Complementary spirits seeking harmony

Mary Everest was born in England, but spent a considerable portion of her childhood in France.Footnote ¹⁰ Her father, the Reverend Thomas Everest, left his Gloucestershire parish owing to poor health and settled with his family in Paris when Mary was five. There the entire family came under the influence of Samuel Hahnemann, the German physician who originated homeopathic practices. The family followed a strict regime influenced by the hygienic philosophy he espoused. While her chronicles of some of the more formidable Hahnemannian directives observed in her youth, including bathing in ice-cold water and taking snowy, wintertime walks by lantern-light before breakfast, testify to a matter-of-fact reserve, Mary effusively recounted memories of her experiences with private tuition in a short memoir entitled ‘A child's idyll’. She was particularly fond of her mathematics teacher, Monsieur Déplace, and his non-pedantic approach to lessons, through which she developed a lifelong interest in mathematics and began to conceive of the mind's ability to establish an unmediated connection with its concepts.Footnote ¹¹ These childhood episodes made a lasting impression on her, and hygienic and educational concerns occupied much of her thoughts later in life.

Other recurrent themes and interests detailed in Mary's extensive collection of written works owe much to her father. By her accounts, Thomas Everest was an unconventional and controversial figure. She acknowledges that he was a ‘learned occultist in days when occultists were few … [who] called attention to the dangerous [reaction] which must come if the clerical and medical professions persisted in ignoring the phenomena of Mesmerism, Trance, and Clairvoyance’.Footnote ¹² He further alarmed fellow clergymen by adopting a pastoral doctrine that envisioned his role as servant to the ‘desires of his most serious and wise’ parishioners rather than that of priest. According to Mary, her father was also particularly disdainful of attempts to convert Jews to Christianity, believing that Judaism deserved to be respected for its traditions and interpretations of sacred texts.Footnote ¹³ The Reverend Everest's ecumenism, which embraced aspects of spiritualism as it legitimated an inclusive perspective on spirituality, had a profound effect on Mary, who developed a freethinking and inquisitive approach to religious topics.

George Boole may well have shared many of Mary's nonconformist inclinations, but, unlike her, he steadfastly refrained from espousing these publicly.Footnote ¹⁴ Although he was baptized in the Church of England and seriously considered joining the clergy, he eventually identified himself as a dissenter. Shortly thereafter he began to question both the veracity of the Bible and the legitimacy of Trinitarian doctrines. At the same time, George was attracted to monism, largely in response to numerous conversations with a Jewish friend who taught Hebrew in his hometown of Lincoln. In particular, the Judaic tradition of referring to God as the great All encouraged him to consider the ‘Infinite Unknown as Unity’.Footnote ¹⁵ In later life, reading widely sated his eclectic religious appetite. The thinking of the French theologians Ernest Renan and Auguste Gratry as well as of the Anglican cleric Frederick Maurice and the British rabbi David Woolf Marks influenced him significantly. With interests that spanned a variety of beliefs and practices, it is unsurprising that George personally dedicated himself to religious impartiality, finding it impossible to accept any particular doctrine or condone idolatry of any kind.

Setting aside his clerical aspirations, George developed his interest in mathematics largely through his own initiative and personal contacts. He studied symbolic algebra with Edward Bromhead, president of the Lincoln Mechanics Institute and former member of the Cambridge Analytical Society. His first research paper was published in the Cambridge Mathematical Journal in 1841. D. F. Gregory, who then served as the journal's editor, provided both assistance and mentorship in this undertaking. At the time George was a respected teacher in Lincoln, his professional options being constrained by his refusal to sign the Thirty-Nine Articles. In 1849, however, he moved to Cork, where he took up his post as the first mathematics professor at the recently established Queen's College. Shortly thereafter George met Mary during a visit to her uncle John Ryall, then vice-president of the college. While their opinions on certain matters were markedly divergent, in many respects it must have seemed that each had found a complementary partner in the other. Mutual interests in spirituality and mathematics figured prominently in their courtship as well as in their later life together. Indeed, their relationship evolved partly as a consequence of time they spent together devoted to Mary's private tuition, George proving particularly helpful to overcoming her various difficulties with differential calculus.Footnote ¹⁶

Sometime before their marriage in 1855 George promised Mary that he would seek her assistance in his work.Footnote ¹⁷ Despite her considerable intellect, Mary felt incapable of contributing to his mathematical research. It is therefore no surprise that it took some years for the two to discover a way through which George's promise might be kept. Regardless of the extent to which her self-perception as a mathematical practitioner was justifiable, Mary had good reason to expect that she would continue to benefit from her husband's mathematical instruction.Footnote ¹⁸ For his part, George's ambition to involve Mary challenges the notion of separate spheres of influence by fostering an essential melding of personal strengths. This likely owed much to a theological position advocated by Frederick Maurice, whom George greatly admired. Specifically, Maurice held that men and women are wedded to each other in the same way as the complementary virtues of manly truth and womanly obedience were wholly integrated in Christ's humanity.Footnote ¹⁹ Eventually a publisher's suggestion offered a resolution apparently amenable to both sets of expectations, which Mary modestly described:

The new book was A Treatise on Differential Equations (1859), and this working relationship implies that Mary would have had to acquire a thorough grounding in the subject in order to be an effective, if unofficial, editor. As further evidence of her dedication to their shared task, she attended his lectures (before her womanly presence was deemed an inappropriate distraction) and worked alongside his students in preparing examples for a future edition of Differential Equations. Some time later, their mathematical attention turned to the subject of finite differences. As Mary recalled, ‘We had no text-book, we created one as we went on.’Footnote ²¹

