In her Memoir of Augustus De Morgan, Sophia De Morgan reported being struck by the ‘almost perfect moral nature’ of Robert Leslie Ellis when she met him when he was a student at Cambridge.Footnote 1 This may seem a somewhat odd assessment of the man. Ellis’s biographer, Harvey Goodwin, certainly wanted to cast him in a positive light, but the man he described manifested few of the traditional external signs of holiness. Goodwin described Ellis as a man, who inherited a ‘highly nervous constitution’ from his mother.Footnote 2 As a child, he had few if any friends of his own age, which translated into a kind of earnestness that led him from the age of twelve to close diary entries with judgments of whether the day had been well spent.Footnote 3 By the time he was twenty, when he was not recording how many pages of mathematics he had read he was finding ultimate meanings, as when after a morning spent in town he came home alone – ‘like many a man who goes to his long home – after losing all the sets of companions he was with in life’ (4 March 1836). Among his fellow students at Cambridge Ellis displayed a ‘kind of elderly sobriety of manner’, that Goodwin was anxious to explain did not amount to ‘stiffness,’ but which certainly did not translate into warmth.Footnote 4 As an adult, he was an intellectual snob, who had no ‘taste or any special fitness for imparting knowledge to average minds’ or any minds except those of his small group of ‘intimate friends’.Footnote 5 Goodwin hastened to add that among his small group of friends, Ellis ‘possessed one of the most gentle of hearts’ but that gentleness was not clear to most of the world.Footnote 6 What they saw was a man whose ‘sense of honour and propriety was perfect; nothing shabby or mean could exist in the same place with Leslie Ellis’.Footnote 7 Try as he might, Goodwin found it difficult to portray Ellis as man who emanated transcendent love.

Sophia De Morgan certainly would not count among Ellis’s “intimate friends”; she knew him only second-hand, as a colleague and occasional correspondent of her husband, Augustus De Morgan. Her assessment seems to have been based on the sickliness that characterized Ellis throughout his life. While still in his teens his education was repeatedly interrupted by illness, and his entrance into Cambridge was delayed for a year because of his health. While at Cambridge his eyes were so weak that he had to hire someone to read mathematical texts to him. At celebration after he finished first on the Tripos Ellis’s fragility was on full display. Ellis described being led between college dignitaries, who ‘made a lane with their maces’, while ‘my good friends of Trinity & elsewhere, two or three hundred men, began cheering most vehemently, and I reached the Vice’s chair surrounded by waving handkerchiefs & most head rending shouts’ (15 January 1840). Having been blessed and taken accustomed oaths of allegiance he then ‘turned back, & walked slowly & stiffly down the Senate House’ (15 January 1840) accompanied by more waves of applause. In response Ellis became so pale that he had to sit down and accept a bottle of salts sent to him by a young woman in the admiring throng crowd.Footnote 8 At the moment of his greatest triumph Ellis distinguished himself by almost fainting. Sophia’s response to the sickly young man was typical of the ways Victorians responded to invalids like John Keats or Elizabeth Barrett Browning. But Ellis was neither a poet nor a woman. In the case of Ellis, the evidence of his moral character arguably lay chiefly in the facility with which he practiced mathematics.

Sophia’s judgment draws attention to the interaction between the mental and the material that was singularly important for the English in the early nineteenth century. Andrew Warwick recognized this and explored the relation of the physical and the mathematical in a piece entitled ‘Exercising the Student Body. Mathematics and Athleticism in Victorian Cambridge’.Footnote 9 Here, Warwick points out that the Tripos examination that marked the culmination of the undergraduate career in Ellis’s Cambridge was as much a physical as a mental examination. Preparing for the examination was a gruelling process that required students to hire private tutors, who drilled them relentlessly in ways to solve the kinds of problems that appeared on the Tripos. The students learned a great deal of mathematics, but memorization and speed-writing were equally part of the preparation for a Tripos examination that rewarded those who could finish as many problems as possible in the time allotted. The examination itself was as much a physical as a mental challenge. The Tripos required students to be physically able to endure on the order of several (the number increased over time) successive many-hour days of examination in over-crowded under-heated halls in January. Under these conditions, Warwick has shown, the idea of a sound mind in a sound body took root in Cambridge. Some of the men who were strategizing with their coaches about how to answer questions quickly were also doing much to keep their bodies strong. Cambridge was filled with strapping young men like James Clerk Maxwell, who routinely ran up and downstairs as a break from his studies, or William Thomson, who was the stroke of the first Trinity college boat. These generations’ experience of Tripos preparation formed the foundation for the development of athletics as essential support for high academic achievement.

Ellis certainly did not display a sound mind in a perfectly sound body. Nonetheless among his contemporaries he was universally hailed as a model of the type who would triumph on the Tripos. Goodwin, who came in second behind Ellis, remembered a friend commenting: ‘If I had seen him before, I could have told you you could not beat him’.Footnote 10 Another, who observed his post-Tripos ceremony remarked, ‘he looked very pale and ill, but this perhaps enhanced the intellectual beauty of his countenance’.Footnote 11 In the eyes of this observer, Ellis’s sickliness was a mark of mathematical brilliance. Rather than marking him as feeble, his frailty served as the sign of a powerful intellect.

The mismatch between Ellis’s world of ethereal intellect and Warwick’s world of athletic wranglers runs deep. Ellis was cut from a different cloth than the strapping Maxwell and Thomson, not only corporeally and but mathematically as well. He was a representative of an eighteenth-century tradition that was only slowly being overtaken by the problem-oriented work of the Tripos. Warwick’s athletic wranglers went on to pursue a form of mathematical physics that is characteristic of the late nineteenth-century English, but Ellis did not. For him, mathematics was more closely related to theology than it was to physics.

