David Lindsay Roberts: Republic of Numbers: Unexpected Stories of Mathematical Americans through History. Baltimore: The Johns Hopkins University Press, 2019.
One might say that David Lindsay Roberts has written a ‘lost’ chapter of Tocqueville’s Democracy in America. Tocqueville does write generally on American intellectual life of the 1830s, especially education. He judges that New England remained at the forefront of American regions at that time, having been founded two centuries before by a population of immigrants among whom “there was a greater mass of enlightenment… than within any European nation of our day” (emphasis added). That is, despite the ‘secular’ Enlightenment of the European eighteenth century, still a matter of contention in Tocqueville’s lifetime, New Englanders of the 1630s were on a whole a better-educated group. The Puritans established a publicly-supported school system dedicated to teaching the Bible. “In America, it is religion that leads to enlightenment; it is the observance of divine laws that guides men to freedom.”
Being mostly Protestants, Americans then and now don’t restrict Bible learning to a spiritual aristocracy. “Primary instruction there is within the reach of each,” although “higher instruction is within the reach of almost no one.” All Americans “can readily procure for themselves the first elements of human knowledge.” These included reading, writing, arithmetic; the doctrines and proofs of one’s own religious sect; the history of the United States and of one’s own American state; and the federal and state constitutions. And New England was not alone. Although the southern and western Americans didn’t put as much emphasis on education as New Englanders did, they too took care to establish schools and churches, read newspapers, and participate in civic life (inasmuch as “genuine enlightenment arises principally from experience”). Even on the western frontier, “I do not believe that so great an intellectual movement is produced in the most enlightened and populated cantons of France.” In all this Americans reinforced their regime: “One cannot doubt that in the United States the instruction of the people serves powerfully to maintain a democratic republic.” Indeed, “in the United States, the sum of men’s education is directed toward politics,” whereas in largely undemocratic, unrepublican Europe education’s “principle goal is to prepare for private life.” With broad-based participation in public life precluded, education can have no other object.
In all the many pages of Tocqueville’s Democracy one finds no mention of American mathematics or mathematicians. As Roberts shows, there wasn’t much for him to write about at that time. Of the twenty-three “mathematical Americans” he considers, only two had come to public prominence by the time Tocqueville visited in 1831-32. Mathematics in America wouldn’t begin to mature until after the Civil War. When it did, it exhibited so many of the marks of the American regime Tocqueville had described that it is easy to imagine how unsurprised Tocqueville would have been at the worries and opportunities Roberts’s mathematicians faced. He had, after all, titled one of his chapters, “Why the Americans Apply Themselves to the Practice of the Sciences Rather than to the Theory.” “It is evident that in democratic countries the interests of individuals as well as the security of the state requires that the education of the greatest number be scientific, commercial, and industrial rather than literary.” America is a commercial as well as a democratic republic.
During the years of the American founding, mathematical education in America didn’t amount to much, if judged by European standards. There was no advanced mathematical research. Americans who used math were surveying land and navigating ships—engaged in conquest and commerce, acquisition of property. John Adams and Thomas Jefferson saw the beauty of higher mathematics, but did not commend its appreciation to the average citizen. Their younger contemporary, Nathaniel Bowditch, grew up in Salem, Massachusetts when it was the sixth-largest city in the country, a center of merchant shipping that already took New Englanders as far as Asia. Having left school at the age of ten to work in his father’s barrel shop, Bowditch taught himself math and science thanks to Salem’s library, which had a collection of science books captured at sea in 1780. Bowditch himself went on five voyages on the merchant ships around the turn of the century. Captains on such expeditions needed to calculate their longitude and latitude, typically consulting The New Practical Navigator by the Englishman John Hamilton Moore. Bowditch made substantial corrections to this book—so many that the American publisher “took Moore’s name off and put Bowditch’s name on, while altering the title to New American Navigator.” (“Copyright in the early United States was only casually observed, especially for books originating in the country from which the United States had so recently emancipated itself.”) The mathematical topic navigators need is trigonometry, which enables the navigator find his location at sea by measuring the distance of the ship from its port in relation to the North Star, the one fixed point in the sky. Given the curvature of the earth, this can work precisely “for short distances,” but Bowditch introduced refinements that overcame the problem, for practical purposes.
