematician, and at the age of twenty-seven, Isaac Barrow resigned as the Lucasian Professor of mathematics in favor of Newton. Newton held the position for thirjty-nine y e a rs • Newton actively engaged in church discussions all his life, and for the last years of his life did very little in mathematics. His successor in the Lucasian chair, Whiston, was expelled from the University and lost his professorship in 1 7 1 1 for his outspoken Arianisra* Newton had followed a dangerous path in some of his work on the prophecies of Daniel and the Apocalypse. Newton always h e s ita te d to p u b lish his fin d in g s. Some controversy had arisen over one of his earlier works in mathematics, and it was only after great pressure, and financial support from the astronomer Hailey that the Principia was produced in 1687. Whereas Kepler had explained the motions of the heavenly bodies as related to the sun, Newton developed a more complicated system in which all bodies in some degree govern each other. Although the sun is so large among the other bodies that the influences of the smaller ones did not amount to much, the 156 theory had to take all that Influence into account. His greatest difficulty in the early years was due to the lack of understanding that it was mathematically correct to act upon the assumption that the whole mass of an object was concentrated in a point, Newton called upon God for the solution of problems which are explained by others in scientific berms. He believed that Cfod kept the fixed stars in their positions by active effect. Although gravity explained planetary motion, it didn’t explain the precise orbits, and Newton felt that God directly determined these paths. In this way, he used the theological affirmation of divine activity to solve scientific problems which demanded scientific answers. It is difficult to claim an active force of ’’attraction" between bodies, and remain scientifi cally objective. The Christian of the twentieth century may s till believe with Newton, that "In Him all things consist," or hold together, but he must also understand that the laws of nature can be described in mathematical formulas and scientific theories. The world was believed to operate like clockwork, implying a clock-maker, Leibniz said it looked like a rather bad clockmaker, if Newton needed a God to supply the initial amount of energy, and to constantly replace the loss. Further investigation disclosed the great 157 skill of the Divine creator, and the infinite perfection of his work, and this left less and less adjustment to be made. Newton’s religious writings were attempts to reconcile Christianity with reason. But his ultimate conclusion was to help men discard the doctrines that transcend reason. The end result was about the same as deism. For Newton, the fact that the work of creation was so perfectly executed that once completed, the God of Creation could rest, only showed the greatness of God more and more. Writing at the end of the seventeenth century, Newton was faced with the problem of setting Christianity upon a place of rational certainty. He tried to demonstrate the existence of God, and the reasonableness of Christian d o c trin e • He fell heir to a t raditional concern with natural religion that flourished among the earlier leaders of the s c ie n ti f i c movement in England. The p u b lic a tio n of Hobbe’s writings had raised the possibility that mechanical philosophy might conclude in atheism; the English scientists, led by men such as Robert Boyle and Rohn Rey, had taken up the challenge, seeking to demonstrate that natural science with its mechanical philosophy in reality offered conclusive proofs of the fundamental truths of Christianity. When Newton declared that the main business of natural philosophy is not to unfold the mechanism of the world but to re v e a l the F i r s t Cause, "And on th is account is to be highly valued," he was following the example set by Boyle and Rey.*6 3^Richard S. Westfall, "Isaac Newton: Religious Rationalist or Mystic?" The Review of Religion, XXII (March 1958), p. 167. 158 The basic question which faced Newton is one which is faced by scientists in the twentieth century# "Can a man who believes in both the method and the conclusions of modern natural science continue to be a Christian?" There are hundreds of Christian men teaching science and mathematics in the colleges and universities of the United States, who have answered "Yes," to that question. Dr. Henry M. Morris, Professor of Hydraulic Engineering and Chairman of the Department of Civil Engineering at the Virginia Polytechnic Institute is one who is actively engaged in answerin this question. In one of his latest books, he stated: The writer has spent a total of twenty-two years on the faculties of five universities (Rice, Minnesota, Southwestern Louisiana, Southern Illinois, and Virginia Polytechnic Institute). At each of these places, there were a number of men who were not only conservative Christians but who did not believe in evolution. At V.P.I., for example, the writer knows p erso n a lly more th an tw en ty -fiv e fa c u lty members who fit this description. None of these schools is a Christian school in any sense of the word; four are state supported universities and the fifth, Rice, is private, and its tone may be described by noting that Julian Huxley was on the faculty there for the first four years of its existence#37 Though it eventually set the stage for materialism, the idea of a mathematically designed and ordered universe was always the strongest argument of the existence of God, 37Henry M. Morris, The Twilight of Evolution (Presbyterian and Reforme’d Publishing Co., 1963), p. 27. 159 and thus formed the backbone of an unshakable and absolute body of thought and ethics. If man is just an accidental freak of nature, there seems to be little more left on the stage of the life drama than the pursuit of whatever the moment can offer. It is striking how far mathematical developments determined the attitude of men toward nature, society, and the meaning of life. And this has been more recently illustrated by the theory of relativity. For some people this means that what is "good” and "true" for one person may be "bad" and "false" for another. Religious bearings in China The bearings of religion upon mathematics during the seventeenth century extended beyond Europe. In the year 1603, the Catholic church had attempted missionary activity in China, but had been unable to enter the country. Loyola told St. Francis Xavier that China would welcome anyone who could further their knowledge of mathematics and astronomy. Mathematics is not commonly thought of as a missionary prerequisite, but it is not entirely strange even in the twentieth century. Ricci was the first graduate of a Roman College selected for China, and he became there a very successful mathematician, geophysicist and missioner. Among other things he computed the radius of the earth and translated the first six books of Euclid into Chinese. 160 The In te rn a tio n a l Geophysical Year (1958) was desperately in need of seismologists to man the seismographic observatory planned for Antarctica. So three Jesuits are therefore carrying out the suggestion of their missionary prototype Xavier, that one of the best keys with which to open missionary doors is a geophysical one.^8 Joseph Needham is acclaimed for his recent writings on the development of science and mathematics in China. He gives frequent mention to Ricci and his companions, and describes the mathematical work which was accomplished in the seventeenth century in Chinese by these men. The extent of the appreciation with which Matteo Ricci and his companions were received may be gauged when we remember that they were almost the only persons of foreign birth who ever attained the distinction of having their biographies admitted to the Chinese official histories. The translation of the first six books of Euclid into Chinese was undertaken by Ricci (Li-Ma-Tou), and Hsu Kuang-Chhi; it was completed in 1607.^9 This was followed by a treatise on European arithmetic in 1614, mathematical methods of the new calendrical system, and eclipses, and the first Chinese logarithm tables together with a discussion of them in 1653. Needham frequently points out that in the earlier ages much of the mathematics which was produced by Ricci had at least been indicated in Chinese works, but the ^®J. Joseph Lynch, "Jesuits and the IGY,” Thought, XXXIII (June 1958), p. 248. Joseph Needham, Science and C iv iliz a tio n in China (Cambridge: University Press, 1959), p. 52. 161 decay of the fifteenth and sixteenth centuries had made all of that nearly forgotten. The reason for the development of mathematics in Europe and the lack of in te r e s t in the O rient is d escrib ed by Needham as due to the extent to which such a development depended upon a belief in a "Supreme Creator and Upholder Deity*" The absence of the idea of a creator deit^, and hence of a supreme lawgiver, together with the firm conviction (expressed by Taoist philosophers in high poetry of Lucretian vigor) that the whole universe was an organic self-sufficient system, led to a concept of all-embracing Order in which there was no room for Laws of Nature, and hence few regularities to which it would be profitable to apply mathematics in the mundane sphere,"*0 This conclusion verifies the fears of Newton, Pascal and others in Europe who felt tint it was important to constantly reaffirm their religious faith along with their scientific activity, Man was prone to sink into a m aterialistic philosophy, which in the end would weaken the motives which had been responsible for the development of learning in the first place. Summary. The seventeenth cen tu ry was a period of intense mathematical and scientific activity. It was so well developed and so successfully described, that much of the work that was accomplished during that century set the pattern for the mathematics and science taught in schools until the present time. ^°Needham, o£. c i t ., p. 66. 162 Much of this developments •was carried out by men who felt that their chief r e s p o ixsibility was to their church and to God, They were s - Peking to use their God-given powers of o b s e rv a tio n s and reason to explore and to describe the earth and the .iverse, Thus the period of intense mathematical a c tiv i.- ■ and of equally fervent religious conviction, produced . great amount of information which can enrich a study of r e H_ _z: .gious bearings in the history and in the content of .thematics, The eighteenth century ■— n>ntinued the work of the seventeenth, and for a time, t ' s purpose of honoring the Creator was carried forward, : :>on, however, the science and mathematics became of g re a .' importance for their own sake. The universe seemed t o : sed no continuing force to keep it in smooth o p eratio n Mathematics production became almost completely a s e c i_ar matter. Religious bearings were almost lim ited t : « the teaching of mathema- tics in the schools. CHAPTER VI THE MODERN AGE The mathematical explanation of the universe which had been so assiduously sought by Galileo, Kepler, and Newton, seemed to be near completion. A few minor d iffi culties remained, but they would soon be cared for. The system of the imiverse had been mathematically interpre ted. This was the situation at the beginning of the eighteenth century. Descartes had prepared a mechanical view of nature, yet he had asserted that this perfect machine proved the eternal certainty of God’s will. Leibniz had sought to show by a similar mechanical pattern that man could find a single glorious design which would enable him to interpret the universe of thought as perfectly as he had interpreted the universe of nature. Christian Wolff, a mathematician of some note, was distressed by the conflict between Protestant and Catholic groups, and sought to demonstrate theological truth so that it could not be contradicted. He understood that this would be possible by the methods of mathematics. But in his determination to find truth in the broad mathematical, 163 164 rational form, he met the opposition of the orthodox theologians, and lost his position as mathematics profes sor at Halle for some years. Newton had credited God with the initial act of creation, and had found it necessary for God to make constant correction of the unexplained irregularities in the motions of the heavenly bodies. During the eighteenth century, Lagrange and Laplace showed that the supposed irregularities in the motions of the moon and planets were really periodic motions, and would not ultimately disrupt the universe. But while this reinforced the perfection of God*s creation for 3ome, it removed for others the need for the creator and sustainer. If the corrective measures of Newton were no longer necessary, perhaps the universe had existed forever, and the laws which explained the motions were eternal laws. Mathematics had been a comforting and a safe science for the Christian. The devout had felt that their religious convictions strengthened as they delved deeper and deeper into the secrets of the created world. A true man of learning was customarily well versed in several fie ld s and somewhat imformed in many others, and of these fie ld s mathematics was in many respects the preeminent one. The rationalists saw in it the discipline which displayed to perfection the power of the human mind as a deductive organ, while the empiricists were impressed by its indispensability for the interpretation of the data of experiment. It was generally regarded as the "true" science, and was 165 Indeed the only heritage from the ancients which was considered acceptable beyond a ll question. You will realize In the light of this why no man of learning would admit to an Ignorance of mathematics and since the subject was furthermore believed to have great bearing upon theology* and In fact* upon all meta physical questions* It Is understandable that the great mathematicians were objects of universal awe* admiration* and deference•^ The order which the mathematicians had found In nature was a proof of the existence of God. Euler felt that God's existence might even be provable by means of the principles of mathematics directly without using the design of the universe. But the theories which resulted from the work of these men seemed to other men to discredit belief in God. If determinism ruled nature* and man's reason was adequate to explain that rule* reason might be sufficient for all problems. Rationalism became the method of the new age. For the next two centuries and more, controversies between theology and science raged. The intellectuals had a religion, but it was deism, and to the Christians that was just a form of atheism. Many of the early leaders of the American government were d e ists, among them Thomas Jefferson* Benjamin Franklin, and several of the early presidents. •^Rudolph E. Langer, "The Life of Leonard Euler," Scrlpta Mathematics III (1935), p. 62. 166 There were also other men who attacked a ll religious beliefs. If everything could now be explained In terms of matter and mathematical principles, there could be no soul and no God. Deism does easily lead to atheism. Bishop George Berkeley sought to oppose the growing materialism which depended upon the existence of an external world consisting only of matter. Since man perceives only sensation and ideas, he said, there was no reason to believe that anything was external to him. According to the rationalist, religion was the product of human Ignorance and fear. But the age of reason attempted too much. Man has no assurance that reason alone is adequate to interpret moral^and ethical values. In the interpretation of physical science, a detached and objective view has been extremely productive. Man has been enabled to investigate fearlessly in many directions, and with great success. But such a detached and objective view is not so easily taken in areas other than science and mathematics. The story of the mathematics of the eighteenth century begins with Euler in Switzerland and in Russia. Years before, violent persecutions of Protestants in Antwerp had caused refugees to flee the country. One of these refugees was Jaoques Bernouilli, who fled to Frankfort. In 1622, his grandson settled at Basel, in 167 Switzerland, and the Bernouilli family, at least nine of them, were to achieve some fame in mathematics or physics. Leonard Euler, son of a Calvinist olergyman, lived in the neighborhood of Basel, and studied theology and Hebrew, because his father wanted him to be a clergyman as he was. Under the influence of the BernouilliTa, he chose rather to study mathematics. At the age of seventeen, Euler earned his master's degree with a paper comparing the Cartesian and Newtonian systems. Years la ter, in Germany, he met de Maupertius, the president of the academy in Berlin, who preferred the Newtonian philosophy to the Cartesian. Because of the con flict which arose between Newton and Leibniz over the right of priority in the invention of the calculus, the Continent and England had little to do with each other mathematically for many years. Leibniz had been unchallenged as the inventor of calculus for fifteen years, when i t was stated to the Royal Society that he had seen Newton's work earlier, had changed the symbolism, and published it as his own work. Leibniz complained of the accusation, and an a rticle appeared claiming that Newton had used the term fluxions to describe the differences of Leibniz, indicating that Newton had got the idea from Leibniz. After additional incidents, the Royal Society investigated, and reported that Newton had 168 Invented calculus first. They made no direct statement concerning Leibniz' claim or guilt. The conflict con tinued, John Bernouilli supporting Leibniz, and Keill writing in favor of Newton. Leibniz and Newton had been friends, but this argument separated them, and the result ing feeling held up the development of B ritish mathematics for one hundred years. England remained to ta lly ignorant of the brilliant mathematical discoveries on the continent. It was not u n til 1820 that England renewed work in d iffer ential calculus. It is generally agreed that Newton's notation was poor, and Leibniz' philosophy wa3 poor. Proper progress demanded the genius of both men. Berkeley, the theologian, joined the attack in England against the calculus, as Newton had developed it. Berkeley had said that the mind evokes ideas in the sequences described by physical laws that seemed to describe the * physical world. And obviously, God caused these sequences in man's mind. But mathematical ideas also seemed to predict the course of the external world. Berkeley.was disturbed by the report of a friend of his who had refused spiritual consolation when on a bed of sickness. The great mathema tician, Hailey, had convinced the friend of the inconceiva b ility of the doctrines of Christianity. Because of this incident, Berkeley wrote the Anatlyst in 1734. He addressed 1 6 9 It to an Infidel mathematician and attempted to show that the principles of fluxions were no clearer than those of Christianity* He called fluxions the "ghosts of departed quantities* " and added "Certainly* . • he who can digest a second or third fluxion* • • need not* methinks, be squeamish about any point In Divinity*" Euler systematized and unified vast theories which had been cluttered with partial results and Isolated theorems for years* He cleared the ground and organized many valuable topics In mathematics* Much of the ordinary college mathematics of the present day is as Euler left it. Just as Euclid had gathered and unified the many bits of information which had been developed before his day, and had synthesized them Into his Elements, so Euler systematized what had been done before his day* One of his equations is so concise that Felix Klein said all analysis was centered there. It is the expression e^-l® 0. This expression contains the numbers one and zero, the chief mathematical relations plus and equals, the impossible square root of minus one, and ei* the base of Naperlan logarithms* To one who expects to see evidence of a Designer In nature, this illustration is a good example from mathematics of such evidence* The rapid development of mathematics in the eighteenth century did not take place in the universities* 1 7 0 Ever since the sixteenth century, the universities had not been interested in science or other mundane subjeots. Theology, with an adequate preparatory arts course, became accordingly the chief concern of the universi ties, and to train and send forth the well-instructed divine, learned in the original tongues of the Old and New Testament, and completely read in the most author itative patristic literature became for the next three centuries almost the sole professed aim of either Oxford or Cambridge,2 It was in the academies, such as those at Berlin and Petersburg, where at this time mathematics was supported, but even so, both Russia and Germany had to import their mathematicians at that time. Euler from Switserland, and Legendre, Laplace, and Lagrange from Prance were the out standing mathematicians of the age, Newton was England*s only outstanding one, and he belonged to the previous century. Gauss was to be Germany's great mathematician, and he would appear later in the century, and on into the next one. These named men are the only men that David Eugene Smith classified as mathematicians of greatest importance for this period of time. It Is of passing interest to note that these men all lived past the age of seventy-six, a coincidence that would be hard to duplicate in the history of any special field, Lagrange and Laplace, though not directly related to religious ideas, are milestones along the path to the 2 J, Bass Mullinger, A History of the University of Cambridge (London: Longman's, Green & Co,, 1888/, p, ll7 . 1 7 1 complete secularization of mathematics and science* They are therefore of some importance to th is s tudy of the religious bearings in the development of mathematics. Lagrange was nineteen years old before he took an interest in mathematics* He thought out his great work, mecanlque analytlque at an early age, and concluded that a ll of mathematics had then been invented. Although he later became a great mathematics teacher, he did no more original work* According to Bell, he was an agnostic* He made the first complete break with Greek tradition, when he wrote his great work without using a single geometric drawing or diagram. He noted that fact in the preface to his work. Newton and the earlier mathematicians had often used diagrams to explain their work* Lagrange used only numbers and equations, and made his work entirely analytic. He is credited with the choice of the decimal base for the metric system, when others had planned to - use the duodecimal base. As a mathematician, he saw some advantages to using a prime number base, but this did not secure popular favor, Laplace is said to have had his first successes in theological disputes, but if so, he must have had some defeats also, for he became atheistic in his later years* His most famous work is the mecanlque celeste, in which he solved the general problem of stability of the solar 1 7 2 system. When he attacked this problem, there were experts who believed it couldn’t be done. Newton had believed that divine Intervention might be necessary from time to time to put the solar system back in order, and prevent It from destruction and dissolution. Some had doubted that Newton had actually accounted for the motions of the planets and satellites at all. It looked as though Saturn might leave the plane tary system, Jupiter might fall into the sun, and the moon might fall into the earth. The system as it was then known, was solved by Laplace, and avoided these difficulties. He proved the solar system was stable. In a sense, he showed that the solar system could be explained as a gigantic perpetual motion machine. With present day knowledge of the immensity of space, and the great number of facts of which Laplace was unaware, his explanation may not answer the entire question of the stability of the solar system to everyone’s satisfaction. If Laplace and those who believed in his results were right, It should be possible to use modern computers not only to predict the future, but also to recover aspects of past history of which we have no knowledge. Struik quotes Laplace in his summary of eighteenth century mechanical materialism: An intelligence which, for a given instant, knew a ll the forces by which nature is animated and the respective position of the beings which compose it, 1 7 3 and which besides was large enough to submit these data to analysis* would embrace In the same formula the motions of the largest bodies of the universe and those of the lightest atom: nothing would be uncertain to it, and the future as well as the past would be present to its eyes* Human mind offers a feeble sketch of this intelligence in the perfection which i t has been able to give to Astronomy.3 To Laplace the only reason for indeterminancy was lack of an intelligence large enough to grasp the whole picture* or a computer big enough to analyse the infor mation* Modem science* at lea st the physics of phenomena where quanta play a part* is no longer deterministic* But science is not sure that this may be only a transition period of indeterminancy resulting from lack of information rather than a lack in nature* It is possible that our present inability to foliar the thread of causality in the microscopic world is due to our using concepts such as those of corpuscles* space* time* etc*: these concepts that we have con structed by starting with the data of our current macroscopic experience* these we have carried over into the microscopic description and nothing assures us, but rather to the contrary, that they are adapted to representing reality in this field.4 Some sc ie n tists believe there are factors which render a basic Indeterminancy in reality unsolvable. But it is more scientific to say it is yet an unsolved problem in physics* Certainly* theologians and philosophers should not attempt to build on the present status of the Dirk Struik, A Concise H is to r y of Mathematics (New York: Dover Publications, 1948), p* 196* 4Louis de Broglie* The Revolution in Physics (New York: The Noonday Press* 1953), p* "517* 1 7 4 problem, nor assume Implications which are wild and fanciful. At present man's attempt to build a determin istic model gives puny results compared with the perfec tion of the universe. An editorial in the Saturday Evening Post makes the comparison: A frog has almost a perfect set of reflexes. If a flying object which he oan eat comes into view, his tongue flicks out faster than the eye can follow, and the Intruder is consumed. If on the other hand, some thing which can eat him appears, he makes a sudden leap to safety. Both actions are Instantaneous and automatic. With SAGE in its underground sh elters, ready to defent, and Mlnuteman missies in their buried sites, loaded and triggered to strike back, we'll finally have, after years of effort and billions of dollars spent, a fair imitation of the reflexes that God, in His wisdom, gave a frog. After that, perhaps, we can get on with the business of emulating another small creature—the mole. Unless, of course, there is some breakthrough-- not in science, but in the realm of the human spirit-- toward a greater understanding.® Laplace and Euler find their names In the history of mathematics, many times because of an incident which is told of each of them. As the incidents each relate to a religious view or expression, they are included here. The first concerns Euler's meeting with Diderot at the Russian court. Diderot was an atheist, but he was also somewhat of a mathematician, so it may be that the story told here has been contrived. The story appeared in DeMorgan's Budget of Paradoxes, and has reappeared in books from time to time, with modifications. Diderot 5Saturday Evening Post, (December 5, 1959), p. 25. 