The foundation for Ellis’s kind of theological mathematics lay deep. Within five years of the publication of the Principia, Newton told the Bishop of Westminster, Richard Bentley: “When I wrote my Treatise about our System, I had an Eye upon such Principles as might work with considering Men, for the Belief of a Diety’.Footnote 12 This was not an idle claim. Newton was so fully convinced that the laws he had discovered mirrored the thought of the designer of the universe that at one point he claimed that the space he understood so completely corresponded to God Himself. ‘Does it not appear from Phaenomena’, he asked in the 28th Query to his Opticks ‘that there is a Being incorporeal, living, omnipresent, who in infinite space, as it were in his sensory, sees the things themselves intimately and thoroughly perceives them, and comprehends them wholly by their immediate presence to himself?’.Footnote 13 Thus identifying the divine mind with physical space was not a position Newton could hold with any precision, but it shimmered in Anglican theology for centuries. Throughout the English nineteenth century, to think mathematically was to enter the mind of God.

This linkage between Newtonian mathematics and Anglican theology arguably constituted the justification for the Cambridge curriculum throughout the eighteenth and early nineteenth centuries.Footnote 14 When Ellis was a student, the Tripos was well on its way to being recognized as the most competitive mathematics examination in the world, but it was not designed to produce mathematicians. The competitive apparatus of training that Warwick so well described was very real, but at the same time, it was never truly embraced by the Cambridge establishment. The private tutors, who coached the most successful students through the examination, were not always members of the university. The athletic training that paralleled their efforts lay equally outside of the academic world. The Cambridge education was intended to produce clergymen. Whatever the reality of their Tripos experience, the mathematics that students learned was understood and defended by many as a means to this end.

Sophia knew the spiritual implications of Cambridge mathematics. She was not a mathematical thinker herself, but she was married to Augustus De Morgan and through her husband well aware of the basic parameters of the mathematical landscape in which Ellis worked. She was also aware that many challenges were threatening the stability of that landscape. In the first half of the nineteenth century, as the English were negotiating an industrial revolution that was leading them to the heights of imperial power, the nature of mathematics became a pressing issue. Throughout the eighteenth century, the success of Newtonian cosmology served the English as a guarantee that in mathematics they could align their thoughts with the mind of God and by so doing truly understand the world in which they lived. As they moved into the nineteenth century, however, this assurance of unity between the human and the divine was being challenged on many fronts. In the decades before Ellis entered Cambridge, George Peacock, who was to be Ellis’s tutor, devoted himself to reaffirming the strength of the link between mathematical symbols and their referents in the context of algebra. By the time Ellis got to Cambridge in 1837, William Whewell and Augustus De Morgan were scrambling to respond to new French ideas of mathematical rigor that were further threatening to divide mathematical from divine understanding. In the decade after Ellis graduated, Whewell and De Morgan found themselves grappling with divisive continental understandings of the subjective and objective, while Ellis was pushing back against the Laplacian interpretation of probability theory that was driving a wedge between humans and their decision-making processes. All of these conversations might be said to have been part of a concerted effort to defend a world in which the human and the divine could work in concert with one another. Sophia’s judgment of Ellis’s character recognizes the role he and his life played in this larger discussion of the nature of mathematics and reality that gripped both of their worlds.

1 The Transcendence of Mathematical Knowing

That Cambridge mathematics was the centre of a theological education significantly shaped the subject there supported.Footnote 15 Newton’s geometry was a representation of the space that was both the substratum of Euclidean geometry and the container of the universe. All of the definitions, axioms and proofs of geometry were essentially descriptive. Their validity lay in the absolute clarity with which they could be understood. Ultimately then, the truth of Newtonian mathematics rested on its meaning.

Newton’s valorization of geometry essentially distinguished his work from that of his great rival, Gottfried Wilhelm Leibniz (1646–1716). Newton and Leibniz properly share the honours for having developed a set of techniques for calculating derivatives and integrals on the Cartesian plane, but their understandings of their work were radically different. Newton understood the calculus as the science of motion within a world of absolute, true mathematical time and space. In it, what we would call the “derivative,” was the “ultimate velocity” of a “fluent,” at a particular instant. Explaining how anything could be said to have a velocity at a particular moment—that is, could be moving without a change in time—was all but impossible. Newton nonetheless tried, in famously opaque passages like the one in which he explained that the derivative is to be understood as ‘the ultimate ratio of evanescent quantities’, that is, as ‘the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish’.Footnote 16 Newton represented this “ultimate velocity” or “fluxion” (or “derivative,”) by placing a raised dot over the moving variable or “fluent” to produce .

Leibniz, on the other hand, lived in an essentially non-continuous, atomistic world. For him, then, the spaces between discrete numbers on a continuous curve were filled with infinitesimally small quantities, which he called “differentials,” and represented as dx, dy, etc. Leibniz’s differentials were so small that no amount of addition or multiplication could ever render them finite. However, since they comprised a universe of tininess in which relations were structured just like those among finite arithmetic numbers, the result of dividing one of Leibniz’s infinitesimals by another one, was a finite number. For Leibniz the derivative was the ratio of two discrete infinitesimals, dy/dx.Footnote 17

Over the course of the eighteenth century, the different sets of symbols Newton and Leibniz produced came to reflect a basic difference in the ways mathematics was perceived. Newton’s dot notation remained as opaque as his explanations of it, and his symbols simply stood as markers for the ideas they represented. In this way, Newton’s dot notation was very well suited to the theological context in which the eighteenth-century English pursued mathematics. For those to whom the calculus provided a glimpse into the mind of the Creator, the clumsiness of the dot notation could be seen as an advantage that required investigators to keep their focus on the transcendent reality that the symbols or the words were pointing at. The English embraced meditations upon the nature of “fluents,” “evanescent quantities,” or “ultimate velocities” as powerful spiritual exercises.

