1. Introduction: The Age of Wonder.
A giddy craze was sweeping across Europe at the turn of the 17th century. The wealthy and the well-connected were hoarding things—strange things—into obsessive personal collections. Starfish, forked carrots, monkey teeth, alligator skins, phosphorescent minerals, Indian canoes, and unicorn tails were acquired eagerly and indiscriminately. Associations among these objects, if they were made at all, often reflected a collector's personal vision of an underlying natural "order". Critical taxonomy was rarely in evidence.
The enthusiasm that characterized such collections was captured by Francis Bacon [1, p. 247], who ironically advised "learned gentlemen" of the era to assemble within "a small compass a model of the universal made private", building
These "cabinets" were proudly displayed, and selected visitors were allowed to wander through the accumulated "shuffle of things" to express their awe and admiration. Other expressions, such as perplexity and incomprehension, were circumscribed by what was, in pre-Enlightenment Europe, a well-practiced deference to presumptive authority. Indeed, the cabinets can be seen as a sort of physical manifestation of centuries-old religious and philosophical states of consciousness. These first "museums" presented visitors with an opportunity to place the era's flood of information within a comfortable psychological order. As the century progressed, patrons would pay for the privilege of a carefully arranged understanding of the world's troubling uncertainties.
Figure 1. The museum of Manfredo Settala, Milan, ca. 1600.
Europe in 1600, one must remember, was crossed by many unsettled intellectual currents. The Renaissance, which had begun in Italy in the 14th century, had spread throughout Europe by the end of the 16th, reviving the continent's slumbering passion for art and learning. As the Renaissance progressed, however, it had transformed Europe into a much more complicated place. By 1600, the consequences of so much concentrated inquiry were beginning to show in the troubled creases across many European brows. Explorers were regularly bringing back incredible, inscrutable objects from the New World and the Far East; objects whose existence brought into question the centrality of Europe and the primacy of its culture. Simultaneously, the Scientific Revolution, led by the likes of Copernicus (1473-1543), Bacon (1561-1626), and Galileo (1564-1642), was challenging the centrality of the world itself. As the extraordinary scope of Geography and Nature began to emerge, many traditional, comforting notions seemed to shrink in comparison. It was a bit too much for some, and the profound unease stirred up by such revolutionary changes gradually fueled a reaction. In extreme cases, science was labeled Heresy. (Giordano Bruno was burned at the stake in 1600; Galileo was imprisoned by the Inquisition and threatened with torture in 1633.) Scientists, still in their early days of mixing powders and picking apart anatomies, were not yet in a position to respond with assurance.
What is remarkable about this era is that, even when every boundary seemed to be expanding beyond the known horizons, and even when the objects from those far horizons suggested a world that was larger than the imagination, there was nevertheless a desire to reach further. It was daring, for what lay beyond was completely unknown. Commerce, politics, and religion, each with its own agenda, tried to temper the excessive enthusiasm. Nothing, however, could direct the collective fascination with the New World's dark tumult. Curiosity cabinets proliferated across Europe. Eventually, these cabinets became the first museums of science, natural history, medicine, and art, depending on how curators chose to arrange their accumulated treasures. What these first museums showed their patrons was that the New World—whatever it was—was undoubtedly wonderful. The collections became known as wunderkammern, or "wonder cabinets".
The emerging scientific community wasn't entirely pleased with the craze for wunderkammern. Those who took the Revolution seriously found the curiosity cabinets not only frivolous, but reactionary—an impediment to the properly measured progress of scientific reason. Wonder, they argued, ought to be lavished on the technical advances in navigation that had allowed explorers to sail back and forth to the New World time and again, not merely on the gewgaws that they had brought back. Galileo [15, p. 459] wrote with apparent distaste of the "curious little men" who would gather
Descartes [16, p. 20] gave the curiosity cabinets an even more peremptory wave: "What we commonly call being astonished is an excess of wonder that can never be otherwise than bad."