Although she acknowledged no direct involvement with its development, Mary later proclaimed her greatest intellectual affinity for George's Investigation of the Laws of Thought: On Which Are Founded the Mathematical Theories of Logic and Probabilities (1854). Perhaps the most significant aspect of this influential work is its attempt to formalize logic in terms of symbolic – or algebraic – expressions and manipulations.Footnote ²² This certainly owed much to the nineteenth-century view of algebra as a notational language abstracted from arithmetic that applied to solving numerical problems. However, his particular innovation accepted that consistent systems beyond those whose referents related to quantity might be productively recast algebraically. As his title suggests, he was preoccupied with the algebraization of laws he associated with correct reasoning. He foreshadowed his intentions in ‘The claims of science’, an address delivered at Queen's College in 1851, when he asserted ‘the moral and the intellectual constitution of man [as] proper objects of scientific inquiry’. Indeed, he was convinced that ‘there are … rigorous, … scientific laws of thought and reason’.Footnote ²³

Central to the calculus of reasoning on which George sought to ‘establish the science of Logic and construct its method’ were the unity law and Boole's equation.Footnote ²⁴ The former law incorporates his novel assignment of unity to any universe of thought, as expressed by a relationship symbolizing the law of the excluded middle: x+(not-x)=1. George applied his algebraic notation to mental acts that referred to both classes of objects and propositions. In the former context,+signifies the union of two mutually distinct classes within a particular universe of thought; that is, x+y denotes the class of objects that belong either to the class x or to the class y. For situations in which the class x contained all members of the class y, he employed the notation x – y to denote the class of objects that belong to x, but not to y. The unity law is therefore expressible as x+(1−x)=1. George also adopted the juxtaposition xy to denote the class of all objects that are simultaneously members of the classes x and y as well as the numerical symbol 0 to signify the class containing no objects. With these, Boole's equation can be represented algebraically as x(1−x)=0, which, among other things, encodes the principle of contradiction; that is, no object can simultaneously satisfy and fail to satisfy a particular condition.

Despite the originality and significance of his research, Mary claimed that George ‘cared for [logic] chiefly as a means of clearing the ground of doctrines imagined to be proved, by showing that the evidence on which they were supposed to rest had no tendency to prove them’.Footnote ²⁵ This appraisal, considered alongside religious references contained in the final chapter of Laws of Thought, is an indication that his motives at the time were not strictly constrained by disciplinary concerns.Footnote ²⁶ As a private man whose educational and professional history served to keep him at the margins of the mathematical mainstream, however, he was understandably reluctant to attract undue attention to himself by connecting his work with broader themes. Apparently his feelings changed over time, for he eventually began work on a manuscript intended for a popular readership that was to de-emphasize the mathematics of the Laws of Thought in favour of philosophical, psychological and religious discussions.Footnote ²⁷

Shortly before his sudden death George signalled to Mary some other shifts in attitude and suggested intellectual challenges that she might undertake on her own.Footnote ²⁸ His work on a more accessible sequel to the Laws of Thought was apparently hindered by a conceptual problem. According to Mary's recollection, he could not ‘see his way’ to approaching his subject matter in a suitable manner.Footnote ²⁹ The two had many conversations about this problem, intertwined as it was with his religious attitudes and beliefs, and how he might resolve his dilemma. These discussions were almost certainly the subject of George's admission to Augustus De Morgan, with whom he maintained a long correspondence, that ‘there is absolutely no person in [Ireland] except my wife with whom I ever speak on subjects like [logic]’.Footnote ³⁰ During one period of reflection Mary offered a biblical passage that to her mind provided a possible solution to some of her husband's difficulties, specifically those concerning the topic of inspiration. Mary's particular focus was the verse John 3:8: ‘The wind bloweth where it listeth, and thou hearest the sound thereof, but canst not tell whence it cometh, and whither it goeth: so is every one that is born of the Spirit.’Footnote ³¹ However, she interpreted the wind described therein as the ‘geometric figure of the dust-whirl’.Footnote ³²

The spiral behaviour of a whirlwind, mathematically describable in terms of components that act tangentially and perpendicularly to its motion, compelled Mary to consider the powerful result of forces that act orthogonally. In this way she understood the verse as an allegory for the inspiration that could be produced by harnessing attitudes and intellectual inclinations apparently at odds with each other. Appreciating its significance, George encouraged her to pursue doggedly the clue she had uncovered.Footnote ³³ In another exchange she spoke sympathetically of the ‘social excommunication … which often follows on the publication of … (real or supposed) heresy’. In reply, George pleaded that she should

Perhaps the most significant change of heart, especially as it affected Mary, involved George's complicated feelings about Frederick Maurice.Footnote ³⁵ There is little doubt that he esteemed this charismatic and controversial cleric for, among other things, the ecumenical positions Maurice typically adopted. George fundamentally disagreed, however, with his Trinitarianism and the idolatry it fostered, yet adamantly refused to broach this subject with him out of respect for the good work he was doing. After years of encouragement Mary finally persuaded her husband to allow her to contact Maurice and express the wish that they might finally meet in order to start a mutually beneficial dialogue. George died before the letter of invitation was sent. In her first year of widowhood, Mary belatedly contacted Maurice out of a desire to understand better his position on matters with which George disagreed but upon which he refused to expound publicly. Responding to her family's straitened financial conditions following George's death, she also noted that she sought employment in an ‘educational institution’.Footnote ³⁶ Learning this, Maurice helped her to secure a position as a librarian at Queen's College, London, one of the first institutions of higher education for women, with which he was then affiliated. Mary supplemented her primary duties by devoting her time and energies to various ancillary activities, which included offering pastoral advice and tutorials to students. Alongside these various commitments to the institution and its students, she also maintained a theological dialogue with Maurice, one that caused her much personal distress. In particular she recounted his agitation over a biography of George that she had drafted; he also strongly advised against the publication of another manuscript she was then preparing.Footnote ³⁷ Both works were temporarily abandoned, and Mary left her employment with Queen's College in 1873, most likely by mutual consent. Her first attempt to sketch and interpret the intellectual life of George was published in the (Dublin) University Magazine in 1878, appearing in four instalments under the title ‘Home-side of a scientific mind’.

