Mathematical Evolutions

This volume contains 27 of the 34 articles that have appeared (some in two parts) in the American Mathematical Monthly column "The Evolution of..." since its inception in January, 1994. From 1994 through 2001 the column appeared five times a year, in every other issue. (Since then there has been only one column, in March, 2002: an article on the continuum problem by John Stillwell.) Articles missing — for unspecified reasons — are Freeman Dyson's "Scientist as Rebel" (March 1996), Detlef Laugwitz's two-part " ... Historical Development of Infinitesimal Mathematics" (May and August/September 1997), F. A. Medvedev's "Non-Standard Analysis ..." (August/September 1998), Laugwitz's comments on two Luzin/Vygodskii letters (March 2000), Walter Felscher's "Bolzano, Cauchy, Epsilon, Delta" (November 2000), and "Weierstrass, Luzin, and Intuition" by Jeremy Gray (November, 2001).

So why publish a collection of articles that are readily available in libraries and faculty offices? Well, some of us put articles aside for lack of time, with every intention of reading them later (and you know what they say about good intentions). Also, some of us have trouble finding that unread article in the jumble of our offices. More importantly, though, instead of being arranged in chronological order, the articles have been grouped thematically by editors Shenitzer and Stillwell into categories of Analysis, Algebra and Number Theory, Geometry and Topology, Logic and Foundations, and Applications, preceded by one "general" article, and concluding with three "miscellaneous" articles. Even those who read the articles when they first appeared may gain new insights by re-reading a cluster of articles organized by topic. Moreover, several references to Andrew Wiles and Fermat's Last Theorem suggest that now may be an appropriate time to gain some hisotrical perspective on the current state of mathematics.

The General article is Sir Michael Atiyah's provocative "Mathematics in the 20th Century," based on his Fields Lecture at Toronto in June of 2000. He organizes his thoughts into eight themes: 1. Local to Global; 2. Increase in Dimensions; 3. Commutative to Non-Commutative; 4. Linear to Non-Linear; 5. Geometry versus Algebra; 6. Techniques in Common; 7. Finite Groups; 8. Impact of Physics. In discussing the fifth theme, for example, he sees a continuing dichotomy between these "two formal pillars of mathematics," putting Newton, Poincaré and Arnol'd in the geometric tradition, as opposed to Leibniz, Hilbert, and Bourbaki in the algebraic camp. Atiyah leaves no doubt about his own geometrical leanings, comparing the offer of algebra as an aid to geometry to the devil's offer to Faust: " ... when you pass over into algebraic calculation, essentially you stop thinking ... about the meaning." But he takes some of the sting out of that remark in his sixth theme, where he describes homology theory, K-theory, and Lie groups as successful meldings of the geometric and the algebraic. The article ends with a historical summary, attempting to tie the various strands together, and offering some guesses about what the 21st century might bring.

The Analysis section begins with N. Luzin's "Function," a 1930s article translated by editor Shenitzer. (If my count is correct, Abe Shenitzer translated 10 of the 27 articles: 2 from German, 2 from Polish, and 6 from Russian.) The article traces the evolution of the function concept beginning with the debate among Daniel Bernoulli, d'Alembert, Euler, and Lagrange about the vibrating string problem. The appropriate definition of function was seemingly settled by Fourier's determination of the coefficients of a trigonometric series (with convergence properties cleaned up by Dirichlet), but the debate resurfaced in a discussion that contrasted Dirichlet's definition with Weierstrass' narrower definition in the context of complex analysis. The ensuing developments involve ideas of Borel, Baire, Lebesgue, and others.

"Two Letters" by Luzin to M. Vygodskii were written in the early 1930s in support of Vygodskii's approach to calculus (in his "Foundations of Infinitesimal Analysis"): beginning the study with an intuitive notion of infinitesimal, and only later presenting proofs using the precise notion of limit. Vygodskii thought of his approach as replacing a formal-logical scheme with a historico-logical scheme. Luzin recounts the rebuffs he himself met as a student when he attempted to investigate notions of infinitesimal analysis.

