## Section 5.4 Themes of Lectures

We end the chapter with a mathematical classification of the lectures sponsored by the Philadelphia Section at annual meetings from 1942 to 1955. Analysis continued to be the leading choice of fields during this period, though the topology/geometry union presented a serious challenge to its dominance. The present period is also notable for the emergence of foundations as an attractive subject. However, two fields with little or no history in the section garnered even stronger interest – applications and education/pedagogy.

The reader is urged to keep in mind that it is often difficult to place certain mathematical papers into one category or another, even when the entire paper is available. This task is further complicated when only an abstract, or, in some situations, only a title, appears. The interested reader is reminded that the sources for the secretaries’ official reports in the Monthly are listed in Table 5.1.1.

*Analysis*

A compelling reason why analysis continued to be a major attraction at sectional meetings was the existence of the internationally strong Penn School of Analysis. Chapter 4 noted earlier contributions of Hans Rademacher, J. A. Shohat, and J. A. Clarkson. During the period 1942-1955 the section heard five talks by four new members of the Penn School, beginning with the 1945 lecture by Antoni Zygmund, “Some unsolved problems in the theory of trigonometric series”. Even though Zygmund accepted a position at the University of Chicago the next fall, the Penn School continued its tradition of keeping the section abreast of recent developments. Indeed, the object of N. J. Fine’s 1947 talk, “On Walsh functions,” was to complete a system of functions described earlier by the School’s reigning leader, Hans Rademacher. Five years before, I. J. Schoenberg gave a talk “On a theorem of Jensen” that dealt with circles in the plane and roots of polynomials but which we have classified under analysis because of a familiarity with the speaker’s work. We drew the same conclusion about Schoenberg’s 1949 paper “On smoothing operations,” even though section secretary C. O. Oakley did not provide an abstract of this talk. There can be no doubt that the two lectures delivered in 1950 and 1951 by Bernard Epstein concerned analysis, however. The first one dealt with “The classification of Schlicht functions,” a topic that concerns smooth (analytic) functions. Epstein’s second talk had a similar theme, “An infinite-product expansion for analytic functions”.

Talks on analysis were not given by the Penn School alone. Penn State’s H. J. Curry, who was assigned to the Frankford Arsenal in 1942, spoke about “The Heaviside operational calculus”. It would be 30 years before Curry would return to deliver another lecture to the section. In 1950 Albert Wilansky spoke about his emerging specialty, functional analysis, in a talk titled “The essential roughness of mathematical objects”. Wilansky’s results were based on earlier findings by S. Banach and A. Zygmund.

There were also two lectures during this period on ordinary differential equations. In 1948 W. R. Wasow of Swarthmore College described the effects of omitting terms whose coefficients are very small, especially when such equations are being used to describe physical phenomena. Seven years later, in the penultimate lecture of this period, William Feller took a more abstract approach to differential equations by adopting an approach via operators.

Three lectures in this period dealt with numerical analysis. In the first one, delivered in 1942 and titled “On modern methods in the numerical solution of linear problems,” Hilda Geiringer took a classical approach to solving systems of linear equations. Such was not the case with Herman Goldstine’s two lectures, which reflected the nascent stage of the computer revolution. In 1949 his lecture, titled “Some problems in numerical analysis,” described one of the four major sources of errors that occur in numerical computations – rounding errors – which arise from the “noise” in the computing machine itself. This theme was continued in his talk five years later on numerical stability. At the conclusion of the lecture that completed the 1954 meeting at Princeton, Goldstine was able to demonstrate his results on the computer at the Institute for Advanced Study. This must have been a real treat for the audience!

*Geometry/Topology*

Eleven talks were devoted to either geometry or topology during the period 1942-1955, seven of which dealt primarily with topology itself. Alexander Doniphan Wallace, who taught at Penn 1941-1947 before going to Tulane and Florida, opened the 1943 meeting with a lecture on “Fixed point theorems”. In his talk Wallace applied Brouwer’s fixed-point theorem to a surprising realm, the theory of matrices. Two years later Wallace’s colleague, W. H. Gottschalk, known for his book on topological dynamics, spoke on “Continuous flows and AP functions”. Here AP stands for almost periodic.