Leibniz’s dys and dxs developed very differently. Virtually no one accepted Leibniz’s interpretation of their meanings, but few could deny their effectiveness. Leibniz’s algebraically nimble dys and dxs could skip through all kinds of situations that were all but inaccessible to Newton’s s. Continental analysts eagerly pursued them into whole areas—from multivariate calculus to differential equations—where the English could not follow.Footnote 18

As the eighteenth century gave way to the 19th, the French revolution fundamentally changed mathematics in France. After the revolution, mathematical thinkers from all over Europe were gathered together to teach their subjects in the newly formed École Polytechnique, which was an engineering school in which the focus of mathematics lay in solving problems, not communing with God. in reading the scriptures.Footnote 19 Establishing a clear foundation for the calculus became a central concern for those who were trying to teach their subject to a new generation. In 1810, Sylvestre-Franc̨ois Lacroix (1765–1843) adopted a hybrid approach in his Traité élémentaire de calcul differential et de calcul integral. There he used Leibniz’s dy/dx notation for solving problems, while using Newton’s ideas of motion in an absolute time and space for justifying its results. It was not a totally satisfying position, but it worked, and more than 100 years of successful mathematical development supported his decision to trust to the basic solidity of the calculus without worrying too much about the strength of its foundations.

Even as the English closed ranks against the French politically and militarily, there were some who were intrigued by the ideas that emerged in the wake of the revolutionary upheavals. Among these was a Fellow of Gonville and Caius College, Robert Woodhouse (1773–1827), who was fascinated by the French discussions of mathematics. In 1803, Woodhouse published The Principles of Analytic Calculation, in which he tried to find a way to keep the mindfulness of English mathematics while embracing the power of dy/dx symbology. Mathematics he there declared, was best understood as an inductive science; just as scientific laws were formed by generalizing individual experiences, mathematical laws were found by generalizing individual results. It is more natural to treat ‘the steps by which we ascend to expressions, more and more general, merely as so many improvements in the language of Analysis’, he explained.Footnote 20 For him, the move from writing (x + x + x) to writing 3x was an inductive one, as was the move from (xxx) to x3. All of these symbolic shifts could be useful as a way to streamline thought, but their validity rested always on the specific cases from which they were generalized. When approached in this way, the successful development of the dy/dx symbology was itself enough to legitimate its use.

Woodhouse’s Principles put him in the somewhat odd position of entering a French discussion by means of an English book. But he was much more interested in opening his countrymen to continental conversations than he was in getting the French to engage with his arguments. Introducing his English colleagues to the Leibnizian dy/dx notation was essential to this part of his project because it was the only way that they could read French mathematical papers. Surely, he argued, it was ‘desirable to have the same notation universally adopted, in order to facilitate the communication of science between different nations’.Footnote 21 Whatever one’s view of the foundations of the calculus, Woodhouse was convinced that using the dy/dx notation was the best way to bring together the worlds of English and continental mathematics that he believed had been divided for far too long.

Woodhouse’s Principles was a far too radical a work to have any effect on the ways mathematics was taught at Cambridge without considerable political manoeuvring of a kind Woodhouse was not suited for. The credit for actually introducing the dy/dx notation to the English goes to a group of young men who arrived at Cambridge eight years after he had abandoned the effort. Charles Babbage, John Herschel, and George Peacock were the founding core of what they called the Analytical Society; the slightly younger William Whewell joined them in 1812. Their goal was to advance ‘the Principles of pure D-ism’—that is, the Leibnizian dy/dx notation— ‘in opposition to the Dot-age of the University’—that is, the Newtonian dot notation.Footnote 22 The young men of the Analytical Society were determined to bring French mathematics into Cambridge and, thereby, to challenge some of the theological parameters of their alma mater, preferring analytical rigour over traditional authority.

Translating Lacroix’s Traité élémentaire for the benefit of Cambridge undergraduates was one of the group’s central ambitions. Babbage and Herschel left Cambridge before it was accomplished, but in 1816 Peacock published an English translation of Lacroix’s text. There was a difference, however. Although Babbage once proclaimed Lacroix’s text to be “so perfect that any comment was unnecessary,” the English version of the work was attended by copious revisionary commentary. In footnote after footnote, Peacock insisted on the neo-Baconian generalizing of Woodhouse’s Principles of Analysis as a corrective to Lacroix’s approach.Footnote 23 Peacock’s translation of Lacroix is an early example of a what was to be a whole series of negotiations in which the English insisted on engaging French ideas on their own terms.

In the case of Peacock’s Lacroix translation, the result is at once strange and an accurate reflection of its moment. At the beginning of the 1820s, no one—whether French or English—was entirely clear about how to understand the nature of the mathematics that they were pursuing. Peacock was uncomfortably aware of this, and having done what he could to explain the nature of the calculus, he turned his attention to an understanding the foundations of algebra, which was ground zero for the problem of symbols in mathematics.

