Extract from a paper written for a graduate course on Scholarly Editing & Textual Criticism at the University of Virginia, Spring 2015. Other sections include a critical edition of a chapter of John Maynard Keynes’s General Theory
I. THE APPLICATION OF TEXTUAL CRITICISM TO MATHEMATICS
“Mathematics may be defined as the subject where we never know what we are talking about, nor whether what we are saying is true”
Bertrand Russell, “Recent Work on the Principles of Mathematics”
“Textual criticism is not a branch of mathematics, nor indeed an exact science at all. It deals with a matter not rigid and constant, like lines and numbers, but fluid and variable; namely the frailties and aberrations of the human mind … It therefore is not susceptible of hard-and-fast rules”
A.E. Housman, “The Application of Thought to Textual Criticism”
As a discipline in the humanities, textual criticism exists in a singular category: it is a distinct field of study, but its fundamental aims and methods of analysis are far-reaching, applicable to texts of not only literature, but also religious studies, history, and sociology, to name a few. In a similar way, mathematics occupies a unique place in the sciences. It is a well-defined subject, but its methods are relevant to most, if not all, scientific fields. There is, therefore, a parallel of sorts between mathematics and textual criticism. But between textual criticism and mathematics, there is a deep chasm, representative of a still all-too present, broader divide between the sciences and the humanities, expressed a half-century ago by C.P. Snow in his buzzy and oft-quoted catchphrase of the “two cultures.”
And yet, if we consider textual criticism as a general method of analysis for analyzing texts, and if we further consider that mathematicians produce texts and documents, physical manifestations of “works,” then it becomes evident that it (textual criticism) is relevant indeed to mathematics. Written literary texts use language; our representations of language, an abstract concept, are based on an entirely arbitrary system of signs, or “words.” Mathematics is based on its own, equally arbitrary system of representation, signs such as “numbers” and “formulas.” It is, in addition, mediated by language in texts that are invariably subject to the “usual” textual problems.
In this short essay, I argue for the application of textual principles to works that have for the most part fallen outside its scope, namely those pertaining to mathematics. I distinguish between categories of mathematics: “pure” mathematics, “applied” mathematics, and “appropriated” mathematics, which all use the same “language of representation.” The goal of this essay is to begin to develop a theory that argues that the broadest aims and principles of textual criticism can, and should, be applied to mathematics. The argument, broadly, is as follows:
Because mathematics is:
1) Composed of abstract concepts that make up intangible “works”
2) that are represented by arbitrary systems of signs that make up physical documents and texts
3) mediated, and thus dependent on, but existing independently of, language,
the principles of textual criticism are relevant to it.
Each point above will be treated separately below.
I. Mathematics is composed of abstract concepts that make up intangible works
Mathematics is a discipline shrouded in a particular mystique, a coinciding of scientific rigor and a strain of surprising, almost romantic, idealism. The great mathematician Carl Gauss, who was admittedly not without bias, is said to have called it the “queen of the sciences.” This is not only because it is, in its “purest” form, extremely difficult, but also because it is seen as a vehicle for arriving at some sort of elemental truth, apart from, and superseding, the limitations and subjectivity of language and human experience. Bertrand Russell, one of the early twentieth century’s preeminent English scholars, who was a logician, philosopher, and mathematician, stated that mathematics, “rightly viewed,” was not only a source of truth, but also “supreme beauty – a beauty cold and austere, like that of sculpture.”
And yet, mathematicians are the sources, or “authors,” of the theorems and proofs in math, which are ultimately human constructions, existing in physical form in textbooks, conference presentations, and published papers. Mathematicians, particularly those working in pure mathematics, may argue that they are working with something “larger”; that they are “discovering” and relating that which exists and has always existed in time or space. While this view is disputable, its validity is ultimately irrelevant to the argument – whether we view mathematicians as discoverers or creators, we can understand “works” in mathematics to mean human creations in historical moments, not unlike thoughts, allowing us to trace how specific individuals have attempted to express at one point in time things essentially abstract. For our purposes, we, in turn, will view works of mathematics in the context of intellectual history: as authored, historical products.
It is useful, at this point, to step back for a moment and distinguish between the branches of mathematics already alluded to in this essay. At the risk of oversimplifying scandalously, mathematics is divided into two sub-fields: pure and applied. I add an additional category, “appropriated” math, which arguably falls outside the canon of traditionally established mathematics. Pure mathematics, which is intuitively the most relevant to our discussion of mathematical works thus far, is a deductive undertaking, almost wholly abstract. Theorems are proven based on established propositions; works exist as entire systems within themselves. Straying from the most pure forms of mathematics – analysis, topology, etc. – we enter applications of mathematics that no longer qualify as “discoveries,” but rather innovations. Applied mathematics is inductive and empirical; for instance, relationships are posited between variables, and these are quantified or validated based on scientific experimentation and observation. Although there is, very broadly speaking, a fairly rigid division between these two branches of mathematics, it is often the case that the findings of “pure” or “abstract” mathematics are found to have remarkable applications to “real world” problems; the division is not always, in theory and practice, so clear cut.