1 7 5 bad been rather free with his views, and the Czarina wished to suppress him and his views. Diderot was Informed that a learned mathematician was In possession of an algebraic demonstration of the existence of God, and would give It to him before all the court, If he desired to hear It. Diderot gladly consented; though the name of the mathematician Is not given, It was Euler. He advanced toward Diderot, and said gravely, and In a tone of perfect conviction: "Monsieur, (a+bn)/n=x, done Dleu existe; repondez I" Diderot, to whom algebra was Hebrew, was embarrassed and disconcerted: while peals of laughter rose on all sides. He asked permission to return to France at once, which was granted.6 Although th is is the story as DeMorgan stated i t , Brown in checking the source of the account found that at least three additions had been made to the original story: the formula had been changed, Euler was Identified, and the phrase "to whom algebra was Hebrew," had been added. Whether the Incident occurred or not, it does not cast any great credit on either Euler or Christianity. It reminds one only too readily of the many times when scorn and ridicule have been the chief weapons of the enemies of Christianity In sim ilarly unfair b attles. The other Incident concerns Laplace after he had presented a copy of mecanlque celeste to the emperor Napoleon. Someone had told Napoleon that Laplace had written a book explaining the universe, but had not used 6Requoted in B. Brown, American Mathematical Monthly XLIX (1942), p. 302-3031 1 7 6 the Idea of God as creator in the book* When Napoleon asked him w hy he had not done so, Laplace replied: "Sire, je n^vals pas besoln de oette hypothese*" The story goes on to say that when Napoleon told th is reply to Lagrange, the la tte r exclaimed, "Ah! c ’est une belle hypothese; ca explique beaucoup de choses."^ There have been many attempts to guess In what spirit Laplace made his response to Napoleon. It must not have been flippant. Strictly within the domain of mathematics and physics it is of course possible to say that one had no need of the hypothesis of God, just as his name does not appear in modern algebra books, to justify the binomial theorem, for example* Laplace probably meant more than that. If he were making a philosophical affirmation of the changes in the positions of the bodies of the solar system occuring by chance, he was departing from the Biblical account, and therefore expressing atheistic views. But he may have had Newton in mind. Newton had not been able to explain by his law of gravitation all the questions which had arisen in the mechanics of the heavens. He, being unable to show that the solar system was stable, expressed the opinion that special intervention ^"Sir, I did not need that hypothesis.11 "Ah, it is a good hypothesis, it explains so many things." 1 7 7 from 'time to time of a powerful external force was necessary to preserve that order. Laplace thought that he had proved the solar system to be stable, and In that sense, may have felt no necessity to refer to the Almighty. After Laplace, the work of the mathematician no longer was directed solely to the revealing of design in the universe. It was sufficient if mathematics could find inner designs within itself. But at the same time mathematics was being commanded to be very practical. Most of the mathematicians of the next decades were to be found in the military schools, particularly the Polytechnic school of Paris. Here the problems of mapping, laying trajectories, navigation, and the build ing of fortifications occupied more attention than the perfecting of astronomical theories. Although the geometric method would be tried again and again In matters of ethics and theology, this was not to r e str ic t the development of mathematics along secular lin e s. Cardinal Newman cast some of his arguments in a mathematical mold, Riemann attempted to use the method of Spinoza on the Book of Genesis, and similar attempts were made occasionally by others. The greatest advocate for the synthesis of mathematics and religion in the eighteenth century was Novalis, or Friedrich von 1 7 8 Hardenberg, a German poet and philosopher, who died in 1801, when not quite twenty-nine years of age. According to Novalis, there is no reason why God could not manifest Himself through mathematics. If God had permitted man to turn speculation on the nature of the universe into physics, alchemy into chemistry, quackery into medicine, astrology into astronomy, and more generally speculative branches of knowledge into systematic sciences with logi cal structures, why should it not be possible to continue this process and mathematize religion , magic, music, and even poetry. He said The purest form of science Is mathematics. The purest form of dogma is religion. Consequently, a reconciliation of science and dogma could be best achieved by a fusion of mathematics and religion . In anticipation of such a fusion, Novalis c&lls his Interrelated universal encyclopedia of knowledge a Scientific Bible, and boldly links various religious concepts with mathematics in his hymns to mathematics.8 In few cases did mathematics and religion come in direct contact with each other. Laplace considered his mathematics to be a tool, neither e v il or holy in it s e lf . The incentive back of the use of the mathematics determined whether the results would be good or e v il. Practically a ll the mathematics which has been produced since the time of Laplace has been secularly oriented. Much of it has been accomplished by religious men, but not directly for 8Martin Dyck, Novalis and Mathematics (Chapel H ill: University of North Carolina Press, 1960), p. 109. 1 7 9 religious alms. Religion did not provide the driving force for the development of new mathematics, but neither did it present any inherent restrictions. There is simply l i t t l e that is moral or religious about the manipulation of symbols, or the solving of numerical problems of a technological society. Fam iliarity with Laplace's mecanlque celeste proved for many years to be the distinguishing mark of a mathema tician, One was judged by what he could do with the understanding of it, Laplace had not oared for rigor, nor for mathematical precision. If he felt a conclusion to be correct, he just noted, "II est aise a voir," and went on. His assistants later confirmed that when he was revising his own work, he often could not trace the details of his reasoning, and failing to do so, but sure of the conclusion, would simply write again th is disarming phrase. Bowditch translated Laplace into English, and wrote a commentary on it to attempt to make it less difficult, Bowditch said, "I never come across one of Laplace's 'Thus it plainly appears,' without feeling sure that I have hours of hard work before me to f i l l up the chasm and find out and show how it plainly appears."^ America received its early mathematics from England, during the time when England and the continent were not speaking, mathematically. After England accepted the ® Smith, History of Mathematics, p. 487, notation of Leibniz in 1820, America began in 1824 to adopt much of the French mathematics. A comparison of the number of pages devoted to French contributions, with the number devoted to English contributions, gives evidence of the importance of the French mathematics education in the United States. Cajor^s The Teaching and History of Mathematics in the United S tates, shows one hundred seventy-nine pages for the French, to only fifty-four for the English. By the middle of the nineteenth century, it was reported that the United States had no mathematician, and only one astronomer, Benjamin Pierce. He was a student of Bowditch, and was professor of mathematics at Harvard for nearly fifty years. Original mathematics was slow in being developed in the United States. There were no established secularly- oriented universities, or sponsored royal societies to encourage such work. About the only mathematics in existence was the simplest forms of arithmetic and algebra, taught by divinity students, whose preparation was in the classics. Harvard established the Hollis professorship of mathematics in 1726, and Yale had its first chair of mathematics and natural philosophy in 1770. The first professors to occupy these chairs were clergymen, as were their several successors. The very first mathematics 1 8 1 professor 3 hip in the new world was at William and Mary, which had such a chair from it s founding In 1688, Rev. Hugh Jones is the ea rliest professor of mathematics whose name has come down in history. One of the contributions of a clergyman to the teaching of mathematics is of special interest: Of Rev. Samuel J. May, of Boston, it is said that nTo the work of teaching a public school he then brought one acquisition which was novel in that day, and which i t has taken a half century to introduce into elementary schools, private and public—a know ledge of the uses of the blackboard, which he had seen for the f ir s t time in 1813 in the. mathematical school kept by Rev. Francis Xavier Brosius, a Catholic priest of France, who had one suspended on the wall with lumps of chalk on a ledge below and cloth hanging on either side."10 Some of the mathematics taught in the early days of college work in the United States included evidences of the religious bearings in the subject. John Farrar, who was the first Amerioan to place translations of the continental writers in the hands of students in the new world, was educated in theology. But he became so involved in the work of improving the teaching of mathe matics, he did not enter the ministry for over thirty years. One of his former students, himself a mathematics professor, reported, I recall distinctly a lecture in which he exhibited in its various aspects the idea that in mathematical lOpiorian Cajori, The Teaching and History of Mathe matics in the United States (Washington: Government Printing O ffice, 1890)', p. 117. 1 8 2 science, and in it alone, man sees things precisely as God sees them, handles the very scale and com passes with which the Creator planned and built the universe; another in which he represented the law of gravitation as coincident with, and demonstrative of, the divine omnipresence; another, in which he made us almost hear the music of the spheres, as he described the grand procession, in infinite space and in immeasurable orbits, of our own system of the (so-called) fixed stars* R* C* Archibald gave this account taken from reminiscences by Prof. Emerson W. E. Byerly of relating to the teaching of Benjamin Piercej Pierce in the middle of a lecture on celestial mechanics said, "Gentlemen, as we study the universe we see everywhere the most tremendous manifestations of force. In our own experience we know of but one source of force, namely w ill. How then ean'.iws help regarding the forces we see in nature as due to the w ill of some omnipresent, omnipotent being? Gentle men, there must be a God. At another time he said, "May I close with the remark, that the object of geometry in all its measure and computing, is to ascertain with exactness the plan of the great Geometer, to penetrate the veil of material forms, and disclose thoughts which lie beneath them."12 In England, and on the continent, the teaching of mathematics began to appear in the u n iversities. For several centuries, theological requirements for degrees at Oxford and Cambridge caused some d iffic u ltie s , but in 1871, Cambridge removed any such restriction s. The University of London had been founded at least in part to by-pass these theological tests. DeMorgan was ■^Cajori, op. cit., p. 128. 18R. C. Archibald, Benjamin Pierce (1809-1880) (Oberlin: Mathematical Association of America, 1925;, p. 3. 1 8 3 professor of mathematics for some time at th is University* His mother had wanted him to be a clergyman, but he had taken up mathematics. He got his bachelor’s degree at Cambridge, but objected to the theological test necessary for the master’s degree, and did not receive it. In 1866, the chair of mental philosophy was vacated, and a Unitarian clergyman recommended for the position. When he was refused the position, DeMorgan considered the school was no longer religiously neutral, said resigned. Sylvester faced similar problems. He was educated at St. John’s College in Cambridge, and was second in the mathematics examinations. But he was not allowed to take a degree nor to receive a fellowship because he was a Jew. He went to Dublin for his degree, and in 1872, after Cambridge had abolished the theological tests, he was given both the bachelor’s and Master’s degree. He was unable to obtain a good position in England, so he came to America and was professor of mathematics in the Univer sity of Virginia in 1841. Following another extended stay in England, he returned to America in 1877, and with his work at Johns Hopkins University, made a great contribution to graduate work in mathematics In the United States. About one hundred years later, the United States was to become the adopted country of many mathematicians from the Old World, who sought freedom to continue their 184 research. A r n o ld Dresden*®*® liste d the names of one hundred twenty-nine mathematicians who came to America between the years 1933 and 1941. Among these names are Einstein, Weyl, Courent, Gamow, Teller, Godel, Stern, Neugebauer, and Polya. At a meeting, Dresden introduced the names with a statement from Priestly, who many years before had come to America because he knew that in America there would be "no objection to a person on account of political or religious sentiment s .11 Bell14 gives a striking summary of the great rate at which mathematics has been produced since the begin ning of the nineteenth century. He describes Cantor*s work of 3600 pages which outlined the work of mathematics up to the year 1799. It is estimated that the mathematics of the nineteenth century alone would require five times as much space to record, as it took for the whole of pre ceding history. There had been 17 mathematics periodicals during the 17th century, 210 during the 18th century, and there were 950 in the 19th century. Mathematical associa tions and societies were established in rapid succession in England, Prance, Scotland, Italy, America, Holland, ^Arnold Dresden, "The Migration of Mathematicians," American Mathematical Monthly XLIX (1942), p. 415-429. l4E. T. B ell, Men of Mathematics (New York: Simon and Schuster, 1937), p. l'TT 1 8 5 India, and Spain, from 1865 to 1911. The rapid develop ment could not be solely due to the technological demands raised by the problems of the new industries, England, the center of the Industrial Revolution, remained mathema tically unproductive for several decades. The greatest growth was in Prance and later in Germany, where the ideo logical break with the past was most sharply felt, and where sweeping changes were being made in all areas of l i f e . The new mathematics emancipated I ts e lf from the tendency to see in mechanics and astronomy the fin a l goal of it s in tellectu a l power. Throughout the history of mathematics i t has been thus; from the time of Plato, to the present time, the most lasting effects have been produced by ideas which have transcended immediate problems. The widespread production of mathematics could lead only to speci&lzation, and fragmentation. The last mathematician who could work in every field , was Gauss. He made outstanding contributions in both pure and applied mathematics. He also is described as a firm believer in '*an eternal, just, omniscient, omnipotent God.”15 1&G. Waldo Dunnlngton, Carl Friedrich Gauss (Dunnington, N. Y.: Exposition tress, 1955), p. 33. 186 Gauss kept his mathematics apart from his theology, for he said, There are problems to whose solution I would attach an In fin itely greater Importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lie s wholly beyond us and completely outside the province of science.16 This appears to be the prevailing view among mathe maticians. There were many Christian men who made great contributions to mathematics learning. Riemann was a sincere Christian who requested that Roman 8:28 be placed on his tombstone, "All things work together for good to them who love God." He developed a non-Euclidean geometry and performed some original work on Integrals In calculus. Georg Cantor established the infinite In mathematical terms. He was an expert theologian, but rejected the attempts of the Jesuits to use his mathematis for theolo gical purposes. Bell says, Cantor’s theory of the Infinite was eagerly pounced upon by the Jesuits, whose keen logical minds detected in the mathematical Imagery beyond th eir theological comprehension, indubitable proofs of the existence of God and the self-consistency of the Holy Trinity with its three-in-one, one-in-three, co-equal and co-eternal. . . . It is only fair to say that -Cantor, who had a sharp wit and a sharper tongue when he was angered, ridiculed the pretentious absurdity of such "proofs," devout Christian and expert theologian though he himself was.1? George Boole became a learned theologian and classic. Later on he gained an European reputation by 16Bell, op. cit., p. 240. 1?0p. clt. . p. 559. the originality and power of his methods in mathematical research* He showed how the laws of formal logic, which had been organized by Aristotle and taught for centuries in the universities, could themselves be made the subject of a calculus. His w ife, Mary E. Boole became famous in her own right as a teacher of arithmetic, and it is said of her, Of the more common subjects there remains now only mathematics, at first sight a most unpromising subject to deal with ethically* Yet Mrs* Boole has shown that not only can arithmetic be taught thus, but that it cannot very well be taught effectively in any other way* In general, the selected examples might largely be of an ethical character, and the need for precision in all thought might be illustrated by applying mathematical subject methods to what are ordinarily considered non-mathematical subjects.18 Thus, although the development of new mathematical content continued in a secular pattern, the teaching of mathematics could make use of opportunities to press for ethical and moral, and religious values. Though the development of mathematics became secular, the enumeration of the subject matter of higher mathematics reveals that many of the concerns of mathema tics are theological concerns as well: infinity, limit, change, the continuum, dimension, interpretation of phenomena, and explanation of creation and origins. But mathematics is only a tool which together with science 18Gustav S p iller, Report on Moral Instruction and Moral Training (London: Watts & Co., I'So'd), p* 33. 188 can tell how and what. They are not able to explain why things are as they are. The division In emphasis between mathematics research and religious persuasion is a natural result of the expanding body of information, and the Increased rise of specialization. The division should not result in conflicts, such as have arisen between religion and science. To the extent that sciences march forward only as they use mathematical processes, mathematics is involved in such controversy to a limited degree. Bernard Ranm suggests some of the reasons why the conflict between science and religion has resulted in such an apparent defeat of religion, at least in the circles of intellectual thought, and particularly in some formal educational circles: 19 1. The revolt of man from the authoritarianism of the Roman Catholic Church in its medieval expression. 2. The psychological and social advantage which the radical and the critic had over against the orthodox. 3. The amazing advance of science, both theoreti cal and technologically, while theology had no new dramatic product. 4. The cleavage of Eastern Orthodoxy, Roman Catholicism, and Protestantism, at the same time science was developing a measure of unanimity. 5. Twentieth century science education and the literary and classical clerical education which were developing people who were strangers to each other. 6. Science was being developed by non-Christians in increasing numbers. 7. The orthodox had no philosophy of science, but fought the battle on each narrow strip, one bit at a time. l^Bernard Ramm, The Christian View o f Science and Scripture (Grand Rapids: Eerdmans P ub. Co., 1 9 5 5), p.l8. 189 Twentieth century science is a great deal different from eighteenth century science. At least four great discoveries have changed Laplacefs stable world with the answers all neatly tucked away in the right files, into a world of uncertain conclusions, Heisenberg^ principle of indeterminancy, the non-Euclidean geometries, EinsteinTs theory of relativity, and Godelfs proof of the inconsistency of any mathematical system, are all matters which can be of significant mathematical import. But if they are placed on the philosophical or religious plane, and direct appli cation made, the results may be invalid or exaggerated. One might then deny all determinancy, discard all abso lutes, count all standards of morals and ethics relative, and deny that man can be assured of anything. Thi3 is extending these principles beyond the function of mathema t ic s . It is neither necessary nor desirable to separate theology and science, nor to divorce mathematics and religion. Man can observe what occurs in one area with out having those events become normative in all areas. The new facts and theories of science at least do not automatically give the scientist a right to exclude God from the world on the basis of what they now know of the universe. Differences of opinion will lead to differences In conclusions and in actions. For some mathematics and 190 religion are so different from each other, that there is no likelihood of there being any interrelationships. For others, these areas of thought are two separate matters, and it is only to the extent that the same person studies in both of them, that there will be integration or separ ation of the principles as he sees them. Perhaps for yet others, the two areas have reciprocal relationships which can be made e x p lic it, and which can in th is way provide enrichment for each other, without robbing either of any of its integrity. Summary. From the time of Newton and the seven teenth century, mathematics has followed a more secular development than at any earlier time. Lagrange and Laplace in the eighteenth century showed that the supposed irregularities in Newton*s explanation of the solar system were really periodic motions, and they taught that the universe could therefore be interpreted as a perpetual motion machine. Divine intervention was not denied, but mathematically speaking, was not included in the equations. The rapid increase of production in mathematics in the twentieth century, leading to specialization and frag mentation of learning, has continued to be almost completely secular in nature. Nearly all teaching of mathematics, which was formerly carried out by clergymen, is now in the hands of professional mathematicians and educators. Their it 191 task Is to assist each new generation In attaining mathema tical literacy at a time when great changes are taking place in science and technology. The invention of new systems of mathematics, and the development of new applications of much of this new mathematics, takes place completely outside of the influence of the church. But the teaching of mathematics, and the study of the history of its development, is, or should be a part of the educational background of all intelligent citizens of the twentieth century. Every Christian school and college will continue to teach mathematics. There is yet a place for the consideration of the religious bearings in the development of that mathematics through the earlier periods of history. PART II RELIGIOUS BEARINGS IN THE CONTENT OP MATHEMATICS 192 CHAPTER VII NUMBER No examination of the religious bearings of mathematics would be complete without a careful consid eration of the aura of mystery which has surrounded the use of number in the Scriptures. It is no longer possible to determine who first became delighted with the fascinat ing interplay of numbers and numerical values supposedly hidden within the words and the structure of the Bible. But some of the story of that interplay can be told. This interest and the study which has resulted from it, have been rewarded with at least a threefold success. In the first place, the Bible made use of many exact and rounded numbers. Comparisons of the uses of the same number as it appeared in different places, led to suppositions of the purposes of such numbers, and their symbolic meaning. Quite a large literature has been developed descriptive of the individual numbers, such as one, two, three, seven, ten, twelve, forty, and many others. In the second place, the numerical value of the letters forming a particular word led to the pseudo-science 1 9 3 194 £emjatrla, with almost unlimited ramifications. The Old Testament was written originally in Hebrew, and the New Testament in Greek. Both languages used the symbols of the alphabet as numerical symbols. Thus a group of symbols can represent either a word or a number, or in fact both at the same time. Although gematrla as such is not indicated in the Bible, the idea has so intrigued the minds of men, that much effort has been expended to find examples of such uses and to attach meanings to them. A great deal of study has also been done in the religious usage of gematria outside the scriptural words themselves. In the third place, there has been a symbolism developed by some who would seek to bolster the Divine origin of the Bible by setting forth a multiplicity of structural patterns, or coincidences. This has been carried out with an Intense fervor and much tedious work. In addition to these three ways in which the numerical relationships of the Scriptures have been sought and applied, there are isolated passages which have been given more than an ordinary amount of attention, because of some numerical value which occurs in the verse. Each of the three special patterns or forms will be discussed, and some survey given of the interest In only two of the passages of Scripture which mention 195 numbers of special interest. These two passages will be John 21:11, and the miraculous draught of 153 fish, and Revelation 13:18, and the number 666, the number of the anti-chrlst. Number symbolism in the Bible. In the religion and philosophy of ancient nations, numbers occupied a far more important position than they do in our modern civilization. This was especially true in the Eastern countries, as the records of Babylonia and Egypt, India, and China, show. From th is, it may be imagined that the comnentators on the Bible have only been in keeping with that mode of interpretation which they found given to other writings. •However, the commentators themselves disclaim th is ref erence to mysticism and numerology. They consider their work a sacred examination and interpretation of the Holy Scripture. The Bible is not a textbook in science, nor is it a textbook in mathematics. Textbooks in these two areas of knowledge are notorious in the speed with which they are outdated. When one considers the times during which the Bible was recorded, it must be considered a miracle indeed that the errors and incomplete scientific know ledge of that day was not included in i t . The message of the Bible is as applicable to the twentieth century as it was to the second century, even though many of the commentaries may be fanciful and transitory. Although the Bible was not intended to be an arithmetic textbook, for some students, it has provided material for arith metical investigation. The significance of individual numbers has been written up in great detail many times. Karpinski tells of one of the active workers in this type of investigation Mtystlcae Numerorus by Petrus Bongus of Bergamo, Italy, which appeared in 1583-84 and enjoyed seven editions. Some 400 pages are devoted to the dis cussion of the numbers from one to ten, in each case with copious reference to Biblical material. A beginning was made along the same lin e about a thousand years earlier by Isidore of Seville, born in the year 570 A.D.1 The writings of St. Augustine include an elaborate enumeration of the specific qualities of the various numbers. More recently, there have been equally large publications which attempt to explain the significance of Individual numbers in the scriptures. E. W. Bullinger in 1894 wrote a book, Number in Scripture, which explains In some detail the numerical design In "The Works of God," and "The Word of God," and then In some 250 pages, gives the spiritual significance of about forty- fiv e different numbers which appear in the Bible. In 1897, P. W. Grant produced the Numerical Bible, a commentary in seven volumes. This covered most of the ^J-Louis C. Karpinski, The History of Arithmetic (Chicago: Rand McNally & Co., 1§25), p. 63. 