2 Maintaining Transcendence Through History

Mathematics is rightly a plural word, and Newton’s identification of mathematical thinking with the mind of God was rooted in only one part of it. His meaning-based understanding of mathematics came naturally to the classical study of Euclidean geometry, but was seriously strained in the new subject of algebra that was developed over the course of the seventeenth century. The whole power of algebraic mathematics lay in the flexible ways it could move over different subject matters. The English held to the view that algebra was a meaningful mathematics, whose subject matter was number. At the end of the eighteenth century some radicals carried this conviction so far as to deny the validity of negative numbers, which were not part of the classical Greek concept of positive counting numbers. Few were willing to go to that extreme, but the problem it highlighted was very real. In the years before Ellis arrived in Cambridge, his tutor, George Peacock, had struggled with the problem of negative numbers in algebra. By the time Ellis matriculated, Peacock had found a way to legitimate negative numbers while preserving the essential link between mathematical symbols and their referents that was the cornerstone of English religion-inspired mathematics.

Peacock took an empirical approach to his challenge. Over the course of the 1820s, he carefully examined reports of different mathematical systems that English explorers were encountering as they confronted various cultures around the world. Some sense of the breadth of his thinking may be surmised from the titles of the papers he presented to the Cambridge Philosophical Society, which had in 1819 arisen from the ashes of the Analytical Society: “Greek arithmetical notation”; “On the Origin of Arabic numerals and the date of their introduction in Europe”; “On the numerals of the South American languages”.Footnote 24 Peacock’s explorations of non-western views of number changed his understanding of mathematics dramatically. In an 1829 article “Arithmetic” published in the Encyclopedia Metropolitana, he showed how all of the diverse mathematical systems he had read about fit into an enormous pattern of progressive development. From that perspective, the concept of number articulated by the ancient Greeks represented just a particular slice of time in an enormous historical process.

Having devoted a decade to mapping the ways the concept of number had developed all over the world, Peacock was ready to turn his attention to the nature of the algebra that had grown out of that concept. In A Treatise on Algebra, published in 1830, he resolved the problem of negative numbers by embedding it in the same kind of progressive narrative he had so carefully mapped out in arithmetic. Algebra, he there declared, had developed through two major stages. In the first, arithmetic equations like 42 – 32 = (4–3) (4 + 3), were generalized into the form a2 – b2 = (a + b) (a – b) for a ≥ b. In the second stage, such equations were further generalized by dropping the restriction that ab. Removing this restriction immediately opened the possibility that b was larger than a, and thereby created negative numbers. That these numbers had consistently yielded true results over the course of 200 years supported his conclusion that the negative numbers were valid.

Having laid out this historical narrative, Peacock enshrined his conviction that it reflected a universal truth in what he called the “Principle of Equivalent Forms.” He worded this principle somewhat differently at different times, but the following may be taken as a basic statement of it: ‘Whatever equivalent form is discoverable […] when the symbols are general in their form though specific in their value, [e.g. a2 – b2= (a+b)(a – b) for a ≥ b] will continue to be an equivalent form when the symbols are general in their nature as well as in their form [e.g. a2 – b2= (a+b)(a – b)]’.Footnote 25 Translated into everyday language, Peacock’s “Principle of Equivalent Forms” asserted that any algebraic equation that had been generalized from a legitimate arithmetic one was valid, even when it was not immediately clear what that equation meant.

Peacock’s principle fundamentally shifted the English view of mathematics. Accepting it entailed resting the validity of algebraic results on the process through which equations were generated, rather than on the meanings of the symbols in those equations. For Peacock, the pattern of development he had traced through number systems all over the world more than adequately demonstrated that algebraic meanings would eventually become clear. In the meantime, he offered his principle as a guide and an assurance of the validity of algebraic results.

Peacock was Ellis’s tutor at Cambridge, and Ellis completely embraced the older man’s vision of algebra. In 1838, while still an undergraduate, Ellis joined with the recently graduated Duncan F. Gregory, who created the Cambridge Mathematical Journal, in which Gregory published an impressive series of articles on algebra. Gregory, it appears, was as physically frail as was Ellis. When Gregory died in 1844 at the age of thirty-one, a saddened Ellis wrote the “Biographical memoir” that serves as the introduction to Gregory’s collected works.

Ellis’s presentation of Gregory’s work powerfully captures one aspect of the view of mathematical knowing that flowed from Peacock’s algebra. Ellis there characterized Peacock’s insight as recognizing that ‘theorems proved to be true of combinations of ordinary symbols of quantity, might be applied by analogy to the differential calculus and to that of finite differences. The meaning and interpretation of such theorems would of course be wholly changed by this kind of transfer from one part of mathematics to another, but their form would remain unchanged’.Footnote 26 At first, mathematical thinkers regarded these analogies to be merely suggestive, Ellis explained. But over time, he continued, mathematical thinkers moved from the sense that results generated in this way were merely suggestive, that they required further proof, to the recognition that ‘these theorems are true, in virtue of certain fundamental laws of combination, which hold both for algebraical symbols, and for those peculiar to the higher branches of mathematics’.Footnote 27 As a result, he went on ‘each algebraical theorem, and its analogue constitute, in fact, only one and the same theorem’, and ‘therefore a demonstration of either is in reality a demonstration of both’.Footnote 28 Ellis did not here explicitly refer to Peacock, but the position he was laying out may be seen as a rewording of Peacock’s Principle of Equivalent forms.Footnote 29

Ellis remained as vague as Peacock had been about the basis for his conviction that results established using algebraic symbols generated from arithmetic, would be equally true using the symbols of calculus or differential equations. The closest he could come was to attribute their apprehension to ‘a peculiar faculty—a kind of mental disinvoltura which is by no means common’.Footnote 30 Ellis’s use of the Italian term disinvoltura here, instead of self-assurance or lack of constraint, signals that he is reaching to capture something that transcends the bounds of everyday English. In this case, Gregory’s “mental disinvoltura” meant that he ‘at once perceived the truth and the importance’ of the identity of the forms of algebra with those of higher subjects and charged ahead ‘with singular facility and fearlessness’.Footnote 31 From Ellis’s perspective, the results Gregory developed were important, but seeing his thinking in action was even more so. ‘The steady and unwavering apprehension of the fundamental principle which pervades all these applications of it, gives them a value quite independent of that which arises from the facility of the methods of solution which they suggest’.Footnote 32 For Ellis, Gregory served as a living example of a mind who had the grace to see truth immediately. The gleeful disinvoltura with which he approached his work was itself sufficient evidence of the truth of his results.