Propagating and fulfilling a legacy: the law of pulsation

If frustrated in her earliest literary efforts to promote George's legacy, Mary was not distracted from what she clearly considered her primary mission. Her Collected Works, including books, essays and lectures produced exclusively in the last forty years of her life, amount to over 1,500 pages of material. In these she examined mathematical, scientific, spiritual, social and political concerns. Moreover, these themes rarely appeared in isolation from each other. The project that essentially inspired all her efforts was defined by her commitment to making intellectual connections, to engaging in dialogue with those who likely harboured contrary opinions and to encouraging syntheses that would in turn inform praxis. As modern scholars have rightly noted, Mary was often maddeningly repetitious in her exposition and frequently eclectic in her choice of subject matter; nevertheless the connections she invited her audiences to make more often than not had their impetus in a consistently identifiable theme that she fashioned from elements found in the Laws of Thought.

The ‘law of pulsation’ constitutes the essential message Mary drew from her privileged insights into the many strands of thought that informed her husband's investigations.Footnote ³⁸ Although it is not explicitly discussed in George's treatise, this law is central to Mary's appreciation of that work and, consequently, to establishing a context for many of the intellectual interventions she would undertake in the future. She traces George's understanding of the law of pulsation to a vision he had as a young man of seventeen. In his rendering, which emerged through private conversations between husband and wife, this vision was fundamental to his efforts to recast logic as an algebraic system. Reflecting on their exchanges, Mary saw it as the moment when he first understood a great mystical secret that underpinned the law of pulsation:

In this account Mary identified the basis for George's belief that ‘1−x (or not-x …) must belong to, and be included in, the same “Universe of Thought”, as the x itself’.Footnote ⁴⁰ Equally evident is an essential interrelationship between the unity law, which relies on this idea, and Boole's equation. The latter, algebraically formulated as x(1−x)=0, underscores the conscious apprehension of differences while the former, symbolically depicted by x+(1−x)=1, suggests that ‘the mind … “functions normally towards Monism”’ in responding to an unconscious imperative to synthesize various pieces of information into a comprehensible whole.Footnote ⁴¹ Correct thinking results from pulsations between these two mental attitudes: the 0 representative of opposition (or the act of compensation) and the 1 of unification. According to the law of pulsation articulated and cultivated by Mary, maintenance of mental elasticity simultaneously requires the regular reversal of intellectual perspective as much as it legitimates the proliferation of associations.

The doctrine of pulsation is inextricably linked with other themes that Mary incorporated into many of her works: mental hygiene, Boole's method and mathematical psychology. The first of these is typically employed in relation either to the healthy state achieved or to the regime undertaken by faithfully adhering to the practice of pulsation, which is to say by adopting Boole's method. Its significance very likely as much reflects her personal experience of growing up according to Hahnemannian principles as it does broader Victorian concerns with hygiene. At the same time, the doctrine of pulsation occupied a special place within Mary's cultivation of mathematical psychology, or what Grattan-Guinness has neatly summarized as ‘a theory of mental acts’.Footnote ⁴² According to her characterization, pulsation is related to mathematical psychology insofar as it describes a ‘sequence of mental attitudes’ that facilitates setting ‘mental machinery to work on [a] subject as to which [one] desires [inspiration] in the same order or sequence as that followed by the mathematician’.Footnote ⁴³ Scholars have found much to discuss in considering the extent to which the Laws of Thought either can or should be read as a treatise that illuminates interrelationships between logic, mathematics and psychology. This valuable approach notwithstanding, it is clear that Mary, in her portrayal of George as a mathematical psychologist par excellence, worked from a simple premise: the act of connecting her husband's work to psychological concerns was both natural and replete with intellectual possibilities that she could develop.Footnote ⁴⁴

Another consistent feature of her propagation of George's legacy is its portrayed relationships with the Catholic theologian Auguste Gratry and his treatise Logique (1855). The title is somewhat misleading. As its English translator noted, the work is ‘more an essay on induction and related problems, including the methods and philosophy of deduction itself as compared to induction, than a handbook for the beginning student in traditional logic’.Footnote ⁴⁵ Gratry's interest in addressing inductive reasoning stemmed from the fact that, unlike its deductive and syllogistic counterpart, it produces general principles from discrete experiences and observations. In attempting to explain the familiar yet transcendent aspect of induction, Gratry sought to correct any notions that this had to do with a ‘vague process of groping’.Footnote ⁴⁶ Instead he argued that the process ‘consists … in searching for God and God in nature: for to seek law behind facts, unity and stability in the multiple and moving, is to look for God without knowing it’. Kepler, whose investigations Gratry highlighted, discovered unifying laws through finite observations by relying on the belief that ‘the eternal ideas of God govern the world’. In the case of its use in Leibniz's development of the calculus, also prominent in Gratry's exposition, induction turns on the belief that ‘the perfections of God are those of our souls, minus our limitations, and … affirms that these infinite unlimited perfections, of which this elimination gives us some idea, are true and subsist in God’.Footnote ⁴⁷

Mary often linked the names of Gratry and Boole in her writing, but among her most concerted efforts are Logic Taught by Love (1890) and the Mathematical Psychology of Gratry and Boole (1897). According to her accounts, George was introduced to Logique not long after its publication and was immediately impressed by it. Certainly the mystical revelation that eventually inspired the unity law and Boole's equation, part of the inductive analysis of reasoning itself, resonated with aspects of Gratry's metaphysics. Consequently his treatise provided Mary with a context for developing themes upon which George chose not to elaborate. Moreover, Logique offered the potential to extend her mission so as to incorporate a recuperative dimension: harmonizing her husband's investigations with other currents of thought.Footnote ⁴⁸ For example, Gratry ended his treatise with suggestions for those who wished to foster the kind of revelatory insights associated with inductive reasoning, a key element of which is the regular observance of silence and relaxation. Indeed, he asserted that sleep, correctly prepared for, could double the time one devoted to useful work.Footnote ⁴⁹ In this regard Mary's expression of the law of pulsation, which speaks to the hygienic benefit of alternating periods of (synthetic and unconscious) reflection with periods of (analytic and conscious) mental tension, signals that her husband and Gratry shared common intellectual sympathies.