1 9 7 s Bible, the work being shortened by the death of the author. This commentary investigates the arrangements of the words, books, divisions, and verses, and then seeks to find the i number symbolism which is contained in them. His volume on the Psalms includes three Appendixes: ’'The Witness of Arithmetic to Christ," "A Study of the Numerical Symbolism of Scripture," and "The Numerals in Relation to the Six Days' Work." In 1939, LeBaron W. Kinney wrote a book entitled The Greatest Thing in the Universe, The Living Word of God. This book attempts to bring to the reader some of the beauties of the design and interpretation of the Word of God, through the study of its numbers. At first glance, this may seem to be a waste of effort, and a hopeless exercise in mystical numerology. The writers recognize this tendency and carefully point out that the symbolic meanings and the numerical structure found in the Bible have not been found in any secular writings. In general these authors have been devout men first of all, men who found in this study an additional manifestation of design in the universe, which pointed to the presence of a designer, Kinney tries to make this clear: We hope that no one will associate the patterns found in Scripture numbers with what is known as "numerology." It would be just as reasonable to connect the practice of the witchdoctor with the science of medicine, or the fortune-telling of the 198 astrologers with astronomy. Satan seems to have an imitation of almost everything that is true and help ful. No doubt his purpose is to turn people away from the true things. There are, however* designs in Nature and in everything God has made.5 The variety of meanings, and the length of the discussions in these writings need not be duplicated in this study. It is of interest to mention a few of the relationships which are thus indicated, by way of example. The number One, stands for solen&ss, unity, primacy; two indicates relation, second, difference, and division; three is harder to Interpret, but is generally considered geometrically, that is, it takes the third dimension to complete so lid ity or so lid measure, and it takes a third line to complete the enclosure of a space. Therefore three Is the symbol of reality, and of fulfillment• By application, One stands for God the Father, Two for God the Son, and Three, for God the Holy S p irit. Four, being a composite number is a sign of weakness, because it can be divided into two times two, but it can also be thought of a s weakness yielding to God when considered to be three plus one. So numbers may be m ultiplied, or added. The number seven, which is four plus three, is often related to the 2LeBaron V/. Kinney, The Greatest Thing in the Universe (New York: Loizeaux Brothers, 1939), p. 1 1 . 199 number twelve, which is font* m ultiplied by three* In cases such as this, the operations themselves enter into the symbolism. Six is the second number of true division. This division again represents manifestation of evil, but this evil is subject to God, as the number of conflict, two, yields to sanctification and the glory of God, three. Most often, the number six is spoken of as the number of man, as being short of the number seven, which symbolizes the perfection of God. Sixty-six is of course a mare certain expression of secular or human perfection, and the number six hundred sixty-six, is the trinity of human perfection or imperfection. This number will be considered in more detail a little later. The men who have written thus have recognized the dangers of letting the imagination run wild. There is always a danger that such a feature as this w ill be abused by some whose zeal outruns their discretion, but enough we trust has been exhibited to enable the student to observe and appreciate this remarkable feature of Holy Scripture.® It Is impossible to include all references and Illustrations which the mind can imagine or discover. In fa ct, It would be considered blasphemy by some of these investigators to imagine that anyone could ever complete such a work. God has provided an in fin ite ®C. H. Welch, An Alphabetical Analysis (Surrey, England: The Berean Publishing Trust, 1958), Vol. 3, p. 109. 200 variety of relationships for manfs continual and ever lasting search, and there is sufficient use of number to challenge the best intellect. On the other hand, enough lies near the surface to interest the simplest soul. Bullinger describes his purpose in writing his book: Many writers, from the earliest times, have called attention to the importance of the great subject of Number in Scripture. It has been dealt with, for the most part, in a fragmentary way. • . There seemed, therefore, to be room, and indeed a c a ll, for a work which would be more complete, embrace a larger area, and at the same time be free from the many fancies which all, more or less, indulge in when the mind is occupied too much with one subject. Anyone who values the importance of a particular principle w ill be tempted to see it where it does not exist, and if it be not there will force it in, in spite sometimes of the original t e x t .4 To those who are not in sympathy with such argu ment, the entire matter is irrational, and deserving of little consideration: From the abundant evidence of astrology and number mysticism it is easily seen that a quite fabulous and mystical—really, a non-rational—view of mathematics has frequently occurred in history. The notion of number or form was intriguing even in the early history of thought. Strange powers and divine influences were attributed to number or form either under the guise of some religious rite or mystical philosophy. This Interpretation of mathematical entities derives from the vagaries of an uncontrolled or excessive use of the imagination. As a consequence, even though this attitude has occurred frequently in the history of thought, exerted much influence and ^E. W. Bullinger, Number in Scripture (London: Eyre and Spottiswoode, 1894), p. vi. 201 even exists today, it can hardly be classified as rational, and will not concern us here.5 There is no question that numbers appear frequently in the Bible, and that man will continue to seek for some way to interpret their presence. After all, the argument goes, the Holy Spirit did not need that particular number to tell that story. If it were used, it must have special significance. Kennedy points out that the Bible i3 not concerned with the mathematics of man so much as it is with the relationship of God to man, and the plan of salvation. He draws some general conclusions: With what has been written as background, it would not be amiss to draw a few conclusions on the inter pretation of numbers in Holy Writings. The general principle of hermeneutics apply to all areas of Sacred Scripture: . . . (1) Numbers are to be taken in a real sense when there is no reason for under standing them in a symbolic sense. . . If for example, one desires to write to say that ten men walked down the road, he simply states that ten individuals walked down the road. Generally he makes neither mental nor verbal reference to the supposition that ten is a perfect number.' We cannot conclude that the sacred writer is always using a symbolic number, (2) Numbers may be taken symbolically whenever the writer directly or indirectly wishes to signify that his words are not to be taken strictly.6 His careful warning loses some of its force with the second point, for a reader can interpret for himself when the sacred writer intends a symbolic meaning. ^Edward A. Maziarz, The Philosophy of Mathematics (New York: Philosophical Library, 1950), p. 143. ^Gerald T. Kennedy, "The Use of Numbers in Sacred Scripture," American E cclesiastical Review CXXXIX (July 1958), p. 33. 202 One additional religious relationship with number has been of special interest to this writer. In the Hebrew language, the number fifteen would most naturally be written with the symbols for ten plus five, and sixteen would be written with the symbols for ten plus six. How ever, fifteen and sixteen are expressed in symbols as nine plus six, and nine plus seven, and appear this way on the modern Hebrew calendar, and in the paging of a book written in the Hebrew language. The reason is not hard to find. The symbols for ten and five taken in order represent the f ir s t part of the sacred name Jehovah, a word so sacred that once written it dare not be erased, and for hundreds of years was not even pronounced by orthodox Jews. The number sixteen is written as nine plus seven for a Aimilar reason. However, the numbers fourteen and seventeen are expressed in the form ten and four, and ten and 3even, because this is the normal form, and there is no religious reason to do any differently. Gematria. The second way in which numbers have been associated with religious activity is in the method of gematria. Every letter in the Hebrew and Greek alpha bet had the double meaning of a sound and of a number. The sum of the numbers represented by the letters of the word was the nuniber of the word, and two words were equivalent if they added up to the same number. This 203 method, of numerical reasoning appears in Greek stories, and according to Dantzig, is a part of the curriculum of the devout Hebrew scholar of today. The Talmudist will offer to call out a series of numbers which follow no definite law of succession, some running as high as 500 or more. He w ill continue this perhaps for ten minutes, while his interlocutor is writing the numbers down. He will then offer to repeat the same numbers without an error and in the same succession. Has he memorized the series of numbers? No, he was simply translating some passage of the Hebrew scriptures into the language of gematria.*1 The most contnonly considered suggestion of gematria in the Bible is that of Genesis 14:14. "Abraham took his trained servants, born of his own house, three hundred and eighteen, and pursued them unto Dan." The name of Abraham's steward, Eliezer, the only one of his servants named in the Bible, by gematria is equal to 318. But the Rabbi traditions have found hundreds of instances in which they could apply the principle. A few examples w ill su ffice for the purpose of th is study. Genesis 26:5. "Because that Abraham obeyed my voice, and kept my charge, my comnandments, my statu tes, and my laws." Abraham was 175 when he died, the numerical value of hearkened is 172, so Abraham is presumed to have known God at the age of three, and to have hearkened for 172 years. The numerical value of the name for Satan is 364, corresponding to the number of the days of the year during which he has the power of slandering Israel, the Day of Atonement being excepted. 7 'Tobias Dantzig, Number, the Language of Science (New York: Macmillan Co., 1933), p. 40. 204 In the book of Esther the word for "the silver" and the word for "the tree" each is equal to 165, a hint of Hainan fs ultimate end* It is no exaggeration to say that there were hundreds of such interpretations, and many rules by which letters could be interchanged or substituted, to force a word to have the number desired* It was even permissible In some cases to add or subtract one to make the word have the right value* From the early church fathers there is evidence of the Gematria ideas in the Gnostic heresy. Tertullian charges Marcus with the statement that Christ, in calling himself the Alpha and Omega, authorized the search for numerical values. One can imagine the satisfaction with which one of these dabblers in number symbolism and mysticism would record the fact that the name Jesus, or saviour, is equal to 888. And then he would go on to note that the sum of the names of Daniel, Hananiah, Mishael, and Azariah also added to 888* Or consider the coincidence that the word Satan in Hebrew Is equal to 13 x 28, and In Greek Is 13 x 13 x 13; while the word Dragon in Greek is 13 x 75, and the word serpent is 13 x 60. This would give the number 13 a significance which then could be applied to Judas as the thirteenth one at the last supper. And anyone in the twentieth century knows of the superstitions 205 that surround the number 13. Recently Time and Life magazines reported the construction of a new building* The one magazine told of the new forty-six story build ing, and the other magazine mentioned the forty-seventh floor* When an observant reader asked about the discrep ancy, he was informed that the building had no thirteenth floor* Most scholars look on gematria as a mnemonic device* They imagine that the rabbi did not consider the mystical values of importance, but used the numerical values to help in memorization* In olden times Jewish law was divided into the written and the oral. The former was represented by the Pentateuch which served as the constitution upon which all subsequent legal decisions were based* By means of the oral law the written law could be so interpreted as to make it applicable to progressive social development and thus insure the continuation of the law of Moses under all conditions and at all times. * . The interdiction of writing down the halakic decisions (3rd century A.D.), made it necessary to devise methods for preserving those decisions and interpretations for retaining them In memory* One of these methods was the numeric*11 There are many patterns in the Bible which illu s trate mnemonic devices used in the oral tradition. The 119th Psalm contains 176 verses, of which there are 22 sections of eight verses each. Many English Bibles head these 22 sections with the 22 letters of the Hebrew alphabet, ®Henry K eller, "Numerics In Old Hebrew Medical Lore," Scrlpta Mathematics I (September 1932), p. 133. 206 but most English readers do not know that in the original Hebrew, each group of eight verses begins with the same Hebrew letter, as a sort of memory device. That is, the first eight verses begin with aleph, the next eight with beth, and so on. This mnemonic device is accredited by some for the order of the alphabet which we have today. Anyone who has studied a foreign language knows that the most irregular verbs are the ones used most frequently, such as the verb to be, or to have. Also, one can identify books and tools which are most frequently used by the degree of wear they show. Consequently, one would expect something which has been used as frequently as the alphabet to have undergone many sh ifts of letters and positions or orders, since the letters were chosen. However, the aleph, beth, gimel; the alpha, beta, gamma; and the A, B, C, of the different alphabets have retained the same order. We must therefore assume that the early attachment of numerical values to the letters played a decisive part in the preservation of the alphabetic order. In a ll probability the arrangement of the Semitic alphabet was in a state of continuous change and flow, Pinal sta b ility and order was brought about through out the numerical values. By receiving the numerical values, the letters obtained the character of numerals and their order was as firmly established as that of the numerals. Thus the survival of the order of the letters is due to their numerical values, which had the effect, as it were, of a chain,9 9Soloraon Gandz, “Hebrew Numerals, “ American Academy for Jewish Research Proceedings V (1932-1933), p. 66. 207 The order of the letters of the alphabet brings to mind the complaint of the letters of the alphabet for the honor of forming the beginning of creation. The honor went to the beth for the word bereshith in Genesis 1:1. Aleph complained, and was told that the Torah of Sinai, the object of creation, would begin with aleph, (anoki—I am).10 One is further reminded that present day scientists in seeking for a universal space language by which they might identify intelligent life on other planets have suggested the sequence of the natural numbers as the proper key. What kind of signals may we expect to receive or should we send out? Probably the most abstrace and the most universal conception that any intelligent organism anywhere would have devised is the sequence of cardinal numbers: 1, 2, 3, 4, and so oh. The most likely signal would be a series of pulses indi cating this sequence repeated at regular intervals. Such a signal may upon f ir s t consideration appear to be too simple for the sophisticated task of cocmunicating with other beings far away among the stars. It would sound lik e baby talk . But after all, interstellar conmunication is surely still in the baby-talk stage.11 It is difficult to tell where science begins and mysticism ends when one begins to deal with the signifi cance of numbers. Edersheim t e lls how gematria was used 10Jewish Encyclopedia, I, p. 310. ^Su-Shu Hsiang, "Life Outside the Solar System," Scientific American (April 1960), p. 63. 208 to explain away objectionable passages In the Scripture, by substituting words or phrases of equal value. Thus i f in Numbers 12:1 we read that Moses was married to an "Ethiopian woman" (in the original, "Cushlth"), Onkelos substitutes instead of this, by gematria, the words, "of fa ir appearance"— the numerical value both, of "Cushith" and of the words "of fa ir appearance" being equally 736. By this substitution, the objectionable idea of Moses marrying an Ethiopian was at the same time removed.12 The editors of Scrlpta Mathematics have found an indication of the number mysticism to be found in the number £ i: As an example of such an interest we quote from a work en titled "Key to the Hebrew Egyptian Mystery in the Source of Measures Originating in the British Inch," the following remarks: "The numerical value of the Hebrew word Shaddai (The Almighty), equals 300 plus four plus ten, or 314, or with the decimal point in the proper place, the value of £ i correct to two decimal places. The numerical values (gematria) of the letters of the Hebrew word for God (Elohim) are 1, 30, 5, 10, 40, or dropping out the zeros, 1, 3, 5, 1, 4, which when put in the form of a circle can be made to read 3.1415.13 Almost no part of the Scriptures or of the other religious literature escaped the interpretation of the Cabala, or number mystics. The variety and amount of gematria evidenced in the Talmud is very great, and is available to any interested reader. Solomon Gandz suggests why the numerals took on such sacred meanings. ^Alfred Edersheim, Sketches of Jewish Social L ife. (New York: Hodder 8c Stoughton), p. 5§§0. 15Soripta Mathematics, VI (1939), p. 246. 209 It was because the letters of the alphabet were closely connected with the religious ideas which were expressed through their use. The international numerals (aramalc) were rational, secular, commercial. The national numerals became holy and sacred, participating in the religious character of the alphabetic letters. In religious life there prevails a certain law of contagion. Everything that touches an unclean object becomes impure and defiled. What touches the sacred thing becomes sacred. In the case of the alphabetic numerals, the profane numbers had touched the sacred letters, the national numbers had touched religious words. Thus the numerals and numbers had become sacred things, and in the gematria, a new sacred arithmetic was developed, a mysterious cabalistic theory of numbers.14 Numerical structure of Scripture. The third way in which number has been applied to religious literature, particularly the Bible, is the examination of its struc ture. In 1890, Dr. Ivan Panin discovered what he termed "The Numerical Structure of the Bible." He quit his work in philosophy to the dismay of William James who is said to have stated that it was a pity that Mr. Panin was "cracked" on religion . A great philosopher was spoiled in him. While reading John 1:1, Dr. Panin noted the definite article appeared before the word God in one place, but not in the other. He began to study, and spent over fifty years on research, which culminated in over forty thousand pages of notes, and the publishing of the Numeric Greek New Testament, and the Numeric ■^Solomon Gandz, op. c it . , p. 104. 210 English Now Testament, His work Involved not only a collection of many Items of numerical interest, but the prior production of tools with which to carry out his study, and organize his findings. He f ir s t prepared his own concordance of the Greek New Testament, and then a concordance of a ll forms of the New Testament Greek words. He prepared a sc ie n tific vocabulary in which each word was listed, together with eighteen columns of information about each word. On the basis of the structure revealed by his study, he claimed to have settled every one of the alternative readings le f t by Westcott and Hort, He determined by the "sevens" resulting from the number of times a word was used, or the number of words in a verse, whether one reading or another was in the original manuscript. For example, he found fourteen evidences of "seven" in the first verse of the Old Testament• Some very slight differences in the original text of the Hebrew can make drastic differences in the meaning. For example, in Deut. 6:4, if the Hebrew le tte r for four were replaced by the one for two hundred, and they look nearly alik e, the verse, "The Lord our God is one Lord," would read, "The Lord our God is a false Lord." And again, the early manuscripts were written without word spacing, and it is important to be able to divide the 211 letters at the right place. Consider the English sentence, GODISNOWHERE. This can be divided into words to read GOD IS NOW HERE, or GOD IS NOV/HERE, certainly very different meanings being derived in the two instances, McCormack in 1923 wrote a 400 page book en titled The Heptadic Structure of Scripture. He stated, Some indeed suggest that the sacred signification of seven probably originated with the week of seven days and the Divine rest on the seventh. But the opposite is more probably the truth, that God stamped the number seven on time because of the sacred significance of the nuraber.15 The number of "sevens" which Mr. McCormack found in his work is amazingly large. At the risk of being petty, the writer notes that in one instance, when he arrived at a number fifteen, which could not be heptadically connected, he stated, tlfifteen--a number strongly suggestive of fourteen." He went on to give several ways to restore the number to fourteen, and concluded with the statement, "The matter requires further investigation." There are many other number references which might be cited. For example, after Cantor had chosen the aleph as the symbol for infinity, a mathematical society in Italy placed an aleph before the name of each life-member of the society to signify permanence of membership. Some numbers have received a great deal of special attention. 15R. McCormack, The Heptadic Structure 0f Scripture (London: MarsTiall Brothers, Ltd., 1923), p. 43, 212 The numbers 153 and 666. There remains yet a consideration of two of the numbers which have had great significance to commentators since the early days of Christianity, The first of these two numbers is in John 21i l l , HSimon Peter went up, and drew the net to land full of great fishes, an hundred and fifty and three: and for all there were so many, yet was not the net broken,” A great many b ib lica l commentators ignore the number, and others simply see in its use, the fact that John was an eyewitness to the event sixty years Before, He recalled that number, just as he had recalled the 3ix waterpobs at the wedding recorded in John 2:6, and the fiv e loaves and two fish es which fed the five thousand, recorded in John 6:9-13. He even remembered in the latter case, the twelve baskets of fragments left over, St, Augustine, in the fifth century recorded some fanciful meanings of this number. Since that time, there have been about twenty interpretations of the number. It is not possible to consider the theological implications of these twenty interpretations, but only to mention a few of the ways in which the number was obtained. If one takes the number of the ten commandments, and adds to that the seven gifts of the spirit, he gets seventeen. The seventeenth triangular number, or the sum 213 of the numbers from one to seventeen is 153* Or, one may multiply the seventeen by three, the number of the trinity, and multiply the result, fifty-one, by three again, to get 153. If one multiplies seven by seven, and adds one, and then multiplies by three, the number of the trinity, and then adds three, he will get 153. Some have suggested the number refers to the 153,600 strangers mentioned in II Chronicles 2:17, or to a popular misconception that there were 153 varieties of fish in the waters of the sea. By gematria the name of Simon Jonah was made to equal 153, and the names of the two cities between which the fishermen will spread their nets when the Dead Sea has been reclaimed, are of value 17 and 153. These cities are mentioned in Ezekiel 47, and are En-Cedi, and Eta-Eglaim. One person squared twelve, the number of the church, squared three, the number of the tr in ity , and added the resulting 144 to 9 and found 153. McCormack, on the lookout for 11 sevens" suggests that if one adds the 153 fish in the boat to the one which Jesus was cooking on the beach, he will obtain 154, which is divisible by 7. The expression "Sons of God," in the Hebrew language, by gematria totals to 153. A Lt. Col. checked the total number of people who were specifically blessed 214 by Christ when he was on earth, and found the number to be exactly 153. The registrar of St. Joseph College of Rensselaer, Indiana, recently wrote a facetious article in which he played around with the numbers in an attempt to show that more fanciful things could be done with them than St. Augustine had thought. The other verse which merits some special consider ation is Revelation 13:18. "Here is wisdom. Let him that hath understanding count the number of the beast; for it is the number of a man; and his number is six hundred three score and six." This is the famous 666. There have been as many different applications of th is number as there have been of the number 153. In using the Greek alphabet for numerals, there is no letter in the present alphabet for the number six. One substitution was an obsolete letter "stigma," which has been thought by some to represent a serpent. Three of these together would represent the nunber 666. The usual way of representing the number would be by using the le tte r s for 600 and 60 and 6. The most common interpretation for th is number has been to apply it by gematria to someone whom you expect to prove is the antichrist, McCormack states ^Charles J. Robbins, "A Fish Story," The Priest XVI (December 1960), p. 1091.. 215 that he had gathered about forty names for whom the number 666 had been figured in one language or another. Most scholars agree that one can apply It to Nero in this way. Generally it is considered to be the three fold use of the number of man, that is, six. Six is one short of seven, which is the perfect number, and the sacred number assigned to God. 666 Is the sum of a ll the numbers from one to the square of six, or thirtysix. It Is the sum of the letters used by the Romans to represent numbers, DCLXVI. This fact has been interpreted to mean the number of the enemies of God is legion, or many. Stifel, when reading the 13th chapter of Revelation, was struck with the thought of the Pope, Leo X, being the a n ti-ch rist. He chose the Roman numerals in the le tte r s of Leo Decimus, and found the sum to be too great by M, and too little by X; but he took the fact that there were ten letters in the name, and that the M stood for mysterium, and the number came out perfectly. This discovery gave him such unspeakable comfort, that he believed his interpretation must have been an immediate inspiration of God. The same general principle was applied to Martin Luther by Peter Bongus, in both Latin and in Hebrew. In the twentieth century, Gerald B. Winrod assigned the 216 number to Franklin Delano Roosevelt, by adding the word Rex, and choosing the letters which had values in the Roman numeral system. One very ingenious writer applied the number to the Ku Klux Klan, by taking the three letters K.K.K. As k is the 11th letter of the alphabet, if one uses the periods as multiplication signs, he can get 11 x 11 x 11, or 1331, Add the founder of the organ ization and get 1332. As there are two bea&s mentioned in Revelation, divide 1332 by 2, and the answer is 666* Summary. Numbers and the numerals which represent them in mathematics are in themselves secular, and man- made. The numerals are symbols or designs which are used to represent quantities. They do not have any religious significance. But it is easy for a devout person, as well as for a mystic cultist, to see in the quantities represented, religious connotations. The great number of times the number seven appears in the Bible cannot help but strike the attention of even the casual reader. The emphasis on the Trinity suggests it s e lf whenever the number three is given, and the significance of such repetitions as twelve tribes, twelve apostles, and twelve gates of the New Jerusalem, seems to ask to be discovered. Serious students of the Bible occasionally find in the use of some such significance, an interpretation which adds to their feelin g of worship and devotion. How much 2 1 7 consideration should be given to this study, and such a feeling, is an individual matter. Certainly it is not the subject of arithmetic as it should be taught in the schools. The symbolic meaning of particular numbers, the numerical value of words by gematria, and the numerical structure of the Scriptures has been presented in this chapter. Is there anything of value in such study which could be taught in school? Should children in school be given problems in which they add the number of prayers said, or the number of pennies given to missions? Or should a conscious effort be made to make every study of arithmetic so secular that no fanciful ideas would have any consideration? The writer believes that a real understanding of numerical relationships must involve the student in a lively and meaningful consideration of number. In a Christian school, the normal classroom discussions w ill include the religious significance of the ideas of this chapter, as they arise. CHAPTER VIII MATHEMATICS—RELATIVE OR ABSOLUTE TRUTH Mathematics as absolute truth. It is commonly believed that if truth is anywhere existent in the secular world, it exists in mathematics. Very often, it is asserted by someone, that mathematics possesses not only truth, but the absolute truth. For example, a recent book in science compared two kinds of truth: Many would say that the objective of science is to discover truth, to find out fact*. We must be very careful here about the meaning of words. "Truth" is popularly used in two senses. It may indicate a temporary correctness, as in saying, 11 It is true that ray hair is brown." Or it may indicate an absolute, eternal correctness, as in saying, "In plane geometry, the sum of the angles in a triangle is 180 degrees This statement appeared in a textbook published about one hundred years after the introduction of several non-Euclidean geometries in which the sum of the angles of a triangle could differ from 180 degrees. It does help to refer in the context of the statement to plane geometry, but one questions whether this limitation permits the designation, "absolute, eternal correctness." ^•Paul B. Welsz, The Science of Biology (New York: McGraw-Hill Book Company, Inc., 19597, p. 10. 