2. Mathematics in the Age of Wonder.
Mathematics, as might be expected, was an eager participant in the excitement that marked the beginning of the 17th century. By this time, a tremendously fruitful interchange had developed among mathematics, science, engineering, and commerce. Practical problems that arose in the burgeoning complexities of international trade, in navigation, in the construction of cities, and in the logistics of warfare were leading to fundamental new developments in algebra, geometry, and computation. As the century progressed, mathematicians such as Harriot and Oughtred would rediscover the algebraic techniques that Arab mathematicians had employed during Europe's Dark Ages, rewriting them in a new language of symbols (much to the chagrin of typesetters); Desargues, Pascal, and Descartes would completely re-invent the way in which geometric problems were investigated; Napier would introduce an efficient new tool for computation called the logarithm; and Fermat would apply infinitesimal methods to the problem of finding the maximum and minimum values of a function. By the century's end, all of these developments would coalesce into one of mathematics' most lasting contributions to practical science: The Calculus.
These contributions were widely noticed and appreciated by the 17th century public. Indeed, "mathematical" apparatus such as calculating machines, drawing instruments, astrolabes, sundials, and the recently invented sector and slide rule found their way into many of the cabinets of curiosity. Unnoticed by much of the public, however, was mathematics' slow turn inward. This was, after all, the "early modern" period of bold new experiments in abstraction. Mathematical advances gradually became much less generally understood, even by those who considered themselves to be active participants in the era's intellectual life. The logic and systematic discipline within mathematics, as well as the many practical applications that it produced, continued to impress, but outsiders could no longer hope to share in the excitement of fundamental new treatises the way they might have only half a century before. As a result, there was not a rush to learn logarithms the way that there was a rush to taste new teas from China.
Many mathematicians and scientists of the Revolution found this to be an entirely appropriate state of affairs. Mathematics and Science, as they were being newly imagined, were disciplined endeavors proceeding along ineluctable paths of reason. Mathematics, in particular, was not advanced through the wondrous contemplation of numerical or geometrical gewgaws. Descartes [4, p. 47] saw mathematics as an exercise in intellectual discipline, necessary for discovering unimpassioned truth. He wrote:
Descartes was removing himself as far as he could from the unrestrained enthusiasms that characterize wonder, and his sentiments were typical among mathematicians of the time. This view of mathematics, as a strictly logical construct with little tolerance for wanderers, would persist for centuries to come.
Descartes' resolve "to observe constantly ...an order" must be contrasted with the obsessive "observations" of the wunderkammern enthusiasts. Pascal [4, p. 58], who like Descartes cultivated more pious enthusiasms, remarks that:
In other words, mathematicians, when doing mathematics properly, keep their heads down and their vision focused: A conception of mathematics as a terribly sober business.
It was, however, a conception that was undeniably productive. The historian Morris Kline [27, p. 391] has written:
Many mathematicians were eager to see these accomplishments included among the "wonders" of the day. Pascal, for example, did not object to having his adding machine appear among the crocodile skins of many wunderkammern.
Most of the objects on the mathematical horizon in 1600, however, had been in sight for some time: algebraic equations, rows of calculations, trigonometric constructions, smoothly varying plane curves. Mapping the mist-covered routes between these objects, and negotiating the rocky shores that appeared when the mists began to clear, surely made for many satisfying mathematical voyages. The existence of the routes themselves, however, could not have been in serious doubt, and the voyages must not have seemed especially risky. Much of the time in mathematics, it was if western routes to the Indies were being found as expected.
It is thus difficult to describe the mathematics of the early 17th century in quite the same terms as those used to describe the revolutionary intellectual discoveries shaping the larger culture. The new era in mathematics that would come to pass by the end of the century was the result of a long, deliberate surveying of the mathematical landscape. There was no Magellan, Drake, or Hudson to bump into calculus; no Bacon, Hobbes, or Locke to repudiate the entire inherited intellectual system; no Copernicus, Galileo, or Kepler to re-imagine the mathematical universe. The mathematical discoveries of the early 17th century, though profound, did not explode ontological categories.
In the Age of Wonder, mathematicians and non-mathematicians alike looked past their astrolabes and into the heavens; beyond the era's navigational achievements and into the dark forests of the New World. Whatever Pascal may have wished, his adding machine could not compare with the awe, astonishment, surprise, and fear—the wonder—that these places evoked. Mathematicians focused on the well-established pleasures of their craft, and it would be some time until their cabinet was filled with curiosities that could capture the imagination in the same way as horned fish, fluorescent birds, and the artifacts of entirely unsuspected parallel cultures.