R. H. Fox, known as an excellent speaker and expositor, opened the 1945 meeting with a lecture titled “Homotopy groups”. Although that subject tends to be rather abstruse – Fox discussed algorithms for calculating homotopy groups – he almost certainly presented the material in terms understandable to an MAA audience. Three other talks during this period dealt with topology: Edwin Hewitt (1946), “Generalizations of the Weierstrass approximation theorem,” J. C. Oxtoby (1949), “Minimal sets,” and S. T. Hu (1950), “Topological properties of spaces of curves”.

Four talks during the period were concerned with assorted topics in geometry. In 1943 the section’s founder A. A. Bennett interacted with the section on a formal basis for the last time when he delivered an address titled “Some modern viewpoints on euclidian [sic] geometry”. Bennett’s talk “arose from discussion of an elementary Hermetian [sic] geometry of the general simplex,” suggesting a topological connection. We do not know the reason for the nonstandard orthography in Major Bennett’s title and abstract.

All three talks at the 1946 meeting at Penn dealt with geometry or topology. T. A. Botts opened the program with a lecture on “Convex sets” in the Euclidean plane, C. B. Allendoerfer followed with “Slope in solid analytic geometry,” and E. Hewitt gave the third one mentioned above.

The remaining two talks on geometry occurred in the 1950s. In 1954 A. S. Besicovitch applied methods from topology to solve problems with “Area and volume”. The following year R. J. Wisner of Haverford College reported on joint work he had carried out with his colleague C. O. Oakley on “Flexagons,” geometrical objects related to n-dimensional polygons for \(n \ge 3\text{.}\)

*Algebra/Number Theory*

Eight lectures dealt with topics in the algebra/number theory category, but only two concerned topics in algebra proper, thus continuing the limited offerings in this field. In 1951 Emil Artin lectured on “Constructions with ruler and divider,” where the divider, as opposed to a compass, was fixed and not free. The following year Russell Remage lectured on “Matrix inversion by partitioning”. The talk by Ernst Snapper in 1954, “Coordinates of algebraic varieties,” lay at the boundary of algebra and geometry.

Five of the papers delivered during this period dealt with number theory, two of which dealt with continued fractions: G. C. Webber (1943), “Transcendality of certain continued fractions,” and V. F. Cowling (1947), “Convergence criteria for continued fractions”. Cowling taught at Lehigh from 1949 to 1961 before moving to the University of Kentucky. One of his first doctoral students there, Jerry King, has been extremely active in the section since coming to Lehigh in 1962 with a newly minted Ph.D.

In 1945 D. H. Lehmer applied his specialty, computing, to arithmetical theorems concerning compositions and partitions of numbers in an address titled “Some graphical methods in the theory of numbers”.

As we noted at the end of Chapter 3, one of the section’s founders, Howard Mitchell, presented a lecture at the 1932 meeting on “The life and work of Ramanujan”. Twenty years later the section heard a talk on Ramanujan’s work again when N. J. Fine extended some of Ramanujan’s results in a talk titled briefly “The Ramanujan identities”. Fine subsequently moved from Penn to Penn State. In Chapter 3 we noted that one of Fine’s colleagues, George Andrews, lectured on this same subject in 1990.

Hans Rademacher delivered the final talk on number theory in 1955, titled “Dedekind sums and classes of modular substitutions”. He had planned to lecture on this same subject when he accepted the invitation to deliver MAA’s Hedrick Lecture in 1963, but illness prevented him from giving the talk. However, his doctoral student Emil Grosswald, who would become the section’s governor 1965-1968, delivered the lecture in his place. Grosswald further memorialized Rademacher’s work in the field by editing and expanding Rademacher’s notes into the 1972 Carus Monograph

*Dedekind Sums*.*Probability/Statistics*

We mentioned that the period 1942-1955 was notable for sponsoring the first talks on foundations, beginning with a talk by J. B. Rosser in 1943 titled “On the many-valued logics”. Five years later Lehigh’s Theodore Hailperin discussed “Recent advances in symbolic logic,” while two years after that Penn’s C. D. Firestone described “Systems of axiomatic set theory”.

In addition to talks on foundations, the period 1942-1955 is notable also for its strong emphasis on curriculum/pedagogy and applications.