218 219 This very problem in geometry is one which brought about a great change in viewpoint toward "truth” in mathematics. For hundreds of years, mathematics had been looked upon as containing unfailing truth, or as being in itself, absolute truth. It had provided a stronghold for those who taught that the laws of mathematics, as they revealed design in nature, proved the existence of a Creator. Edward Everett, eloquently voiced this view in the latter part of the nineteenth century when he said: In the pure mathematics we contemplate absolute truths, which existed in the divine mind before the morning stars sang together, and which w ill continue to exist there, when the last of their radiant host shall have fallen from heaven. They existed not merely in metaphysical possibility, but in the actual contemplation of the supreme reason.2 Mathematics deprived of absolute truth. With the understanding of the validity of non-Euclidean geometries, and their application through the law of relativity, the place of truth in mathematics was subject to re-examination. Bertrand Russell, in 1901, summed up the new approach with his famous definition: Pure mathematics consists entirely of assertions to the effect that if such and such proposition is . true of anything, then such and such another propo sition is true of that thing. It is not essential to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true. • . If our hypothesis 2Edward Everett, Orations and Speeches, quoted in Robert Moritz, On Mathematics and Mathematicians (New York: Dover Publications, Inc., 1958), p. 45. 220 is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is tru e.3 But Russell also stated in 1907, ’’Mathematics, rightly viewed, possesses not only truth, but supreme beauty, . . . sublimely pore, and capable of a stern perfection such as only the greatest art can show.”4 Fifty years later, he explained how he had modified his views: All th is, though I s t i l l remember the pleasure of believing it, has come to seem to me largely non sense, partly for technical reasons and partly from a change in my general outlook upon the world. Mathematics has ceased to seem to me non-human in its subject matter. . . . I cannot any longer find any mystical satisfaction in the conteinplation of mathematical truth. . . In th is change of mood (World War I) something was lost; though something also was gained. What was lost was the hope of finding perfection and fin a lity and certainty. What was gained was a new submission to some truths which were to me repugnant•5 One result of non-Euclidean geometry and the law of r e la tiv ity was that mathematics was deprived of its status as a collection of truths. In fact, it seems that many scientists and mathematicians felt that they 3Bertrand Russell, Mysticism and Logic (New York: Barnes & Nobel, 1954), p. 75. 4Bertrand Russell, "My Philosophical Development," The Hibbert Journal, LVII (October 1958), p. 2. ^Loc. c it. 221 had been robbed of th eir moat respected truths, and per haps even of the hope of ever attaining certainty about anything. The non-Euelidean geometry of Lobachewsky was called a landmark in the search for truth, for "It eman cipated us from the absolutism of Euclidean geometry and made clear that all truth is relative."6 Kline discusses some of the effects which these ideas had on the relation of truth to mathematics. Before 1800 every age had believed in the existence of absolute truth; men differed only in their choice of sources. Aristotle, the fathers of the Church, the Bible, philosophy, and science all had their day as arbiters of objective, eternal truths. In the 18th century human reason alone was upheld and this because of what it had produced in mathematics and in the mathematical domains of science. The possession of mathematical truths had been especially comforting because they held out hope of more to come. Alas, the hope was blasted. The end of the dominance of Euclidean geometry was the end of the dominance of all such absolute standards.7 The conclusions in the last two quotations go too far. Mathematicians could no longer' find absolute truth in their mathematics, and concluded, "All truth is rela tive." James Russell Lowell expressed the impossibility of attaining absolute truth through man-made ideas: New occasions teach new duties; Time makes ancient good uncouth. They must upward s t i l l and onward Who would keep abreast of truth. 6Mimerographed class notes, Ohio State University, 1949. ^Morris Kline, Mathematics in Western Culture (New York: Oxford University Press, 1953), p. 430. 222 Actually, mathematics had Just been shorn to be <• another in the long line of hopeful pretenders to the throne of absolute truth* True, mathematics might need now to settle for something other than absolute truth* But that gave no assurance that absolute truth did not at all exist. One was now sure that Euclidean geometry did not have a monopoly on truth* There may be areas containing equal truth; and certainly there was no valid reason to deny that the possibility of absolute truth existing in some other realm of thought was as great as ever. Mathematics was incapable of doing the whole job. E. T. B ell, in the Development of Mathematics, makes a dozen or so references to the fact that mathematics once claimed to be absolute truth, and that it no longer made that claim. Some of his statements are simple state ments of fact, others are charged with argumentative over tones. Two examples are given as illustrations: p. 182. Ultimate realities and eternal truths, at least in mathematics and science, suffered an eclipse in the 20th century. p. 330. It seems improbable that our credulous race is likely ever to get very far away from brute- hood until it has the sense and the courage to discard its baseless superstitions, of which the absolute truth of mathematics was one.8 8E. T. B ell, The Development of Mathematics (New York: McGraw H ill Book Company, Inc., 1945), p . 182, 330. 223 Mathematics was declared to To e independent of reality, and as a result independent of any science to which it may be applied. Kline points out that Because the twentieth century must distinguish mathematical knowledge from truths, it must also distinguish between mathematics and science, for science does seek truth about the physical world.9 Mathematics was not able to offer absolute truth to science, nor could mathematics provide religion with absolute verification. But these conclusions need not lead to another error, namely, that mathematics would deny the privilege of the attainment of truth to other of manfs great concerns. The fact that mathematics can do valid calcula tions with dimensions beyond the four of our space- time universe is not the sligh test proof that the Christian heaven e x ists. It is , however, proof that scientists and philosophers cannot say dogma tically that the Christian heaven does not exist.10 Mathematics did not lose any of its effectiveness by being deprived of its claim on absolute truth. Rather, it has been freed to explore, at any length, the possibil ities in any question, and with any set of axioms, so long as the investigation is of interest to the investiga tor. It may or may not apply to the physical world. That fact is no longer necessary for it to be valid 9 Kline, 0 £. cit•, p. 9. ^Rachel H. King, "My Pilgrimage from Liberalism to Orthodoxy." Christianity Today (December 6, 1963), p. 7. 224 mathematics* If the calculations and conclusions in a system can be matched with some aspect of nature, it is addition good applied mathematics* Teachers of elementary mathematics generally do have the physical world in mind when they select undefined terms, and define other terms. The postulates are stated as accurately as possible in terras of the observed facts about the physical environment* It is therefore general ly expected, that most of the mathematics with which students have to do, and with which scientists bperate, applies to the physical world quite well. It is not so certain that the results will be equally fruitful when the mathematician selects postu lates which are not intended to apply to the physical world. Many times the results are barren, and many times they are fr u itfu l only after the passage of time* That non-Euclidean geometries did eventually have applications was one of the surprises of mathematical history, and a bombshell to philosophy and science. 1 The arguments for and against the premise that truth can be found in mathematics, are usually related to the development of those non-Euclidean g eometries in the nineteenth century* This development showed that space was not limited to a single explanation. Since men had ^iHollis Cooley, and others, Introduction to Mathe- matics (Boston: Houghton Mifflin Company, 1949), p, 607. 225 for so long equated Euclidean geometry with truth, the question was raised, "which of all the geometries that man has developed, is the true one?" Riemann showed that there existed an infinite number of geometries that seemed to be free from contradictions. Later, Klein was able to show that any or all of these many geometries are "true." These geometries Were like the languages spoken by people. To ask which one of these geometries is true, is, for all the world, like asking the questions "Which, of all the languages of the world, is the true language?" The question really has no answer.12 Or, one might ask which is the true rainbow in the sky? The maindrops which enable one person to see a rainbow, do not form the rainbow for any other person. Yet they all may stand near each other and see the rainbow in a ll its glory. The fact that these different types of geometry have been developed in modern mathematics shows clearly that mathematics cannot be said to assert the truth of any particular set of geometrical postulates; a ll that pure mathematics is interested in, and all that it can establish, is the deductive consequences of given sets of postulates and thus the necessary truth of the ensuing theorems rela tiv e ly to the postulates under consideration. . . and the postulates of geometry cannot be considered as "self-evident truths," because where no assertion is made, no self-evidence can be claimed.13 ^W illiam MacMillan, The Doctrine of Uniformity in Religion. (Chicago: Patton Reporting Service, 1935), p. 4. 13Carl G. Hempel, "Geometry and Empirical Science," Our Mathematical Heritage-(New York: Collier Books, 1963), p . 1 § 1 . 226 However, it should be made clear, that within a particular mathematical system, an absolute truth of a very real sort can be built up. It is a finite system of thought, but within its bounds, there is security and certainty, Man is able to define mathematical systems in such a way that for the study of properties of number and spatial relations, the conclusions have absolute truth within the system. The mathematician speaks of closure: If a set of numbers is closed under certain conditions, it is like a self-perpetuating corpor ation, or better, like a walled city, whose gates are closed and guarded. There is no way out for him who obeys the prevailing laws; there is no way of getting out of the set of numbers so long as we limit ourselves to the operations with respect to which the set is closed. It is comfortable and conservative; no surprises in the form of unsolv- able pnoblems arise, but neither is there an inkling of new worlds beyond the walls, • • • Extension is made through the inverse of an operation with respect to which the set of numbers is closed. Here is significance far beyond the domain of mathematics. . • the Golden Rule, Alice in Wonderland , international relations, and so on. ^ Most elementary and secondary school teachers work with mathematics within such closed systems. They are not involved in the production of new systems of pure mathema tics, nor the building of abstract mathematical structures. The mathematics with which they have to do may not represent the whole of absolute truth, indeed, it doesn't represent the whole of mathematical truth. The search for truth in • G. Jacobi, "We Must Always Invert" in Arnold Dresden, An Invitation to Mathematics (New York: Henrv Holt & Co., 1^36), p. 40, 227 the physical world is best conducted according to mathe matical methods. And the understanding of these methods may be worth as much or more as the content of the system of thought. The process of the search for truth is an important process, and may be one of the greatest contri butions which mathematics can make. It is not enough that students learn the mathematical content, and the rules of calculation whereby it may be manipulated. It is also imperative that the postulational structure of mathemati cal thought be made clear. Both content and method are of great practical value, and have proved their dependa bility in their application to the physical world. Truth in mathematics in Christian schools. Nearly all expressions of objectives in the study of mathematics in the publications of Christian schools relate mathematics and truth in some way. Generally the statements are cautious, and carefully lim it the extent to which truth is to be found in or through mathematics. Only rarely is mathematics referred to as the truth in itself. For example, a Christian Reformed group affirms: Mathematics, then, is a product of nan's cultural activity. The system man developed from those mathe matical principles everywhere present in creation is his interpretation of a part of the whole body of truth which is God’s general revelation to mankind. And teaching mathematics becomes a calling to impart to the child this part of God’s revelation in a clear, meaningful way according to his ability to receive it. Mathematics is a medium in which the school forms a maturing Christian—a child who, 228 first learning only to group and count objects, grows Into one who uses numbers in every part of his devel opment, who begins to see mathematics as a part of the whole pattern of truth, and learns to praise the God of truth, and in whom there is a growing sense of responsibility to place this gift along with all others in the service of the Giver. & In a recent teacher’s guide for an arithmetic series of textbooks prepared for use in Catholic schools, the statement of the "Purpose of the Program," included the expression: The authors of an arithmetic program for Catholic schools must always be conscious that the learner is a child of God, endowed with intelligence and free will. This child must be thoroughly instructed in the science of quantity, basic number facts and processes and their relationships, basic measure ments, and problem-solving. Like a strand running through this instruction is the brightness of mathe matical truth. This truth is sacred and the sacred ness of this truth must be woven into the mind and heart of the child. Usually truth is spoken of as a goal toward which all subject matter areas are to be directed. And of course, this includes the study of mathematics. Through the order and relationships of the num- system, children can be given a beginning glimpse of the infinite truth and beauty of the God from whom all that is good derives.1' 15Mathematics Curriculum Guide (Grand Rapids: National Union of Christian Schools, 1958), p. 9, 10. 16Finding Truth in Arithmetic. Primer and Book O n e , Teacher’s Edition TBoston: Ginn and Company, 1958), p . 1 ^■^Slster Mary Stanislas, New Ways in Numbers (Boston: D. C. Heath and Co., 1962), p. 1. 229 But mathematics arrives at truth, science arrives at truth, theology arrives at truth, and if truth is one, and if truth is a reflection of True (another way of saying God), there must be some way to tie all loose ends together.18 Most often, the teacher is thought to be the key element in any religious education setting. Two educators, one Protestant, and the other Catholic, speak from a life time of work in Christian education: Surely the conclusion is inevitable that a subject so close to the way in which God works and—we say it reverently--so close to the way in which He thinks, is well inside the pattern of His truth. Consequently, it follows that the Christian teacher of mathematics m ust know su ch th in g s as th e common ground of un- pro vable knowledge shared by mathematics and Christ ianity, the presence of number and order throughout nature and art, and the perfect congruity of the stars with mathematical calulation. Out of these facts end others like them, he can show his pupils more than is generally realized. Young people can wonder at the wisdom of the God of mathematical truth quite as much as they marvel at the Creator of the great mountains, the restless oceans, and the star- decked heavens.19 If, however, out of his devotion to the truth, without special pleading or forcing the note, he, (the teacher), happened to say something rather more luminous than most of his contemporaries are saying, that also I would think, might emerge from the nature of the case. But here we should be above the level of any sectarian controversy-- shouldn’t we? Because truth is not for or against anything. Truth simply is.80 18J. P. Whalen, "Integration of Theology, Science, and Mathematics,11 (Catholic Education Review, LV (October 1957), p. 470. 19Prank E. Gaebelein, The Pattern of God's Truth (New York: Oxford University Press, T&5477 P* 63. 80Aelred Graham, "Toward a Catholic Concept of Education," Harvard Educational Review XXXI (Pall 1961), p . 412. 230 Mathematics is not asked, to give absolute truth* nor to supply the whole of truth. But it is asked to contribute to the whole program of education truth which it can best supply. In this way it can help to develop greater capacity and appreciation for the identification and understanding of truth which is found in other areas of thought. Truth in mathematics in non-religious education. It is not only the Christian educator who speaks of truth in mathematics. The Bulletin of the National Association of Secondary School Principals in 1954 stated: Algebra puts arithmetic into a challenging differ ent setting. It distinguishes between "truth” that is relative to postulates and the "absolute truth" of the pupilfs religious beliefs. The public schools in the statements of objectives for the teaching of mathematics sometimes include as an aim "The search for truth." The Board of Education of the City of New York state In 1956 that "Both science and mathematics* by their very methods, lay the groundwork for developing a devotion to tru th ,"^ The Los Angeles City Schools state that the schools should find in mathematics education opportunities for developing values, ^"Mathematics In Secondary Schools," XXXVIII (May 1 9 5 4 ), p . 12. op The Development of Moral and Spiritual Ideals in the Public Schools (New York: New YorkCity Board of Education, l3S6), p. 8. 231 among which are "Faith— through realization of the unchanging nature of truth--the existence of eternal v e r i t i e s 1.12^ Einstein has explored space in a way that Newton never thought of. Whether his explanation of space con tains a greater degree of "truth" than Newton*s is a scientific question yet incompletely solved, Man felt in Newton's day that he had found absolute truth about space. After Einstein's theories had been submitted to testing, and had been shown to be experimentally valid, they were given the allegiance which had been given to Newton's t h e o r i e s . In the person of this modest mathematician, man has advanced a long step forward into the mysteries of existence which surrounds him on every side. For the first time, Einstein has suggested a means where by the several riddles of time and distance, of space, of gravity, inertia and matter itself, of light, electricity and magnetism, may be included in one comprehensive exactitude of coordinated law. . . Here is no chance guess by a gambler in truth.24 What is truth? It is evident that the problem of truth is somewhat a mathematical problem. It is likewise evident that the problem of truth is a religious issue. After Jesus had said "For this cause came I into the world, that I should bear witness unto the truth," Pilate par Moral and Spiritual Values in Education (Los Angeles City Schools, 1954), p. 96. 2^P. W, Wilson, Man*s Long Road to Scientific T ru th (New York Times M agazine), (May 26, 19&9), p . 11. 232 asked Jesus "What is Truth?" Theologically, one needs only turn to other verses, such as "Thy Word is Truth," and "I am the Truth," in the same book of the Bible to find that Jesus Himself and the Bible are presented to the Christian as the absolute truth. The problem of absolute truth is therefore not a mathematical problem, but a religious one. Most mathematicians would agree, at least in part, with E. T. Bell: Thanks to him (Euclid), I am (I hope) immune to all propaganda, including that of mathematics itself. Mathematical "truth" is no "truer" than any other, and Pilatefs question is s till meaning less. There are no absolutes, even in mathematics. b It is possible that one might become so certain of the superiority of the methods of mathematics, and the scientific method of investigation, that even though he denies that absolute truth exists, he acts as though he imagines that he has found it. According to the mathematical models of postulational thinking, a man can build up a consistent system of truth without mak ing provision for absolute truth, or for God. This will be valid mathematically, because a system does not conclude anything that is not implied in the postulates, and the postulates may not have included the possibility of absolute truth. ^E ric Temple Bell, "What Mathematics Has Meant to Me," Mathematics Magazine (January-February, 1950), p . 161. 233 As mathematics has its systems which are valid within bounds, so other areas or knowledge have equally valid systems, if they use similar methods. And it may even be that different methods of reasoning can lead to truths beyond the reach of the mathematical methods, I may be unenlightened and naive, but it seems to me that I can detect one or two weak spots in the attitude of the typical university person. He objects to taking anything on faith or authority, yet he himself takes on faith the very foundation of his technique, namely, that only experimentation or quantitative research can lead to truth. Again, he condemns a person who accepts doctrine on the authority of a preacher. Yet he himself expects lay persons to accept his own research conclusions on the authority of ’'science," even when the lay person is utterly unable to look into or understand his researches,2o Summary. Mathematics no longer lays claim to the possession of absolute truth. The removal of this claim, resulting as it did in the freeing of mathematics from impossible restrictions, enabled the mathematician to explore a wider universe of thought. Mathematical systems and structures could be examined and constructed, without having to know in advance whether the conclusions would be either "true" or "useful," In a Christian school, truth is the all-important end of all instruction. Mathematics, through Its content, its explanation of the created universe, and its methods, J. Freund, "The Place of Religion in Engineer ing Education," Religious Education, L (Hovember-December 1955), p. 365. 234 is one of the best vehicles for directing one into an understanding of the nature of truth. Mathematics is one of the disciplines which has a major aim, the search for truth. Within a given structure or system of mathe matics, there exists a certainty or "truth" which cannot be denied within that system. Christian schools do not look to mathematics for the same kind or degree of truth which they seek for in the Bible. But they do look to mathematics to supply these systems or models of thought which can help to explain the laws of the universe, and which can contribute to the understanding of an educated p e rso n . « Not all students will learn from their study of mathematics all that they might learn. Beside the con tent of number and spatial relations, there are desirable insights which can be nourished by a proper consideration of mathematical truth, as presented in this chapter. Some of these insights have great value, and should not be ignored any more than the fundamental operations and facts of elementary arithmetic should be ignored. The awareness of, and insistence upon clarity and precision in definition and statement; the ability to discriminate between a mere assertion and an inference; the habitual testing of inferences for consistency with known or given conditions; awareness of the nature of postulational 235 thinking, of the arbitrary nature of hypotheses and defin itions; and the ability to eliminate emotional or preju dicial factors from argument, are all values strengthened through a proper study of mathematics. Mathematics does not claim the absolute truth of religious faith, nor yet the approximate truth of science, as its own. It lays cia ira on yet a third kind of truth, hypothetical, or postulational truth. It claims nothing more, and is satisfied with nothing less. Mathematics is neither a description of nature nor an explanation of Its operations; it is not concerned with physical motion or the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such, it is concerned with neither approximate or absolute truth, but only with hypothetical truth. That is, mathematics determines what conclusions will follow logically from given premises. The conjunction of mathematics and phil osophy, or mathematics and science, is frequently of great service in suggesting new problems and points o f v ie w .27 27Carl Boyer, The Concept of the Calculus (New York: Columbia University Press, 1§39), p. 3^8. CHAPTER IX MATHEMATICS— INVENTION OR DISCOVERY? In the beginning God created the heaven and the earth. Genesis 1:1, Thus in the beginning was created the universe with the spiral of the nebulae, the elliptic paths of the planets, the geometric structure of the crystals, and the physical laws expressible in mathematical eq uations. One thus should be able to conclude: The history of mathematics may be considered as a record of the discovery of existing laws in this science and of the invention of better symbols as needed from time to time for their expression.^ However, one may find w riters who hold that man has invented more than the symbols. This view states that mathematics, its concepts, axioms, and theorems, are all invented or created by man. Mathematics, according to this view, is a humaa creation in every respect. It Is a consequence of what human beings are and how they think rather than what the physical world or some objective ideal world really contains.2 1-D. E. S m ith , H is to ry o f M athem atics (New Y ork: Dover Publications, Jhc.,195X7, P« 3* 2Morris Kline, Mathematics, A Cultural Approach (Reading: Addison-Wesiey Publishing Co., Inc., 1962), p . 