There were, however, important distinctions that Mary was careful to delineate. In general, her celebration of George's contributions to logic recast and extended Gratry's discussion of inductive reasoning in such a way as to deny that inspiration relies on any particular theology or religious doctrine. It was of little consequence, according to Mary,

Whereas one could identify Gratry's Catholicism as the impetus for his notion of prayer as a means by which finite minds make their appeal to the Infinite, Mary's articulation of the law of pulsation redirected attention by stressing the dialectical interplay between analytic (conscious) and synthetic (unconscious) phases of thought. Gratry also claimed that logic could acquire wings by accepting the relevance of faith to inductive reasoning. Mary disagreed with his assessment insofar as she felt that something more was required if logic was to be rendered ‘a more powerful instrument than it had hitherto been’. She maintained that George, in moving beyond the structured syllogisms of Aristotle by encoding laws specific to mental processes within an algebraic system, deserved his share of credit for giving wings to logic – for making it ‘freer and more vigorous’.Footnote ⁵¹

Education, pedagogy and reform

Mary once wrote of her own efforts that ‘I should be proud if I could convince a single teacher that the isolation of any mode of thought is misleading; and that no system of Logic can be valid unless it is able to focus together various rays of truth.’Footnote ⁵² While broadly employed here, the word ‘teacher’ indicates an important facet of the promotion and fulfilment of George's legacy: the demonstration of its significance to educational reform. Most of her writing on this topic dates from 1900. It includes both books, the most notable being Preparation of the Child for Science (1904), and articles published primarily in the Journal of Education and Parents' Review. Many of the ideas she formulated were also presented at conferences of the Parents' National Education Union. One of the most salient features of her commentary on education is that it reflects her husband's thinking informed by her personal experience. The Laws of Thought undoubtedly provided a framework, but it is also evident that she drew on her own education and involvement with tutoring students in mathematics when developing her pedagogical attitudes and suggestions. Moreover, in disseminating these, she took full strategic advantage of all aspects of her roles as a confidant to genius, parent and educator in orchestrating a presentation of George's work for an audience that would otherwise likely remain unaware of it. Were it not for her efforts at educational reform Mary might well have been consigned to historical footnotes as the rather eccentric wife of an eminent Victorian mathematician. Even so, it is surprising that while most modern commentators have rightly noted their significance, very few have explicitly acknowledged or elaborated on the various ways her ideas related to encouraging an appreciation of the laws of thought.Footnote ⁵³

Along with other late Victorian critics, Mary disdained teaching that demanded rote memorization or study undertaken solely for the purpose of passing examinations. Moreover, she believed it was her patriotic duty to help ‘improve the development of constructive imagination’ in the sciences that some then felt was seriously impaired.Footnote ⁵⁴ In response to these impulses she advocated an approach to education that, rather than demanding that students learn the ‘last new theory’, would best allow students to ‘extract the truth from a series of impressions and statements’.Footnote ⁵⁵ Learning, for Mary, had little to do with the passive absorption of fact; rather, the process of learning was paramount. As she once encapsulated her position: ‘Education is the integration of brain and nerve action.’Footnote ⁵⁶ To this end, Boole's method of encouraging a rhythmic pulsation between analysis and synthesis signposted the way by which education could simultaneously enhance its efficacy and reorient its goals. Interestingly, in undertaking this project, Mary performed the kind of intellectual reversal that characterizes the law of pulsation. By her accounts, George had developed mathematical insights that would extend the scope of psychological knowledge. The time had come for mathematical psychology – as she understood it – to make a contribution to teaching science and mathematics.Footnote ⁵⁷

Perhaps the most prominent feature of the recommendations in Preparation of the Child for Science is the emphasis placed on developing the unconscious mind and the importance of physical manipulations to stimulate and reinforce mental processes. According to Mary, pedagogies embedded in traditional education had subordinated activities that tended to feed the unconscious imagination, considering these manifestations of idleness. Serious combat against an entrenched position could not be left to educators alone. Mary therefore spoke directly to parents about how, mindful of the expectations of a formal education, they could best provide environments in which their children's unconscious minds might thrive. The central tenet of her recommendations involves fostering what she called a ‘Sabbatical rhythm in science; a clear distinction between the time when [the child] is being taught by a man and the time when he is free to investigate or experiment as he pleases.’Footnote ⁵⁸ Elaborating on this prescription, Mary discussed the ways in which caring for animals and tending gardens could prime the unconscious for later education in the natural sciences. Further, her suggestions included domestic activities that would prepare children to appreciate formal lessons in mathematics and the physical sciences, including topics such as electricity and magnetism, hydrostatics and mechanics.Footnote ⁵⁹ Reform-minded teachers could reinforce the work undertaken by parents by striving to maintain a balance in their four-fold obligation as lecturers, laboratory teachers, textbook authors and cultivators of an inner voice that could be called upon when the child ‘retires into silence to think out the meaning of apparently conflicting evidence’.Footnote ⁶⁰

As part of her appeal for nurturing the unconscious mind Mary paid special attention to the significant lessons learned by making mistakes. Parents who might be inclined to coddle their children or unnecessarily constrain this crucial component of the learning process failed to understand that among the intellectual hallmarks of a scientific mind ‘none is more important than the habit of learning when there is no [guidance], of profiting by … past errors, of rising on stepping stones of our mistaken selves to correct judgments’.Footnote ⁶¹ While she stressed the importance of an appropriate degree of supervision, lest children inflict serious injuries upon themselves, Mary was equally adamant that they be protected in the classroom from the damage that might be done by ‘directing educational care too exclusively to the conscious mind’.Footnote ⁶² In this regard, she considered Euclid's Elements the prototypical example of an inappropriate text for children who lacked the benefit of ‘geometric instinct and the habit of geometric observation’. Its very danger was that it represented a masterfully distilled compendium of (geometric) knowledge; by her reasoning, such texts stifled the heuristic impulse that best engages the unconscious mind. Attempting to revise textbooks so as to disrupt hierarchical and linear narratives was not a solution for Mary. Indeed, she insisted that the ‘remedy is not to substitute for Euclid some inferior and less thorough textbook, but to precede and supplement the use of textbooks by some gymnastic calculated to restore normal vitality to the paralysed natural faculty’ associated with the unconscious.Footnote ⁶³ The reforming work required was not to be achieved by providing more honest historiographies of scientific and mathematical investigations nor by diluted presentations of accumulated knowledge. The changes Mary advocated were extra-textual.