3. Wonderful Mathematics.
Henle  has pointed out that mathematics seems to pass through its own great Ages, corresponding in spirit to great Ages in art and culture, but often with a considerable delay. Descartes' influential belief that mathematics proceeds by its own internal logic, on its own time, independent of a posteriori input, may help to explain this apparent lag behind the Zeitgeist. Mathematics did eventually experience a comparable Age of Wonder, but it was some 200 years after Europe had been so transfixed by wunderkammern.
It was the 19th century when mathematicians really began to wonder. The beginning of that century was marked by the appearance on the mathematical horizon of the first ship from what was undeniably a New World: Fourier's 1807 publication of Theory of the Propagation of Heat in Solid Bodies [13, vol. 2]. The objects within its hold, which Fourier called "functions" but which did not seem to be anything of the sort, provoked an intense period of re-examination among mathematicians. Fourier's functions seemed to be perfectly well-defined by a method that was by then common practice—infinite series—and yet they exhibited properties that could not be reconciled with the predominant mathematical worldview. The series he used were trigonometric series, rather than the familiar Taylor series. The functions they defined did not seem to be determined by their derivatives, as Taylor series were. They were defined on closed intervals, yet they behaved as if they were periodic. Their graphs did not even look like functions—they jumped from one place to another without passing through intermediate values. All of these properties were very abnormal, and probably, it was thought, the result of some fundamental misconception. The new objects were dismissed by Descartes' disciples, who felt certain that this infidel mathematician and his ungodly "discoveries" could be explained away. Excruciatingly, however, they could not. Despite the best efforts of the Mathematical Inquisition, the terms of damnation could not be agreed upon. Instead, mathematics worked itself into a very non-Cartesian state of distraction. As the threads of Fourier's arguments were pulled apart, they tangled around the feet of everything that was holy. Edicts and proclamations were drawn up, but they could not dispel a profound anxiety that was spreading throughout the whole of mathematical culture.
It is a credit to the great mathematicians of the era that they, like their forebears from centuries before, responded to the challenge with a sense of wonder. Lagrange, Abel, and Cauchy began to dissect the new functions with a spirit reminiscent of the early anatomy theaters. Before long, other explorers, map-makers, and curiosity collectors would follow them on long voyages to a fertile new place called Analysis. Younger mathematicians would come to their museums to ponder the skeletons of strange new structures. They would wander through the curiosity cabinets of Cauchy's Cours d'analyse [6, series 2, vol. 3]. When it was their time, this next generation would follow curiosity's trade winds to an entirely New Mathematical World.
Undoubtedly, the second half of the 19th century was a time of tremendous mathematical treasures—wonderful treasures. Cantor, returning from his solitary voyages, brought back the varieties of infinity; Riemann sailed geometry to the higher dimensions where it blends seamlessly with analysis; Weierstrass charted the ever-stranger continuous but nowhere-differentiable functions, and Frege explored the deep caverns of logic itself. What mathematics was, at this point, became a thing of wonder in itself. The subject no longer seemed to proceed along Descartes' ineluctable paths of reason, but rather through flights of the imagination, inspired by dreams of what might be. Hans Hahn [17, p. 1956] later called this time a "crisis in intuition", and indeed it was; but it was a crisis only for those who could not leave behind the solid ground of Cartesian certainty. For those who were willing to test the wild waters of the great new sea and give up their minds to the uncharted, it must have been a wonderful, heady ride.
Consider, for example, the following testimonial from Sylvester, delivered in an 1869 address [45, vol. 2, p. 654]. This was before many of the greatest explorations of the era, but already it reflects a significant sea change away from Descartes and a strictly a priori view of mathematics. The tone suggests the contentiousness of those uncertain times.
As analysis began to mix inextricably with geometry and the other branches of mathematics, the curiosities multiplied. New results stretched the limits of imagination. Consequently, mathematicians began to build intricate models out of wood, string, and plaster, transforming the far horizons of their explorations into finite forms that others might contemplate.