665. 236 237 This difference in point of view is of some import ance in a study of the religious bearings in mathematics* The content of mathematics in school textbooks is made up of the relationships that exist between numbers, the fundamental operations of arithmetic* and the other processes of elementary mathematics* Often the work in arithmetic seems to resemble the playing of a game, in which, if one abides by certain rules, certain results are sure to happen* And if one uses the right rules and works accurately, he gets correct answers* The rules and the symbols of mathematics can be traced back through history for hundreds of years. As man has invented new symbols, and discovered new relation ships he has pondered the origin of this powerful ally. For centuries the controversy has waxed and waned— is mathematics invented or discovered? Is its structure purely a creation of the mind of man, or is it superhuman, awaiting discovery by man? Whichever view we take, it is impossible to rule out the role of man in the pioneering and developmental activities associated with mathematics.^ If one makes a list of writers who have considered this question, and groups them into two groups according to their decisions, he is led to some fairly obvious conclusions: the religious education writers, and those mathematicians sympathetic to a religious viewpoint have ^Roger Osborn, and others, Extending Mathematics Understanding (Columbus: Charles E. M errill Books, Inc., 1§6T), p. 143. 238 generally held that mathematics is discovered. Those who have sought to disparage religion, or who have held that religious thought and mathematical thought are unrelated, have taught that mathematics is the invention of man. The converses of these statements are not necessarily so, and there are some who would claim mathematics to be the Invention of man, who hold religious views. An attempt to find a synthesis between the two views must begin with a definition of terms. For it may be that the two groups are debating different views of the same general position. In fact, it is difficult to be sure when reading some authorities, whether they distinguish between discovery and invention. Not a few mathematicians are agreed that these characteristics are summed up in considerable measure in the word invention. Some of the things in mathe matics one may think of as being discovered; but others, and the more fundamental things, seem to have been created by the human spirit. After their crea tion many of their properties have been discovered. This relation between invention and discovery per vades most of the mathematical literature. Mathema tical space has been created, not found in nature, as is shown by the fact that the mathematician has several kinds of three-dimensional space as well as numerous spaces of higher dimensions. . . We may assert that the process of discovery in mathematics is primarily that of invention. Sarton makes two statements in which he uses both words, and seemingly makes a distinction. But the context ^Robert D. Carmichael, "The Larger Human Worth of Mathematics," Our Mathematical-Heritage (New York: Collier Books, 1963), p. 223. 239 seems to indicate that he is speaking generally of the acquisition of new mathematical knowledge in both state ments : There is no doubt that mathematical discoveries are conditioned by outside events of every kind, political, economic, scientific, military, and by the incessant demands of the arts of peace and war. Mathematics did never develop in a political or economic vacuum. The main sources of mathematical invention seem to be within man rather than outside of him: his own inveterate and insatiable curiosity, his constant itching for intellectual adventure E. T. Bell in Men of Mathematics, after having emphasized that mathematics is invented, not discovered, in the closing paragraph of the Introduction, gives the reader his choice: "Today, mathematics invention (discovery, if you prefer), is going forward more vigor ously than ever."6 The religious bearings in the development of mathematics as outlined in the earlier portions of this study were produced for the most part by men who believed that they were discovering eternal laws. Plato recognized and taught relations between numbers, and spoke of some of the principles of geometry. He believed that the truths of these relations were obviously there, forever, and that the relationship existed before and 5George Sarton, The Study of the History of Mathe matics (Cambridge: Harvard University Press, 1936), p. 15. 6E. T. Bell, Men of Mathematics (New York: Simon and Schuster, 1937), p. 18. 240 would exist after man thought of it# Mathematical truth was tim eless, and did not come into existence when man discovered it, even though its discovery seemed to be a very real event. This view is echoed in the twentieth century by a philosopher, who spoke of mathematics discovery as one of the greatest accomplishments of man. It is not to art, nor to theology, nor to religion, nor to philosophy, nor yet to natural science, inesti mable as their achievements have been and are, but it is to mathematics that we owe the most precious of certitudes, the knowledge, namely, that there exists an ideal cosmos, a universum of ideas and relations, perfect in its order and harmony, pure in its beauty however austere and cold, infinite in the wealth of its spiritual content, and everlasting. The discovery and progressive exploration of that world is the supreme triumph of what I have called the supreme enterprise of the human spirit. The ideal cosmos will abide. That world is the great invariant.7 St. Augustine believed that there existed certain necessary and immutable truths which are neither created nor altered by the human mind. He taught that the truths of mathematics have been discovered, not invented, by man. They lead the mind to study the relationship between the changeable and the unchangeable, and thus the soul, provided it traces this relationship to its ultimate, the love of God, will be conducted to the springs of wisdom. 7Cassius J. Keyser, "The Spiritual Significance of Mathematics," Religious Education. VI (April 1911), p . 387. 241 These views declare that mathematics existed prior to the mathematician. Before man calculated, nature multiplied and divided, and arranged her products in geometric form. Objects fell in accordance with the law of gravitation before Newton framed the law in mathematical terms. Thus, mathematics in its ultimate reality is not a creation of the mathematician. But the mathematician has invented an intellectual machine so to speak, with which he can better accommodate himself to his surroundings, or better still, adjust his environment to himself, Man does not invent the idea of transportation or communication, but he invents the gasoline engine and the radio, and continually seeks ways to refine his invention, to serve him better. And man does not invent the idea of mathematics, but he may invent calculus or algebra by which he can understand nature. He develops a machine which can be applied to natural phenomena, but the natural phenomena comes equipped with its laws ready built, and man can only develop and perfect his explana tion of those laws. What man calls mathematics is at present his best interpretation of these laws. To the extent to which one refers to these rules, one may speak of invention. To the extent to which one refers to the laws which are explained by the rules, one must 242 speak of discovery* There really is no conflict between the two views of the origin of mathematics* Concerning the title of this study, two remarks are useful. We speak of invention: it would be more correct to speak of discovery. The distinctions be tween these two words is well known; discovery concerns phenomena, a law, a being which already existed, but had not been perceived* Columbus discovered America; it existed before him; on the contrary, Franklin invented the lightning rod: before him there had never been a lightning rod. Such a distinction has proved less evident than appears at first glance. Torricelli has observed that when one inverts a closed tube on the mercury trough, the mercury ascends to a certain determinate height; this is a discovery; but, in doing this, he has invented the barometer.® Bell, referring to the discoverer of wireless waves, notes that Heinrich Hertz remarked of his mathe matical expressions: One cannot escape the feeling that these mathema tical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser than their discoverers, that we get more out of them than was originally put into them.® Bell says that any competent mathematician will understand Hertz1 feeling, but will also incline to the belief that whereas continents and wireless waves are discovered, dynamos and mathematics are invented and do what we make them do. ®Jacques Hadamard, An Essay on the Psychology of Invention in the MathematicalH^ield"TPrinceton: Prince- ton University Press, 1949),p. xi. ®E. T. Bell, o£. cit., p. 16. 243 The publications of Christian schools, when refer ring to mathematics, usually adopt the discovery view: The laws and principle s of mathematics have been set down by God from all eternity. He has, and is s till allowing man to discover these relationships. Dependence has been basic to many mathematical discoveries as well as to many inventions and sci entific discoveries, (an attitude to be developed through the teaching of Dependence in mathematics is) A thankfulness to God for having permitted us to discover laws which help us describe so many of the physical phenomena about us.1^ Another account, from one who teaches that all of mathematics as well as all other subjects are for the purpose of revealing God and his attributes, makes an interesting application of the idea of discovery: That man does not make the mathematical proposi tions, but instead discovers them, is evident from the following historical fact: Euclid was a great Greek mathematician who lived about 300 B. C, This highly talented man of antiquity formulated a large number of geometrical propositions. In the 17th century A. D., Pascal, a great French mathematician, discovered with no one teaching him, the thirty-two propositions of Euclid. Note, the one did not copy what the other had fabricated. Both were extraordi narily gifted by God to discern the footsteps that God had taken through the devious ways of mathema tical thought,11 1(^Board for Parish Education, The Lutheran Church, Missouri Synod, Resource Units for Lutheran High Schools, Mathematics, Unit IV Dependence, p,~T ^Mark Fakkema, "Christian Way of Teaching Mathe matics in Upper Grades," The Christian Teacher (March 1958), p. 18. 244 Leslie A. White quotes from a long and varied list of simultaneous inventions and discoveries in the fields of chemistry, physics, biology, and mathematics. He explains the coincidences by a culturological means. That is, each individual is born into a pre-existing organization of beliefs, tools, customs, and institutions. If Newton had been reared as a sheep herder, the mathema tical culture of England would have found other brains in which to achieve its new synthesis. He concludes his discussion of the role of Invention versus discovery with these words: There Is no mystery about mathematical reality. We need not search for mathematical ’’truths11 in the divine mind or in the structure of the universe. Mathematics is a kind of primate behavior as languages, musical systems and penal codes are. Mathematical concepts are man-made just as ethical values, traffic rules, and bird cages are man-made. But this does not invalidate the belief that mathematical propositions lie outside us and have an objective reality. They do lie outside us. They existed before we were born. As we grow up we find them in the world about us. But this objectivity exists only for the Individual . . . Ideas interact with each other in the nervous systems of men and thus form new syntheses. If the owners of these nervous systems are aware of what has taken place they call it invention or creation. If they do not understand what has happened, they call it a ’’discovery” and believe that they have found something in.the external world. Mathematical concepts are independent of the individual mind but lie wholly within the mind of the species, I. e. culture. Mathematical invention and discovery are merely two aspects of an event that takes place simultaneously in the cultural tradition, and in one or more nervous systems. Of these two factors, culture is the more significant; the deter minants of mathematical evolution lie here. The human 245 nervous system is merely the catalyst which makes the cultural process possible.12 The scientist and mathematician have operated with the conviction that the outward universe is intelligible, and shall some day be understood. If this had not been their faith, they would not have continued their work. The fact that their discoveries and descriptions of nature enabled them to make great contributions to civilization was reason enough to make them believe that they were on the right track. The order and design was there, and it was up to man to discover that design by Inventing tools which would help him. Aaron Bakst has stated: Some have the mistaken notion that nature is endowed with mathematical characteristics. Nature knows nothing of or about mathematics. The fact that we are able to express (and only approximately) the laws of nature In mathematical terms should be Interpreted that the creator of nature, in His Supreme Intelligence devised order in His design. The intelligence of man follows the 3ame orderliness and, by relying on fundamental beliefs, it manifests the phenomena of this orderliness by developing mathematics. Mathematics is that instrument which enables us to lift the curtain which hides the secrets of the Creatorfs designs. Will man succeed in perfecting the mathematical instruments so that all the secrets will be revealed to him? Probably n o t . 1^ Two writers of recent years whose works have been of the greatest value to the humanizing and the 12Leslie A. White, "The Locus of Mathematical Real ity: An Anthropological Footnote," The World of Mathema tics (Simon and Schuster, 1956), p. 2363-64. 13Aaron Bakst, "What Is Mathematics?" The Science Counselor, XV (December 1952), p. 144. 246 popularizing of mathematics have unfortunately also both taken delight in making slighting remarks at religious viewpoints# In spite of this predeliction, they have both been more than generous in their accounts of the religious background of the men and the ideas of the times in which mathematics developed# These men are E, T. Bell, and Morris Kline. Both insist that mathematics is the inven tion of men, and not their discovery, and yet both recog nize the greatness of the contribution of mathematics to the discovery of scientific laws. Bell states that In the f u t u r e , Mathematics will be recognized for what it has always been, a humanly constructed language devised by human beings for definite ends prescribed by themselves # ^ In another place, Bell clarifies the difference between the idea of invention and that of discovery, as this difference applies in the world of mathematics today: These two ways of speaking divide mathematicians into two types: the "we can" men believe (possibly subconsciously) that mathematicians produce purely human inventions; the "There exist," men believe that mathematics has an extra-human "existence" of its own, and that "we" merely come upon the "eternal truths" of mathematics in our journey through life. . . Theologians are "exist" men; cautious skeptics for the most part "we" men. "There exist an infinity of even numbers, or of primes," say the advocates of extrahuman "existence," "Produce them," say Kronecker and the "we" men. ^E . T. Bell,' The Development of Mathematics (New York: McGraw Hill Book Co., Inc., 1945), p. 530. 247 That the distinction is not trivial can be seen from a famous instance of it in the New Testament, Christ asserted that the Father ’’exists"; Philip demanded "Show us the Father and it sufficeth us,"15 Morris Kline agrees that mathematics is more a consequence of what human beings are, and how they think, than what the physical world or some objective ideal world really contains. It may be that man has introduced some limited and even artificial concepts and only in this way has managed to institute some order in nature, Man’s mathematics may be no more than a workable scheme. Nature itself may be far more complex or have no inherent design. Nevertheless, mathematics remains the method par excellence for the investigation, representation, and mastery of nature. In those domains where it is effective it is all we have; if it is not reality itself, it is the closest to reality we can get,15 It was predicted in the nineteenth century that if mathematics were to be considered as a game, in which man is privileged to invent new rules and symbols, and make changes at w ill, the results would be barren, and it would be the end of the production of mathematics. It is evident however, that this, is not the case. Pure mathematical research has proved to be of great value and productivity. It is not expected that pure mathema tics invention will ever exhaust the realm of mathematical possibility, for inherent in any scheme of mathematics are some lim its which make it impossible to have a complete 15E. T. Bell, Men of Mathematics, p. 573. 16Morris Kline, Mathematics. A Cultural Approach (Reading: Addison-Wesley Co., 1962), p. 676. 248 system. Godelfs proof showed that there will always be truths that can be discovered outside any mechanical mani pulation of rules. Martin Gardner, in his review of Godel1s Proof for the advertising circular of the Library of Science for December 1963, says Godel, himself, believes in a "Pla tonic realism in which mathematical truths are external, objective; discovered rather than created by the human mind." Godel's undecidable propositions can be decided only by expanding the deductive system; the larger system is certain to contain new undecidable proposi tions which of course can be decided in a still larger system. For all we know even a simple arithm etical conjecture, such as Fermat's last theorem, may be true, but impossible to prove. Among Christian schools, mathematics will be taught as a system of thought whereby man has discovered more and more of the pattern which God put in nature. One may state that a man has discovered a new prin ciple. He might equally well state that that man was the first to create or invent the statement of the principle. There may be no conflict between the two ideas. A most productive approach for the development of new ideas is to think of them as the invention of man. One one is surprised when these inventions help explain in explicit terms some phenomenon of nature. Invention in mathematics assists in the discovery of more accurate descriptions of nature. Far from being contradictory, the ideas of Invention and discovery are complementary. PART III RELIGIOUS BEARINGS IN THE TEACHING OP MATHEMATICS 249 CHAPTER X PRESENT PRACTICES IN THE TEACHING OP MATHEMATICS This chapter proposes to review the extent to which Christian schools are aware of the religious bear ings in mathematics. It can deal only with the external evidences of such an awareness as reflected in textbooks, official statements, and articles in periodicals. As the result of correspondence with many of those who are work ing in these schools, the writer recognizes that most of these schools claim far more integration of religion and mathematics than such external evidences indicate. Almost without exception, these Christian schools claim that they depend upon the teacher, rather than the textbook or other visible source of information, for the inclusion of relationships which may be emphasized between religion and mathematics. It is fairly certain that mathematics, as one of the three R’s, is taught in nearly every one of these schools. There is a sizable number of such schools in the United States. The Catholic church operates nearly twelve thousand schools, with four million elementary school children, about one million secondary school 250 2 5 1 children, and a third of a million college age students. The Lutheran church, Episcopal, Seventh-Day Adventists, Christian Reformed, and the Jews have established school systems. Since 1947, the National Association of Christ ian Schools, has served as an independent, non-sectarian association of Christian schools, and now includes about 215 schools in more than thirty st&tes. Most of the other Protestant groups, and many of the local congregations of the denominations mentioned above, do not operate their own schools, but support the public schools. Mathematics in Christian elementary schools. There has been a great deal more activity in parochial elemen tary schools in mathematics, than there has been in the high school. That is, more special materials have been prepared. In the very first volume of the Catholic Periodical Index, in 1930, was listed an article, "Mixing Primary Arithmetic in Religion." However, there have not been half a dozen similar articles in all the Catholic educational magazines in the ensuing thirty-four years. Early arithmetic books, like the early books in reading, contained religious content. Cajori tells that: One of the earliest references to arithmetic in the United States is of an arithmetic book published at Ephrata (Pa) in 1786. The Ephrata publication is an exceedingly curious compound of religious exercises, and exercises in arithmetic. The creed, Lord's prayer, hymns, and texts of scripture, are strangely intermixed with problems and calculations in the simple parts of arithm etic.-*- This type of effort to include religious ideas in the arithmetic problems themselves, has continued to some extent to the present day. For example, in 1834, there appeared in an arithmetic book the problem: "Charles learndd 10 chapters in the Bible, and Lydia learned 18; how many more did Lydia learn than Charles?" And in 1959, this problem appeared: "Mother had five flowers for Baby Jesus. Two of them died. How many flowers were left for Baby Jesus?" There are not many of this type of problem printed in arithmetic textbooks today, even in textbooks printed in collaboration with Christian schools. And there seems to be a definite trend away from such inclusion of relig ious material in arithmetic books. The following quota tions from correspondence with three leading publishers in July 1964 indicate that this is so for each of them: I might add that all preparations regarding mathe matics, English, science, and social studies are getting away from special editions for various denom inational schools. These worktexts have not incorporated special religious concepts to any great degree. Wherever normal concepts relating to good conduct and virtues applicable to right living for e v e r y o n e co u ld be interwoven, of course the authors made it a point t o do so . Florian Cajori, The Teaching and History of Mathe matics in the United States (Washington: Government Printing Oi'fice, 1890), p. 12. 253 (Our arithmetic book) has no bearing on this thesis, you will see that there is no Religion whatever in this book; it is a straight arithmetic book. Although there are not many arithmetic problems with religious content, printed in textbooks, there are likely to be a large number of such problems used informally in the classroom. Such use is not surprising, when one con siders the philosophy under which such schools operate. A classroom is an extension of the child's world, and as such, both shares in and contributes to the child’s total growth and development in his environment. The teaching of arithmetic to Catholic children should not be confined only to knowledge that pertains to quantative thinking, but also should present appli cations of this knowledge in situations which are normal to them. . . Finding Truth in Arithmetic presents situations which are normal to the Catholic child.2 What is normal in the atmosphere of the Christian elementary school is the language of religion. It is as natural to use Biblical references in teaching, as it is for a literature teacher to speak casually of Shakespeare or for a science teacher to refer to Galileo. And it is normal for the children in that environment to respond to, and with, similar illustrations. It does look and seem incongrous to include that type of problem in print, and almost every teacher and administrator with whom the writer has spoken or corresponded, has disavowed their ^Finding Truth in Arithmetic (Chicago: Ginn and Company, 1959)7 P« 18. 254 dependence upon such formal prepared material. Attempts such as the following are not looked upon as making a great contribution to the field of mathematics for the elementary school children in the Christian school. In a publication prepared for use in a Protestant school, and used by second grade pupils, page one begins with pictures of various animals, and some questions to be answered. (The format is different from this, and the following is spaced in the workbook throughout a full page). ”For now thou numberest my steps” (Job 14:16) Mr. Brown had a farm. He had many animals on the farm . He th a n k ed God f o r th e farm . He th a n k ed God for each animal. 1. Count the 1 cow, 2. Count the 2 goats, 3. Count the 3 pigs, 4. Count the 4 hor ses, 5. Count the 5 chickens, 6. You counted just l ik e God co u n ted . God made colors. He gave each animal the right color. Color the animals as God has colored them.3 This way of presenting arithm etic examples does not appeal to many teachers, at least in print. On the other hand, in the classroom, elementary school teachers in Christian schools use such examples because they are normal to the situation at hand. The most recent listing of textbooks for Catholic schools includes a few series of arithmetic books which have been published in recent years, and which have been 3Doris Anicker, Finding God in Arithmetic (Spokane: privately printed, 1956), p. 1. 255 developed in cooperation with the Catholic church. These have not been developed to teach religion, but arithm etic. The content is not expected to be primarily of a religious nature, and the problems sim ilar to those described above are either not included, or exist at a minimum. There was a methods book in elementary school arithmetic for Catholic teachers' published in 1926, In the introduction to this book, the statement Is made that a valuable feature of the book is the successful attempt made to show how arithmetic should be t aught to bring out to the full its Cultural, disciplinary, and spiritual values, instead of being treated as a merely utilitarian subject. Closely allied to this feature is the attempt made In several places to correlate arith metic with the other subjects of the course of study, notably with the religious instruction,^ With a few outstanding exceptions, the book is devoted to a thorough explanation of the teaching of elementary school arithmetic which would be acceptable to the public school, De Noue lists seven ways in which religion can be integrated into the academic material in textbooks, and he classifies them according to the subject matter area in which they are likely to occur. Three of these categories include mathematics in some way. These ^Sister Mary Eberharda Jones, A Course in Methods of Arithmetic (Boston: D, C, Heath and Co., 1^26), p, vi. 256 are not criticisms of the use of subject matter, but objective indications of such use. 1. Religious symbols and subjects used in examples for arithmetic and grammar d rill. (Common in mathematics and language). 2. A general theistic Christian approach to all matters. (Common in all subjects). 3. Selective emphasis on (Catholic) institutions and contributions to the culture and on facts favorable to the Church and omission of insti tuting and contributions to the culture by non-Catholics and of facts unfavorable to the Church. (Common in mathematics, science, and language)•® The latter point is also one which causes difficulty for the Christian teacher who uses secular books and articles. Many times there are unfavorable comments which appear against religious ideas and institutions in text books, and the church feels that these are unfortunate, and in some instances, unfair. There have been no special arithmetic books for Protestant schools, produced by the usual school publishers. There have been some teacher’s methods books produced by Christian school organizations. In 1937, the Lutheran Church prepared a Curriculum in Arithmetic for Lutheran Schools. In recent years, the General Course of Study for Lutheran Elementary Schools has appeared, with each revision including a section on arithmetic. The National Union of Christian Schools has produced a sim ilar book 5George R. LaNoue, "The NDEA and ’Secular' Subjects," Phi Delta Kappan, XLIII (June 1962), p. 384-386. 