The imperative to look beyond texts is intimately related to the significance Mary placed on mastering advanced concepts by way of experience with commonplace objects. The principle underpinning her commentary is, in her words, ‘that actions which are artificial should be practised in connexion with ideas which are familiar; and new ideas should be learned by means of actions which are natural’.Footnote ⁶⁴ In this scheme, regular engagements with natural objects and processes, as well as tools and simple machines, pay dividends when the child confronts concepts in the sciences. Likewise, experience with aesthetic forms and draughting tools and techniques prepare young minds for appreciating geometrical and mathematical knowledge.Footnote ⁶⁵ Undoubtedly, Mary's most notable personal contribution to teaching mathematical concepts in conjunction with physical activities, one probably known to many who are unaware of her connection with it, is found in her promotion of ‘curve stitching’ on sewing cards. By this Mary referred to the process of creating geometric patterns by means of holes punched into cards with thread passing amongst these according to some chosen scheme. This is her description of curve stitching:

Her own childhood discovery of this form of amusement enabled Mary to provide a compelling personal account of the benefit curve stitching had when it came time to absorb and assimilate advanced concepts essential to calculus: describing tangents to curves and conceiving of curves as comprising infinitesimally short line segments.Footnote ⁶⁷

Psychological functionalism, ethical development, gender and suffrage

As much as they are of a piece with sentiments expressed by others then advancing the New Education in Britain, Mary's recommendations on educational reform evidence strong connections to psychological functionalism.Footnote ⁶⁸ It is not surprising, then, that her ideas came to the attention of William James, who, as one of its early and influential proponents, facilitated the publication of her 1902 essay on ‘Suggestions for increasing ethical stability’ in the Monist.Footnote ⁶⁹ Partly through James, the efforts of both Booles garnered a new audience of American academics and editors. The Open Court Company, then under the leadership of the prolific and polymathic Paul Carus, who also served as editor of the Monist, republished the Laws of Thought in 1916. At that time the most recent British edition was over sixty years old. As a graduate student, the University of Chicago anthropologist Milton Singer came to appreciate Mary's writing and considered her efforts to be a contribution to logic in their own right.Footnote ⁷⁰ The preface to the published proceedings of a 1927 Symposium on the Unconscious acknowledges the relevance of both George and Mary Boole's thoughts on the subject. This further indicates that her interpretative efforts effectively communicated and developed psychological aspects of her husband's innovation.Footnote ⁷¹ Indeed, the twentieth-century revival of interest in George's work ultimately owes much to Mary's contributions to psychological discourse.

As signalled by its title, ‘Suggestions for increasing ethical stability’ is dedicated to offering a resolution to the problem of heterogeneous and subjective nomenclature. According to Mary, ‘the whole subject of ethical training is in chaos owing to words being used in an ambiguous manner, and much confusion is cleared up when the material is first algebraized and retranslated in the vernacular’.Footnote ⁷² Consequently she endeavoured to effect praxis by applying Boole's method to psychological subject matter with the aim of forming a neutral system of notation that could be applied to a variety of specific situations. The law of pulsation implicitly permeates much of her discussion of promoting the ethical quality of a life, which is described as best achieved by requiring a degree of wilful elasticity that could forestall the formation of inappropriate, fixed habits. The ‘qualities of reverence, justice, etc.’, she wrote, ‘depend essentially on the habit of suspending action till there has been time for a mental act of conscious sympathy with facts’.Footnote ⁷³ It would be difficult, however, to recover anything that might be strictly identified as the algebraization of psychological knowledge in this piece. What can be said is that she presented a scheme for systematizing ‘an ethically stable nervous system’ that makes use of symbolic notation representing mental states, actions and motives and suggests functional relationships between these by outlining several examples.Footnote ⁷⁴

Another, perhaps more notable, aspect of her treatise on ethical stability can be found in the introductory material that Mary appended to her original essay as part of its 1909 publication in book form. Its dedication indicates that this was attached to her existing essay as a response to a personal request to articulate ‘in terms of Logic the reasons [she maintained] for not joining in the demand for suffrage’. The preface clearly defines both her audience and her intentions:

Her introductory comments convey sentiments indicative of thinking most closely aligned with psychological functionalism. ‘The laws of logic’, she wrote, ‘were worked into animal organization along with dawning consciousness of the relation of the individual to environment and especially to other individuals. They are in reality the laws of self-preservation from delusions; of economy of nerve-force; of maximum efficiency with minimum waste of nervous energy’.Footnote ⁷⁶ The law of pulsation still had a crucial role to play in her exposition; however, she now approached this from an anthropological perspective that facilitated a gendering of the laws of thought.