D. Laugwitz's "Riemann's Dissertation ..." presents a brief overview of the contents of the dissertation (which included the idea of a Riemann surface and the Riemann Mapping Theorem), and notes the delay by many in recognizing the importance of Riemann's ideas.

"The Evolution of Integration" by Shenitzer and J. Steprans outlines high points in the history of integration, from the quadratures of Hippocrates and Archimedes, through the polynomial integrations of Cavalieri and Fermat, to the definitions of definite integral due to Cauchy (for continuous functions only), Riemann, and Lebesgue, and finally the HK-integral (Henstock 1955, Kurzweil 1957). The article concludes with an explanation, involving results of Gödel and Ulam, of why there can be no total measure on the reals that is both countably additive and translation invariant.

The final article in the Analysis section is E. Kreyszig's "On the Calculus of Variations ... ," which takes as the subject's beginning Johann Bernoulli's brachystochrone challenge. The article highlights the work of Euler and Lagrange, and traces the development of sufficient conditions for the minimum of a function by Legendre, Jacobi, and Weierstrass. The subject's impact on functional analysis and the (sometimes non-rigorous) application of Dirichlet's principle are outlined, and the article concludes with discussion of Plateau's problem and of Morse's calculus of variations in the large.

The Algebra and Number Theory section begins with "The Evolution of Literal Algebra" by I. Bashmakova and G. Smirnova. Mention of Babylonian numerical algebra and Greek geometric algebra is followed by description of the decline of science in conjunction with Roman conquest, and then the "Greek Renaissance" of the 2nd century A.D. Movement from a geometric orientation to an arithmetization of mathematics is illustrated by Heron's formula for a triangle's area in which segments are thought of as numbers. Diophantus is described as the founder of algebra (and of course of Diophantine analysis), and the authors interpret the first book of Arithmetica as tantamount to an axiomatic construction of the field of rational numbers. They also see general methods in Diophantus' specific problem solutions, arguing that it was the best he could do with limited notation. The second part of the article notes Viète's use of literal notation for parameters as well as unknowns, thus enabling him to obtain general forms of equations and identities. Viète retained the homogeneous magnitude idea of Greek geometry, which was then rejected by Descartes, who established conventions for constants and variables that are in use today.

"The Evolution of Algebra 1800-1870" by Bashmakova and A. Rudakov is a somewhat rambling account of algebraic matters in the time period specified, beginning with Gauss' Disquisitiones. The work of Abel and Galois is mentioned, and group-theoretic ideas begin to appear in results of their successors. Both group theory and linear algebra are visible in the writings of Cauchy and Jacobi, Hamilton's quaternions are mentioned, and several other threads are seen as coming together to form what we today recognize as abstract algebra.

Stillwell's "What Are Algebraic Integers ..." confesses that its title is stolen from Dedekind's "Was sind und was sollen die Zahlen," and proposes to present a sketch of a book that Dedekind would have written, given time. He notes Euler's ("not exactly rigorous") proofs that 27 is the only cube exceeding a square by 2, and of FLT for n = 3. A key result of Eisenstein leads to an appropriate definition of algebraic integer: a root of a monic polynomial equation having ordinary integers as coefficients. Subrings of algebraic integers having unique prime factorization are seen as giving rigor to some of Euler's results, and Dedekind's ideal theory as laying the foundation for modern algebraic geometry (and thus as essential for Wiles' proof of FLT among many other results).

I. Kleiner's "The Genesis of the Abastract Ring Concept" is an article complementary to Stillwell's. It traces the title concept from its origins (algebraic number theory and algebraic geometry for commutative rings, quaternions for the noncommutative case), through Dedekind's ideal concept, to Noether's extension of decomposition theories and E. Artin's generalization of the Wedderburn structure theorem.

The first part of Kleiner's "Field Theory ..." has much of the same structure and material as the commutative ring part of the preceding article. But Gauss' congruences with prime modulus, Galois theory, and British symbolical algebra are also seen as precursors of Weber's abstract definition of field that opens the second part of this paper. This is followed by discussion of Hensel's p-adic numbers, of Steinitz's comprehensive axiomatic study of fields, and of subsequent developments such as class field theory.