257 In its Courses of Study for Christian Schools. These schools expect the teacher to supplement his teaching as necessary to carry out the objectives of the school* Special objectives are given for each area of the school curriculum, including both academic and religious ideals and a im s• _ Mathematics in the Christian high school. There have been no textbooks written to adapt algebra or geom etry to any special program of religious education. There has been no thought of such a program. Most educators look with strong disfavor on any attempt to so "dilute" the mathematics program. However, statements of object ives of the several church groups do include references to mathematics, and suggest that the study of mathematics can contribute to the total program of religious e d u c a tio n . The Lutheran church emphasizes that mathematics though a product of man’s reason, is a gift of God. In each of the five resource units in mathematics, which have been prepared for secondary school teachers, there is a paragraph which relates the subject matter of the unit to the Christian view of life. Included in the special objectives in each of the units, Is a list of attitudes to be developed. In each instance, there are some attitudes which direct the teacher's attention to a 258 spirit of thankfulness and appreciation for those gifts of God which are evidenced through mathematics* But the content of the unit is strictly mathematics, and not r e l i g i o n . The National Union of Christian Schools has pro duced a curriculum guide in mathematics for the high school. In the introduction to this guide, the teacher is made aware of the fact that ideas of quantities and exact relationships have their source in God. The teach ing of mathematics is to be a calling to impart to the child that part of God's revelation in a clear, meaning ful way according to his ability to receive it. There is a striving in man, placed there by God, to search out this wonderful universe and bend its forces to his control. This is the task God gave at the beginning of history. . . God also placed within man the ability to grasp the principles of mathematics and develop them into a science to use as a tool in fulfilling this task. This too is a revelation of God and a gift from rtim. Although the content of secondary school mathema tics is strictly secular, there are different philosophies which motivate the instruction of any subject. The Christian schools believe that this is a legitimate situation in any classroom, and that instruction in any field will reflect the philosophy and beliefs of the instructor. If there were no differences between the 6Mathematics, Curriculum Guide (Gradd Rapids: National Union of Christian Schools, 1958), p. 9, 10. 259 geometry classroom in a Christian school and in a secular school, perhaps the teacher, the pupils, and the admin istration of the school are disappointing their clientele, 7 and not fulfilling their stated purpose. In 1948, Ibrahim showed that the philosophy which one held would color his teaching of mathematics. The National Council of Churches recently stated that the regular school subjects should teach facts, and that teachers should be prepared to deal with the religious aspects of these subjects. Some may y e t th in k t h a t m athem atics i s so s e c u la r in content and method that one’s philosophy will not be apparent. To select an anti-Christian example, according to "Education in the USSR" subjects are taught in Russia in conformity with the prevailing interpretation of Marxist doctrine favored by the Communist Party. This includes the "Postering of an atheistic attitude toward unexplained material phenomena,"2 the area in which mathematics is particularly of importance. The teacher does have a philosophy which determines the spirit in which he approaches his teaching, and which results in his parti cular mode of teaching. In the review of a recent ^Al-del Ibrahim, "Philosophies of Education—Their Implication for Mathematics and Classroom Procedures," (Unpublished doctor's dissertation, The Ohio State Univ ersity, 1948). 8U. S. Department of Health, Education, and Welfare, Educ at ion in the USSR, Bulletin 1957, No. 14, p. 58. 260 Russian book on the history of mathematics, E. S. Kennedy of the American University of Beirut stated: It Is not surprising and entirely understandable that the book should have a strong flavor of Marxism, especially since, as.the author remarks In the first lecture, students of the fourth course, begin the study of dialectical materialism simultaneously with that of the history of mathematics.9 In the lists of approved textbooks for Catholic schools,-1-0 there has been no mention of any publication for secondary school mathematics which is not a standard textbook for the public schools. Statements of objectives for secondary school mathematics in Catholic school maga zines have frequently had no special religious objectives 'V included at all. The writer visited a large Catholic high school In Columbus, Ohio, in July 1964. All of the recent work of the curriculum committee in mathematics was examined, and there were no special religious objec tives mentioned. The textbooks in use were all of current public school adoptions. Religious influences in other publications. There is no lack in redent years of books devoted to the human izing of mathematics for the general public. Many of the books listed in the bibliography, and particularly many of those from which references have been selected, should 9Review of Istorlya Matematiki by K. A. Rybnikov, in Scripts Mathematics, XXVI (December 1963), p. 363-364. l°Annually in the Catholic Educator, for example, XXXII (January 1962), p. 469-484. 261 be of help to one who wishes to understand the religious bearings of mathematics. The method’s books and curriculum guides for the teacher of mathematics, with their special lists of objectives, generally include some references to moral, spiritual, or ethical values. Many of these values are of common concern to all teachers. Summary. It is difficult to say what are the present practices in the teaching of mathematics in Christian schools. Prom the responses to the question naire, referred to in the first chapter of this study, it is evident that some confusion exists in the minds of the teachers. The administrators place their confi dence in the teachers, in the school atmosphere, and in the special classes in religion. They are confident that proper integration of mathematics and religion can and does take place. In published materials there is not much directly relating to mathematical content which differs from that of the public schools. Almost unanimously, the teachers agree that this is as it should be. Within the classroom, the teacher sets the environment, and it becomes normal to count pennies for Sunday School as well as to count pennies for milk. It Is no more abnormal to speak of the twelve disciples than It is to speak of the twelve months 268 of the year. And when one is teaching probability, it is normal to discuss the probability of two, three, or even of no seeds germinating out ofr*a group of ten seeds. Just as normally, the teacher observes, "Suppose one of the seeds fell upon stony ground?" The arithmetic need not be lost, and the atmosphere is enriched through such religious reference. In conducting a status study of practices in the teaching of mathematics in Christian schools, one is limited to the external evidences of those practices. There is an almost complete agreement among Christian educal-ors that in mathematics classrooms, the subject is strictly mathematics. It is generally agreed that there is insufficient time to prepare the student thor oughly in the concepts and processes of mathematics, if too much time is taken for consideration of all areas with which it comes in contact. In the elementary school, this may not be so groat a problem, and more attention can be given to the religious bearings to be found in the study of number. In the secondary school, and in the college, the extent to which such religious bearings will be evident in the mathematics curriculum w ill depend upon the interest and effort of the teacher. CHAPTER XI SUGGESTED PRACTICES IN THE TEACHING OP MATHEMATICS Summary of the study to th is poinE P a rt I of this study has been an attempt to show th a t relig io n has been a strong influence in the history and development of mathematics. Recognition of this in flu en ce might well be a part of the contribution of the m athem atics department of a Christian school to the total curriculum of the school. P a rt II has shown the religious bearings of some of th e content of mathematics. Some consideration was given to the intense interest displayed in the religious use of number symbolism for hundreds of years. Attention was draw n to the great common concern of mathematicians and relig io u s leaders in the pursuit of truth, and the sig n ific an c e of the factors of invention and discovery in th e origin and development of mathematics. P a rt III is an attempt to display some present p ra c tic e s in the teaching of mathematics in Christian sch o o ls, and to explore additional ideas and practices w hich may give added direction to such teaching. It is th e purpose of the present chapter to suggest some 263 264 of these ideas, as a point of departure for those who would like to make use of the religious bearings of mathematics in their teaching. It is hoped that much of the historical material of part I, and at least some of the ideas of part II, will contribute to such a purpose. General background Great movements in the teaching of mathematics have been developing in recent years; the changes have been of so great a nature that they have been termed revolutionary. The basic questions of content and method are yet being re-examined, and a multitude of varied answers are being proposed. Many different experimental programs have appeared, and these have covered the entire spectrum of mathematics, from the arithmetic of the first grade, to the symbolic logic of the graduate school. Many of the topics which were included in college courses of fifteen years ago have been either deleted, or the space and time alloted to them has been greatly reduced. A recent revision of a book in methods of teaching arithmetic reported that 1 0 % of its contents had been rewritten in the four years since the previous edition, in order to keep abreast of new developments. Christian schools have participated in these move ments, recognizing that the early conclusions of these new programs were admittedly tentative. To close one’s 265 ©yes to what is being done, and to continue with nothing but traditional mathematics would be educational foolish ness, It was during these same recent years, that the few teacherTs guides and resource units mentioned in the previous chapter were produced, and the arithmetic series for Catholic schools were developed and published. In attempting to look into the future to consider how mathematics may be used in Christian schools, some assu m p tio n s may be made: 1. Christian, schools w ill continue to Increase in number, and their enrollment will also increase, 2. The majority of the Christian schools will continue to use secular textbooks in mathematics, 3. Teachers In Christian schools will be expected to recognize and include some religious bearings of mathematics in their teaching. 4. The variety of approaches to the solutions of problems in mathematics education, provide a large measure of freedom to any school which may wish to include more of the cultural and religious bearings of mathematics in their curriculum. Principles of operation Any development of a major program of mathematics for Christian schools will need to be religiously advan tageous and educationally desirable. To the extent to which the program includes the production of textbooks and other supplementary teaching materials, it will need to be economically practicable as well. Re1igiously advantageous. The values to be sought and taught in the study of mathematics must be made 266 explicit. The role of method, content, and teacher, need to be clearly defined. In the Chri3tian school, the role of the teacher is of paramount importance. The method and content are often determined in large part by the teacherTs preparation in a secular school, and through the use of secular textbooks, both in the teacher's prepara tion and in her teaching. Therefore, special direction will need to be given to the preparation of in-service education materials, in the form of articles and resource publications. All factors in the teaching enterprise should contribute toward the objectives of the school, and in a Christian school, the religious objectives cannot be ignored in any classroom. Educationally desirable. Textbook competition in the United States if very great. The best mathematics educators, and the most experienced textbook publishers, have joined to produce the several arithmetic series on the market. In almost every instance, and in nearly every detail, the content of the mathematics is secular, and thus completely non-objectionable to religious groups. The effort required to produce a textbook series that could compete with these expertly prepared and produced commercial series is prohibitive for any relatively small group. It is impossible to provide comparable material by means of mimeographed or sim ilarly processed worksheets. 267 It Is inconceivable that a small group would attempt to produce books competitive with those of the large pub lishers, particularly in a non-religious subject such as mat he mat ic s « Christian schools have adopted modem fluorescent lighting, thermostatically controlled heating and cooling, and the improvements In architecture and furnishings. The administration and teachers are just as eager to be up-to-date educationally, in their use of teaching mater ials, including the best educationally-sound textbooks. Economically practicable. In a consideration of i moral and spiritual values, this monetary concern is of least importance In theory. In practice, economic factors cannot be avoided. Given a certain amount of money, a school board must make decisions on the basis of some set of values. The board can either paint the auditorium, or add seating to the gymnasium. They can hire teacher assistants for over-crowded first and second grades, or they can hire additional teachers for split-groups of first and second grade overflow. There would be little sentiment for the outlay of a large sum of money for non secular books in mathematics, when there is little which is objectionable in the books now on the market. The Christian school faces these problems and more. Money is not available to hire the scholars capable and 2 6 8 willing to produce the materials which each denominational group might wish. Neither could such scholars be spared in sufficient numbers, if indeed there are sufficient numbers, to perform such a service. It must be conceded, that with these schools, mathematics textbooks of their own design are low on the priority list, if they are even considered at all* Assuming that the textbook approach is out for the immediate future, there remains the matter of resource units, teacher units, in-service training materials, and magazine articles. Any or all of these and similar aids might be so prepared to give possible direction to the teacher who wishes to emphasize religious bearings in his teaching of mathematics. Values in a secular mathematics program in Christian s'chooYs Mathematics can be taught, and taught well, with little or no concern for its religious bearings. The study of mathematics in a Christian school can and will contribute to the goals of that school, even if there are no stated or Implied religious goals in the mathema tics content or method. The courses in mathematics stand alongside the other courses in other fields as a part of the integrated program of the school. Some of the other courses may be more intensely humanizing, and may reach a 269 larger portion of the student body. But each, co u rse has its contribution to make, and the teaching of © very subject must be adequately strong. The entire program of instruction in a Cfctristian school is like a chain--only as strong as its w eakest link. If such a school cannot recognize relig io tis hear ings in the subject of mathematics, it is s till n o t justi fied in relegating mathematics to a position o f little importance. A strong academic program of m attiem atics is essential to a well-rounded curriculum. C h ristia n schools, , however, have a commitment to teach all su b jects in such a way that the religious bearings of each one of t inem will contribute to the whole program. A conscious e f f o r t to relate mathematics to the total program and re lig io u s philosophy of education, should enable the te a c h in g of mathematics to contribute even more than ever to the aims and objectives of Christian education. Some suggested practices Elementary school. The key word here is tfcxe word normal. Elementary school teachers need firs t to under stand the atmosphere in which they teach, and th e educa tional and philosophical bounds within which th e y may operate. Every school, public or parochial or p rivate, has its set of rules and objectives to which tlae teachers owe some allegiance. Elementary school teachers are 270 sensitive to the needs and abilities or young children. In a Christian school, the normal atmosphere is one of reverence for God, and one in which religious ideas are as natural as the ideas of science and history. It is neither strange nor f orced when the teacher and pupils alike make use of religious ideas in working with ordinary number concepts and relationships. Over-much use of mathematical figures to symbolize religious truth, or over-concentration on numerical refer ences to religious things, is not good mathematics. The normal use of concrete materials for the introduction of number Is encouraged, and only occasionally shouH relig ious objects be the subject for this part of the work. Oral work in socially adapted exercises and d rill should be emphasized in order to make arithmetic meaningful to the children. Most of this work will be done with non religious objects and terms. Religious ideas will not be forcibly injected into the mathematics content. But neither w ill such ideas be avoided or forcibly ejected from consideration when they naturally contribute to the matter at hand. Mathematics and religious training can be of assistance to each other, as each fulfills its role in the total school program. But mathematics is not religious instruction, and religious instruction is not mathematics. 271 Over-emphasis in the use of religious terms and ideas in arithmetic is not a sensible approach. The Christian teacher will not expect to use the Bible as the chief source of arithmetic problems. Nor will she expect the arithmetic book, or class period, to be the chief vehicle for teaching explicit religious ideas. For example, the Christian virtues of justice and charity are taught more by the teacher’s attitudes toward each child than by words. The call for accuracy, or insistance upon values of precise and accurate answers, do not instill accuracy unless the pupils sense in the teacher the pre cision and controlled intent and enthusiasm of the teacher for this precision and accuracy as a way of doing things. Children live and do, they do not merely sit and listen. They observe, and copy what they see* The moral education of the school is accomplished largely through the play of the teacher's righteous judgment upon the subject matter of the curriculum--and the teacher is always the indispensable agent, without whom all else is but idle instruments. Growth is slow, and the child must participate in the activities himself, if growth is to t ake place in a naturally directed pattern. The objective is mathematical literacy, keeping in mind the total picture of Christian education. The ultimate goal is the strengthening the child’s awareness of, and his relationship tp God, the church, his class mates, his family, and the world about him. If the teacher Is a conscientiovis Christian teacher, her normal impulses will add to the secular textbook, those religious over tones which are real to her, through arithmetic problems and incidental comments. These overtones w ill enable the arithmetic class to contribute to the development of the whole child in the Christian school. If the atmosphere encourages this approach, the teacher will unconsciously and consciously make use of her opportunities in the arithmetic classroom. For example, when she counts the days to Christmas or Easter vacation, the spiritual significance of the days w ill be a matter of normal comment and concern. There need be no loss of attention to the arithmetic involved In related problems. Ingeneral, the decision as to whether there should be a great number of problems with religious content in number drills, with discussion of religious ideas, will depend upon the type of school, the Immediate situation, and the particular teacher involved. Secondary school. On the secondary level, an appreciation of some of the religious bearings of the development of mathematics can be encouraged. There may not seem to be much time, nor opportunity, for a class in algebra or geometry or trigonometry to take what might 273 be termed a religious approach to mathematics. An attempt to permit side-issues and philosophical background inform ation in one of these subjects might appear to dilute the mathematical content beyond the lim it of common sense. The student may be able only to obtain an insecure under standing of the material of the course, if much time is lost in considering related matters. However, the school day is the same length for every teacher, and everyone chooses what shall be included in his course on the basis of some set of values. The Christian teacher is often on the defensive with the secular world. He is constantly apologetic when he attempts to include religiously oriented material. But every teacher is selective and places varying emphases on his choices of content, method, time allotment, and concomitant learning. The Christian teacher must exercise judgment in deciding how much consideration can be given to such matters so that the mathematical goals of the course will be helped, and not hindered, by his choices. The chief justification for mathematics in the curriculum is that the world in which we live is of such complexity that no one can really expect to find a satis factory place in it without a good deal of science and mathematics knowledge. It is paradoxical, expecially to a mathematics teacher, that in this complex world in 274 which science and mathematics play so large a p a r ti that a great majority of the citizens can function w ith such a small deposit of mathematics or scientific know ledge, Naturally, the future scientist, engineer, a n d applied mathematician needs more. The content of c o u rs e s in mathematics is bound to be dictated largely Toy th e society of today, and the most probable society of th e Jfnture. When the extent and scope of the subjects a r e "being deter mined, a teacher’s selection might not be a f f e c te d much by the fact that he is a Christian, but his ap p ro ac h to the teaching of them might be. He is of course, concerned with more than the mere acquisition of skill; he w ill demand from the student those moral qualities n e c e ssa ry for good academic work. There may not be Christian a lg e b ra , or religious geometry, but the Christian te ach e r m ay often find himself placing a different emphasis izpon some part of the subject matter than would one with a d if f e r e n t philos ophy and purpose. Textbooks and courses of study inclixcLe a. definable body of concepts and skills, and a teacher i s n o t honest if he robs the student of a fair opportunity to attain mastery of that content. Integration of religious and m th e m a tic a l ideas in secondary schools will take the form of o c c a sio n a l short lectures, or incidental remarks. These rem ark s should be 275 timed to fit the nature of the work at hand, and need not be ao involved that they destroy either the religious idea, or the mathematics being taught. For example, a teacher might suggest that the fact that man is able to match almost all of his scientific work with either a linear, quadratic, or cubic curve, gives some evidence of sim plicity of design which can help to show the wisdom of the Creator. David Eugene Smith, who was continually sensing the religious bearings throughout the study of mathematics, quoted Voltaire as saying "One merit of poetry few will deny; it says more in fewer words than prose." Smith went on to say: With equal significance we may say, "One merit of mathematics few will deny; it says more and in fewer words than any other science." The formula e /fi — _ i expresses a world of thought, of truth, of poetry, and of religious spirit, for "Cod eternally geome- t r i z e s •"1 Some very fanciful ideas have been proposed in an attempt to find religious significance in mathematics. This tendency to excess leads to criticism and ridicule, and is not advocated by this study as a legitimate method of teaching mathematics. However, as a note of interest and information, or as a matter of exciting appreciation for some values beyond the skills of mathematics, some extreme examples are useful. Every teacher has his ^David E. Smith, "The Science Venerable," Mathe- matics Teacher, XLV (1952), p. 348. 276 collection of illustrative materials which may serve no greater purpose than to ease tension in the classroom, or to provide easyt ransition from one subject to another. For example, it was suggested that one might use the laws of progression to show the value of missionary work. If one person were to make a convert in one year, and if he and his convert were each to make a convert the next year, and this pattern continued, all the people in the world would be converted in x years, find x. This illustration would likely confuse a student who is attempt ing to learn the formulas and applications of progressions. But it could be used as easily as many of the customary problems of compound interest for an application. The teacher should not expect this example to teach the formulas for geometric progression, nor a real appreciation of the process of evangelism. The student might become involved with the difficulties of transportation, communi cation, language barriers, and so on, and lose out.complet ely on the value of the example. Or consider this example in geometry: it was pro posed by a teacher that one could demonstrate the existence of God easily and confidently. Consider the proof of the "theorem (?), Through a given point outside a line, one and only one line can be drawn parallel to a given line." There are two possibilities: a. either the two lines are 277 parallel, or b. they a re not parallel. One assumption is proved incorrect. Therefore, the other is true. Then the argument is applied to the statement: "Since you exist, you either exist of yourself, or have had existence con ferred upon you by another being.” The first is impossible, and therefore eliminated, so the second must be true, and another being exists. This would add confusion to a student who has been told that the study of geometry will help him to think more logically, and who has this argument presented as a model of a proof. It has been shown in earlier studies^ that by the use of demonstrative methods in geometry and algebra, children can be taught to think more intelligently, and can learn to detect fallacies obscured by irrelevant words. They can learn to discriminate between sound and unsound arguments. The goal of a Christian school is certainly to produce more intelligent and interested citizens of the world and of the church. Mathematics can help develop mature students who can reason and make decisions on the basis of logical processes. An official from the Bell Telephone laboratories makes an interesting comnent on this matter: It is because the mathematician is expert in analyzing relations, in distinguishing what is essential P. Fawcett, The Nature of Proof (New York: Columbia University Press, 1938), 278 from what is superficial in the statement of these relations, and in formulating broad and meaningful problems, that he has come to be an important figure in industrial research teams. Obviously, these traits are most needed at the early stages where a situation is being studied and plans are being laid. He plays much less part in the execution of these plans, NIKE is a case in point: the study and plan ning of this project required a team of experts from many engineering areas—propulsion, aerodynamics, radar, digital circuitry, etc,, as well as from the m ilitary. But the central members of the team were the mathematicians. This was not merely because the theory of games was involved. It was primarily because they, better than other scientists, are adept in detecting the essential thread that lies obscured by the irrelevant details and divergent languages of the other scientists. Later, as the development progressed, they played a less promi nent part in its activities,3 The benefits for religious training which the students in a mathematics class will derive from their study, w ill not depend upon how many specific problems or references they employ which explicitly use religious phrases. Benefits are likely, however, to be in direct proportion to the efforts of the teacher toward the end of developing Christian character, and an appreciation of God’s gifts to man. Man has been placed in an intricately designed universe and has been given the intelligence and desire to Investigate it. Thus far mathematics is one of man’s best ways to subdue and have dominion over the earth, Man could have been entrusted with a simple universe. ^Thornton C. Pry, "Mathematics as a Profession Today in Industry," Mathematics Magazine (February 1956), p . 