However much Ellis admired the exuberant self-confidence with which Gregory approached mathematics, he did not display it himself. His contemporary biographer, Goodwin recognized the deep connection and friendship between Ellis and Gregory, but noted that ‘neither in mind nor in manner was there much likeness between them’.Footnote 33 There is certainly no hint of disinvoltura in the picture of Ellis that Goodwin there painted. Ellis was no less engaged in Peacock’s transcendent world than Gregory was, but he did not fearlessly rush into it. He did a certain amount of algebra, but was always more interested in connecting his work with that of the past, than he was to ‘press forward knowledge upon that one line’.Footnote 34 What delighted him was ‘to discuss the principles of investigations already known, to trace the history of processes, to examine the philosophy of a subject, to hunt up its literature, or to simplify its treatment’.Footnote 35 In this Ellis was picking up on a different facet of Peacock’s mathematics. Ellis’s joy in mathematics lay the subject’s essential connections with the whole history of humankind.

Ellis was not alone in his world-opening view of the history of mathematics. Peacock was lifted out of Cambridge to become Dean in Ely Cathedral while Ellis was still a student, but another of the Analytics remained. William Whewell devoted his life to developing Peacock’s kind of historical view of mathematics both in philosophy and in the Cambridge curriculum. By the time Ellis arrived in Cambridge almost 20 years had passed since Peacock published his peculiarly English version of Lacroix’s text. In the interim the dy/dx symbology had become standard on the Tripos examination, but the understandings of the subject remained mired somewhere in the conceptual mud of Newton’s evanescent quantities, the empty symbolism of Leibniz’s dy/dx s, or some poorly defined ‘mixture of the two’.Footnote 36 In 1836, the year before Ellis arrived in Cambridge, William Whewell returned to the question. In “Thoughts on the study of mathematics as a part of a liberal education”, he declared himself ready to break out of this box by asserting that the ‘real fundamental principle’ on which the Calculus rested was ‘the conception of a Limit’.Footnote 37 Neither Lacroix nor Peacock would ever have claimed this. But much had happened in the decade and a half that stood between Peacock’s translation of Lacroix’s text, and Ellis’s entry into Cambridge. Whewell’s proclamation was pointing the way to a major change.

Whewell’s talk of the limit signaled his recognition of the work of the Frenchman Augustin Cauchy, who in an 1821 textbook entitled Cours d’Analyse (Course of Analysis), claimed to have finally established the true foundation of the Calculus. Key to Cauchy’s approach was a breathtakingly crisp three pages, in which he formulated a series of definitions. There, “numbers” became ‘the absolute measures of magnitudes’, “quantities” became numbers preceded by the signs + or -‘; and “variables” became ‘quantities that can be considered as able to take on successively many different values’.Footnote 38 In this context, “the limit” became the “fixed value” that is approached when ‘the values successively attributed to a particular variable end up by differing from it by as little as we wish’.Footnote 39 The limit thus defined established the critically important definitions of the infinitesimal as a variable whose successive values fall below any given number, and the infinite as the limit obtained ‘when the successive numerical values of a given variable increase more and more in such a way as to rise above any given number’.Footnote 40 Cauchy’s definitions did not capture the richness of the ways the words had been used for centuries, but they had the advantage of being both individually clear and mutually dependent. They were, in addition, briskly practical; so, for example, his definition of the infinite provided a straightforward way to decide whether a variable was infinite or not. Tightly interwoven with one another, these clear, operational definitions formed a net strong enough to support virtually all of the analytic results of the previous century. And for Cauchy, that was enough. ‘In establishing precisely the meaning of the notation that I will be using, I will make all uncertainty disappear’, he crowed.Footnote 41

Others were distinctly less enthusiastic. Cauchy’s French colleagues clearly recognized that embracing his definitions entailed abandoning the enlightenment view of reason that had been modelled on the clear thinking of Euclid’s geometry throughout the eighteenth century. Cauchy knew this, but he didn’t care. He was a leader of the post-Napoleonic reaction that swept a generation of Enlightenment mathematicians from their posts at the École Polytechnique. In the “Introduction” to his Cours, the counter-revolutionary dispensed with his predecessors’ enlightenment ideals in a sentence as brief as his definitions: ‘Let us cultivate with ardour the mathematical sciences, without wishing to extend them beyond their domain; and let us not imagine that we can attack history with formulas nor that we can offer the theorems of algebra or of integral calculus as moral training’.Footnote 42 Thus relinquishing all of mathematics’ claims to be an essentially humanistic subject did not trouble him. He was an ultra-conservative Catholic who through truth was to be found through revelation and the authority of the Church, not through some form of mathematical reasoning.