*Curriculum/Pedagogy*

Whereas five lectures in the period 1933-1941 dealt with either curricular or pedagogical issues, the present period sponsored seven, highlighted by the three- person presentation “Mathematics through the television lens” at the 1954 meeting. Since the presentation consisted of three people describing their experiences in designing new ways to communicate mathematics using an emerging technology, the presentation serves as a preview of one that might be conducted today under the title “Mathematics on the Internet”. First, F. G. Fender reported on a 13-week series of talks presented on TV at his home institution, Rutgers. The aim of this series was to help the general public gain a clearer understanding of “the story of the modern mathematician and his work”. Next Bryn Mawr’s Marguerite Lehr described a 15-week series of half-hour talks presented on one of Philadelphia’s three major channels. (For details on the series see her article in the January 1955 Monthly, pp. 15-21.) In the final part of the presentation, Delaware’s R. F. Jackson bonded two emerging technologies when he outlined the general philosophy and specific plans for a TV series titled “Thinking machines – From fingers to flip-flops”.

A prevailing folklore asserts that the launching of Sputnik in 1957 not only began the space age but also launched mathematics education in the United States into the 20 th century. Activities in our section throughout the decade prior to the Russian launch belie that assertion. Indeed, activities at our section’s annual meetings show that American mathematical educators were quite concerned about the status of their offerings and were in the process of making changes even before the historic lift-off. In addition to the series on television the section sponsored six other lectures that dealt with high school and college curricula. In 1947 F. D. Murnaghan strongly advocated the use of vector methods in teaching trigonometry and analytic geometry. The meeting four years later included two talks that dealt with educational issues, “The stimulation of interest” by R.C. Yates and “Articulation of secondary and college mathematics” by P. J. Kiernan. As noted above, Yates described topics that stimulated interest in mathematics courses taken during the first two years of college, including “linkages, cams, paper folding” and more. The prep-school teacher Kiernan noted, “Aside from the curriculum, the major point brought up was that of reading ... the college teacher must continue the training of the student in the intensive type of reading that mathematics requires”. Kiernan reported that the issue arose from a panel discussion conducted under the auspices of the New Jersey Committee on the Articulation of Colleges and Secondary Schools.

In 1953 C. O. Oakley presented a talk titled “A new approach to freshman mathematics” in which the master expositor reported on an ambitious freshman course developed over the previous seven years at Haverford College. How ambitious? The syllabus included topics in logic, groups, the number system, fields, functions, analytic geometry, calculus of polynomials, probability and statistics. Two more lectures on educational themes were delivered at the final meeting of the present period, held at Bryn Mawr College in 1955. H. W. Brinkmann began the day’s program with “A report on the Ford Foundation study on the integration of high school and college mathematics”. In this lecture Brinkmann summarized the results of mathematics examinations administered that spring by the Conference of Mathematics Teachers in coordination with the College Entrance Examination Board. As noted above, in the final lecture of the day Morris Kline strongly recommended that in all freshman courses, “mathematics be tied intimately to physical problems and to its cultural setting and significance”.

*Applications*

Speaking of applications of mathematics, our section never offered so many talks on this topic in its 75-year history. Their titles indicate the broad scope of the applications covered. Not surprisingly, many of these talks took place during World War II, including three of the four lectures given in 1944: G. E. Raynor (1942), “Exterior ballistics,” Peter van de Kamp (1943), “Photographic astrometry,” Marguerite Lehr (1944), “Mapping problems in aerial photography,” F. L. Dennis (1944), “Spherical triangles on a slide rule,” and F. D. Murnaghan (1944), “The uniform tension of an elastic cylinder”. The two post-war talks on applications were A. D. Hestenes (1949), “Some observations relative to mathematics in research and development organizations” and Jan Tinbergen (1953), “Mathematical techniques used in economics theory”.

Overall, one can see that from 1942 to 1955 the Philadelphia Section (1) was kept abreast of developments in analysis and topology/geometry on a regular basis, (2) remained somewhat in the dark on topics in abstract algebra, (3) was apprised periodically of work done in foundations and in probability/statistics, (4) showed an interest in applications, especially during World War II, and (5) began to take a close look at educational issues relating to the high-school curriculum and the first two years in college.