Mathematics without Apologies: Portrait of a Problematic Vocation
In 1940, at a time when war led many to introspection, G. H. Hardy wrote A Mathematician’s Apology to reflect upon and to justify his work in mathematics. A classic for the field, the essay provided a glimpse into the mind and motivations of a mathematician. To Hardy, mathematics was a creative art using the world of ideas, and beauty was the primary test of good mathematics. The pacifist Hardy proposed that mathematics should be pursued for its own sake rather than for its potential, and perhaps destructive, applications.
Seventy-five years later, Michael Harris has attempted to provide an updated version of the research mathematician’s life in Mathematics without Apologies: Portrait of a Problematic Vocation. Rather than Hardy’s ethical standards of moral realism, Harris takes an eclectic approach by taking into account social and cultural influences. He does not argue systematically from first principles, and he states up front that he will not arrive at definite conclusions. He does not claim to be objective, and he uses his personal story to inform his description of the actions and practices of the profession rather than focusing on epistemology.
Some in other disciplines may wonder what the fuss is in taking such an approach. After all, these conversations have been taking place for decades. The discipline of mathematics comes from a tradition of logic and deductive reasoning heavily influenced by Platonism. There have always been books containing mathematics and books about the mathematical process, but they have often been foundational in nature. This book appeals to contemporary academic thought and culture. It is engaging and provocative, challenging and deconstructing the usual narratives of the discipline. Harris describes mathematical research as not so much about solving problems but as challenging perspectives and raising more questions. And his book surely does so.
For example, Harris is not overjoyed with the idealized constructs of philosophers and sociologists whose claims are divorced from how mathematics is actually done. He argues they often ignore or misrepresent the field of mathematics as an afterthought to the scientific process. The philosophy of mathematics is often grounded in addressing truth claims presupposing an outsized concern for logical formalization, which matters little to research mathematicians. In other words, the norms of mathematical practice do not line up with philosophical norms.
Be that as it may, the problem of how to communicate that mathematical practice remains. While applied mathematicians examine problems arising from reality, pure mathematicians are not bound by external forces, but by curiosity, exploration, and discovery. Because pure mathematicians deal with abstract concepts in a virtual world and sometimes need to create language to describe the work, they have trouble explaining to their colleagues in other disciplines what they do and why they do it. Harris tries to accommodate by providing a layered picture of his own experiences along with the views of other contemporary mathematicians. He also valiantly endeavors to explain his own field of number theory in an approachable manner. Titled “How to Explain Number Theory at a Dinner Party” and numbered with Greek letters, the 4.5 chapters (yes 4.5) include a series of witty dialogues between a mathematician and a performance artist. Readers will still not understand number theory, but Harris’s playful approach gently introduces the motivations and issues surrounding the mathematical process.
One issue is whether or not pure mathematics is warranted. Mathematicians, as well as those in other academic disciplines, are having to defend their work and their professional autonomy. Rather than war, as in Hardy’s time, dwindling university budgets and government demands for economic output are calling for such justifications. Harris refuses to develop his own apology—hence another allusion to the book’s title. Before justifying why mathematics ought to be done, we need to understand what is being done. That is, what motivates a pure mathematician? Instead of providing an idealized version argued from first principles, Harris deconstructs each of the usual platonic justifications given for mathematical research: its potential for applications, its relationship with truth, and its beauty.
Harris calls the potential for applications the “golden goose” argument. The usual story is that throughout history abstract mathematical ideas have turned out to have important applications. For example, mathematicians often point to Hardy himself for the unpredictable benefits. Hardy ended his essay claiming that he had never done anything useful, but, ironically, Hardy’s area of number theory is essential to electronic commerce today. Harris suggests these claims of potential benefits are overblown. Wary of privilege and sensitive to power, Harris claims the term “useful” “provides a marker to divide its users according to priorities that differ radically, depending on the positions they have chosen to occupy, or that have been chosen for them, in the social panorama” (279). The question becomes not just what are those few benefits, but who benefits and in what ways. Furthermore, it “is not only dishonest but also self-defeating to pretend that research in pure mathematics is motivated by potential applications” (56). Not only are pure mathematicians not oriented in this way, but to focus on potential applications hampers research, hinders motivation, and causes ethical dilemmas. Harris includes an entire chapter critiquing the 2008 financial meltdown rocked by misunderstandings of the mathematical models, unethical behavior, or both. While departments and mathematicians may have received additional funding for research into mathematical finance, the Faustian bargain came with compromises of autonomy and ethics.