Practical purposes animated Bowditch throughout his life, as he retired from seafaring and entered the insurance business, where his skills were equally useful and more lucrative. Two decades later he joined the Harvard Corporation and assisted in righting the College’s shaky finances. And he hired a Harvard student named Benjamin Peirce to translate an important French math text; Peirce would go on to teach astronomy and mathematics at his alma mater, “recognized as a major national figure in science and mathematics in the United States, a status to which Nathaniel Bowditch, for all his talent, never seems to have truly aspired.” In his day, government and commerce were simply more needed, and consequently more prestigious, than academic studies.
If Bowditch represents the commercial side of American mathematics in the decades after the Founding, Sylvanus Thayer represents its military side. Thayer had two undergraduate degrees, one from Dartmouth and the other from West Point. He helped to design coastal fortifications during the War of 1812. In 1815, in the aftermath of the war, Secretary of War James Monroe sent the Thayer, now a brevet officer, to inspect military facilities in France. He returned to take command of West Point two years later. During the eighteenth and early nineteenth centuries military engineers “were the intellectuals of the battlefield,” not only designing fortifications but measuring distances and “pondering the angles of impinging forces”—”the ones who defined the environment” of battles. Roberts recalls that no less a commander than Napoleon, “himself an accomplished student of mathematics,” collaborated with the mathematician Gaspard Monge, throughout the General’s career, “a rare and possibly unique relationship between a first-class mathematician and a powerful political leader.”
While Monge was a theoretical pioneer in mathematics—effectively inventing the field of descriptive geometry, whereby three-dimensional objects can be depicted in two-dimensional figures—the less brilliant but eminently practical American followed not Monge’s example but the example of the École Polytechnique in Paris, which he’d visited during his stay in France. “Whereas Frederick the Great’s Berlin Academy… had employed mathematicians to glorify the sovereign and sometimes to provide technical advice to the government, the Polytechnique explicitly gave mathematicians the mission of teaching and examining the new generation, thus exerting an influence beyond the achievements of any one person, or any one generation of scholars.” The despotic Enlightenment of Frederick’s Prussia deployed mathematics in service to a monarchic regime; the Polytechnique, “founded in 1794 by the Revolutionary government,” served the purposes of a democratic republic. Thayer, along with almost all other Americans, and very much in line with the democratic-republican Madison and Monroe administrations, set the American military academy on a mission to teach. As it did: West Point graduates would go on to write math textbooks and to teach in many colleges and high schools across the country. At the behest of President Monroe’s Secretary of War, John C. Calhoun, West Pointers began to design the roads, bridges, and canals needed not only in military expeditions but in western expansion generally. Civil engineering worked with military engineering to carry Americans west to the Mississippi and beyond. Thayer stayed at the Point or sixteen years, longer than any other superintendent before or since. “Consequently, his impact lingered long,” long after he resigned in irritation at the rather anti-academic tendencies of President Jackson’s administration. One of his most prominent hires, civil engineering professor Dennis Hart Mahan, named his son after his patron. Alfred Thayer Mahan went on to “become the greatest theorist of naval power of his time,” effectively the founder of the American school of geopolitics. Geopolitics requires precise mapping, which requires precise measurement or mathematical calculation. In this sense, Sylvanus was the step-grandfather of American geopolitics.
By far the most famous American Roberts recalls is Abraham Lincoln. No mathematician (having received altogether about twelve months of formal schooling—somewhat beyond the average U.S. citizen of his time), Lincoln was nonetheless profoundly influenced by mathematics. His first non-manual employment was as a surveyor in the years 1833-36. He was “respected for his work,” but his ambitions far transcended it. He began his political and legal careers at the same time, and eventually found a way of bringing the three kinds of knowledge together. By the 1840s, now an experienced lawyer, a former state legislator, and a former member of Congress, Lincoln began to reflect on what it means to prove an argument, and to do it in a way that will convince juries and voters. “He had gleaned that such certainty was a central feature of mathematics in general, and that the Elements of Euclid in particular was considered by many to be the epitome of demonstrative reasoning.” And so he taught himself the Euclidean proofs, which consist of definitions, postulates (things to be done), axioms (things to be thought), and theorems (the results of syllogisms constructed of postulates and axioms). To be certain, the truths of plane geometry must be founded upon postulates and axioms that are ‘self-evident’ or undeniable. Lincoln then saw that the Declaration of Independence was a logical syllogism, analogous to a Euclidean proof, drawing its conclusion (“these colonies are, and ought to be, free and independent states”) from self-evident propositions (“all men are created equal,” that is, “endowed with certain unalienable rights”). Hence Lincoln’s celebrated phrase in the Gettysburg Address, that by their Declaration Americans dedicated themselves “to the proposition that all men are created equal”—a striking example of using mathematical language to vindicate the principles of the American regime.
