Equations from God: Pure Mathematics and Victorian Faith

CHAPTER FIVE: Earthly Calculations:Mathematics and Professionalism in the Late Nineteenth Century

chapter five Earthly Calculations Mathematics and Professionalism in the Late Nineteenth Century For one it is the high, heavenly goddess; For another it appears as a capable cow, which supplies him with butter. —Friedrich Schiller Augustus De Morgan’s determination that mathematics should separate itself from other realms of Victorian intellectual life, although shrewd, was not a self-fulfilling prophecy. To achieve this distinction required an aggressive promotion of the new isolationist ideal within mathematical circles as well as in the public sphere. As a boisterous critic and the first president of the London Mathematical Society, De Morgan lobbied brusquely, but effectively , toward this end during the last decade of his life. Other prominent mathematicians, aware of the opportunities and dangers associated with their professional standing within society, soon joined him. As their concerns increased in the 1860s and 1870s, a secular, research-oriented mathematics held the promise of a removed, “scientific,” usefully incomprehensible sphere of experts. In a sure sign of professionalization, mathematicians began to reexamine and redefine their discipline’s foundations, categories, and relationships to other fields. Academic researchers, in particular, became more outspoken about the definition of mathematics and who should be called a mathematician. They forthrightly emphasized their training and certification , and formed more exclusive organizations and journals. Moreover, they worked diligently to distinguish their attitude and work from that of amateurs, and began to chide other mathematicians for “unprofessional” behavior, such as including theological rhetoric in technical papers. What may reasonably be called a “professional superego” developed—a set of be137 havioral rules internalized by mathematicians regarding what was proper and improper within their occupation. Useful for this shift was a diminished sense of mathematics’ intellectual and cultural ambition, anticipated in De Morgan’s later thought. New advances in the understanding of mathematical symbols and concepts made the discipline seem less transcendental and less absolute in its proclamations . Scholars investigating number theory, algebra, geometry, and the calculus found weaknesses in the foundations of their fields. Once-secure mathematical structures now appeared to be more fragile than mathematicians had believed. Ancient texts such as Euclid’s Elements, which remained central to mathematical education, as well as seminal modern works like Isaac Newton’s Principia, seemed vulnerable for the first time. Yet mathematicians were not obliged to abandon the traditional, religiously tinged philosophy of mathematics merely because new discoveries challenged the accepted wisdom. After all, they could have viewed advances such as non-Euclidean geometry in the same way some religious idealists viewed Darwin’s theory of evolution—simply as further proof of the inexhaustible genius of the divine Mind. Furthermore, non-Euclidean geometry had been around for many years before mathematicians seized on it and broadcast its supposed impact on the ideology of mathematics. Why was a nontranscendental mathematical philosophy so attractive in the late Victorian era, and why were mathematicians so eager to remove religious innuendoes from their works? Why did this shift in the philosophy of mathematics occur in the second half of the nineteenth century, and not earlier or later? Although this transformation certainly emerged in part for internal reasons related to mathematical research and forms an important chapter in the progression of modern scientific thought,1 the correlation between the ideological change and mathematical professionalization is conspicuous . A new generation of mathematicians who followed in the footsteps of Augustus De Morgan wrestled with issues surrounding their professional standing at the same time they struggled with the nature of their discipline. Circle Squarers versus Professional Mathematicians The fractious debate over the seemingly unworthy topic of circle squaring illustrates the difficulties that confronted professionalizing mathematicians in distinguishing their enterprise. Along with the duplication of a cube (finding a cube with double the volume of the original) and the trisection of equations from god 138 an angle, producing a square with precisely the same area as a given circle was an ancient, and highly mystical, mathematical problem.2 All three problems were bewitching because—unbeknownst to those who encountered these puzzles before the advances of modern mathematics—they involve the strange and elusive numbers called irrationals. Irrational numbers cannot be expressed as fractions involving whole numbers; put another way, they have endless decimal places with no pattern in sight. In the case of circle squaring, since the problem requires pinpointing the ratio between a circle ’s diameter and circumference, the irrational number the investigator bumps into...