76, 279 He could have been created an automaton* All knowledge could have been written down by the time of Abraham or the early Babylonians. There would be no more challenges for modern man greater than that illustrated by the act of rebuilding a fallen pile of blocks for a five-year old c h ild .' Or, man could have been entrusted with a universe complex enough so that it would have taken until Newton's day to discover all of its secrets. Again, the excitement of discovery and creativity would have disappeared before the present day. But man has been entrusted with a universe which every day gets bigger and bigger, and which presents challenges to the best minds of every generation. Man used to imagine that knowledge was like a box in which a balloon was expanding. The size of the balloon represented man's acquisition of knowledge. Soon the baloon would fill the box, and man would know all there was to know. But man has continued to enlarge the balloon of knowledge, only to discover that the larger he expands the baloon, the greater contact is made with the unknown. There seems to be no lim it to the mystery. Only through the freedom of extension which mathematics has given to the intelligence of man, has he been able to advance so far. Mathematics is a gift of God to man for this task. 280 College preparation of the teacher. Thus far, It has been stated rather emphatically that the key to a program of integration of religion and mathematics is the teacher. The textbooks in mathematics will continue to be secular in content, the mathematical preparation of the teacher will be secular in content, and most of the method’s books and lesson materials being produced w ill likewise be secular in nature. It has been suggested, therefore, that if consideration is to be given to the inclusion of the religious bearings of mathematics in the classroom, it will be most likely the result of the efforts of a teacher, whose aim is to enrich the learn ing experiences of his pupils in this way. Outside the mathematics classroom there may be opportunities to further the integration of religion and mathematics in other history classes, through topics cf discussion in mathematics club meetings, and in occasional lectures or chapel talks which would include references to ways in which academic areas and religious ideas are r e l a t ed. Inclusion of religious ideas in the study of mathematics in the classroom may be accomplished by several means: through a consideration of moral and spiritual values which are included in the purpose of the school, and which form a natural concern of every teacher and every department of the curriculum; through 281 historical references, noting incidents of interest, or following trends of thought over a longer period of time and through a comparison and contrast of mathematical ideas with religious ideas such as infinity, lim it, absolute truth, and authority. Moral and spiritual values. Teaching for values is often vague and idealistic. In the past decade there has been a concentration of effort to discover ways and means through which these moral and spiritual values can be defined and transmitted to the pupils. This may be by means of religious exercises, or through secular measures which excluded religious expressions. The Los Angeles schools proposed that a proper consideration of mathematics could strengthen such values as appreciation, integrity, inspiration, respect for law, cooperation, responsibility, reverence, faith, perspective, tolerance, and humility. In presenting the claim for the development of reverence through a study of mathematics, some use was made of David Eugene Smith’s presidential address delivered before the Mathematical Association of America, September 7, 1921. At that time Dr. Smith asked some important questions, and gave some answering thoughts: What bond of concord, if any, is there between mathematical knowledge and religious faith? What influence can an exact, abstract, reputedly frigid science like ours have upon the religious nature 282 of man? What, in fact, is the soul of mathematics, and to what wave lengths must our own soul be tuned to catch its message?4 It is commonly thought that science rests upon facts and experimentation, but religion must rest upon '‘blind faith" and is therefore less real and stable. Smith has in a very interesting parallel manner outlined certain postulates of mathematics and theology. In so doing, he admits that both science and religion rest upon postulates (faith), and that in both fields, truths rest upon faith in those postulates, A few of these parallel postulates are: MATHEMATICS RELIGION 1, The infinite exists 1, God e x i s t s 2, Eternal laws exist 2. Eternal laws exist 3, The laws relating to 3, God's laws are so finite magnitudes do different from ours not hold respecting as to be absolutely the Infinitely large non-understandable or the infinitely by u s . s m a ll, 10, Mathematics Is a vast 10, Religion Is a vast storehouse of the storehouse of the discoveries of the discoveries of the human intellect. We human spirit • We cannot afford to cannot afford to discard this material, discard this m a te r ia l. 11, It is not necessary that a solution of a problem 11, It is not necessary by lim ited means—say that the solution of the trisection of an the problem of religion 4David Eugene Smith, Religio Mathematici. The Poetry of Mathematics and Other Essays. (New York: Scripta Mathe- raatica, 1934), p. 30. 283 angl®--should bo found by our lim ited moans In o rd o r t h a t wo may should bo found In feel certain that the order that we may fool problem canbo so lv e d certain that tho pro by some means. blem can bo solved by some means. And what is tho conclusion? Does mathematics make a man more religious? Does It glvo him a basis for ethics? Will tho individual lovo his fellow man more certainly because of tho square of tho hypotenuse? Such questions are trivial; they aro food for the youthful paragraphor. Mathematics makes no such claim. What we may safely assert, however, Is this--that mathematics increases the faith of a man who has faith; that it shows him his finite nature with respect to the Infinite; that it puts him in touch with Immortality In the form of mathematical laws which are eternal; and that It shows him the futility of setting up his child ish arrogance of disbelief In that which he cannot see. . We should teach it primarily for the beauty of the discipline, for the "music of the spheres," and for the faith that it gives in truth, In eternal law, in the Infinite, and in the reality of the imaginary, and for the feeling of humility that results from our comparison of the laws within our reach and those which obtain in the transfinite domain.° The full richness and depth of human life can never be sounded by mathematics. It is impossible to prove some things by the processes of mathematics. Some things can not be proved with the clearness with which one solves a relatively simple mathematics problem. Thera is mare than one road to conviction and certitude. The deeply-rooted convictions of religious faith are necessary to give com pleteness to the search for truth. It has been.said, "Education is what is left over when one has forgotten what he has learned," This la a negative definition, and one which would not in Itself be 5Ibid.. p. 46-48 284 sufficient to define education. But it carries the thought that education is more than the memorization of rules and formulas, even in the mathematics classroom. If the student is a whole being, his study of mathematics affects his character, and it is his same will which will guide his achievements and actions, whether moral or intellectual. The content of mathematics is not, of course, transferred to an alien field of thought. If this were so, it would be a magical transformation. Principles of mathematics do not suddenly become principles of ethics and government when the thinker turns his attention from mathematics to religion or politics. The National XJnion of Christian Schools, in the Mathematics Curriculum Guide, devotes most of its forty page booklet to the study of mathematics content. At the end of the booklet, there is a list of items which are suggested to the teacher who wishes to appraise the student’s progress toward God-centered thinking in mathematics. It is admittedly a difficult task, and cannot be accomplished by a pencil and paper test, or a single examination. Periodic discussion of questions like the following are given as suggestions to help the the teacher make the proper evaluation: Why is it important to learn mathematics? Responses should relate to the idea that mathematics reveals God. 285 Why should a studentTs work be neat, accurate, and h o n est? Responses should relate to the idea that mathematics is a useful tool for work and service, and must be done according to God’s sta n d ard s* What would be the basis on which you would establish a business? Responses should relate to Christian ethics, stewardship, and usefulness. How are number ideas used in making things? Responses should relate to mathematics as a tool for our creative activity. How were mathematical ideas used in the creation of the world? Responses should relate to God the Creator and indicate the observation of form and order in creation. Where does the idea of numbers come from? Responses should relate to God as the source of mathematical principles and of all know ledge. What part did man play in revealing God through mathematics?. Responses should deal with the history of the number system. On the basis of the student’s contribution to such discussions the teacher can obtain some evaluation of the student’s thinking. Care must always be exercised that the discussions though planned are not forced. Of course, the grade in which the particular discussion is being held will determine the topics to consider and the formulation of the questions. The impressions received from such discussions and the result s of tests should give a fairly accurate appraisal of the extent to which the objectives for the course are achieved.® Historical references. In a course in the history of mathematics, there should be sufficient opportunity to National Union of Christian Schools, Mathematics Curriculum Guide (Grand Rapids, 1958), p. 38. 286 consider the religious influences which have affected that development. The incidents and the trends which have been outlined in the early chapters of this s tudy will supply suggestions for such consideration. Additional materials can be gathered through the assigning of term papers relating to specific influences, people, and historical events. A study of the religious writings of some of the famous mathematicians, a study of the develop ment of the number symbols, the number system, or of some particular part of mathematics can be made, with the idea in mind of seeing how that development was affected by being associated with religious ideas. Mathematical and religious ideas. There are many subjects which have been of common concern to both mathe maticians and religious leaders. The concepts of infinity, continuity, and lim it, have been difficult for the mathe matician to explain in explicit terms. There are some mathematicians s till who do not feel comfortable away from the ideas of discrete, countable, finite mathematics. The basic considerations of the church are those of eter nal, immutable, and unlimited measure. Dantzig, Bell, Smith, Keyser, and others have spoken on the relationship of these ideas in mathematics to similar ideas in religion. Pierre Conway completed in the summer of 1964 a monograph on modern mathematics, in which he presented some objections 287 to some of the ideas which, in his opinion, are in opposition to the views of Catholic education. He takes issue with the idea of "set," which leads to the division into finite and infinite sets, and immediately into the erasing of distinction between absolute and relative t r u t h . The idea of limit is another one which involves thinkers in both areas. Genuine ideals are not goals to be reached, but perfections to be endlessly pursued. They are like mathematical lim its, and religious ideals, which may be approached more and more nearly, but never attained to perfection. Other ideas of common concern would include such matters as dependence and freedom, unity and diversity, variance and invariance, relationships, proof, the inter pretation of natural and supernatural law, the problem of origins, the basic question as to whether it is possible to harmonize mathematics and religion, or whether these two areas of concern are to be two separate parts of man's thinking. One student may use an answer book in mathematics to find an answer to a problem in order to work the pro blem, The wise student will use the answer book only as a response verification source, checking the answer to confirm his own work. One student will search in some 288 source such as archaeology, mathematics, or biology, for ways to prove the Bible is true, or that God exists. The wise student will accept God and his religious faith, and w ill see when science and mathematics have spoken, the verification of his faith. Suggestions for teacher training. The historical material included in the early chap ters of this study will provide a panoramic view of the religious influences in the history and development of mathematics. The information in Part II should illustrate the topical manner in which religious bearings in mathema tics can be studied. The bibliography at the close of Part III will identify many sources of information for fertile investigation. Books on the history of mathema tics; magazines such as Isis» Osiris. Scripta Mathematica. and the Mathematics Teacher; and the publications re-issued by Dover Publications, Inc., will be found of great value. Lectures by the instructor, research assignments and reports by the students, and class discussion on the various topics assigned should provide a real source of interest and insight in relating mathematics and religion. In a Christian school, it is anticipated that a teacher will have a special interest in making explicit the relationships which exist between religion and the aca demic areas. The following suggestions are included at 289 the close of this study to assist such teachers In this worthwhile endeavor* These questions could be used in discussion groups, as topics for mathematics club meet ings, or as suggestions for term papers and oral or written reports* 1* Use of number. What numbers have been con- sidered of specTalTreligious significance because of their use in the Bible? Study can be made of individual numbers, or patterns in numerology which have been developed by w riters. How has number been used in the study of the great pyramids? How are numbers used in the everyday work of the church? 2. Humeral systems. How did the use of the Greek and Hebrew alphabets for numerals lead to the develop ment of gematria? What effect did gematria have during the middle ages? Are there evidences of this approach to religious study and numerology at the present time? Are there examples in the Bible and in the Talmud of this principle? How did the use of an alphabetic numeral system hinder the development of a place-value system? 3. Numerical operations. Can you find examples of addition, subtraction, multiplication, division, counting, fractions, interest, ratio, the use of dimensions, measurement, and other mathematical ideas in the Bible? 4. Pythagorean number religion. What parallels exist between the number philosophy of Pythagoras and a religion, as you view it? What are figurate numbers, prime numbers, perfect numbers, and what religious significance have they had? What caused the problem with the rational philosophical system of Pythagoras? Does a sim ilar situation ever arise in religion? 5. Humber patterns in nature. What is the phyllo- taxis principle? How are the ffbonacci sequences related to this principle? Are there other such number designs in nature? . 290 6. Geometric patterns in nature. How did Plato and other early philosophers make use of the limited number of regular polygons in a religious sense? What geometric patterns exist to illustrate design in creation? What is meant by divine proportion? What are some of the illustrations of its use in . nature? What is the structure of the honeycomb, the snow flake, sunflower head, pine cone, and rain drop, to mention only a few such shapes? 7. Religious use of geometry. What use has been made of geometric shapes in the Church? How has architecture changed over the years? What significance has the circle, the equilateral triangle, and the c ro s s ? 8. Values in the study of mathematics. Dr, Fehr suggests three classifications of values: a. Non related ones such as neatness and cleanliness, self- reliance, making wise decisions, cooperation, and leadership, expression of ideas publicly, and the infinitude of God. b. Related values such as information, logical thought, operational tools, cultural enrichment, and the search for truth. 9. The search for truth. (See Chapter Eight). Does absolute truth exist in mathematics? How does an understanding of closure help explain the mean ing of truth in mathematics? Did the invention of non-Euclidean geometry and the principle of relativity end the search for absolute truth? What effect has non-Euclidean geometry had on religious thought? 10. Science and religion. Why do so many people think there is an impassable gulf between science and religion? Why does education lead to secular views and often to non-religious thinking? Does 2 9 1 such a sequence justify the church's ignoring of new discoveries and inventions? Must abstract study be non-religious? Can one accept the scientific method and be a Christian? 11. Concepts related to both mathematics and religion. How are lim its in mathematics similar to ideals? What is the relationship between infinity in mathematics, and infinity in religious thought? Are there ideas dealing with the continuous in mathematics which are similar to the idea of eternal life in religion? What role does intuition play in mathematics learning, and in religious experience? 12. New ideas. Is it necessary for the Church to accept new ideas of science when they are first pre sented as being proved? Was the church wrong in suppressing the Copernican theory? Can there be any conflict between religion and pure man-made mathema tics which is admittedly hypothetical? Will it be possible for man to follow any ideas mathematically for which there are no physical applications? 15. Existence of God. Doe3 the perfected theory of the universe as successively presented by Ptolemy, Copernicus, Kepler, Newton, Laplace, Einstein, and others exclude a necessity for a Creator? Or does this study make the evidence for a Creator more assuring? Pascal said one does not know God until he has already found him. The Bible says It Is by faith that we understand that the worlds were created by the word of God. Does Godel's proof help demon strate the existence of God through the impossibility of man's ever finding all propositions by reason alone? Are there lim ts of scientific investigation beyond which man will not be permitted to go? What mathematical arguments have been used to demonstrate the existence of God? 14. Religious influences a help or a hindrance to the advance of civilization. Was the secrecy o t th e priests in Egypt and Babylon a help or a hindrance? Was the establishment of monasteries a help or a hindrance? Was the organized church a help or a hindrance? What was the relative position of the Protestant and Catholic church leaders toward science in the seventeenth century, or in more recent times? How d id th e mohammed movement a f f e c t th e developm ent of mathematics? What has been the contribution of 292 Jewish scholars to mathematics? What contributions have been made by oriental religions? Of what value is the flowery language of Omar Khayyam and others who open and close their mathematical treatises with elaborate religious expressions? What affeot did Charlemagne’s adoption of Christianity have on the study of mathematics? 15. Invention and discovery. Is mathematics a creation of the mind of man, or is it a discovery of eternal laws, and man’s best development of a symbolism by which these laws can be understood? What difference will it make in one’s teaching of mathematics whether he accepts the one or the other? 16. Early mathematics. If symbols were not used until the alphabet was invented, how did Moses and David count and record the numbers found in the Old Testament? Wens the early priests really religious leaders or public officials in charge of calendars, standards, banking, surveying, and other practical m a tte rs ? 17. Printing and paper-making. Why have the Bible and Euclid been the first books printed in new situa tions? What affect has printing and wide-spread dissemination of knowledge had on the relation between religion and mathematics? 18. Origin of mathematics. Have the motives for the development of mathematics been religious, socio economic, magical, or mystical? Has the West devel oped mathematics and a modern civilization because of the religious belief of its leaders, that there was a Creator, and upholder of creation, and thus a definite pattern which was worthy of investigation? Have architecture, art, and other mathematical applications depended greatly upon religious influences for their perfection? Does it seem that men would wish to create better things for their church than they would be satisfied for for themselves? Who were the first mathematicians, priests or sribes? 19. Religious persecutions. How have religious persecutions spread mathematical knowledge? How was this accomplished In the first centuries of the present era? In the twentieth century in recent years In Germany? During the time of the Crusades? During the time of the French Revolution? 293 20. The calendar. What is the religious signifi cance of the calendar? Who developed and kept the calendar in the early days of history? How was the time of the birth of Christ chosen as the dividing date.for the recording of history? How do other religions date their important events? Has the necessity of determining the date of Easter had a great effect on mathematics and the calendar? Would the new World Calendar being advocated be acceptable by church leaders, since it would destroy the seven-day cycle one or two times a year? 21. Astrology* How was it possible for Kepler and other church leaders to accept astrology? What value is there in this study today? 22. Mathematical biography. What events in the lives of mathematicians are of interest because of some religious significance? Think of Leibniz and the binary system. Leibniz and the imaginary number. Swedenborg and the base-eight system. Thomas H in»s mathematical sermons. The universal plans of absolute argument, and irresistible evangelism as proposed by Leibniz, Novalis, and Lully. The arguments of Descartes, Spinoza, Russell, and Godel for a mathematical method of thinking religious thoughts. David Eugene Smith included more than one hundred men who were related to the church, and who made some contribution to mathematics, in his History of Mathematics. 23. Present day movements. Are mathematics teachers in Christian schools arore of or interested in religious bearings in mathematics? What use can be made of some of these ideas in areas other than mathematics? Can these ideas be introduced in mathematics classes other than in the class in the history of mathematics? What is the status of the idea of producing mathematics textbooks for Christian schools which differ from those for public schools? Is there any material in secular textbooks which is objectionable from the religious viewpoint? Is there any principle in modern mathema tics which will lead to atheistic or agnostic views? What groups are interested in applying religious emphases in mathematics in the classrooms? 24. Appreciation. What use has been made of mathe matics in the writing of great literature? What poetry contains mathematical allusions to describe lofty ideas? 294 In what ways can mathematics act as one of the humani ties? How does mathematics contribute to the esthetic se n se s? 25. Memorabilia Mathematica. A book by this name was prepared by Robert Edouard Moritz, containing 1140 anecdotes, aphorisms, passages by famous mathematicians, scientists, and writers. Students may find many relig ious inferences in this book. Additions may be made to this collection from the daily newspaper or periodicals. For example, a mortician advertisement quotes William E. Gladstone as saying "Show me the manner in which a nation cares for its dead, and I will measure with mathematical exactness, the tender mercies of its people, their respect for the laws of the land, and their loyalty to high ideals." When President Eisenhower sent the first message across the country by satellite, he spoke of "Peace on Earth." He was taken to task by the International Council of Churches for the absence of reference to God, in the light of Morse's first message "What hath God wrought." This can remind a class of Laplace's encounter with Napoleon mentioned on page 176 of this stu d y . BIBLIOGRAPHY Books Abbott, Edwin A. Flatland—A Romance of Many Dimensions. New York: Dover Publications, Inc., 1952. Abrahams, Israel. Jewish Life in the Middle Ages. Philadelphia: The Jewish Publication Society of America, 1911. Allman, George Johnston. Greek Geometry from Thales to Euclid. London: Longmans, Green & Co., 1889. Anicker, Doris. Finding God in Arithmetic. Spokane: Privately published, T §56. Archibald, R. C. Benjamin Peirce, 1809-1880. Oberlin, Ohio: Mathematical Association of America, 1925. Armitage, Angus. The World of Copernicus. New York: The New American Library of World Literature, Inc., 1951, Arons, A. B., and Bork, A. M. Science and Ideas. Englewood Cliffs, N. J .: Prentice-Hall, 1964. Ball, W. W. Rouse. History of Mathematics. New York; Macmillan Co., 1925. ______• A Primer of the History of Mathematics. London: Macmillan Co., 1906. ______. A Short Account of the History of Mathematics. New York: Dover Publications, Inc., 1960. Barnes, Albert. Notes, Explanatory and Practical on the New Testament. New York: Harper & Brothers, 1868. 295 296 Bell, Eric T. Mathematics. Queen and Servant of Science. New York: McGraw H ill Book Company, In c ., 1951 ______. Men of Mathematics. New York: Simon and Schuster, 1937. ______. The Sea? ch fo r T ru th . B altim ore: The W illiam and Wilkins Co., 1934. Bengel, John Albert. Gnomon of the New Testament. Philadelphiaj Perkinprie and Higgins, 1860. Blau, Joseph L. The Christian Interpretation of the Cabala in the Renaissance. New York: Columbia University Press, 1944. Board for Parish Education, The Lutheran Church, Missouri Synod, Resource Units for Lutheran High Schools. St. Louis! Curriculum Commission of the Association of Lutheran Secondary Schools, 1953. Board of Education, The Development of Moral and Spiritual Ideals in the Public Schools. New York: New York City Board of Education, 1956, Branford, Benchara. A Study of Mathematical Educ ation. Oxford: Clarendon Press, 1921. Bronowski, J. Science and Human Values. New York: Julian Messner, Inc., 1956. Brubacher, John S. Modern Philosophies of Education. New ' York: McG-raw Hill Book Co., 1950. ______• (ed) The Public Schools and Spiritual Values. New York: Harper and Brothers, 1944. Buchanan, Scott. Poetry and Mathematics. New York: The John Day Co., 1929. Bullinger, Ethelbert W. Number in Scripture. London: Eyre and Spottiswoode, 1894. B u tte r fie ld , H erb ert. The O rigins of Modern S cien ce. New York: Collier Books, 1962. Buttriclc, Geo A. (ed. ) The Interpreter1 s Bible. Vol. Ill: Luke, John. Nashville: Abingdon Press, 1958. 