As a professional historian of mathematics in a country that has come to deny fixed principles derived from the Laws of Nature and of Nature’s God, Roberts is quick to jump in to object that Lincoln, “like almost all Americans then and now, had no conception of mathematics as a living, growing activity.” “He was learning what appeared to him as a fixed body of incontestable knowledge.” But in Lincoln’s lifetime “a small network of advanced mathematicians in Europe were indeed questioning” Euclidean geometry, doubting that Euclid’s fifth postulate—that “two straight lines, which intersect one another, cannot be both parallel to the same straight line”—is necessarily so, genuinely self-evident. A non-Euclidean geometry was possible; “Euclid’s geometry might not necessarily offer the best description of physical reality.” Albert Einstein “would fully exploit this new standpoint in his general theory of relativity.”
Further, the discovery/invention of non-Euclidean geometry shows that mathematics doesn’t stand still; it changes. It has a history. At the Johns Hopkins University, founded in the decade after Lincoln’s murder, historicism and the mathematics of a nature that changes would be brought together, issuing politically in Progressivism, the claim that human rights derive not from permanent natural principles but from the process of evolution, and in the valorization of government by bureaucracy—an administrative state, staffed by experts, wielding knowledge often derived from the mathematical field of statistics, guiding the direction of historical progress and the attendant evolution of rights produced in its course. Whether the undeniable (one might say, almost self-evident) progress of mathematical and scientific knowledge means that the principles of nature are themselves ‘progressive,’ evolutionary-historical; and whether those aspects of nature than (again, undeniably) do change over time alter the principles of human right remains a vexed question—one that Roberts doesn’t address here, having other fish to fry.
Roberts exhibits one of the characteristics of the changing conception, if not reality, of rights by including Catherine Beecher in his survey. Because he didn’t consult Tocqueville, however, he underestimates the status of women in the America of Beecher’s time. “In almost all Protestant countries,” Tocqueville writes, “girls are infinitely more mistresses of their action than in Catholic peoples.” Moreover, “in the United States, the doctrines of Protestantism come to combine with a very fee constitution and a very democratic social state; and nowhere is the girl more promptly or more completely left to herself.” Even as a child, the American girl “already thinks for herself, speaks freely, and acts alone,” quickly coming to consider the world “with a firm and tranquil eye.” “The American woman never entirely ceases to be mistress of herself”—as much a model of self-government in her own way as an American man is in his way. “Although Americans are a very religious people, they have not relied on religion alone to defend the virtue of woman; they have sought to arm her reason” as well, providing “a democratic education to safeguard woman from the perils with which the institutions and mores of democracy surround her.” Miss Beecher exemplified the type.
Daughter of prominent Boston clergyman Lyman Beecher, Catherine Beecher founded the Western Female Institute in 1833 in Cincinnati, where her family had moved the previous year. She later returned to New England, founding the Hartford Female Seminary in Hartford, Connecticut. For classroom use, she published a math textbook, along with other books “she could then use in her school.” Most of her books were not widely adopted, although her 1841 Treatise on Domestic Economy enjoyed many reprintings; in it, she aimed at “put[ting] a woman’s work in the home on the same footing as academic subjects,” and she undoubtedly succeeded in advancing what would later be called ‘home economics’ as a longtime staple of secondary-school education. Overall, however, it must be said her relatives and acquaintances far exceeded her own influence and renown. Not only her father but her sister, Harriet (author of Uncle Tom’s Cabin), her friend, William McGuffey (author of the famous Reader), and even her husband, Alexander Metcalf Fisher, (professor of mathematics and natural philosophy at Yale and “a leading light of the as yet tiny community of American academic mathematicians), surpassed her own respectable but modest achievements. But Roberts understands that to write history in the United States in the first quarter of the twentieth century brings with it a solemn obligation of race-class-gender ‘inclusiveness,’ an obligation he does not neglect to fulfill.