According to this text, general use of the word ‘induction’ mistakenly obscures two distinct forms of reasoning. To quote Mary,

Mary immediately related the duality she illuminated to the distinction between inferences that give rise to general truths and those that lead to the articulation of fundamental laws such as those developed by George.Footnote ⁷⁸ Men had practised the first kind of induction for millennia. It gave rise to classical systems of logic upon which legal and legislative procedures had been established. In attempting to systematize the laws of thought, George and others ‘were formulating the laws of a faculty [then] not recognized as having any law’. Even as she claimed the nascent ‘new logic [had] … no conventions as to conduct, no canons of procedures, no precedents’, Mary assured her audience that it reflected synthetic sensibilities ‘weak in men and very strong in women’, those who were more adept in the second form of inductive reasoning.Footnote ⁷⁹

Fully cognizant of the inevitable associations that would be drawn, Mary chose to adopt labels rooted in anthropology rather than gender to distinguish between the two forms of inductive reasoning. The hunter's mode of induction is shaped by its preoccupation with contrast and tension, the mental imprint of which is encoded by Boole's equation. Alternatively, brooders structure inductive reasoning around a tendency to harmonize information, a synthetic inclination encapsulated by the unity law and encouraged by their status as non-hunters. Complete accord with the law of pulsation required the integration of these two modes of thought, albeit on a macroscopic, sociocultural level rather than the microscopic level of individual minds. To this end Mary reminded her readers that ‘suffrage, the whole apparatus of voting and legislation, belongs on the whole to the male line; while silent influence is on the whole more appropriate to the female line’.Footnote ⁸⁰ Both attributes were essential to ensure social stability; ‘they must take counsel together … [for] there is nothing more sterilizing and disastrous than premature attempts at fertilization’.Footnote ⁸¹

As one acquainted with her later observed, the conservative position Mary adopted regarding suffrage was predicated on a belief that women are particularly well suited to bringing forth ‘the new truth that is usually waiting to be born of polar opposite opinions’.Footnote ⁸² At the same time, one can look to her work at Queen's College, London, to her commitment to tutoring women in mathematics and to her essays directed specifically at schoolgirls for evidence that her attitudes on female education, although perhaps separatist, were considerably more progressive. Consequently she can be counted among those identified in Julia Bush's work of historical reclamation, which examines late Victorian women who promoted higher educational opportunities for their sex, yet failed to embrace suffrage. Even so, Mary stands apart from many of the women Bush considers, since her opposition to suffrage was founded primarily on the significance she attached to the new logic of Boole's method and its relation to mental hygiene as opposed to appeals to, among other things, domestic stability and status, explicitly Christian sensibilities, or imperialistic imperatives.Footnote ⁸³ Moreover, she was not suggesting that embroilment in suffrage represented a direct threat to the nervous dispositions of women. Rather, anything that might encourage women to forsake the duty associated with their own special mental strength would likely compromise the health and well-being of society. Ultimately Mary amplified her agency in imploring women to fulfil their obligations to the law of pulsation by deftly combining her roles as both messenger and model practitioner.

Mathematical knowledge

The functionalist inclinations and anthropological perspectives that shape her commentary on ethical stability and the women's question also pervade much of Mary's discourse on the nature of mathematical reasoning. Central to the promulgation of her own ideas in relation to George's legacy, this theme explicitly emerges in essays already considered, specifically the Mathematical Psychology of Gratry and Boole and Preparation of the Child for Science, as well as others, including Boole's Psychology as a Factor in Education (1901), Lectures on the Logic of Arithmetic (1903) and Philosophy and Fun of Algebra (1907). In terms of explaining what mathematics is and is not, Mary consistently argued from a core belief that, although typically applied to basic concepts, suggests the true nature of the discipline. Specifically she maintained that ‘arithmetic corresponds from its very origin to properties which are distinctly human, not numerical. Its processes are, from the first, anthropomorphic, not, as is commonly assumed, purely abstract’.Footnote ⁸⁴ Adoption of this position allowed her to make several points, not the least of which involved aligning certain aspects of mathematics with other scientific enterprises. For example, essentially all scientific investigations are enhanced by the use of ‘implements and methods … of human origin’, such as microscopes; in this mathematics is no different. Its processes, Mary asserted, ‘are well adapted to meet the limitations of our direct faculties, and to assist us in attaining knowledge unattainable by our unaided powers’.Footnote ⁸⁵ By pointing out such commonality she undertook to correct a ‘popular but erroneous conception of mathematics as a science differentiated from most other sciences by being purely abstract, non-human, out of line with vital processes’. Others at about this time proclaimed the kinship of mathematics with the natural sciences by drawing attention to the influence of empiricism on the production of its knowledge. Mary's commentary lays claim to a more expansive role for human agency: mathematical processes are tailored to ‘meet the needs of the human mind in its effort to understand what is beyond its comprehension’.Footnote ⁸⁶

In exposing its more mundane roots, Mary did not mean to deny mathematics special, transcendent attributes. Although the impetus for its methods might well reflect mental limitations, its reasoning makes use of extra-logical powers. It is by appealing to inductive, non-syllogistic modes of thought that one is able to both extend calculations to contexts previously unimaginable and grasp the validity of arithmetic principles, such as the commutative property of multiplication as applied to natural numbers, by reflecting on a few computations rather than relying on repeated validations or authoritative prescriptions. This aspect of its knowledge also means that one is less likely to be led astray owing to the fallibility of common sense or hindered by hidden facts when dealing with mathematical concepts and processes. Some people may not be able to provide an immediate or correct response to a mathematical question, but this is due only to their unfamiliarity with the required techniques. Once appreciated, nothing would prevent them from arriving at the appropriate conclusion.Footnote ⁸⁷

Closely related to the insularity of mathematical practice from the influence of hidden facts is the ‘power of self-correction which the mind possesses in relation to it’. In this regard Mary acknowledged that ‘there seems to be some mysterious court of arbitration within man’ that is essential to detecting and adjudicating disagreements without necessarily appealing to any external authority.Footnote ⁸⁸ Mary claimed that it is a myth to attribute this undeniable feature of mathematical reason to ‘something in the nature of the subjects commonly treated mathematically’, whether number or quantity. If this were so, then how could one account for the fact that we are as susceptible to computational errors as we are limited in our apprehension of large numbers? ‘This weakness of our power of direct cognition with regard to numbers’, she pointed out, ‘may be partly due to lack of exercise … [and it] might be possible to possible to strengthen it by practice. But at least it leaves … no excuse for asserting that numbers, as such, are easier for us to [re]cognize with certainty than facts of other kinds.’Footnote ⁸⁹ This, of course, leaves open the question of whence the power of self-correction. Mary offered no opinion on this, but suggested that it might be philosophically sound to consider mathematics, as a developed discipline, ‘a science not of number and quantity, but of conditions under which man can make his progress towards unknown Truth uniform and safe, and can preserve himself from being seriously misled by the mistakes which he is sure to make on the way’. Perhaps, she conjectured, mathematics is most appropriately considered the science of ‘sane inspiration’.Footnote ⁹⁰