Stillwell's "Elliptic Curves" presents two examples from Diophantus' Arithmetica, one involving a conic and the other a cubic equation, which suggest to the author that there may have been geometrical ideas behind Diophantus' algebraic solutions. Results of Fermat and Newton related to elliptic curve ideas, and of Jakob Bernoulli on elliptic integrals, lead to the inversion of elliptic integrals by Abel and Jacobi. Then we are offered a compressed outline, using ideas of Riemann, of why elliptic curves cannot be parametrized by rational functions.

Yet another Stillwell contribution, "Modular Miracles," notes that "...things modular are back in the news." The author recalls Dedekind's modular function j, Hermite's quintic equation solution using that function, use of j by Gauss and then Kronecker in matters involving class number, a curious near-integer value of the exponential function, and a strange connection with the monster simple group.

H. Weyl's "Topology and Abstract Algebra..." is suitable as the final article in the Algebra and Number Theory section that could also serve as a bridge to the next section. This 1931 address begins with cautionary words about generalizing for the sake of generality, noting that a natural generalization actually simplifies a situation and makes it more easily understood. Weyl then moves through several examples (real line, systems of polynomials over a field, and Riemann surfaces, to mention a few), contrasting algebraic and topological approaches. He concludes that a topological approach, when applicable, is usually more effective in promoting understanding than an algebraic one, and ends the article with a concern that the "mathematical substance" that must be present, before generalization, formalization, and axiomatization can take place, may be "near exhaustion."

The first article in the Geometry and Topology section is "Symmetry," a 1986 lecture by J. Tits. Through examples of increasing size and complexity, the author contrasts easily visible symmetries of figures with "hidden" symmetries requiring considerable mathematical work to discover. Among the examples are the symmetric group S₆, the enormous symmetry group of the Leech lattice, and the monster simple group.

Stillwell's "Exceptional Objects" argues that exceptional objects such as the five regular polyhedra (his first example) are not only fascinating but important, and have "a certain unity and generality" when seen in historical perspective. Polytopes, the higher-dimensional analogues of polyhedra, include regular analogues of tetrahedron, cube, and octahedron in all dimensions, leading the author to classify regular polytopes as falling into three infinite families together with five exceptions (dodecahedron, icosahedron, plus three four-dimensional regular polytopes). The same kind of classification (some infinite families plus a few exceptions) is then seen to be valid when considering regular tesselations, simple Lie groups, finite reflection groups, and finite simple groups. Other exceptional situations examined involve sums of squares and division algebras, and projective planes and division rings.

M. Postnikov's "The Problem of Squarable Lunes" is taken from the 1963 edition of his book "Galois Theory." The problem of finding all constructible squarable lunes is transformed into one of deciding when solving a certain equation reduces to solving quadratic equations. A key tool in the analysis is the Eisenstein irreducibility criterion, and the result is that there are only five such lunes: three were already known to Hippocrates of Chios, and the other two were found by T. Clausen in the 19th century. The proof that there are no more than five was begun by Chebotarev around 1934 and completed by his student Dorodnov in 1947. [There are two glitches on page 184: the binomial coefficients are written upside down, and it is the prime p, rather than the quantity 2(p - 1), that must be 2 or a Fermat prime.]

Another Shenitzer contribution is "How Hyperbolic Geometry Became Respectable." A brief historical introduction mentions attempts to prove the parallel postulate, efforts to deduce consequences in a hyperbolic geometry, and models of the hyperbolic plane. The models of Beltrami and Klein are then discussed. with technical details but in a very compact presentation.