Born in the next generation of Americans, Josiah Willard Gibbs proved another sort altogether. Professor of mathematical physics at Yale, “Gibbs is often considered the greatest American scientist of the nineteenth century and indeed one of the world’s greatest scientists,” a man of whom Einstein himself “spoke glowingly.” In 1861 Yale was the first American college to award a PhD; Gibbs received his two years later. He then went to Europe, where he attended lectures in math and science in Berlin, Heidelberg, and Paris (he was also a linguist), returning to teach at Yale in 1869. There he mentored several prominent American mathematicians and physicists. Gibbs’s specialty was thermodynamics, yet another iteration of the science of change, inasmuch as it is “fundamentally concerned with irreversible processes,” such as entropy. (Roberts aptly illustrates the point by citing one of novelist Thomas Pynchon’s characters, Thermodynamic Officer Chick Counterfly, who remarks, “You can’t de-roast a turkey,” an excellent example of the Second Law of Thermodynamics.) Gibbs showed that the principles of thermodynamics could be expressed with geometric figures, using these “pictorial representations to offer a more comprehensive”—and more comprehensible—”understanding of phase transitions among solid, liquid, and gaseous forms of a substance.” He went on to provide “the foundation of the field of physical chemistry” by positing his “phase rule,” which provides a mathematical formula for understanding the way in which a given set of chemical substances will change their “phase” (that is, their condition as gas, solid, or liquid), given such variables as temperature and pressure.
Gibbs then turned to “his last great project,” statistical mechanics. Everyone can see that matter can be considered microscopically or macroscopically. But how is the “macroscale” behavior of matter (temperature, for example) caused by “the microscale behavior of the tiny particles making up the matter.” You can’t know exactly “what all the individual particles are doing.” However, “if one could estimate the likelihood that a certain proportion of the particles were moving in certain ways, then one might be able to somehow average the whole conglomeration of motions.” This Gibbs proceeded to do, publishing his findings in 1902. His Elementary Principles in Statistical Mechanics won the esteem of Dr. Einstein and of the distinguished French mathematician and physicist Henri Poincaré, who called it “a little book, little read, because it is a little hard.”
Abstruse as Gibbs’s writings may have been, this did not prevent their appropriation by at least one prominent non-mathematician: Henry Adams. In The Education of Henry Adams and in his essay, “The Rule of Phase Applied to History,” Adams proposed that the natural laws of thermodynamics, and particularly entropy, explains the course of events—a sort of anti-progressivism historicism whereby decline not advancement rules the nature of things. There may have been a measure of irony in the puckish Adams’s presentation, but Roberts takes him seriously and is supremely unimpressed. Adams shows “little appreciation for or interest in the productive interplay between precise logical reasoning and shrewd approximation that characterizes modern physical science,” and this results in “a parade of undigested scientific terminology in the service of Adams’s increasingly gloomy view of the human condition. For Adams, words such as entropy, critical point, phase, and equilibrium never achieve more than amorphous content.” Against Adams’s characterization of modern mathematics as naïve idealism, with causation in the material world spurred by “immaterial motion” conceived “only in the hyper-space of Thought,” Roberts ripostes that “this entirely misses the decidedly utilitarian spirit of Gibbs’s approach to mathematics”; “the importance of Gibbs’s rule of phase for science lies not in vague implications but in its explicitly numerical character: if certain simplifying assumptions are made, certain precise results follow, and these results can be used to predict specific useful phenomena in the world.” That Adams may have offered his formulations with the intent of annoying morally earnest Progressives, among whose moral descendants we may count Professor Roberts, does not occur him.
There can at least be little doubt regarding the utilitarian character of Charles H. Davis, a naval commander who also edited the Navy’s Nautical Almanac, a publication no one has ever confused with the writings of neo-Platonists, ancient or modern. In his early career he was a man of action, helping to put down a whale-ship mutiny in the western Pacific and intervening in an ill-judged attempt by the prominent Tennessean, William Walker, to seize control of Nicaragua and turn it into a launching pad for “a great slave empire encompassing the entire Caribbean basin.” He later became served the United States as a planner of and participant in naval operations against the Confederacy, including the capture of Memphis. But his main contribution to mathematics was administrative. Although “a skilled mathematician,” he “made his most significant mark by organizing the mathematical talents of others,” introducing mathematical theorists to the experts in applied mathematics and overseeing the symbiosis during the course of preparing the Nautical Almanac, which appeared in 1852 and went through multiple editions.
