The Specialization of Mathematics and the Rise of Formalism

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The Specialization of Mathematics and the Rise of Formalism

Overview

Mathematics is the study of the relationships among, and operations performed on, both tangible and abstract quantities. In its ancient origins, mathematics was concerned with magnitudes, geometries, and other practical and measurable phenomena. During the nineteenth century, mathematics, and an increasing number of mathematicians, became enticed with relationships based on pure reason and upon the abstract ideas and deductions properly drawn from those relationships. In addition to advancing mathematical methods related to applications useful to science, engineering, or economics (hence the term "applied mathematics"), the rise of the formalization of symbolic logic and abstract reasoning during the nineteenth century allowed mathematicians to develop the definitions, complex relations, and theorems of pure mathematics. Within both pure and applied mathematics, nineteenth-century mathematicians took on increasingly specialized roles corresponding to the rapid compartmentalization and specialization of mathematics in general.

Background

Well into the nineteenth century mathematicians continued to scramble to invent and refine analytical methods that would be of use in solving the seemingly endless list of questions and problems being raised by the emerging European industrial revolution's demand for increased experimentation in physics, astronomy, and engineering. By the middle of the century, however, attention began to shift toward the operations of mathematical logic, and, as a consequence, there was an increased emphasis on the relationships and rules for evaluating axioms and postulates.

Building upon the calculus of the seventeenth-century giants Sir Isaac Newton (1643-1727) and German mathematician Gottfried Wilhelm von Leibniz (1646-1716), nineteenth-century mathematicians extended the accuracy and precision of mathematical calculations. In particular, nineteenth-century mathematicians built applied theory upon foundations laid by Swiss mathematician Leonhard Euler's (1707-1783) work in mechanics, differential and integral calculus, geometry, and algebra.

The large strides being taken in applied mathematics, however, left much theoretical and logical ground untouched. After centuries of emphasis on practical applications, the nineteenth century also proved ripe for the development of pure mathematics.

Impact

The ultimate end of the increasing specialization of mathematics was the schism between pure and applied mathematics. A simplest view of this division rests on the definition of pure mathematics as composed mathematics advanced for theoretical interests while applied mathematics develops tools and techniques used to solve problems in science, engineering, and economics. Such a simplistic definition, however, denies the common history and crossovers between the two divisions.

Starting in the nineteenth century there evolved a profound difference in methodology of pure and applied mathematics. Within pure mathematics, deductions are valid if properly derived from a given hypotheses. In the reasoning of applied mathematics, deductions are grounded in experimental evidence, and the goal is to correctly identify the hypotheses from which the deductions can be properly deduced. In this sense, pure mathematics is a matter of correctly following the laws of reasoning—and applied mathematics is a matter of identifying the hypothesis (e.g., scientific law) and the correspondence of results properly deduced from those hypotheses. Another view of the division between pure and applied mathematics allows pure mathematicians the ability to substitute formal calculation with conceptual analysis of a problem.

During the nineteenth century pure mathematics and its abstract calculations became a method to explore the methodology of deduction itself. In contrast, applied mathematics became the methodology used to understand or predict results, particularly the results of scientific and technological experimentation, produced by objective procedures in terms familiar to already established and agreed upon deductive systems.

At the beginning of the nineteenth century the work of German mathematician and physicist Johann Carl Friedrich Gauss (1777-1855) embraced and embodied both pure and applied mathematical concepts. Gauss's practical mathematical applications included advances in the study of the shape of the Earth (geodesy), planetary orbits, and statistical methodology (i.e., least-square methodology). In addition to this practical work, Gauss established himself as true polymath by also advancing the cause of pure mathematics through seminal works in number theory, representations of complex numbers, quadratic reciprocity, and proof of the fundamental theorem of algebra.

In 1847 English mathematician George Boole (1815-1864) published his Mathematical Analysis of Logic and followed it in 1854 with another important publication, The Laws of Thought, in which he asserted that the development of symbolic logic and pure mathematical reasoning was being retarded by an overdependence on applied mathematics. Boole advocated the expression of logical propositions in symbols to represent logical theorems that could be proven just like any other mathematical theorem. Boole championed pure mathematical reasoning by attempting to dissociate and abstract mathematical operations without regard to their underlying applications.