Cauchy’s rigorous approach was as unacceptable to the subject that lay at the heart of liberal education at Cambridge as it was to the mathematical thinkers of the French enlightenment. Nonetheless, the efficacy of his methods was so impressive that within fifteen years even Whewell was being forced to recognize it. As he did so, however, the Cambridge don rejected the rigorous austerity of Cauchy’s definitional approach. The limit, Whewell explained, may be approached through abstract definitions or intuitive axioms or some combination of the two, but ‘whatever course is taken, the foundation on which our conclusions rest is the idea itself’.Footnote 43 When Whewell identified the limit as the “real fundamental principle” of the calculus, he was claiming that it could be conceived as clearly as a geometrical theorem could.

Whewell did not further develop this idea, but over the course of 5 years, from 1836 to 1841, De Morgan took up Whewell’s challenge to translate the operational power of Cauchy’s limit into a properly English mathematics of transcendent truth. De Morgan may be seen as standing chronologically half-way between Ellis and the Analytics. Like Ellis he was powerfully influenced by Peacock, and resisted the problem-driven mathematics of the Tripos. In response to Whewell’s call De Morgan began work on a series of one-shilling pamphlets that were then compiled into a single 785-page volume entitled Differential and Integral Calculus in 1842.Footnote 44

An essential point that distinguished De Morgan’s Calculus from Cauchy’s Cours lay in the nature of definitions. For Cauchy, definitions were fixed points that defined the limits of mathematical legitimacy; for De Morgan, they were constantly negotiable way-stations on the path to full understanding. De Morgan had learned from Peacock that mathematical understandings grew organically from murky darkness to blinding new insights. This meant that the true mathematician would be always on the look-out for ways to explore, develop, and expand the definitions of his subject to be sure they were large enough to include all possible scenarios. He insisted that allowing definitions to fix the boundaries of the legitimate and the illegitimate was a sure way to impede mathematical progress.

Adjusting the certainties of Cauchy’s rigorous definitions to accommodate the richness of De Morgan’s historically grounded ones was a challenge. It took De Morgan five pages to cover the ideas of magnitude and number Cauchy covered in two sentences, more than ten pages to elucidate the definition of the limit that Cauchy had laid out in two neat lines, and another twelve pages to recognize the many intricately intertwined meanings of infinitesimal and infinity. As De Morgan twisted his way through one tortuous example after another, he explained that the convolutions ‘are not unimportant’ because ‘it is of great consequence that the fundamental notions of mathematics’ should be presented using ‘the rude and unrigorous form in which they are expressed in common life’.Footnote 45 Even when he finally declared on his twenty-seventh page that ‘in future we shall use the theory of limits in all reasonings’, he promised to bracket the paragraphs in which he did so to signal that this was just a temporary nod to convenience.Footnote 46 For De Morgan, as for Ellis, the true mark of legitimate mathematical understanding was the soil of human history that clung to its roots. Learning mathematics was only valuable when it lifted the mind to contemplate a transcendent reality that lay beyond any artificially constructed set of confining definitions.

3 Defending Transcendent Truth

For modern historians, Whewell and De Morgan’s collaboration on the calculus may seem strange. Whewell was a high Tory Anglican who devoted his life to defending the elitist education of Cambridge University, while De Morgan was a radical religious dissenter who taught mathematics at the secular University College London. Their collaboration on the calculus points up that the whole structure of their political and religious differences was rooted in a single English ideal of transcendent mathematical truth. Whewell and De Morgan disagreed, often fiercely, about any number of issues of their day, but it was not at all difficult for them to join forces against threats from abroad. Ellis, who was a moderate Whig, agreed completely.

The depth of these Englishmen’s agreement is reflected in their response to yet another foreign threat. In the decade after Ellis emerged from the Tripos the English confronted yet another, even more fundamental threat to their transcendent view of mathematical understanding. For De Morgan the issue arose in the context of logic. For Whewell it was a philosophical challenge. For Ellis it proclaimed itself in probability theory. For all of them, finding the proper response to an emerging opposition between the individualistic subjective experience and the external objective world brought out their most passionately articulate defences of the transcendent truth of their mathematical worlds.

De Morgan spent the 1840s immersed in the Aristotelian logical tradition in which these two terms were well understood. There the term “subjective” referred to the subject of a proposition like “All men are mortal,” while the term “objective” referred to its object or predicate. In this format, the subject “men” was the most concrete, essentially real part of the proposition, while the object “mortal” was considered less real because it existed only in relation to the subject.

De Morgan was completely comfortable with these traditional meanings, but as he was drawing his thoughts together into a book, he realized they were being undermined in the world around him. So, in October of 1846 he opened a correspondence in which he asked Whewell to provide him with ‘two words to supply the place of subjective and objective’.Footnote 47 The challenge that led De Morgan to this question may be neatly located in an article George Lewes wrote for the Penny Cyclopædia. In the article “Subject, Subjective” Lewes declared his determination to introduce the terms subjective and objective ‘to English philosophical language, through the medium of the German writers’.Footnote 48 He then explained that from that German point of view, ‘the Subject is in philosophy invariably used to express the mind, soul, or personality of the thinker—the Ego. The Object is its correlative, and uniformly expresses anything or everything external to the mind; everything or anything distinct from it—the non-Ego’.Footnote 49 In an effort to appear less foreign, Lewes claimed that these were the original meanings of the words, which had then been muddied and confused by centuries of loose usage He then credited Kant with having developed these definitions in response to a century of continental adjustment to the rise of scientific thinking.Footnote 50

De Morgan was in no position to question the validity of Lewes’ interpretation of German philosophy. ‘I stop at Kant, whom I spell with a c and an apostrophe: I c’an’t get through him’, he quipped.Footnote 51 But he did have to come to grips with the meanings of “subjective” and “objective” that the younger man was insisting upon. Those meanings were so alien to his English world that he said ‘people tell me they are always obliged to find out the meanings of these words afresh, every time they come to the subject. And this I find in my own case’, he complained.Footnote 52 Two days later he was still whining: ‘Is it likely that the ordinary antitheses of language should express an antithesis which people in general never think of?’.Footnote 53 Nonetheless, De Morgan turned to Whewell for help partly because he did not want to appear ignorant by using the words in a way that was outmoded and partly because Whewell was a known word-coiner (e.g. ‘scientist’).