Meanwhile, far removed from Davis’s Cambridge, Massachusetts, South Carolinian West Pointer and math professor Daniel Harvey Hill wrote an algebra textbook with a decidedly Calhounian edge, as in: “A planter hires a slave and the slave’s clothing at a certain annual rate and then returns the slave too his master after only eight months, with a cash payment but minus the clothes. What was the value of the clothes?” Not to neglect balance, Hill’s exercises did not overlook illustrations of Northerners “as cheaters in commercial transactions, cowardly in the face of danger, tolerant of the absurd notion of women’s rights, and hypocritically miserly when given the opportunity to buy the freedom of a slave.” Not to mention the Salem witch trials and “the disloyalty of the New England states during the War of 1812.” Simultaneous linear equations have seldom been taught with such verve. Hill also became professor of mathematics and artillery at the North Carolina Military Institute, interrupting math and science education only for the war, when he “led the entire body of the college, students and faculty, into the Confederate service” and achieved the rank of major-general in The Cause. “Hill retained to the end of his life the belief that better leadership could have saved the South as an independent nation,” although he also saw that slavery in some respects kept Southerners from learning math and science, as the planter class contented itself with master-ship at home and commerce abroad, at the same time promoting verbal and mathematical illiteracy not only among slaves but among lower-class whites.
After the Cause became the Lost Cause, the study of mathematics in America coalesced in the newly-elaborated university system, increasingly modeled on the pattern of the German research universities. Vassar graduate Christine Ladd wrote her doctoral dissertation at Johns Hopkins on “the algebra of logic,” under the eye of that genius, Charles Sanders Pierce, the American pioneer in the field of symbolic logic, which transformed logic into “a part of mathematics and not an odd appendage of philosophy” by substituting mathematical symbols for words in logical syllogisms. “By designating propositions with letters and treating the logical operations of and, or, and not analogously with the operations of multiplication, addition, and negation in arithmetic,” logicians “turned logical deductions into an exercise in rule-based symbol manipulation, like algebra.” Johns Hopkins declined to award the PhD degree to Ladd for some forty years, a dilatoriness Roberts sensibly ascribes to prejudice against women. Married to Hopkins math professor Fabian Franklin, she continued her mathematical studies. When Franklin took a job in New York, she lectured (without pay) at Columbia.
Ladd’s contemporary at Hopkins, Kelly Miller, was an African-American graduate of Howard University and “the first African American graduate student of mathematics in the United States.” He studied physics and astronomy at the graduate level, as well. He never received a degree, but returned to Howard as its only black faculty member and, “for a time, the only black mathematics professor in the United States.” Believing that “Christian faith and mathematically based scientific knowledge were the essential foundation for future advancement of African Americans,” Miller soon turned to the newly-invented discipline of sociology, which he taught exclusively in the last three decades of his career. He published extensively on the race question, bringing his mathematical expertise to bear on bogus claims of then-respected ‘race science’ quacks. And he may have had a hand in hiring Elbert Cox and Dudley Woodard, “the first two black Americans to earn a mathematics PhD,” for the Howard math faculty.
Despite such self-imposed handicaps, pure mathematics emerged “as the dominant concern of academic mathematicians in the United States” between the close of the frontier in 1890 and World War II. Mathematics moved west with the frontier, with the University of Chicago leading the way. The chairman of the Chicago math department, E. H. Moore, worked for separating the study of mathematics from the study of astronomy and physics, giving institutional recognition of mathematics as an independent discipline. On the practical side, the needs of the 1890 United States Census brought Columbia University statistician Herman Hollerith to invent a technique for aggregating data on census tally sheets. By translating the data to patterns of holes punched in cards, and then inventing a machine that could ‘read’ the card “by probing the card with pins, so that only where there was a hold would the pin pass through the card to make an electrical connection,” Hollerith enabled the federal government to present the information collected by its field workers into usable statistical tables.
This first sign of what Woodrow Wilson would call “scientific administration” in the United States was well understood by Tocqueville, who had seen the beginnings of it in European statism. Bureaucracy, he saw, is a crucial underpinning of the centralized state, and this has important implications for public education. “Education as well as charity,” he writes, “has become a national affair among most peoples of our day. The state receives and often takes the child from the arms of his mother to entrust him to its agents; it takes charge of inspiring sentiments in each generation and furnishing it with ideas. uniformity reigns in studies as in all the rest; diversity like freedom disappears from them each day.” If a time traveler were to tell Tocqueville that American students of the twenty-first century were to be instructed uniformly of the benefits of diversity, he would say only that democracy lends itself to such uniformity of opinion, and to the centralization of powers within the administrative state.