Figure 2. Clebsch's Diagonal Surface: Wonderful.
Fourier himself would have appreciated the spirit in which these models were created, for he had always believed in the power of mathematics to elucidate what could not be experienced directly. He had written [14, p. 8]:
The movement by mathematicians to build intricate representations of exotic mathematical destinations was paralleled by a wider 19th century educational movement to promote the use of much simpler models in the universities and public schools. Some of the currents in 19th century mathematics education reform that led to this development have been documented by Kidwell . In the United States, for example, sets of geometric solids were sold to the newly established common schools in order to codify the impression of a common curriculum. The market for these models was maintained well into the early parts of the 20th century with fervent calls for, successively, "Object-Oriented Instruction", "Technical Training", "Art Education", and "Exact Thinking". The business eventually diversified into much more lucrative catalogues of "Mathematical Apparatus", which included everything from finely-crafted orreries and tellurians to the latest in elegant "Pointing Rods".
There was money to be made in outfitting mathematics departments with the instant educational cachet of a fine set of models, but there was also a genuine belief in their inherent instructional value, even among those who were doing the selling. This belief—that mathematical models could help to develop essential intuition about difficult analytic constructions—originated with two very influential mathematicians: Gaspard Monge in France and Felix Klein in Germany. Together, they set the standard for the way that mathematics was taught in Europe and America throughout the 19th century. Monge is known as the father of differential geometry, and his efforts in the early 1800s to classify surfaces by the motions of lines, along with his "descriptive geometry" for representing three-dimensional surfaces in two-dimensions, led naturally to the construction of elaborate models made of tightly stretched strings. One of his students, Théodore Olivier, built some of the most beautiful mathematical models ever made. He also made some money in the process: the models were expensive. Olivier sold them to the emerging technical schools in the United States, which were attempting to emulate the example of Monge and the École Polytechnic. Klein came along later in the century, promoting the use and construction of mathematical models in graduate education, and the first research universities in the United States did their best to follow the European lead. Klein and his colleague Alexander Brill established a Laboratory for the Construction of Mathematical Models in Munich, and the labors of their graduate students were reproduced and sold world-wide by Brill's brother Ludwig.
Figure 3. Klein's Munich Wunderkammern.
It was thus through a combination of free-spirited mathematical exploration, educational idealism, and conspicuous commercialism that the mathematical descendants of the wunderkammern came into being. Schools proudly displayed their newly acquired models, to show that they were up-to-date with the latest mathematical discoveries and the most progressive educational trends. The models were often housed in elaborately crafted cabinets, made of fine wood. The models themselves sat on pedestals within the cabinets, sometimes on lush carpets of velvet. When they were taken from their case and into the classroom, they were presented, by all accounts, with great ceremony. Teachers and students making note of the Cartesian principles that the models were meant to demonstrate couldn't help but marvel at them—the treasures of the New World.
By the turn of the 20th century, mathematical wunderkammern had proliferated across Europe and America. Their spell, however, eventually lost its hold on the mathematical community. Economic realities in the early part of the 20th century made the acquisition of such "treasures" an increasingly difficult proposition, and the market for finely-crafted models, as well as the finely-crafted theories of education that went with them, fell off. Entrepreneurs jumped in, and for a while cheaper knock-offs supplied the diminishing demand. The increasingly clumsy constructions, however, could no longer capture the collective imagination. Mathematicians had gotten down to the hard work of sorting and classifying the accumulated discoveries of their great Age of Exploration, and traditional Cartesian work ethics were once again the mathematical vogue: skepticism, circumspection, and careful linear argument. Educators also gathered themselves together, returning to an emphasis on "fundamental skills". A purposefulness settled on the century, and dust began to settle on the wunderkammern.
Figure 4. Johns Hopkins University, ca. 1895.