Boole's publication was akin to drawing a line in the sand for mathematicians and demarked a trend away from Gaussian-like mathematical universalism toward an increased specialization within mathematics community. In particular there became an increasing divergence between pure and applied mathematics. As a consequence of the popularity of formalism, there also resulted an increasing number of mathematicians dedicated to pure mathematics of minimal consequence to applications to science and technology.

This divergence was not always viewed with favor. With regard to mathematical logic, although initially created as a sub-discipline of mathematics, the field was widely ignored or held in disdain by many mathematicians. By the end of the century, however, symbolic logic, progressed from academic obscurity to become popular entertainment. Lewis Carroll's books on logic, The Game of Logic (1887) and Symbolic Logic (1896), became popular topics of conversation and entertainment both for scholars and laymen in Victorian England. Despite initial resistance, mathematical logic made strong inroads into philosophy after William Stanley Jevons heralded its use in his widely read 1874 publication, Principles of Science. Jevon argued that symbolic logic was of importance to both philosophy and mathematics.

The increasing volume of work relating to number theory also lead to hierarchies within the emerging specialties. Fueled by Gauss's work on the theory of numbers, algebraic theories of numbers took on a preeminent position in pure mathematics.

Some initially pure mathematical theory was met with outright derision and scorn. Among the most controversial of advances in mid-nineteenth-century mathematics was the publication of non-Euclidean geometries championed by German mathematician Georg Friedrich Bernhard Riemann (1826-1866). Riemann asserted that Euclidian geometry was but a subset of all possible geometries. Riemann's expanded concepts of geometry treated the properties of curved space and seemed abstract and useless to nineteenth-century Newtonian physics. In addition, many mathematicians thought Riemann's conceptualizations bizarre. Regardless, Riemann's theories proved of enormous consequence and value to the expansion of concepts of gravity and electromagnetism—and of fundamental importance to the twentieth-century theoretical work of Albert Einstein (1879-1955) regarding a formulation of theories describing the interactions of light and gravity. Topology as a specialization of mathematics was born in the advances of nineteenth-century geometry.

Non-Euclidean geometry was so controversial that it may have prevented Gauss from advancing the concept. Eventually found among Gauss's unpublished works were the essential elements ultimately formulated by Riemann.

Later in the nineteenth century when German mathematician Georg Ferdinand Ludwig Philipp Cantor (1845-1918) proposed his transfinite set theory, many thought it the height of abstraction. Advances in twentieth-century physics, however, have also found use of Cantor's theories.

Not all developments were polarized into the pure and applied camps. Early in the nineteenth century, French mathematician Jean Baptiste Joseph Fourier's (1768-1830) work with mathematical analysis allowed him to establish what is known to modern mathematicians as the Fourier series, which is central to Fourier analysis and an important tool for both pure and applied mathematicians. Late in the nineteenth century, group theory made possible the unification of many aspects of geometric and algebraic analysis.

Although there was an increasing trend toward specialization throughout the nineteenth century, near the end of the century French mathematician Jules Henri Poincaré (1854-1912) embodied Gauss's universalist spirit. Poincaré's work touched on almost all fields of mathematics. His insights provided significant advances in applied mathematics, physics, analysis, functions, differential equations, probability theory, topology, and the philosophical foundations of mathematics. Poincaré's studies of the chaotic behavior of systems subsequently provided the theoretical base for the continually evolving chaos theory of late twentieth-century mathematics.

Mathematical rigor, defined in the early part of the century by the applications of calculus, was broadened near the end of the century by German mathematician Karl Theodor Wilhelm Weierstrass (1815-1897) into the types of analysis familiar to modern mathematicians.

The advancement of elliptic functions principally through the work of Norwegian mathematician Neils Henrich Abel (1802-1829) and Prussian mathematician Karl Gustav Jacob Jacobi (1804-1851) provided mathematical precision in calculations required for discoveries in astronomy, physics, algebraic geometry, and topology. In addition to their use in applied mathematics, however, the development of the theory of elliptic functions also spurred the study of functions of complex variables and provided a bridge between the widening chasm between pure and applied mathematics.

Although there were subtle divisions of mathematics at the beginning of the nineteenth century, by the end of the century there were full and formal divisions of pure and applied mathematics. University appointments and course-work reflected these divisions, and an increasing number of professorial positions were designated for pure and applied mathematicians.

K. LEE LERNER