"Does Mathematics Distinguish Certain Dimensions of Spaces?" is by Z. Pogoda and L. Sokolowski, and divides the attempt to answer the title question into seven distinct parts. (1) The Greek notion of extension was transformed by Ptolemy into a notion of dimension: 3 for the physical world, or 4 for spacetime in a more modern view. Even in superstring theory there seems to be a distinguished 4-dimensional subspace. (2) Anomalies such as Peano's space-filling curve led 20th century mathematicians to develop the notion of dimension in a rigorous way, with the expectation that spaces of higher dimension would display greater complexity of structure. (3) This hope is dashed by study of "regular cells," n-dimensional analogues of regular polyhedra. The number of such is infinite for n = 2, 5 when n = 3, 6 for n = 4, but then exactly 3 for all n > 4. (4) Classification of topological manifolds yields 2 types for n = 1, two infinite classes for n = 2, and no possible classification for n ≥ 4, but the situation for n = 3 is still murky. (5) The Poincaré conjecture is still unsettled for n = 3. (6) Milnor's 1956 discovery of nondiffeomorphic structures on the 7-sphere comes as quite a surprise. (7) ("An attempt at a summary.") The authors' answer to the title question is that mathematics does seem to distinguish dimensions 3 and 4 from the others, in some sense. [Note: This article dates from 1989, so items (4) and (5) display no hedging in light of Grigori Perelman's recent work.]

The last article in the Geometry and Topology section is "Glimpses of Algebraic Geometry" by I. Bashmakova and E. Slavutin. After some preliminaries (definitions of plane algebraic curves and their degrees, including points at infinity via homogeneous coordinates in the projective plane), the notions of birational equivalence and genus of curves are introduced, followed by statements of theorems of Poincaré on birational equivalence of curves of genus 0 or 1. The article concludes with a discussion of the Abelian group structure imposed on the set of rational points on an elliptic curve (Jacobi, 1835).

The Logic and Foundations section contains just two articles, the first being "Four Significant Axiomatic Systems ..." by S. Mykytiuk and A. Shenitzer. The four systems, and some of the issues addressed, are: A. Euclid's postulational and axiomatic version of geometry. B. The hyperbolic alternative to the parallel postulate developed by Gauss, Lobachevskii, and Bolyai, the Beltrami-Klein disk model of hyperbolic geometry, and the growing importance of group theory in geometric study. C. Peano's axioms; the notions of consistency, independence, and completeness; Gödel's incompleteness result. D. The Zermelo-Fraenkel axiomatization of set theory, and Paul Cohen's independence results, traced from Cantor's naive set theory, through the paradoxes and their resolutions via Zermelo's axioms (with refinements by Fraenkel and Skolem), to the proofs that the axiom of choice and the continuum hypothesis are consistent with (Gödel), and independent of (Cohen), the other axioms of set theory. The article concludes with some philosophical comments by the authors.

The other Logic and Foundations article is "Foundations of Mathematics in theTwentieth Century" by V. Marek and J. Mycielski. An introduction covers much of the same ground as the preceding article (with different emphases and style), but also includes comments on Brouwer's intuitionism (viewed quite negatively by the authors) and on Turing's ideas on computability. Subsequent discussion focuses on three areas ("large chapters") of mathematics. The set theory section moves quickly from Cantor to Russell's paradox to Zermelo-Fraenkel set theory. The two versions (ZFC with the axiom of choice included, ZF without it) and some variants are discussed in some detail, with special attention to the work of Gödel and Cohen. The section on logic and model theory discusses Frege's formal logic, as extended by Hilbert, and introduces the notion of a model for a theory, noting Gödel's 1929 completeness theorem for first order logic. Tarski's study of models of first-order theories is mentioned, as is A. Robinson's use of model theory in nonstandard analysis, and the section concludes with Gödel's negative solution to Hilbert's tenth problem. The section on computability theory deals partly with that Gödel result, from his introduction of the notion of computable function to the equivalent alternative definitions of computability by Post, Church, and Turing around 1935. The concept of Turing machine leads to the concept of recursive enumerability, which was used to show that there is no algorithm that will decide if certain Diophantine equations have solutions. The section concludes with discussion of decidability notions.

There are also just two articles in the Applications section, beginning with "The Development of Rigor in Mathematical Probability (1900-1950)" by J. Doob. The author's "informal outline" begins by distinguishing between real world probability and mathematical probability, and the role of measure theory in "mathematical modeling of real world probabilistic contexts." Developments by Lebesgue, Borel, Radon, and Fréchet are mentioned, with the 1930 Radon-Nikodym theorem seen as a crucial factor in Kolomogorov's 1933 exposition of a basis for mathematical probability. Several quotations throughout the paper (from Poincaré in 1912 to Kac in 1959) express ongoing resistance by probabilists to various measure-theoretic developments.

