Classic style and mathematical writing

September 5, 2019

The elements of writing style are usually taken to include grammar, usage, composition, form, structure, tone, and voice, as exemplified by the famous pamphlet of Strunk and White.1 It is tacitly understood that the questions of how best to implement the elements of style have definite answers, applicable to at least most nonfiction writing. So, on the topic of composition, Strunk and White tell us to “put statements in positive form,” “use definite, specific, concrete language,” and, most famously of all, “omit needless words.”

In Clear and Simple as the Truth: Writing Classic Prose,2 Francis-Noël Thomas and Mark Turner offer an entirely different analysis of style. A writing style is not a list of prescriptions on grammar and usage; rather, it is a stand on the five elements of style:

  1. Truth: What can be known and how confidently? Is the truth complete or partial, universal or personal? Who can access the truth, and how?
  2. Presentation: Assuming that we know it, how should the truth be presented? How should clarity, accuracy, and persuasiveness be balanced?
  3. Scene: What is the model scene of communication between writer and reader?
  4. Cast: Who are the readers? What is their relationship to the writer?
  5. Thought and language: How are thought and language related? Which, if either, comes first? Can thought be faithfully articulated?

A style of writing is any coherent set of answers to these five questions. Style does not directly address mechanical and surface features, such as grammar or composition; however, a particular style may lend itself more naturally to some surface features than to others. The surface prescriptions of Strunk and White pair easily with plain and practical style, somewhat well with classic style, and not at all with reflexive or romantic style.

There is not a single style of writing but as many styles as there are stands on the elements of style. No single style is universally best, and different situations call for different styles, possibly even “bad” styles. Consider this statement by Alan Greenspan, former Chair of the Federal Reserve, in response to pressure from U.S. senators to cut interest rates:

While I’ve indicated to you previously that we may well have, probably do have, enough monetary stimulus in the system to create that [economic recovery], I’m not sure that we will not need some insurance or to revisit this issue, and all I can say to you is that we’re all looking at the same set of data, the same economy, the same sense of confidence which pervades it. We’re all making our judgments with respect to how that is evolving with respect to economic activity and where the risks of various different actions are. And there will be differences inevitably.3

As Thomas and Turner argue, what looks to be disjointed, rambling speech is in fact carefully crafted to create the appearance that Greenspan is conveying useful information, when his true purpose is to convey no information whatsoever. No matter how many times you read and reread this passage, you will be no wiser than before as to whether Greenspan intends to cut interest rates. It is a masterful display of a specialized style, the “style of gridlock,” employed by those unwillingly compelled to testify in public.

Classic style

Clear and Simple as the Truth is both a conceptual analysis of style in general, via the elements of style, and an exposition of a particular style, the “classic style.”4 In the book, Thomas and Turner present the classic stand on the elements of style and contrast classic style with the plain, reflexive, practical, contemplative, romantic, prophetic, and oratorical styles (Part One). The other styles are not developed independently, but only by comparison with classic style. The authors then analyze a series of examples and non-examples of classic style, in both fiction and nonfiction, from antiquity to the present (Part Two). In addition, the book is itself a beautiful specimen of classic style. Finally, in the main innovation of the second edition, Thomas and Turner propose speech and writing exercises for learning how to write classic prose (Part Three).

In brief, classic style takes the following stand on the elements of style.

Truth. Although it may be nuanced and complex, the truth is perfectly definite, and although we may easily fall into prejudice and self-interested thinking, anyone can, in principle, know the truth. No special power or revelation is required, only a willingness to acquire the relevant background knowledge and examine the issue disinterestedly. The truth is verifiable, and anyone verify it for themself by accurate observation. “In the classic view, what cannot be universally verified cannot be true. The classic attitude is thus both foundationalist and universalist.”5

The classic attitude towards truth is an “enabling convention”6 of the style, not an epistemological commitment.7 Here, for example, we find Descartes, an originator and master practitioner of classic style, raising a radical skeptical doubt:

I will suppose… some malicious demon of the utmost power and cunning has employed all his energies in order to deceive me. I shall think that the sky, the air, the earth, colours, shapes, sounds and all external things are merely the delusions of dreams which he has devised to ensnare my judgement. I shall consider myself as not having hands or eyes, or flesh, or blood or senses, but as falsely believing that I have all these things.8

Descartes is relating a troubling thought that has occurred to him and that he is deliberately entertaining in order to question the certainty of his commonsense beliefs. But though the thought sprung from Descartes’s mind, not from yours or mine, it remains perfectly accessible to us. Anyone who imagines for themself this malicious demon, invisibly controlling every aspect of their sensory perception, will appreciate the force of the argument. The Evil Demon argument, as we now call it, exerts such a pull that it has been rediscovered again and again, most recently as the fear that we live inside a computer simulation.

Descartes goes on to reject this radical skepticism in much the same style that he introduces it, but for us that is beside the point, which is that classic style is just as suitable for presenting radical skepticism as it for presenting Cartesian rationalism. What classic style cannot accommodate is the stance that truth is uniquely personal, as in romantic style, or that truth comes only by revelation, as in prophetic style.

Presentation. The guiding metaphor of classic style is that prose is a window onto the world. The aim of writing is to convey something that writer has previously seen or understood, so that the reader can see or understand it for themself. Thus, “clarity is the central virtue of classic prose.”9 Classic writing is accurate as concerns all essential matters, and it never attempts to mislead by omission, but it is not pedantic and it does not dwell on irrelevant details or obvious caveats. It presents thoughts that were fully developed before the act of writing. “Classic style is perfect performance, with no hesitation, revision, or backtracking. Its essential fiction is that this perfection happens at the first try.”10

For sheer performance, perhaps the best of Thomas and Turner’s examples is this passage from the Memoirs of François de La Rochefoucauld:

Mme de Chevreuse avait beaucoup d’esprit, d’ambition et de beauté; elle était galante, vive, hardie, entreprenante; elle se servait de tous ses charmes pour réussir dans ses desseins, et elle a presque toujours porté malheur aux personnes qu’elle y a engagées.

[Madame de Chevreuse had sparkling intelligence, ambition, and beauty in plenty; she was flirtatious, lively, bold, enterprising; she used all her charms to push her projects to success, and she almost always brought disaster to those she encountered on her way.]11

La Rochefoucauld reveals the essence of Madame de Chevreuse in a single, perfect sentence. Every word, every phrase, serves a purpose and builds towards a memorable conclusion. The conclusion cannot be anticipated, yet having reached it, it seems inevitable. The whole sentence can be pleasantly read aloud and sounds unforced and spontaneous, even if no spontaneous utterance is ever so well crafted. Having read it, we feel as though we understand Madame de Chevreuse just as well as La Rochefoucauld does.

Scene and cast. The model scene of classic style is a conversation between two people, in which “the writer adopts the pose of a speaker of near-perfect efficiency.”12 Classic prose is “ideal speech.”13 Because it is modeled on conversation between two people, classic style is informal, with little pomp, self-indulgence, or metadiscourse. Classic writing is authentic,14 an impression created partly by informality but more importantly by the writer’s conviction that they know the truth and can express it.

Classic style is aristocratic in one sense and egalitarian in another. The writer is competent and demonstrates mastery of their subject. However, the elitism of the writer “is not the result of natural endowment. It is the result of effort and discipline ending in achievement.”15 The writer has learned a truth and will present it to the reader, but those roles could easily be reversed. Perhaps in the future they will be. The reader, like the writer, is competent. Thus, the writer has no need to belabor the point or appeal to cheap tricks. “The classic writer does not have to persuade the reader. All he has to do is offer the reader an unobstructed view, and of course the reader will see.”16

In this passage, philosopher Michael Huemer asks whether political authority can ever be legitimate and whether citizens have a moral duty to obey to their government:

Questions of this kind are notoriously difficult. How should they be approached? One approach would be to start from some comprehensive moral theory—say, utilitarianism or Kantian deontology—and attempt to deduce the appropriate conclusions about political rights and obligations. I, unfortunately, cannot do this. I do not know the correct general moral theory, and I don’t think anyone else does either… What is the alternative? I shall start from moral claims that are, initially, relatively uncontroversial. This seems an obvious plan… Yet natural as it may seem, this approach is seldom taken up.17

Like all academic writing, Huemer’s book is not entirely classic, but it is more classic than most. This passage in particular takes a classic stand on the relationship between writer and reader. Huemer is going to arrive at a controversial conclusion—that all political authority is unfounded—but not by an abstruse moral theory, which the reader is incapable of verifying, but only by carefully tracing the consequences of commonsense ethical intuitions,18 which the reader is taken to share. Thus, when Huemer presents his chain of reasoning, the reader, who is competent, will verify each step for themself and will reach the same conclusion. The tone is informal and conversational. Frequent questioning gives the reader opportunity to ask themself whether they assent (which, of course, they will, upon sufficient consideration).19

Thought and language. According to some philosophical traditions, and versions of romantic and sublime style, thought is ineffable. The thoughts in the writer’s head cannot be articulated, only vaguely gestured at. Trying to put a sublime thought into words is like trying to hold water in your hands.

Classic style rejects this view as the excuse of an incompetent writer. According to the classic stance, thought and language are congruent. First the writer has a perfect thought, and then they give it perfect expression. Language is always sufficient to the task: “it is always possible to achieve a perfect fit between a thought and its expression just as it is always possible to achieve a perfect solution to a problem in elementary algebra.”20

The congruence between thought and language holds equally for concrete observations, like what I see when I look out my window, and complex abstractions, like beauty or justice. When Strunk and White tell us to “prefer the specific to the general, the definite to the vague, the concrete to the abstract,” they confuse subject matter with style.21 Some subjects are inherently more abstract than others, but classic style embraces them all:

From the classic viewpoint, the distinction between abstract and concrete has no consequence. A writing instructor or consultant who advises us to write concretely and avoid abstractions offers shallow and impractical advice because the distinction is simpleminded. What matters is not the ontological category of the subject but rather the style in which it is conceived.22

Mathematical writing

Although antecedents of classic style can be traced back to antiquity, Thomas and Turner see the style as crystallizing in seventeeth-century France. Masters of classic style from this era include the philosophers René Descartes and Blaise Pascal; the early novelist Madame de La Fayette; and men and women of letters, François de La Rochefoucauld and Madame de Sévigné. It is no coincidence that the most famous of these figures, Descartes and Pascal, were as much mathematicians as philosophers. The classic stand on style can be seen as an extension of mathematical style beyond the confines of mathematics.

Let us distinguish mathematical style from another specialized style, academic style. The latter is a professional style practiced those who make their living writing scholarly books and articles. It has features of several general styles—practical, reflexive, and classic—in proportions depending on the writer’s discipline and temperament. Academic style also has distinctive features of its own. In order to survive and advance in the profession, academics must persuade referees to publish their work. To do this, they must establish the correctness, originality, and importance of their work and fend off hostile criticisms. This practical reality creates a conflict of interest between writer and reader that is incompatible with the disinterested stance of classic style. Thus, academic writing is hardly ever wholly classic.

Mathematical style is what remains of mathematical writing when all specifically academic features are removed. In its pure form, mathematical style is most easily found in textbooks, where the pressure to claim ownership and establish novelty is lower than in research articles. The fundamental features of the style remain basically unchanged since antiquity, even as the standard of rigor has steadily increased. Mathematicians on MathOverflow regard John Milnor, Jean-Pierre Serre, and Michael Spivak, among others, as contemporary masters of mathematical style.

As a device for writing clearly, classic style expands mathematical style beyond its proper scope. In classic style, the “enabling conventions” are just convenient fictions; in mathematical writing, some of them are actually true. Consider the classic stand on truth. In ordinary circumstances, the truth is, if not unknowable, then knowable only in an uncertain, fallible sense, subject to revision at any time. In mathematics, by contrast, the truth is immutable, it is knowable with certainty, and it is the business of mathematicians to find it out.