The third category, “appropriated mathematics,” I will use to refer to mathematical methods and concepts applied loosely, generally in or by other disciplines or non-mathematicians. While employing the language of mathematics, and its systems of representation, works in this category of mathematics would generally not be considered rigorous enough to be categorized as “real” math. They follow mathematical rules, but also may fudge them somewhat. For example, microeconomists working in “social choice theory” might present a well-ordered proof that is self-contained, mathematically correct, and based on “axioms” or “propositions,” but does not map to the real world exactly. In other words, it may seek to describe and predict some facet of human behavior, but in absolute terms, simplifying in order to subject humans to “hard-and-fast” rules; to achieve this, it must make “assumptions” (presented as “axioms”) about human behavior that cannot apply universally. In pure mathematics, a single counter-example to a theorem, axiom, or proposition renders it invalid (Velleman 85); thus a “proof” in social choice theory is “valid” only in some theory-sphere, where human behavior is absolutely predictable. In macroeconomics, the study of the economy as a whole, functions are frequently used to describe, theoretically, the economy in its entirety. Alone, they are mathematically sound, but they are removed from all possibility of convincing empirical validation, as they are difficult, or in many cases outright impossible, to verify or prove.
All three categories of mathematics described above use a system of signs that operate under the same rules or laws. Variables and symbols stand for certain concepts, or directly represent entities. Mathematicians attempt to convey their ideas in texts, but the concepts – and works – remain intangible. We can acquire, for example, an understanding of “counting,” a simple, though fundamental, idea in mathematics. For all its apparent simplicity, counting is highly complex conceptually. It immediately brings in additional concepts, such as order and magnitude, which can each individually be represented many ways. What is a number? Attempting to answer this question is as vexing as trying to pin down what is a word.
II. The concepts and works of mathematics are represented by arbitrary systems of signs that make up physical documents and texts
In literature, a text is, according to Tanselle, something like a guide or set of instructions for reconstructing a thought. In “The Nature of Texts,” he argues that the form of the medium is crucial in establishing the nature of textual problems. Literary texts are closer to “musical scores” than to works of painting or sculpture (23) because they provide “the basis for the reconstruction of works, even though the medium of those works is different.” According to this framework, then, mathematics is much closer to literature than is either sculpture or painting, and the “textual problems” (27) of mathematics ought to more closely approximate those found in literature. Texts of mathematics reconstitute concepts and theorems (or mathematicians’ thoughts); “no text…is the work” (32).
Whether in math textbooks or articles, we are dealing with texts and documents, and should, therefore, be concerned with alterations, just as with literary texts: “Textual critics who focus on literary texts (as most people traditionally called “textual critics” have done) concern them- selves in one way or another with alterations. Whether they present their work to the public in the form of editions or of essays, the force that impels their work — indeed, at one level or another the actual subject of it — is the possibility that received texts are incorrect (according to one of many conceivable standards) and in need of alteration to set them right” (20).
The systems of signs used in mathematics are just as prone to errors, variants, and typos as are the words that make up poems and novels. It is quite evident that many of the same issues are present. In “Reproducing Texts of Documents,” Tanselle says, “The efficiency of a document — written or printed — in performing its utilitarian task is measured (or would be, if such a measurement were possible) by the degree to which the work that we think the document is telling us to create matches the one that its producer had in mind” (40-41). The relevance to mathematical texts is immediately apparent.
In addition, thinking about mathematics textually opens a realm of fascinating, and worthwhile, though largely unexplored, questions. What does it mean to talk about authorial intention in mathematics? Is math a fundamentally collaborative process? Does it manage to arrive at some fundamental truth(s), existing independently and “discovered”?
III. The concepts and works of mathematics are mediated, and thus dependent on, but exist independently of, language
Bertrand Russell was keenly aware of the limitations of language, and aspired to rid logic, and mathematics, of such limitations altogether, by relying exclusively on “logical symbolism”:
It is impossible to convey adequately the ideas that are concerned in this subject so long as we abstain from the use of logical symbols. Since ordinary language has no words that naturally express exactly what we wish to express, it is necessary, so long as we adhere to ordinary language, to strain words into unusual meanings … Moreover, ordinary grammar and syntax is extraordinarily misleading … Because language is misleading, as well as because it is diffuse and inexact when applied to logic (for which it was never intended), logical symbolism is absolutely necessary to any exact or thorough treatment of our subject” (114).
It is easy to agree upon the fact that our understanding of math concepts, even if these concepts exist outside of language, is ill served by language. Words are slippery, insufficient – the inadequate “execution of a statement” does not accurately convey the “potentiality of ideas” behind it (Tanselle). But regardless of one’s view on whether the subject, that is mathematics, does truly arrive at some ultimate truth or not, and whether it exists superior to language, as Russell suggests, ultimately the debate must not distract us from the more practical observation that transmission of the theorems and findings of mathematics is through texts that are entirely susceptible to errors – whatever system of signs employed. In addition, these texts invariably employ language together with symbols.
In this short essay, I have argued for the application of the principles of textual criticism to mathematics. There are particular editorial challenges in bridging the gap this gap; much work remains to be done. Practically, it is often trickier to represent formulas, which might require specialized software, such as LaTeX; in addition, electronic collations of formulas might be problematic or altogether impossible with certain programs. A degree of specialization is necessary; reaching across disciplines necessarily may involve gaps in knowledge and one may encounter resistance. Nevertheless, it is worthy undertaking: the work of the textual scholar can greatly illuminate that of the mathematician, and vice versa.