The other Applications article is "The Evolution of Methods of Convex Optimization" by V. Tikhomirov. After noting a dearth of calculus of variations work involving inequality constraints prior to the 1940s, the author describes a watershed problem proposed in 1939 to L. Kantorovich by engineers at a plywood factory. Kantorovich's 1939 paper on solution methods for linear programming problems was deemed too abstract by Soviet authorities to have economic application. The article then describes Dantzig's simplex algorithm, and also Dantzig's rôle in bringing the work of Kantorovich and Koopmans to the attention of the scientific world (work that won the two the economics Nobel prize in 1975). We are then taken through the central section method of convex optimization due to A. Levin, independently developed somewhat later by D. Newman (both papers published 1965); the 1974 method of circumscribed ellipsoids attributed to A. Nemirovskii, D. Yudin, and N. Shor; the 1979 result of L. Khachian demonstrating that the linear programming problem can be solved in polynomial time; and the acclaimed 1984 polynomial method of Karmarkar.

The first of three Miscellaneous articles concluding the volume consists of six "mini-essays" (in seven pages!) by Abe Shenitzer on: (A) Umemura's 1984 result on solving equations using modular functions and hyperelliptic integrals. (B) Poincaré's late 1800s geometric breakthrough in dynamics. (C) Remarks on the importance of the genus of a Riemann surface. (D) A quotation by Hausdorff on how Cantor's work on Fourier series led to point-set topology. (E) A trip from Gauss' intrinsic geometries, through Riemann's metric geometries and Klein's Erlanger Program, to Levi-Civita's extension of the notion of parallel transport in the plane to more general spaces, with remarks by E. Cartan and H. Weyl. (F) Daniel Bernoulli's guess about vibrating strings and trigonometric series, doubted by Euler and d'Alembert, vindicated by Fourier, with distributions having the last word.

Next is V. Vizgin's "On the Emotional Assumptions Without Which One Could Not Effectively Investigate the Laws of Nature." The fact that good physics and good mathematics have always gone hand-in-hand has been expressed by many thinkers in quasireligious terms. The author points out the Pythagorean-Platonic tradition, the medieval Scholastic idea of a God creating a world in a rational way, and the "preestablished harmony" thought of Leibniz, echoed much later by Hilbert and Minkowski. Special attention is given to Einstein's expression of the physicist's faith in a rationally-structured universe in terms of ""cosmic religious feeling," and Wigner's thoughts on the "unreasonable effectiveness of mathematics in the natural sciences." phrased in terms of an "empirical law of epistemology."

The final article is "The Significance of Mathematics: The Mathematician's Share in the General Human Condition" by W. Magnus. The article begins with brief comments about the place of mathematics in the systems of three philosophers: Plato, considering mathematics as a source of absolute truth and certainty; Leibniz, for whom mathematics was the science that tells us what is possible; and Spinoza, who attempted (misguidedly, in Magnus' opinion) to construct a philosophy in the axiomatic and logical style of Euclid. The author then presents his own attempt at an intuitive understanding of the nature of mathematics: it is subject to the rules of logic (including the law of excluded middle), involves the concept of the infinite, and uses abstraction to increase the number of questions we can ask. His ideas are illustrated using the sum-of-four-squares theorem and Euler's graph-theoretical analysis of the Königsberg bridge problem. He concludes by arguing that the results of mathematics should be appreciated not only for their usefulness, but also as products of pure thought.

This book is a wonderful collection of essays on how all sorts of branches of mathematics developed. The mathematical community owes a debt of gratitude to Abe Shenitzer for the huge amount of work he put into the column beginning in the January 1994 Monthly, and to John Stilwell, first listed as co-editor in the November 2000 Monthly, but obviously involved from the beginning as author and even translator.

David Graves (dgraves@elmira.edu) is Professor of Mathematics at Elmira College, where he is active as a pianist, and has taught courses in cryptology, opera, and history of astronomy as well as the usual run of mathematics courses.