For the second justification of truth, mathematical proofs are becoming increasingly elaborate to be understood fully by an individual mathematician. Harris advises that we should get out of the habit of assuming that mathematics is about being rational as understood by the philosophers of mathematics. Whereas foundational questions regarding logic and ultimate truth are part of the long mathematical tradition, Harris sees foundations as a metaphor and not amenable to mathematical concepts. Foundational questions are a nuisance and a restraint to the imagination. While mathematical proof is the discipline’s standard, proof is not the culmination of intuition, but rather a confirmation thereof.
Third, mathematicians, following Hardy, often compare their work to art and its beauty. What Harris finds ironic is that many contemporary artists are not concerned with beauty. Moreover, while mathematicians may insist they are artists, most artists want to keep their distance from associations with formulas and equations. Meanwhile, only specialists can comprehend contemporary mathematics, let alone make any aesthetic judgments. Harris proposes that when mathematicians refer to beauty, they really mean a particular type of pleasure that comes in discovering mathematical results. Lacking the language to describe the nuances of the experience, mathematicians fall back on the term “beauty.”
If utility, truth, and beauty are not the justifications, what is? This is perhaps the key question of the book with no simple answer provided. In parts, Harris reverts to a form of Romantic Existentialism, in which the primary motivation of the mathematician is the sense of curiosity, wonder, and joy of discovery in the virtual world of mathematics and not the inauthentic narratives usually given. Given the intentions, values, and emotional investment needed to undertake such a challenging endeavor, the mathematician becomes a type of Romantic— not as a lover or as an expresser of emotion, but as one marked by imagination, freedom, and commitment despite all hurdles. In the past, this romantic ideal has been encouraged by many popular mathematical biographies such as E. T. Bell’s Men of Mathematics. Perhaps because it is a declarative statement wrought with linguistic baggage dangerously close to hyperbole, Harris is resistant to state firmly the characterization of the Romantic. Instead of using biography, he uses the perceptions of others outside of mathematics, particularly those in the arts, to depict the characterization.
The chapter “Further Investigations of the Mind-Body Problem” begins with the question of whether mathematicians see themselves in cinematic images since playwrights and screenwriters have increasingly featured mathematicians in such films such as A Beautiful Mind and Pi and plays such as Proof and Arcadia. Like Archimedes running naked through Syracuse shouting “Eureka!,” mathematicians have traditionally been portrayed as misfit geniuses who suffer madness or martyrdom. They are characters who enter into the mathematical world within the mind oblivious to the material world of the body. To Harris, the question is not whether mathematicians are portrayed as deranged (since madness is a common plot device), but in what sense their torment is characteristically mathematical. From this starting point, the chapter incorporates the arts, philosophy, and theology as it takes on the metaphysical mind-body problem through the lens of mathematics.
The chapter frames this exploration through the short film Rites of Love and Math. Wanting to counter the stereotypes of mad mathematicians, California Berkeley mathematician Edward Frenkel co-directed, co-wrote, and starred in the short film which paid homage to the Japanese film Yukoku. In Frenkel’s erotic film, evildoers pursue a mathematician for his mathematical equation for love which can bring great happiness to the world or be used as a powerful weapon of control. In the end, the character realizes his imminent death and tattoos the formula onto his lover so that his formula and their love can continue. Through metaphor, the mathematician’s frenzied and passionate search for a formula becomes like the creative process of creating a poem or a piece of music. To co-writer and co-director Reine Graves, the movie upholds mathematics as one of the last areas where there is genuine passion not sullied by economics, unlike contemporary art and cinema. From here Harris weaves several artistic and literary examples connecting the movie to art and to whether one must lose one’s mind while one’s body remains intact. He includes his own structuralist double binary opposition scheme to illustrate how the wider culture portrays mathematical madness with regards to the material and mental world while opposing material or mental powers.