297 Cailliet, Emile. Pascals The Emergence of Genius. New York: Harper & Brothers, 1945. Cajori, Plorian. History of Mathematics. New York: The Macmillan Co., 1919. Carr, Herbert Wildon. Leibniz. New York:. Dover Publica tions, Inc., 1960. Chelebi-Katib. The Balance of Truth. London: George Allen and Unwin, Ltd./ 1957. Colburn, Zerah, A Memoir of Zerah Colburn. S p rin g fie ld : G. & C. Merriam, 1833. Columbia Associates in Philosophy. Introduction to Reflective Thinking. Boston: Houghton Mifflin Company, 19*23. Conant, James B. Modern Science and Modern Man. New York: Dover Publications, Inc., 1953. Conant, Levi Leonard. The Number Concept. New York: The Macmillan Company, 1923. Cook, P. C. ( e d .) The Bible Commentary. Vol. I I: S t. John, The Acts of the Apostles. New York: Chas. Scribners Sons, 1883. Cooke, Josiah Parsons. The Credentials of Science, the Warrant of Paith. New York: D. Appleton and Company, 1893. Cooley, Hollis R., and others. Introduction to Mathematics. Boston: Houghton M ifflin Company, 1949. Coolidge, Julian Lowell. The Mathematics of Great Amateurs. New York: Dover Publications, Inc., 1963. Conway, Pierre H. Mathematics' Notes: AConfrontation of Modern Mathemat1cs w ith the P rin c ip le s of A ris to tle and S t . Thomas A quinas. Columbus: S t. Mary of the Springs, 1964. (Mimeographed, j Courant, Richard, and Robbins, Herbert. What is Mathematics? New York: Oxford University Press, 1941. Dampier, William G., and Dampier, Margaret. Readings in the Literature of Science. New York: Harper & Brothers, Publishers, 1959. 298 Dantzig, Tobias, The Bequest of the Greeks. New York: Charles Scribners Sons, 1955, ______, Number, the Language of Science. New York: The Macmillan Co., 1933, Dewey, John, A Common F a ith . New Haven: Yale U n iv ersity Press, 1934, ______• Moral Principles in Education. New York: Houghton Mifflin Co., 1909. Dickinson, G. Lowes. The Greek View of Life. Chautauqua, New York: The Chautauqua Press, 1909, D’ooge, Martin Luther. Nicomachus of Gerasa, Introduction to Arithmetic. New York: The Macmillan Co., 1926. Drane, Augusta Theodosia. Christian Schools and Scholars. New York: C. E. Stechert & Co., 1910. Dresden, Arnold, An Invitation to Mathematics. New York: Henry Holt & Co., 1936, Dunnington, G. Waldo. Carl Freidrich Gauss: Titan of Science, A Study of His Life and Work. New York: Exposition Press, 1955. Durant, Will. The Story of Philosophy. Garden City: Garden City Publishing Co., 1926, Dyck, Martin. Novalis and Mathematics. Chapel Hill: University of North Carolina Press, 1960. Edersheim, Alfred, The Life and Times of Jesus the Messiah. Grand Rapids: Wm. B. Eerdmans Publish ing Co., 1958. ______. Sketches of Jewish Social Life. Hodder & Stoughton, New York, 1888. Efron,' Andrew. The Sacred Tree Script. (The Esoteric Foundation of Plato’s Wisdom). New Haven: Tuttle, Morehouse, and Taylor Co., 1941. Eves, Howard. An Introduction to the History of Mathema t i c s . R inehart & Company, In c ., 1953. Feibleman, James K. The Pious Scientist. New York: Bookman A ssocTatlon. Gaebelein, Frank E. The Pattern of God * s Truth. New York: Oxford University Press, 1954. Ghyka, Matila. Philosophie et Mystique du Uombre. Paris: Payot, 1952. Ginsberg, Louis. The Legends of the Jews. 6 vols. Philadelphia: The Jewish Publication Society of America, 1938, Graubard, Mark, Astrology and Alchemy, Two Fossil Sciences. New York: Philosophical Library, 1953. Graves, Robert and Hodge, Alan. The Long Week- End. A Social History of Great Brltalh, 1918, 1939. London: Faber Sr. Faber Ltd., 1940. Gundersen, Borghild. . Cardinal Nev/man and Apologetics. Oslo: I Kommisjoin IIos Jacob Dybwad, 1952. Iiadamard, Jacques. An Essay on the Psychology of Invention in the Mathematical Field. Princeton: Princeton University Press, 1949. Hall, Everett W. A Study in the History of Ideas. Prince ton: D. Van Nostrand Co., Inc., 1956. Harding, Iswry W. "The Place of Value in the Education of Teachers." Unpublished Doctor's dissertation, The Ohio S tate Univo:* s ity , 1941, Hartford, Ellis Ford. Moral Values in Public Education, Lessons from the Kentucky Experience. Harper & Brothers'^ 1958. Ileaton, E. W. Everyday Life in Old Testament Times. New York: Chas. Scribners Sons, 1956. Hegland, Martin. Christianity in Education. Minneapolis: Augsburg Publishing House, 1954, Heisenberg, W. The P h y sic ists Conception of N ature. London Hutchinson Publishing Co., 1922, Hendrikensen, William. New Testament Commentary. Grand Rapids: Baker Book House, 1954. 300 Herford, Wm. H. The Student's Froebel. Boston: D. C. Heath and Co., 1894. Hill, Thomas. Geometry and Faith. Boston: Lee and Shepard, Publishers, 1882. ______. The Life is More than Meat. Portland, Maine: Privately printed by Stephen Berg, 1874. ______• The Three Open Books. Waterville, Maine: Privately printed, 1890. Hobson, E. W. John Nap ier and the Invention of Logar ithms. Cambridge: Cambridge University Press, 1914. Hofmann, Joseph Ehrenfried. The History of Mathematics. New York: Philosophical Library, 1957, Hogben, Lancelot T. Mathematics for the Million. New York: W. W, Norton, 1937, ______. Mathematics in the Making. Garden City, New York: Doubleday & Co., Inc., 1960. Hopper, Vincent Foster. Medieval Number Symbolism. New York: Columbia University Press, 1938. Hovey, Alvah (ed.) An American Commentary on the New Testament. Philadelphia: American Baptist Publication Society, 1885. Howlett, Duncan. The Essenes and Christianity (An Inter pretation of the Dead Sea Scrolls) New York: Harper & Brothers,.. 1957. Humberd, Jesse D. "A Proposed Program in General Mathema tics for the Ministerial Student.” Unpublished M aster's th e s is , The Ohio S tate U n iv ersity , 1950. Hutchins,. Robert Maynard. Great Books of the Western World. Syntopicon. Chicago: Encyclopaedia Britannica, Inc., 1952. Ibrahim, Al-del-Aziz Essayed. "Philosophies of Education-- Their Implications for Mathematics Curriculum and Classroom Procedures,” Unpublished Doctor's dissertation, The Ohio State University, 1948. 301 International Conference on Public Education—Geneva 1956. Teaching of Mathematics in Secondary Schools—A Comparative Study. Paris: Unesco Avenue Kleber 19, Publication No. 172, Jastrow, Morris. The Civilization of Babylonia and Assyria. Philadelphia: J. B. Lippincott Co., 1915. Jeans, Sir James H. The Universe Around Us. New York: The Macmillan Co., 1944. Johnson, Francis R. Astronomical Thought in Renaissance England. Baltimore, 1937. Jones, Sister Mary Eberharda. A Course in Methods of Arithmetic. Boston: D. C. Heath and Co., 1926. Karpinski, Louis Charles. The History of Arithmetic. Chicago: Rand McNally and Co., 1925, Kasner, Edward, and Newman, James. Mathematics and the Imagination. New York: Simon and Schuster, 1940. Keyser, C. J. Mathematical Philosophy. New York: Dutton, 1922. ______• Science and Religion, The Rational and the Superrational. New Haven: Yale University Press, 1914. Kinney, LeBaron W. The Greatest Thing in the Universe-^ The Living Word of God. New York: Loizeaux Brothers, 1939. Kline, Morris. Mathematics, A Cultural Approach. Reading, Mass: Addison-Wesley Publishing Co., Inc., 1962. ______• Mathematics and the Physical World. New York: Thomas Y. Crowell Co., 1959. ______. Mathematics in Western Culture. New York: Oxford University Press, 1953. Koyre, Alexandre. From the Closed World to the Infinite Universe. New York: Harper S: Brothers, 1958. Kramer, Edna E. The Main Stream of M athem atics. New York: Oxford University Press, 1951. Lange, John P. Commentary on the Holy Scriptures. John. Grand Rapids: Zondervan Publishing House, 1958. Leibniz, Gottfried Wilhelm, Philosophical Papers and Letters. (tr. and ed. by Leroy ~E. Loemker). Chicago: University of Chicago Press, 1956, Lenski, R. C. H. The Interpretatioh of St. John1s Gospel. Columbus: The War tburg Press, 1942. Los Angeles City Schools. Moral and Spiritual Values in Education. Los Angeles, 1954. Macfarlane, Alexander. Lectures on Ten B r itis h Mathema ticians . New York: John Wiley & Sons, 1916. Macmillan, William Duncan. The Doctrine of Uniformity in Religion. Chicago: The Patton Reporting Service, 1935. Manschreck, Clyde L. A History of Christianity. Engle* wood Cliffs, N. J.: Prentice-Hall, Inc., 1964. Maziarz, Edward A. The Philosophy of Mathematics. New York Philosophy Library, 1950. McCormack, R. The Heptadic S tru ctu re of S c rip tu re . London: Marshall Brothers, Ltd., 1923. Seven in Scripture. Marshall Brothers, Ltd., 1926. Menon, C. P. S. Early Astroriomy and Cosmology, A Recon struction of the' Earliest Cosmic System. (ed,"by A. Wolff.) London: G. Allen & Unwin Ltd., 1931. Meyer, H. W. W. Critical and Exegetical Handbook to the Gospel of John. New York: Funk & Wagnalis, 1884. Midrash Rabbah, Numbers» (tr. by Judah J. Slotki.) London: Soncino Press, 1939. Miller, Arthur L. Curriculum in Arithmetic for Lutheran Schools. St. Louis: Concordia Publishing House. 1937. Moritz, Robert Edouard. On Mathematics and Mathematicians. New York: Dover Publications, Inc., 1942. Moorman, Richard Herbert. Some Educational Implications of Descartes Synthesis of Mathematics and Philosophy. Abstract of Contributions to Education 278, Nash ville: George Peabody College for Teachers, 1940, 303 Mullinger, J. Bass. A History of the University of Cambridge. London: Longmans, Green, and Co., 1888 . Needham, Joseph. Science and C iv ilis a tio n in China. vol. 3: Mathematics and the Sciences of the Heavens and the Earth. Cambridge: University Press, 1959. Neugebauer, Otto. The Exact Sciences in Antiquity. Providence: Brown University Press, 1957. Newbold, William Romaine. The Cipher of Roger Bacon. Philadelphia: University of Pennsylvania Press, 1928. Newman, Albert Henry. A Manual of Church History. The American Baptist Publication Society, 1899. Newsome, Carroll V. Mathematical Discourses. Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1964. Niblett, W. R. Christian Education in a Secular Society. London: Oxford University Press, 1960, Nicoll, W. Robertson. The Expositor’s Bible in 25 volumes. Neu York: Punk & Wagnalls, Co., 1900. Oakland Public Schools. Report of Committee on Moral and Spiritual Values in the Oakland Public Schools. Oakland C a lifo rn ia , n .d . (Mimeograp h ed )• Oliver, Welch. Improved American Arithmetic. Portland, 1841. (Referred to in Mathematics T0acher 1928). Olney, Helen L. "A Study of the Textbook Situation in Christian Elementary Schools,11 Unpublished MRE thesis, Grace Theological Seminary, 1963. Osborn, Roger, >et. al. Extending Mathematics Understanding. Columbus: Charles E. Merrill Books, Inc., 1961. Pepper, S. C., Aschenbrenner, Karl, and Mates, Benson. George Berkeley. University of California Publica tion in Philosophy, Vol. 29. Berkeley: University of California Press, 1957. Pledge, H. T. Science Since 1500. London: Ministry of Education, Philosophical Library, 1940. 304 Plummer, A. (ed.) Cambridge Greek Testament for Schools and Colleges. Cambridge; University Press, 1882. Ramm, Bernard. The Christian View of Science and Scrip ture. Grand Rapids: Wm. B. Eerdmans Publishing Company, 1955. Reichelt, Karl Ludwig. Religion in Chinese Garment (tr. Joseph Tetlie). London: Lutterworth Press, 1951. Riggs, J. S. The Messages of Jesus According to the Gospel of John. Mew York: Chas. Scribners Sons, 1908. Roback, A. A. Jewish Influence in Modern Thought. 1 Cambridge: Science Art Publications, 1929, Russell, Bertrand. Philosophy of Leibniz. London: George A llen & Unwin, L td ., 1937. Sabiers, M. A. Astounding Mew Discoveries. Los Angeles: Robertson Publishing Co., 1941. Santillana, Giorgio de. The Crime of Galileo. Chicago: University of Chicago Press, 1955. Sarton, George. Ancient Science and Modern Civilization. Lincoln: University of Nebraska Press, 1954. ______. The Study of the History of Mathemati cs. New York: Dover Publications, Inc., 1959. Sawyer, W. W. Prelude to Mathematics. Baltimore: Penguin Books, 1955. Schaff, Philip, History of the Christian Church. 8 vols. Grand Rapids: Wm. B. Eerdmans Publishing Co., 1949. ,______• Nicene and Post-Nicene Fathers. Vol. 1: The Confessions and Letters of St. Augustine. 1886. Schaaf, William L. Our Mathematical Heritage. New York: Collier Books, 1963. Schenectady Public Schools. Religion, Moral, and Spirit ual Values in the Public Schools. Schenectady: Public Schools, 1953. Schrodinger, Erwin. Mind and Matter. London: Cambridge University Press, 1958. 305 Sebaly, A. L. et al. Teacher Education and Religion. Oneonta, New York: The American Association Of Colleges for Teacher Education, 1959. Seiss, Joseph A. A Miracle in Stone or The Great Pyramid of Egyot, Philadelphia: The United Lutheran Publishing House, 1877. Shaw, Charles G. Trends of C iv iliz a tio n and C u ltu re. American Book Co., 1932. Simpson, James. Necessity of P o p u lar Education as a National Object. New York: Leavitt, Lord and Co., 1834, Slotki, Rev. Dr. Israel V/. The Talmud: Erubin. London: The Soncino Press, 1938. Smeltzer, Donald. Man and Number. New York: Emerson Books, Inc. Smith, David Eugene. History of Mathematics. 2 vols. New York: Simon and Schuster, 1958. ______. Source Book in Mathematics. 2 vols. New York: Dover Publications, Inc., 1959. Somervell, Edith L. A Rhythmic Approach to Mathemat ics. London: George Philip Son, Ltd., 1906. Soothill, William Edward. The Hall of Lights, A Study of Early Chinese Kingship. London: Lutterworth Press, 1951. Spence, H. D. M. and Exell, J. S. The Pulpit Commentary. Grand Rapids: Wm. B. Eerdmans Publishing Co., 1950. Spiller, Gustav. Report on Moral Instruction and on Moral Training in the Schools of Many Counties. London: Watts & Co., 1909. Struik, Dirk J. A Concise History of Mathematics. New York: Dover Publications, Inc., 1948. SJullivan, Sister Helen. An Introduction to the Philosophy of Natural and Mathematical Sciences. New York: Vantage Press, 1952. Taylor, E. G, R. The Mathematical Practitioners of Tudor and Stuart England. Cambridge: University Press. 1954. 306 Thorndike, Lynn. University Records and Life in the Middle Age~sT New York: Columbia University Press, 1944. Turnbull, Herbert W. Bi-Centenary of the Death of Colin Maclaurln.1698-1746. Aberdeen: The University Press, 1951. ______. The Great Mathematicians. New York: Simon and Schuster, 1962. U. S. Department of Health, Education, and Welfare. Education in the USSR. Bulletin No. 14, 1957. Van der Waerden, B. L. Science Awakening. New York: Oxford University Press, 1961. Wayland, Francis. The Elements of Moral Science (ed. Joseph L. Blau). Cambridge: Belknap Press of Harvard University Press, 1965. Wedberg, Anders. Plato1s Philosophy of Mathemat ics. Stockholm: Almquist and Wilcsell, 1955. Welch, Charles II. An Alphabetical Analysis of Terms and Texts Used Tn the Study of Dispensational Truth, Surrey, England: The Berean Pubiihsih'g Trust, 1958. Weyl, Hermann. The Open World. Hew Haven: Yale Univer sity Press, 1932. ______• Philosophy of Mathematics and Natural Science. Princeton: University Press, 1949. Wightman, William P. The Growth of Scientific Ideas. New Ka,ven: Yale University Press, 1953. Wilson, Grove. The Human Side of Science. New York: Cosmopolitan Book Corp^, 1929. Wolf, A. A History of Science, Technology and Philosophy in the 16th and 17th Qenturies. New York: Harper & Brothers, 1959. Wolf, A. A History of Science, Technology and Philosophy in the 18th Century. New York: Harper & Brothers, 1961. Wright, G. G. Neill. The Writing of Arabic Numerals. London: University of London Press, Ltd., 1952. 307 A rtic le s Abrahim, Omar Ibn, "Discussion of Difficulties in Euclid," (tr. by All R. Amir-Moez). Scrlpta Mathematica, XXIV (1959), 275-303. Amir-Moez, Ali R. "A Paper of Omar Khayyam," Scripta Mathematica. XXVI (December 1963) 323-337. Archibald, R. C. "Wallis on the Trinity," American Mathematical Monthly, XLIII (January 1936), 35-37. Bakst, Aaron. "What is MathematIcs?S The Science Counse lor, XV (December 1952), 124, 144. B e ll, E. T. "What Mathematics Has Meant to Me," Mathema tics Magazine, (January-Pebruary 1950). Birkhoff, George D. "The Mathematical Nature of Physical Theories," Science in Progress (New Haven: Yale University Press, 1945), 120-149. Boyer, Carl B. "Pascal: The Man and the Mathematician," Scripta Mathematica, XXVI (December 1963), 283-307. Bradley, George. "Another Responsibility for the Science Teacher," Focus on Religion in Teacher Education, (Oneonta, New York: American Association of Colleges for Teacher Education, 1955). Brendan, Brother T. "Ranking Objectives in Mathematics," School Science and Mathematics, XLIII (May 1958), 333-337. Brett, G. S. "Nev/ton’s Place in the History of Religious Thought," Mathematics Gazette, IV, p. 430. Bryden, James D. "Specialization and Secularism in Higher Education," The Christian Scholar, XL (March 1957), 51-60. B u tts, R. Freeman. "What Image of Man Should Public Education Foster?" Religious Education, LIII (March-April 1958) 114-120. Cairns, Grace Edith. "Liberal Theology for a Global World," The Journal of Bible and Religion, XVIII (April 1950), Cajori, Florian. "Leibnitz, The Master Builder of the Mathematical Notations," Isis, VII (1925), 412-429. 308 Carnahan, Walter H. ’’Alternative, Conventions, and Exceptions in Mathematics,” Mathematics Teacher, XXXV (1941), 312. Carver, W. B. "Thinking versus Manipulation," American Mathematical Monthly, XLIV (1937), 359. Christensen, Glenn J. "Attitude, Not Subject Matter," College and University, XXXIX (Pall 1963), 5-12. Clark, Walter Houston. "A Protestant University," Religious Education, XLVIII (July-August 1953), 248-255.. Coleman, A. J. "Faith and Math," Christian Education, XXXII (June 1949), 126-130. Dillenberger, John. "Science and Theology Today," The Christian Century, LXXVI (June 17, 1959), 722-724. Dunning, Frederick A. "Ku Ivlux Fulfills the Scripture," The Christian Century, XLI (September 18, 1924), T207. Efros, Israel. "Saadyah's Second Theory of Creation in its Relation to Pythagoreanism and Platonism," Louis Ginsberg Ju b ilee Volume American Academy fo r Jewish Research (New York: Jewish Research, 1945), 133-143. Elson, Edward L, R. "Toward the Renewal of our Spiritual Foundations," Vital Speeches of the Day, XXV (January 1, 1959')', 163. Emerton, J. A. "The Hundred and Fifty-Three Fishes in John 21:11," The Journal of Theological Studies, IX (A pril 195871! 86-89. Fakkema, Mark. "The Christian Interpretation of Physical Science," The Christian Teacher, No. 90 (June- July 1958), 32, ______. "Christian Way of Teaching Mathematics in Upper G rades," The C h ristia n Teacher, No. 88 (April, 1958), 17-18. Fehr, Howard F. "Values and the Study of M athem atics," Scripta Mathematica, XXI (March 1955), 49-53. Finkel, Joshua, "A Mathematical Conundrum in the Ugaritic Keret Poem," Hebrew Union College Annual XXVI (1955), 109-149. Fraenkel, Abraham A* "Jewish Mathematics and Astronomy," Scripta Mathematical XXV (1960), 47. Freund, C. J. "The Place of Religion in English Education," Religious Education, L (November-December 1955), 368 Fry, Thornton C. "Mathematics as a Profession Today in Industry," Mathematics Magazine, (February 1956), 71-80. Fulton, J, F. "Robert Boyle and His Influence on Thought in the 17th Century," Isis XVIII (1932), 77-102. Gandz, Solomon. "Complementary Fractions in Bible and Talmud," American Academy fo r Jewish R esearch, Jubilee Volume (1945")"", 143-157. ______• "The D ivision of the Hour in Hebrew L ite ra tu re ," Osiris, (1952), 10-34. . "Hebrew Numerals," American Academy fo r Jewish Research, IV (1932-33), 53-111. ______• "The Knot in Hebrew Literature, or From the Knot to the Alphabet," Isis, XIV (1930), 189-214. ______• "The Sources of al-Khowarizmi1s Algebra," Osiris, I (January 1936), 264-266. ______• "Studies in Babylonian Mathematics I," Osiris, VIII (1948), 12-40. ”” "Gematria," The Jev/ish Encyclopedia, (New York: Funk & Wagnalls Co., 1903), 589-591. Ginsburg, Benjamin. "The Scientific Value of the Copernican Induction," Osiris, I (January 1936), 305-313. Ginsburg, Jekuthiel, "An Unknown Mathematician of the 14th Century,"' Scripta Mathematica, I (1935), 9-32. ______• "Rabbi Ben Ezra on Permutations and Combinations, Mathematics Teacher, XV (October 1922), 347-356. Graham, Aelred, "Toward a Catholic Concept of Education," Harvard Educational Review, XXXI (Fall 1961), 408- 412. Heath, Sir T. L. "Greek Mathematics and Science," Mathematics Gazette, XXXII (July 1948), 122. 310 Hendrix, Gertrude. "What Mathematical Knowledge and Abilities for the Teacher of Geometry Should the Teacher Training Program Provide in Fields Other than Geometry?" Mathematics Teacher, XXXIV (1941), 70. Henry, C arl F, H. "The Power of T ruth," C h ristia n T eacher, I (First Quarter 1964), 15-25. Hilgard, Ernest R., and Russell, David H. "Motivation in School Learning," Learning and Instruction. Forty- Ninth. Yearbook, Part I, National Society for the Study of Education (Chicago: The Society, 1950). Hill, Thomas. "The Usas of Kathesis," Bibliotheque Sacre, XXXII (1875), 498-524. Hunt, Rolfe Lanier. "Religion: Its relation to Public Education," Nation1s Schools, (March 1961), 59-65; 106-112. "Kabbalah," Encyclopedia Britannica. Keller, Henry. "Numbrics in Old Hebrew Medical Lore," Scripta Mathematica, I (1955), 32. Kennedy, Gerald T. "The Use of Numbers in Sacred Scripture," American Ecclesiastical Review, CXXXIX (July 1958)', 22-35. Keyser, C. J. "Benedict Soinoza," Scrinta Mathematica, V (1938), 36. . "A Mathematical Prodigy," Scripta Mathematica, “V (1938), 9. ______. "Mathematics and the Question of Cosmic Mind," Scripta Mathematica, (1935). ______. "The Role of In fin ity to the Cosmology of Epicurus," Scripta Mathematica, IV (1937), 33-44. ______. "The Spiritual Significance of Mathematics," Religious Education, VI (April 1911-February 1912), 384-390. Koyre, Alexandre. "The Significance of the Newtonian Synthesis," Archives Internationale d* histoire des Sciences, XXIV (April 1950), 291-311. 311 Danger, Rudolph E. "The Life of Leonard Euler," Scripta Mathematica, III (1935), 61-66. LaNoue, George R. "Thett National Defense Education Act U and fSecular1 Subjects," PhllDelta Kapoan, XLIII (June 1962), 380-387. Leith, T. H. "Some Thoughts on a Christian Philosophy of Science," Journal of the American Scientific Affiliation'll X (June 1958), 14-16, Locke, E. "Teaching of Morality in the Public School," High School Journal, XLII (November 1958), 49-53. Lynch, J. Joseph. "Jesuits and the IGY," Thought, XXXIII (June 1958), 248-254. Maritain, Jacques. "God and Science," University, (Winter 1962), 29-34. McDonald, Louise Anderson, "The Interplay of Mathematics and English," Mathematics Teacher, XVIII (May 1925), 284-295. Merton, Robert K. "Science, Technology, and Society in the Seventeenth Century England," Osiris, IV (1938), 360-632. Mitchell, Stephen 0. "Necessary Truths and Postulational Method," The Modern Schoolman, XXXVII (November 1959), 49-52. Morse, Marston, "Mathematics, The Arts and Freedom," Thought, XXXIV (Spring 1959), 16-24. Muggli, S r. Joanne. "B enedictine C ontributions to Mathema tics," American Benedictine Review, IV (Soring 1955), 34-47. Nickerson, Albert L. "Editorial," The Lamp, (June 1954), 9. "Number of the B east," The S ign, XXI (A pril 1942), 560. "Number," The Interpreter1s Dictionary of the Bible, vol. 2, (Nashville: Abingdon Tress, 196^),561-567. Odescalchi, Edmond P. "Mathematics in Western Thought," Sehool Science and Mathematics, LXIII (May 1963), 379-385. Olds, George B. "Mathematics and Modern Life," Mathema- Teacher, XXI (April 1928), 183-196. Ong, Walter S. "Secular Knowledge, Revealed Religion, and History," Religious Education, LII (September October 1955), p. 33. Osborn, Roger. "Some Historic and Philosophic Aspects of Geometry," Mathematics Magazine (November- December 1950), 77,-62. Parsons, Elsie Clews. "The Favorite Number of the Zuni Indians," Science Monthly, III (1916), 596-600. Patrides, C. A. "The Numerological Approach to Cosmic Order During the English Renaissance," Isis, XLIX (1 9 5 8 ), 391-397. Peacockel, A. R. "The Christian Faith in a Scientific Age," Religious Education, LVIII (July-August 1963), 372-375. Pearson, Roy. "The Give-and-Take Between Science and Religion," Think, XXVIII (March 1962), 26-28. Polachek, Harry. "The Structure of the Honeybee," Scripta Mathematica, VII (1940), 87-98. Preuss, Arthur, "An Excursion into the Realm of Mathe matics," The Fortnightly Review, XXXIV (October 1§32). Quievreux, Francois. "La Structure Symbolique de l'Evan- gile de Saint Jean," Revue D’Histoire et de Philosophie Religieuses^ ("Paris: Presses Universitaires de France, 1953), 123-165. Ramshaw, Walter A. "Can a Christian be a Scientist?" This Day, XXV (January 1959), 37-38. "Religion in Higher Education," a Symposium, Journal of Higher Education, XXIII (October 1955), 3S0- 371. Richardson, E, J. "On Gravity," The Rosicrucian Digest, XXXVII (March 1 9 5 9 ), 33. Richardson, Moses. "Mathematics and Intellectual Honesty, The American Mathematical Monthly, LIX (February 1552')' , 2 2 -2 7 . 313 Robbins, Charles J* "A Fish Story," The Priest, XVI (December 1960), 1091-1092. Russell, Bertrand, "My Philosophical Development," The Hibbert Journal, LVII (October 1958), 2-8. Rybnikov, K. A. Istoriya Matemayikl, I. Moscow: Izdatel*stuo Moskovskogo Universiteda 1960. (Reviewed in Scripta Mathematica XXVI, December 1963), 363-364. Sarton, Ge'orge, "Decimal Systems Early and Late," Osiris IX (1950), 581-001. ______. "Spinoza," Isis, X (1928), 11-14. ______. "The Unity and Diversity of the Mediterranean World," Osiris, II (1936), 430-433. Seerveld, Calvin. "Marks of Christian Education," Christianity Today, (February 27, 1961), 26-27. Seidenberg, A. "The Ritual Origin of Counting," Archive for History of Exact Sciences, II (November 16, 1962), 1-40. ______• "The Ritual Origin of Geometry," Archive for Hi&tory of Exact Sciences, I (1960-1962), 488-526. Seidlin, Joseph. "High Standards: Sacred & Profane," Mathematics Magazine, (Mareh-April 1950), 189-192. Seminar on the School, National Convention Religious Edu cation Association, Religious Education, LIII (March-April 1958), 157. Senn, Milton}. "On Teaching Ethics," The Instructor (May 1961), 3. Sen, Sanat Kumar. "The Method of Spinoza," The Philos- Quarterly, XXXI (July 1958), 53-35. Shaw, J. B. "A Chapter on the Aesthetics of the Quadratic," Mathematics Teacher XXI (March 1928), 121-134. Short, H. L. "Survey of Recent Theological Literature," The Hibbert Journal LVII (October 1958), 76. Sikora, Joseph. "Geocentricism in the Syntaxis Mathematica," The New Scholasticism, XXXII (January 1958), 61-72. 314 Sinclair, J. HI* d. R. S. Tolmar, "An Attempt to Study the E r r -fc of Scientific Training upon Prejudice and IXZL. id eality of Thought," Journal of Educa- tio n a X ^zychology, XXIV (1933), 362-37677 Sisson, Edward. "The High School’s Cure of Souls," E d u c a t 3_ Review, (April 1908), 359-372, Stapleton, H. "Ancient and Modern Aspects of Pytha- g o re a n . ” Oalris, XIII (1958), 12-53, Sukumar, R an jE L ix l s . "Scope and Development of Indian A s t r o n o r " Osiris, II (1938), 197-219, Suter, Rufus. I? X ae Strange Case of Blaise P ascal," The S c i e n t 3_ o Monthly, LXII (A pril 1946), 423-428, Smith, David E \x j m e . "Introduction to the Infinite," Math.enx«L~ o s Teacher, XXI (January 1928), 1-9, "T h .« sson of Dependence," Mathematics "Teaob-on 3CXI (April 1928), 214-218. , " R e ! dLo Mathematica," Mathematics Teacher, "XIV ( T >< -xriber 1921), 413-426, ”Ttx< oience Venerable," Mathematics “Teach.© p ZXLV (1952), 348, Thiessen, Pefcox^ — "The Need for Christian Textbooks," The Cfcur-ix: -fc ian Teacher, No, 94 (December 1958), 41, 4 3 — Thompson, J . A j t * lxxi *. "The Gospel of Evolution," Mat h e m fa. o s Gazette, XV (1931), 125. Tuls, John. ” 07 Place of Mathematics in the Christian S c h o o X -xriculum ," The Calvin Forum, XXI (0 c to t> © 3.955), 25-29. Turner, E. R. — m - lie Hindu-Arabic Numerals," Popular Scienc € >nthly, LXXXI (December 1912), 601-614, W estfall, R i c P x c S, "Isaac Newton: Religious R ational- i s t ox* s t l c ? " The Review of R eligion, XXII (M arctx 58), 155^170. W hitaker, J o h rx "The Position of Mathematics in the Hieraofct: o f ' Speculative Science," Thomist, III (J u ly □ IL), 467-506.