Uncoupling readers' perceptions of mathematics from immediate associations with number and quantity was crucial to Mary's mission. Indeed, any hope of introducing or explaining the nature of mathematical psychology to her audience required that they appreciate the possibility, perhaps even more than the potential, of algebraically formulating the laws of thought. Consequently, she endeavoured to correct what she considered to be two fundamental myths regarding arithmetic and, by extension, algebra: first, that its subject matter is relegated to questions of number and quantity, and second, that its processes exclusively reflect laws associated with these concepts. To address the former she proffered several practices selected to expose the misapprehension that number dictated the scope of arithmetic considerations. In one example, she discussed the rules applied to computations involving negative numbers in terms of the broader conception of ‘balancing … mutually neutralizing operations’, a process she felt was at least equally well appreciated in contexts defined by social interactions.Footnote ⁹¹ The concept of sacrifice also played a role in arithmetic computation, for the ‘whole process of numeration depends on and expresses the power and the need of the human mind to merge a group of its ideal units of one stage into a single higher unit’.Footnote ⁹² The five examples she provided all highlight social or psychological processes. These therefore simultaneously serve to dispel the other identified myth.Footnote ⁹³ By offering these she was also able to reinforce her underlying belief that mathematics reflects dynamics indelibly marked by human actions and emotions.

In his efforts to mathematize the laws of correct reasoning George anticipated, to a considerable degree, foundational questions that would occupy many logicians, mathematicians and philosophers for years. Simply posed, these ask, ‘what constitutes mathematics?’ and ‘how does one account for the certainty of its knowledge?’Footnote ⁹⁴ Psychologism, the belief that psychological experiences provide a framework for explaining the laws of logic and/or mathematics, has often been the focus of discussions of George's philosophical temperament.Footnote ⁹⁵ Nevertheless, his Laws of Thought contains relatively few passages that explicitly address the nature of mathematical knowledge. He certainly reminded readers that such knowledge is not conceptually bound to notions of number. Further, he ended his exposition with the claim that mathematics, while a fundamental component of it, did not exhaust the full scope of human knowledge.Footnote ⁹⁶ In developing her discourses on mathematics, it is not to be suggested that Mary wielded any influence either directly or indirectly on foundational debates. It is, however, undeniable that she was aware of them and willing to present somewhat maverick, psychologistic views to general audiences that would likely have had little exposure to such ideas. Furthermore, through this engagement, she once again fulfilled a function intimately related to Boole's method by providing an ideological inversion of George's sentiments: while he maintained that mathematics is an essential aspect of human knowledge, Mary asserted that humanistic imperatives formed the essential core of mathematical knowledge. In this she anticipated aspects of the constructivist epistemological models later espoused by theorists such as Jean Piaget.Footnote ⁹⁷

Scientific enterprises, practices and authority

Mary made strategic use of her mission to disseminate and interpret the Laws of Thought as a way of giving voice to opinions she held regarding scientific enterprises other than mathematics. While various works contain evidence of her thinking, addresses she delivered in the 1890s are of particular interest. Her most obvious target at the time was the practice of vivisection in biological research and medical training, but the concerns she addressed related to all sciences. The views she espoused undoubtedly reached a smaller audience than some of her published works. Nevertheless they provide the clearest insights into the way Mary formulated her attitudes regarding the professionalization and specialization of science in the context of the law of pulsation. Further, her commentary suggests that she developed her own explanation to account for the fact that George's system, either on its own or in any broader context, was largely ignored by his peers.Footnote ⁹⁸ In rendering and presenting her views on these matters she critically challenged as much as effectively negotiated parameters then typically associated with scientific authority.

Before it offers specific objections to vivisection, Mary's critique begins by acknowledging the impotency of arguments framed around an outdated misconception that moral objections could significantly alter the course or pace of scientific progress.Footnote ⁹⁹ She believed that those inclined to prioritize moral righteousness above intellectual progress also risked underestimating the useful knowledge that biological and medical research might generate.Footnote ¹⁰⁰ Abandoning spiritual rhetoric, she chose instead to frame her objections in terms that scientists claimed as an essential part of their remit: the pursuit of ‘intellectual truth’. Specifically, Mary was able to position herself within this discourse by arguing that anyone who professed such aims must be obliged to learn the laws of thought and practise the doctrine of pulsation. That scientists had largely failed in this duty was evident in the problem of vivisection, which she described as the ‘eruption … of a disease … infecting all of the intellectual and educational life of the world’.Footnote ¹⁰¹ Mary's argument ultimately defines the disease pervading science by symptoms directly related to the improper employment of the law of pulsation, or indeed ignorance of its primacy to logical thought.

Vivisection in biological and medical research provided Mary with a particularly rich context in which to address the risks inherent in the pervasive disregard of good mental hygiene in the sciences. She maintained that its experimental emphasis on dissection and differentiation made excessive demands on the analytic phase of pulsation to the detriment of its synthetic complement. Such predominance suggested that balance must be regained. ‘We might safely give any man leave to do as much vivisection as he cared to do’, she wrote,

This prescription, however, was not meant to be limited to the sphere of a scientist's particular subject matter, where synthetic reassessment after analytic reduction was taken as a methodological requirement.Footnote ¹⁰³ Indeed, she intended that the act of pulsation go further in attempting to bridge professional practices with those outside any special study. To this end vivisectionists should regularly attempt to put themselves or family members in place of their subjects as a means of fostering the mental elasticity that sanity required.