De Morgan’s problem lay in the ways Lewes’ definition separated the individual knower from what was known. He was simply not able to accept a construction that insisted on distinguishing between ideas and experiences in a way that gave no credence to the experience of communion that marked moments of true understanding. After several days of struggle, he decided that the ‘ideal and objective is the important distinction in practice’.Footnote 54 Replacing the ego-limited term “subjective,” with “ideal” was De Morgan’s way of maintaining the ultimate validity of the experience of understanding the true nature of the world.

Whewell had no trouble understanding De Morgan’s resistance to dividing experience into the subjective and objective. By the time De Morgan was asking him about the definitions of these terms, he had come to see that some form of this split was a precondition of the philosophy that was coming into England from abroad. In a paper he sent in response to De Morgan’s questions, Whewell had identified a whole list of oppositions—form vs. matter, theory vs. fact, necessary vs. contingent, subjective vs. objective—as “fundamental antitheses” that lay at the base of continental philosophy. All of these distinctions made Whewell as uncomfortable as they did De Morgan, because one after another they drove wedges between the fusion of words and meanings that supported the Newtonian natural theology. Although continental philosophers considered ‘these elements of knowledge separately, they cannot really be separated’, Whewell insisted.Footnote 55 On the one hand he admitted, ‘Knowledge requires ideas. Reality requires things. Ideas and things coexist’; nonetheless, he continued, ‘Truth’, which entails merging these oppositions, ‘is and is known’.Footnote 56 At a loss as to how to make sense of this contradictory situation, Whewell was forced to conclude that ‘the complete explanation of these points appears to be beyond our reach’.Footnote 57 Thus, even as Whewell tried to help De Morgan understand the modern meaning of the terms subjective and objective, the two men agreed that they were foreign concepts. They saw the necessity of becoming familiar with them, in order to participate intelligently in the conversations of continental philosophers. But in their English world, real understanding transcended the antitheses of form and matter, subjective and objective, ideas and things, words and meanings.

Ellis was not privy to the specifics of De Morgan and Whewell’s struggle with the objective and subjective, but he was equally resistant to continental forms of dichotomous thinking. His particular bugaboo was Pierre Simon-Laplace, who in his 1814 Essai philosophique sur les probabilités essentially located all of epistemology within the theory of probabilities.Footnote 58 Since most of our knowledge is only probable, Laplace wrote in his first paragraph, all of the most important problems we face turn on questions of probability. And so, he concluded, ‘the whole system of human knowledge is tied up with the theory set out in this essay’.Footnote 59 From this beginning, Laplace turned to calculating probabilities in evermore complex situations, in order to reach rational conclusions. As he did so, Laplace recognized that seemingly rational people often differ in their opinions, but he say these individual differences as failures of reason, which could and should be eliminated by the determined application of an increasingly sophisticated probability theory. To reach the right conclusion requires ‘great precision of mind, a nice judgement, and wide experience in worldly affairs’, he explained: ‘It is necessary to know how to guard oneself against prejudice, against illusions of fear and hope, and against those treacherous notions of success and happiness with which most men lull their amour-propre’.Footnote 60 For Laplace, truly rational decision making required standing outside of the situation.

Laplace’s view of decision-making was thus diametrically opposed to the one embedded in the Cambridge mathematical education. Instead of recognizing the power of the human knower, he located decision-making in an extra-human set of mathematical procedures. For the Frenchman, coming to a true understanding entailed a process of separation not of communion.

Ellis devoted one of his first published papers to countering Laplace’s divisiveness. In an 1842 paper “On the foundations of the theory of probabilities” he pointed out that Laplace’s position was ‘in direct opposition to the views of the nature of knowledge, generally adopted at present.’Footnote 61 Laplace’s mathematics did not really provide a platform from which to make decisions, because it was erected on a fundamental fallacy. All of Laplacean probability theory rested on the premise that any particular decision could be separated from its particular circumstances, but this kind of separation was impossible in practice. Ellis found Laplace’s probability theory to be an elaborate construction designed to give a mathematical form to conclusions that were always already embedded in the particular circumstances of the case.

In 1854, Ellis wrote a second paper on probability theory, where he again insisted that Laplace’s effort to divide the knowing person from the things that are known is at once artificial and pernicious. ‘Man in relation to the universe is not spectator ab extra [observer from the outside], but in some sense a part of that which he contemplates,’ Ellis there insisted.Footnote 62 We do not sit outside of life’s possibilities like entrants in a raffle. We do not calculate the odds as we move through our days. Life’s possibilities are us and we are them; we are in their midst at all times.

Ellis moved from this position to confront the forces of separation that De Morgan and Whewell had wrestled with in the opposition between the subjective and the objective eight years before. In an elegant two sentences the younger man laid out the position they had struggled to articulate. ‘The thoughts we think are, it is true, ours,’ Ellis explained, ‘but so far as they are not mere error and confusion, so far as they have anything of truth and soundness, they are something and much more. The veritas essendi [essential truth] […] is the fountain from whence the veritas cognoscendi [known truth] is derived’.Footnote 63 Here again Ellis’s use of Latin signals he was moving beyond the plane of everyday language to embrace the transcendent. The relation between the veritas essendi and veritas cognoscendi was what De Morgan was trying to capture when he suggested replacing the term “subjective” with the term “ideal”. The flowing waters of his fountain captured the essence of Whewell’s mighty, but shimmering “Truth”.