Notably, Harris points to Jesus Christ as the survivor of the spirit despite martyrdom in the flesh. Harris rejects mathematics as a religious experience even metaphorically, but he recognizes discussions about the value of mathematics borrow heavily from religious discourse. Doing mathematics produces a feeling of transcendence which cannot be described with a formula, but is perhaps analogous to love or infinity in representing what is incomprehensible to the human mind. Harris points to Loren Graham’s book, Naming Infinity, and to the Russian Orthodox theologians who see love, knowledge, and truth as essentially personal. The act of knowledge is a relationship, a kind of friendship, between the one who studies and the one who is studied. Just as in the doctrine of name worshipping—repeating the names of God brings one into His presence—naming the mathematical objects brings the mathematics into existence. To translate into Russian the word “truth” in Jesus’s phrase, “I am truth,” one can use istina, which carries the sense of genuineness, or the word pravda, which deals with law, justice, and rules. Harris notes that the Russian Orthodox theologians used istina, as did Frenkel’s mathematician character in Rites. In closing the chapter on mind/body on the mad/martyred/mathematicians in cinema, Harris states, “Our readiness to sacrifice our minds and bodies to our vocation is the ultimate proof that what we are doing is important” (173). But he immediately undercuts the statement’s grandiosity with an ironizing dialogue on the next page.
Chapter 7, “The Habit of Clinging to Ultimate Ground,” is the most philosophical and, by extension, the most theological chapter in the book. The chapter integrates highly specialized and formal areas of mathematical research with Western and Eastern philosophies. Exploratory in nature without foundational first principles, the chapter is focused more on raising issues than on providing pat answers. For example, how do we come to trust mathematics? Referencing Descartes, Anselm, and ontological arguments for the existence of God, Harris makes metaphysical and metaphorical comparisons between theology and mathematics to explore the nature of mathematics, its language, and its trustworthiness. Four of the five section headings are “Incarnation,” “Everything,” “Existence,” and “Second Life,” which might lead someone to conclude (erroneously) that this chapter is slanted toward questions arising from Christian history, but Harris counterbalances these Christian allusions with contributions from Eastern philosophers.
In Harris’s account, mathematics is still going through a paradigm shift since the first part of the twentieth century when the foundations of the field were shaken off of their moorings in relation to concepts of truth, meaning, object, and existence. His interpretation is that the shift was not so much about the epistemological foundations, but rather in orientation arising from category theory and other mathematical frameworks. Whereas foundations of mathematics can mean the first principles or the entire edifice, Harris would rather look toward the future without any concern about the fruitlessness of the quest for ultimate truth. Instead of worrying about developing an airtight logical structure from first principles, researchers should look forward toward the unfolding of problems. The most important and persistent questions about the nature of mathematics are rooted in aesthetics and ethics, not epistemology. He cites the mathematician Vladimir Alexandrovich Voevodsky, whose Univalent Foundations proposes computer verification of mathematical reasoning. Voevodsky has speculated that perhaps mathematics could borrow an idea from the field of artificial intelligence, in which expert systems are designed to take in contradictory input and use them as guides to concrete actions rather than submit to conventional logic.
If the real interest of mathematicians is not with the ultimate truth of mathematics but with its relations and connections, how does one solve or even formulate a problem when the terms or language to describe the problem have yet to be invented? The challenge becomes developing a heuristic to overcome linguistic difficulties, often through metaphors to other mathematical objects. Research mathematicians employ these metaphors but then have difficulty rigorously proving a result whose description is in transition, eluding precise definitions. To try to capture the elusiveness of research, Harris employs the word “avatar.” For mathematicians to make progress, they need to play a psychological trick on themselves to believe that the objects, although tenuous, are being discovered and exist independently. The mathematical research process includes a rhetoric of trying to enhance vividness of the ideas so that research becomes not as much about solving problems as about developing heuristics to reformulate problems to make solutions appear obvious.
Harris raises the possibility of a model which integrates Indian philosophy with the mathematical formalism of category theory. Category theory studies the structures of formal, generalized mathematical objects and the mappings between these collections of abstract objects. It is difficult to communicate the theory and how highly advanced mathematics makes these relations, but a more generalized category may give a better way to think about a particular mathematical problem. As a greater number of formal structures are evolving with a greater number of relationships, mathematicians are climbing a ladder with broadening convictions that the process will never end. Influenced by mathematician Alexander Groethendieck’s relative point of view, Harris’s assessment of his specialized mathematical research area is that “Knowing a mathematical object is tantamount to knowing its relations to all other objects of the same kind” (195). When looking at a mathematical avatar, one does not look back toward a concept’s underlying principles, but looks to a higher category whose reflections are the familiar structures one seeks to understand. In a paraphrase of the mathematician André Weil, the obscure analogy of an avatar allows one to reach knowledge and indifference simultaneously, as the Bhagavad-Gita teaches. Harris states, “If you were to ask for a single characteristic of contemporary mathematics that cries out for philosophical analysis, I would advise you to practice climbing the categorical and avatar ladders in search of meaning, rather than search for solid Foundations” (218). One of the book’s main challenges to philosophers is that they should recognize that it matters to mathematicians what they think their work is about even though it may not matter to the work itself. The formal mathematical definition is only one part of a mathematical concept. It also includes a mathematician’s intentions or senses of the underlying theory, which may not exist, so that the mathematician’s role is “to create the source of the shadows they have already seen on the wall of Plato’s cave” (187).