Mary highlighted other anxieties associated with the continued violations of the law of pulsation in biology and medicine. For example, she was especially concerned that educational programmes structured around vivisection would produce new generations of researchers who would accept the practice as an absolute necessity for research. She was willing to concede that some might profess a preference for vivisection as part of their research. But to claim it was the only means possible for attaining useful knowledge indicated both an unsoundness of mind and an ignorance of intellectual history.Footnote ¹⁰⁴ Furthermore, she worried that a dependence on vivisection in medical schools would only reinforce weaknesses in students with less-developed synthetic faculties, arguing that such deficiencies require less, not more, attention being paid to observational abilities.Footnote ¹⁰⁵ What, she mused, of the student who ‘cannot learn by synthesis, from some facts – if he must see all the facts with bodily eyes, how can he diagnose in the case of a living patient?’Footnote ¹⁰⁶ Ultimately, her concerns extended to all medical practitioners who, while typically ignorant of regimes of sound mental hygiene themselves, were allowed to commit others to institutions and asylums.Footnote ¹⁰⁷

An additional aspect of the general scientific malaise that troubled Mary is notable inasmuch as it shifts focus from particular elements of George's contributions to logic. Instead it draws attention to her efforts at expanding the contexts – whether metaphysical, historical or disciplinary – in which they might be appreciated. A central problem with science, she asserted, manifests itself in

According to the narrative developed throughout her Collected Works, the Laws of Thought at once represented a hallmark in a lifelong quest and occupied a special place in an intellectual lineage that could claim affinity with, among other things, Jewish religious tradition, the theology of Auguste Gratry and the spiritual influences of her father. Additionally, Mary often referred to the support and encouragement George received through his long correspondence with Augustus De Morgan, whom she portrayed as a faithful collaborator and someone appreciative of the full extent of her husband's undertaking.Footnote ¹⁰⁹ She was, however, critical of those who later turned their attention to his work, most notably William Stanley Jevons and John Venn. While acknowledging their improvements to cumbersome notation, she particularly chided Jevons for focusing his considerable talents on specific applications of George's innovation, yet failing to apprehend, and thereby disregarding, the true scope of its power and potential.Footnote ¹¹⁰ Indeed, she once likened the reception of the Laws of Thought among the British intelligentsia to the magnanimous gift of a steam hammer being exclusively employed for the menial task of cracking nuts.Footnote ¹¹¹ The pervasiveness of such blinkered and specialized mindsets not only impeded the recognition she felt her husband was due; according to Mary, these also corrupted the true spirit of scientific investigations. In making this point she sought to (re-)establish her authority in promoting George's ideas.

Mary's commentary on vivisection merges lifelong engagements with health and medical practitioners, which can be traced to the Hahnemannian influences on her childhood, with her later preoccupation with interpreting the laws of thought. Consequently, she undoubtedly considered herself well suited to formulating a critique that attempts to elevate discourses on biological research, in particular, and scientific enterprises, in general, above mere moralistic sermonizing. Indeed, Mary intended to raise the tenor of her contribution by harnessing the new logic encapsulated by the law of pulsation. Certainly hers was an antagonistic voice from outside an increasingly distinctive and persistently male scientific community, but it encouraged harmonizing attitudes even as it highlighted differences of opinion. As such, the Law of Pulsation served to legitimate as much as define her intercession. Another interesting aspect of Mary's lectures on vivisection is her observation regarding the self-gratifying pleasure she claimed scientists took in their particular researches. Indeed, the lustful and sensual terms employed when describing this symptom of the scientific malaise she sought to expose foreshadow a Foucauldian analysis of the medicalization of sexuality.Footnote ¹¹² Finally, apropos of Kathryn Neeley's scholarship, if the notion of a distinctly illuminated female mind provides a meaningful analytic framework for historicizing Victorian women who cultivated scientific enterprises, then Mary certainly represents a worthy subject insofar as she herself maintained that women's mental faculties are particularly attuned to accord rather than conflict. Nevertheless, her comments on selfishness in science illustrate one of the many paradoxes that typically complicate any such gendered perspective, for there she used the intellectual (masculine) platform of the laws of thought to expose and denounce emotional (feminine) motives that undermined scientific research.

Conclusion

In addition to the topics discussed heretofore, Mary's efforts at propagating and fulfilling her husband's legacy extended to commentary on Judaism, Eastern mysticism, spiritualistic mediumship, imperialism and economics. This is an undeniably diverse array of topics. Yet her intellectual mission of framing and disseminating as much as acting in accordance with the law of pulsation was a commitment to fostering new ideas and influencing praxis through the proliferation of associations. It is clear that she also envisioned her continuing engagement with George's contributions to logic as transcending the role of a mere biographer; hers are collaborative narratives. She writes as one who knowingly and tactically occupied a privileged position from which she developed her own ideas by contextualizing her husband's innovative work, embedding it within a rich intellectual history, demonstrating its relevance to a variety of discourses, and articulating its import to audiences outside of purely academic circles.

The constancy of her dedication to the law of pulsation represents the most obvious point of ideological commonality amongst what can otherwise seem a disparate set of attitudes and engagements. It gave shape to her progressive opinions on the nurture of the unconscious mind, on the role of psychology in educational reform and on the significance of human activity to mathematical reasoning as much as it effectively underpinned her conservative position on suffrage. Further, in commenting on the social and intellectual obligations of women, Mary amplified her agency by practising what she preached. She notably exercised the special, complementary mental strength she attributed to women by her gendered presentation of the law of pulsation in several other contexts. This is, for example, evident in the critique she mounted against vivisection and the male preserve of science, in which she suggested that such practices must harmonize their efforts with the laws of correct reasoning. Ultimately, identifying some of the themes, motives and strategies around which she organized her many efforts to promote the laws of thought enables us to reconsider her interventions – like those undertaken by other Victorian women who sought to cultivate scientific enterprises – within a contextual framework expansive enough to include purposeful responses to cultural realities, expectations and possibilities.

Mary often declared that her husband George gave wings to logic. Yet it is she who, in her own ways, devoted herself to promulgating ideas that spoke to their continued appreciation and vitality.