4. Whither Wonder?
The demise of the mathematical wunderkammern was fated, inevitably, by larger historical forces that had been in motion for centuries. The Enlightenment, with its emphasis on positivist inquiry and empirical science, had come rushing through the 19th century and into the 20th. Its great legacy, the Scientific Method, sought to draw careful boundaries around what had once been wonderful, charting intellectual continents of human dimension on a grid of comprehensible design. The ability to map Nature became as much a source of wonder as Nature itself. Commerce became Industry, and the success of the Scientific Method encouraged the belief that everything inconvenient and unruly in Nature could eventually be out-reasoned.
There were differences of opinion about this, of course. Perhaps nothing symbolized the opposing world-view so much as the Romantic movement in 19th century poetry, and perhaps no one symbolized that movement so much as Samuel Coleridge. During a period absent of muse, he wrote in his notebooks [22, p. 301]:
There is, after all, something in the triumph of Science that is fundamentally opposed to the essential requirements of wonder. Wonder requires a diminished sense of oneself and one's capabilities. The historian Adalgisa Lugli has noted [29, p. 123] that:
The great Age of mathematical wunderkammern was only a back-eddy in the larger currents of 19th century Science that were carrying the rest of the world over and around the "problems" in its path. It is not surprising that mathematicians, true to Science and forever beholden to manifestations of Cartesian progression, could not abide by their disorderly collections of curiosities for very long. They sought to understand the wonders, to put them in orderly mathematical contexts. After a century of dissection and classification, the elaborately constructed models now collecting dust in so many neglected cabinets look almost quaint, like the mementos of a faded romance.
Has wonder, then, finally disappeared beneath the flood of Science? The source of the flood, the rush toward Enlightenment and its ideals, has been diverted many times during the past century. The idea of Progress as Destiny has been battered about in political, economic, and social turbulence. In uncertain currents, the belief that Science would always find the true heading—implicit in Newton's mechanics, but not in Heisenberg's—has been called into question. Finally, and irrevocably, the twisted roots of Science were revealed in the brilliant light of the Atomic Bomb. As a result, uncertainty and disillusionment have shaped the past century as much as the ideals of the Enlightenment.
Wallace Stevens, the great philosophical poet of this past century, laments the legacy of living in the grip of a Method that is forever doomed by its inadequacies [44, p. 128]:
There is, however, something essentially new in this kind of uncertainty. Whereas in past centuries there may have been more general unknowing, this was accepted; it was not in conflict with previous experience. Before the Enlightenment, there was not the expectation that everything could be understood and controlled. Now, in the post-Enlightenment, the loss of certainty feels like the result of some sort of guilty excess, and there is an air of contrition surrounding every attempt to re-establish the Cartesian course. Mathematics has not been exempt: Attempts to control perceived excesses have consistently found their way into the debates on mathematics education. If the past century has revealed a fundamentally chaotic dynamic at the interface between Nature and the human mind, then (the arguments go, consciously or not) it must be banished from mathematics, in much the same way that the "misconceptions" of Galileo and Fourier were banished. The efforts are as futile now as they were then.
Like Lagrange, Abel, and Cauchy before them, there are mathematicians at the turn of this new millennium who recognize the fundamental sea change that has taken place, and are willing to accept it with wonder. While mathematics and science have undoubtedly helped to create the late 20th century aesthetic of containment and control, mathematicians and scientists also have a long history of fighting such strictures, especially when they begin to look like hegemony. Einstein didn't like the uncertainties of quantum mechanics, but he could still say that [8, p. 11]:
Richard Feynman noted [12, p. 248]:
As long as there is "unknowing", there will be a recognition of the need for wonder.
In modern mathematics and mathematics education, the pervasive presence of "technological" curiosities is once again calling into question the Cartesian ideal of rigorously linear exposition. Computer investigations are inherently non-linear experiences, tossing the mathematical explorer about in a seemingly limitless sea of information. Even the most timid students of mathematics, however, can ably surf the Web. The numerical and graphical processing powers of computers have taken mathematicians to previously unimagined worlds, and allowed them to return with fantastic new treasures. There will continue to be, as there has always been, time to apply the lessons of history, and to sort, connect, and classify these marvelous new objects. It would seem a shame, however, with such fine galleons harbored on the desktops in almost every modern mathematics department, not to go along on the voyages, open to the wonder of what may lie ahead.
Figure 5. The Modern Wunderkammer.
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