4 Conclusion

Ellis was under considerable duress when he was contemplating about fountain of truth. Eight years before, he had embarked on a disastrous trip to Europe for his health. While there he contracted the rheumatic fever that essentially broke him. After several months in bed, the man who was before pale and wan, described himself as “much bent forward and the head a good deal down on the chest and immovable sideways or nearly so. I drag one foot in walking partly from the knee and partly from general weakness.” Over the course of several weeks of his return by stages to England he struggled not to be ‘repulsive’ to his comrades, ‘not that I see half of them, though I pull my hat off as a mark of respect to the master or mistress of the house of whom I see at first only the skirts or gaiters’.Footnote 64 It was an agonizing journey.

Ellis never recovered. The final ten years of his life were a slow and painful dissolution from a disease that bears many of the remarks of amyotrophic lateral sclerosis, the disease that afflicted Lou Gehrig and Steven Hawking. Ellis maintained a somewhat morbid sense of humour through his remorseless decline—‘Today the dog died. I find that neither flowers nor the thought of them withering cheers me’Footnote 65—and wrote poetry that his sister circulated among her friends. It is difficult to find inspiration in verses like:

  • The flower was cankred [sic] in the bud

  • It was a sickly faded flower

  • What recks it now? however fair

  • It still had died before this hour.Footnote 66

As he reprinted this piece, Goodwin scrambled to assure his readers that as Ellis’s sickness progressed ‘his mind gradually found more settled peace and rested more surely upon the love of God and the merits of the Saviour’,Footnote 67 but this assurance hardly rings true. Goodwin was the Dean of Ely Cathedral at the time he wrote these words, but even he found it difficult to cast Ellis’s experience in traditional Christian terms.Footnote 68

It is a challenge to find the spiritual power that so inspired Ellis’s contemporaries in his sometimes dreary diary, his self-absorbed correspondence, his stiff personal interactions or his insipid poetry, but something more blazes out when he was defending the unity of man and his maker against the claims of Laplacian probability theory. Impending death beat a numbing refrain through the final years of Ellis’s life, but he was focused on the present when he wrote: ‘Only on the horizon of our mental prospect earth and sky, the fact and the idea, are seen to meet, though in reality the atmosphere is everywhere present. Everywhere it surrounds and interpenetrates the [black earth] on which we stand—making it put forth and sustain all the numberless forms of organization and of life’.Footnote 69 Eight years of living in a body broken beyond all repair, had enabled him to realize that the transcendent truths that could seem to lie so far away—‘on the horizon of our mental prospect’—were in fact all around in the atmosphere that was ‘everywhere present’.Footnote 70 Ellis breathed this truth in his darkened room, it blew around the wheelchair when he ventured forth into his garden, it everywhere penetrated the black earth on which he lived although he could no longer stand.

There is no reason to believe that Sophia De Morgan read Ellis’s musings about the immediacy of transcendent truth. She was a remarkably well educated Englishwoman, whose father, the religious radical William Frend, was a pioneer in the effort to defend meaningful mathematics against continental forces of intellectual alienation. Frend taught his daughter a great deal of mathematics, but the subject never really caught her interest. She knew that mathematics served as a portal into the world of transcendent truth, but she preferred approaching that world through the words of the Bible and other sacred texts.

From this perspective, Sophia’s determination to attribute “a perfect moral character” to Ellis may seem to have been grounded simply in his having experienced an illness as bad as that of Job, but there was more to it than that. Like most Victorian women, she lived in a world that was permeated by the experience of sickness and death. At the time, she was first struck by the frail young man one of her younger sisters had just died and she was beginning to contemplate life after death. Her desire to access the transcendent world of the afterlife only increased after the death of her father in 1844, and in 1853 exploded into desperation with the sudden death of her and Augustus’s sixteen-year-old daughter. In the painful years that followed, Sophia desperately sought ways to maintain contact. After 10 years of determined searching she published her findings in From Matter to Spirit: The results of ten years experience in spirit manifestations intended as a guide to enquirers. Sophia’s husband was not completely comfortable with her conclusions, but in a detailed introduction he fully supported her effort. Although the subject matter was not one he would ever enter directly, De Morgan recognized her work as a legitimate contribution to an ongoing conversation about the nature of transcendent truth.

Ellis had been dead for four years (1859) by the time Sophia’s book was published. Peacock had died the year before. Whewell and De Morgan continued to defend the transcendent view of human understanding, but their insistence on pursuing a mathematics of transcendent truth was becoming ever more outmoded. By 1865, De Morgan was admitting that few still adhered to Peacock’s approach in which insight and history grounded mathematical truth, though, he proclaimed, ‘I have entire faith in the future’.Footnote 71 Whewell died in 1866, De Morgan in 1871. These deaths signalled the passing of a generation, and a fundamental shift in the English understandings of the nature of mathematics. Sophia was well-aware of the ways that perceptions of the nature of transcendent truth were changing in the world around her, and a defence of the legitimacy of her husband’s work lay at the heart of her Memoir of Augustus De Morgan. When she attributed “an almost perfect moral character” to the sickly young man, whose mathematical gifts so impressed everyone, she was recognizing another defender of the faith in transcendent reality. Twenty years after Ellis’s death, Sophia hailed him as someone who was on her and her husband’s side in a battle for England’s soul that centered on the nature of mathematics.