The Progressive movement appealed powerfully to school teachers, who formed the core of support for such politicians as Woodrow Wilson and, later Franklin Roosevelt. One way in which Progressives took control of the public schools was to continue and accelerate the longstanding American attempt to democratize mathematics education. Cal Tech math professor E. T. Bell wrote books popularizing mathematics as an attractive activity. His 1937 book, Men of Mathematics, obviously the precursor to Roberts’s book, sold well, touting mathematics as “the queen of the sciences” and calculus as “the queen of mathematics.” Charles M. Austin, first president of the National Council of Teachers of Mathematics, founded in 1920, worried that as “progressive ideas of education” (and therefore of democratization) “were creeping into the schools,” “easier subjects” like social science and civics were being substituted for algebra and geometry. The NCTM was designed to fight the trend, which was given intellectual respectability among educators by John Dewey, for many years the resident philosopher at Teachers College, Columbia University. Dewey urged that “education needed to be totally reoriented to accommodate the industrializing twentieth-century world.” “Since students would be graduating into an ever-changing environment, general thinking and problem-solving skills were more valuable than any fixed bodies of knowledge.” Although Dewey didn’t intend this as a move to cut back on math in public schools, and although modern math had arisen as a challenge to the notion that mathematics and logic were static disciplines teaching fixed bodies of knowledge, his strictures were often interpreted as if justification for beheading the Queen. After World War II, Lithuanian-born Holocaust survivor Izaak Wirszup spearheaded the movement called the “New Math,” which attempted to meet both the Progressives’ desire to democratize education and the concern over ‘dumbing-down’ the curriculum.
Such worries proved somewhat overwrought because, as Hollerith had demonstrated, mathematics proved highly useful to the other dimension of Progressives’ government, the administrative state. If the Progressives talked of democracy, and moved to enhance it with such devices as the popular election of United States senators, ballot initiatives and referenda, and reform movements directed against political bosses, they also moved to establish a new form of aristocracy: a cadre of experts trained in the science of administration. E. B. Wilson, sometime chair of the MIT physics department and professor of public health at Harvard, a former student of Willard Gibbs, eventually came to teach a course in mathematical economics in Harvard’s sociology department, where he mentored Paul Samuelson. ‘Social science‘ as a would-be science would make its most lasting mark in economics, with Samuelson’s famous textbook leading the way for several generations of economics students.
Although social science claimed to be ‘value-free,’ it seldom was. In a sort of mirror-imaging of Confederate stalwart Daniel Harvey Hill, Lillian and Hugh Lieber wrote math books proclaiming that “Mathematics (capitalized throughout) with Science and Art (also capitalized)” revealed the great truth that (in their words) “Internationalism and Democracy are very deep in the human spirit.” This seems to have had something to do with the abstraction from the concrete seen in mathematics, theoretical science, and modern art, all of which take the mind away from such physical facts as armies wielded by national governments and police work at the service of social hierarchies. In a book published after the Second World War, Mrs. Lieber “made a plea for world peace and expressed alarm about the dangers of biological warfare and the atomic bomb.” By the end of the Forties, the Liebers were “an intellectual power couple in New York City”—alas, without much ‘pull’ in the Kremlin.
The war itself and the decades following saw mathematics applied to computers. Grace Hopper developed some of the earliest computers while working for the Navy in a research program at Harvard. She later joined a firm that became part of Sperry Rand. Joaquin Basilio Diaz, armed with a Brown University PhD, pursued his career in the military arm of the administrative state, never dismantled after the war because the Cold War soon followed. At Princeton, the brilliant John F. Nash, Jr. developed “game theory,” initially proposed by John von Neumann of Princeton’s Institute for Advanced Study in the mid-1940s, into a tool usable in both business and military strategy and organization.
Beginning in the New Left’s assault on ‘Enlightenment rationalism’ injured the Progressive ideology of such persons as the Liebers. Former NCTM president Frank B. Allen decried this “anti-intellectual counterculture,” with its “disparagement of science and technology” and its rejection of “rationality in general.” To counter the counterculture, Allen proposed an approach to algebra (the experience of which has turned no small number of high school students away from rationality in math, if not in general) which emphasized the logical character of mathematical proofs. His 1966 book, Modern Algebra: A Logical Approach, was intended to stem the tide, but his own organization, the NCTM, soon became infected by “fads such as cooperative learning and vain attempts to solve the racial, economic, and environmental problems of society through the schools.” The controversy would continue into the next century.
Roberts concludes that “the lives recounted in this book do not suggest a static future” for America mathematics, which may turn increasingly to the science of computer modeling, with results he hesitates to predict.
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