Chapter 10 revisits Hardy’s A Mathematician’s Apology as a mirror image of Harris reflecting back on his life and ideas. By way of contrast, G. H. Hardy firmly believed in Platonism, that mathematical reality lies outside of us, a view held by many since Plato. The mathematician’s function is to discover or observe that reality, and human “creations” are simply our notes on our observations. In rereading Hardy, Harris was struck by the indoctrination persisting today in which washed-up mathematicians spend time in administration, philosophizing, writing, and exploring math history rather than their “proper” job. Hardy claimed in his essay that mathematics was a young man’s game. Even though he had produced top-notch mathematical results in his time, the aging and melancholy Hardy wrote that he was rightly to be pitied for writing “about” mathematics. Harris characterizes such statements as elitist, but goes on to find elements of support for this Romanticism.
Hardy’s essay is often disparaged or lauded for its math-for-math’s-sake stance. Harris provides context for its mathematical aesthetics by tracing beauty as an ethical principle to Cambridge and the time of philosopher G. E. Moore and the Bloomsbury Group. Moore made it possible to find an ethical justification for something as useless as pure mathematics, provided it could be framed in terms of aesthetic enjoyment. Thus Hardy could famously write, “The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.”1 Furthermore, early twentieth-century art critic Roger Fry saw that there was more than the truth of art; there was also the fundamental pleasure in doing art. Fry critiqued as puritanical the view that the life of the imagination was no better or worse than the life of sensual pleasure and thus was rejected by others. He stated that art does not have moral responsibility of action, but frees the artist from the binding necessities of his actual existence. Against this backdrop, Hardy could articulate the aesthetics of mathematics while Harris can build upon the aesthetic enjoyment of mathematics as an ethical principle.
Influenced by Robert Bellah, Harris proposes a vision of the good mathematical life. Mathematics is a relaxed field, not subject to the pressures of material gain and productivity. It values contributions to a coherent and meaningful tradition, carrying on a dialogue with human history and past achievements with an orientation to the future. Mathematical tradition exhibits universality, as it welcomes contributions from all nationalities. At a fundamental level, mathematical research is playfulness pursuing pleasure. Mathematics should resist certain economic demands, in that such a life moves away from the pursuit of pleasure to a stressed life.
Critique and Summary
Harris’s Romantic ideal is appealing. It provides an individual authenticity and seeming purpose, while still allowing individuality and freedom. One’s imagination can roam free without being bound to the necessities of existence. It is a life which emphasizes dignity over importance. Unfortunately, it ultimately lacks purpose and responsibility. It may be part of human flourishing, but Harris may fall prey to the same critiques he makes of philosophers. The actual practice of mathematicians may not line up with his claims. Being human, mathematicians’ desires often go beyond noble ones.
This review has only scratched the surface of the book’s contents. Harris introduces a variety of characters, ideas, and metaphors to unpack. There is literary criticism, pondering the role of computer proofs, the mathematician as trickster, and much more. The book does not have to be read cover to cover; one can pick out individual chapters to read without losing much. While the mathematics is beyond reach, the wittiness of Harris’s writing, along with the devotion and joy, may not be. The dizzying number of cultural references to mathematics might be surprising to some. After all, it is rare to find a mathematics book with 42 pages of footnotes and 12 pages of bibliography but with few equations.
For those in the humanities, Harris’s take may provide fresh insights into the creative process and aesthetics. Hardy’s Apology has attracted a few artists for its appeal to aesthetics and creativity. Harris’s No Apology turns the table and makes an appeal to artists for their perspectives on mathematics. The interaction of a seemingly unrelated discipline to mathematics may indeed unearth new ideas about both.
For the mathematician, the book is recommended reading if only to move beyond a Modernist mindset. Mathematicians often raise problems to the community for others to solve. The problem raised in the book is the justification of mathematics if the old reasons have not kept pace with current mathematical practice and academic thought. Harris has taken stock of outsiders looking in at mathematical culture. While Harris may not provide satisfying answers, he does reveal the challenging questions being asked today.