Mathematical style, like classic style, is perfect performance. All the failed attempts, useless directions, and aimless wanderings, all the hastily scribbled notes and mounds of discarded paper, are removed from view. You would never imagine that the author labored to produce what is written, much less that it rests on two thousand years of investigation by mathematicians across the world. The author is merely giving elegant expression to what they already know to be true. Possibly they are transcribing from a great Book where God keeps all the most beautiful proofs.

Mathematical style shares, in extreme form, both the elitism and the egalitarianism of classic style. To those who do not speak the language, the book of mathematics is closed, and the beauty and utility of the subject are almost entirely inaccessible. But the language of mathematics, unlike ordinary human language, is universal. Anyone can, in principle, acquire fluency in mathematics, given enough self-discipline and access to the right kind of education.

Mathematicians can be said to make “arguments,” but only in a very special sense, because mathematical arguments, correctly executed, do not admit disagreement. Once the reader grasps the flow of the argument and the connection between the ideas, they cannot fail to assent. The psychology of reading mathematics is one of temporary cognitive strain and confusion, followed by the unmistakable feeling of inevitability that accompanies understanding. Mathematical writing comes closer than nearly any other genre to realizing the classic ideal that writing is presenting, not arguing.23

In classic style, the perfect congruence between truth, thought, and language is a conceit, especially in the case of complex abstractions.

When a classic stylist presents an abstraction—cultural reality, heroism, historical causation, the nature of representation, taste—it is first conceived as independent of the writer, exhaustively definite at all levels of detail, visible to anyone competent who is standing in a position to see it, immediately recognizable, and capable of being expressed in direct and simple language.24

Though it may be a useful literary device, presumably no one believes that abstractions like cultural reality, heroism, and historical causation are completely definite, mind-independent, immediately recognizable, and capable of direct expression. Yet the abstractions of mathematics are all of these things. As far as mathematical style and method are concerned, there truly is no distinction between abstract and concrete.

Mathematical style is a purification of classic style; classic style is a universalization of mathematical style.


  1. William Strunk Jr. & E. B. White, 1999. The Elements of Style, 4th ed. 

  2. Francis-Noël Thomas & Mark Turner, 2011. Clear and Simple as the Truth: Writing Classic Prose, 2nd ed. Princeton University Press. Website

  3. Ibid., p. 169. 

  4. Inspired by Thomas and Turner, Steven Pinker summarizes and promotes classic style in Chapter 2 of his style guide (Steven Pinker, 2014. The Sense of Style: The Thinking Person’s Guide to Writing in the 21st Century, Chapter 2: A Window onto the World). In an earlier essay, I discuss Chapter 3 of that book, on the curse of knowledge. 

  5. Thomas & Turner, 2011, p. 29. 

  6. Ibid., p. 29. 

  7. There is an analogy between the classic style and what I have called the scientific stance. Just as the scientific stance is a method for investigating the natural world, not a metaphysical thesis, the classic style is a method for writing clearly, not an epistemological thesis. 

  8. René Descartes, Meditations on First Philosophy, First Meditation, in: John Cottingham, Robert Stoothoff, Dugald Murdoch, trans., 1984, The Philosophical Writings of Descartes, Volume II. Cambridge University Press. 

  9. Thomas & Turner, 2011, p. 32. 

  10. Ibid., p. 34. 

  11. Ibid., p. 159. 

  12. Ibid., p. 37. 

  13. Ibid., p. 42. 

  14. That classic prose presents itself as authentic does not imply that it is authentic. One should never confuse a stand on the elements of style with the purpose or character of the writer. 

  15. Thomas & Turner, 2011, p. 45. 

  16. Ibid., 48. 

  17. Michael Huemer, 2013. The Problem of Political Authority: An Examination of the Right to Coerce and the Duty to Obey. Palgrave Macmillan. DOI. pp. 14-15. 

  18. See also: Michael Huemer, 2005. Ethical Intuitionism. Palgrave Macmillan. DOI

  19. Through its conceit that it is only presenting, not arguing, classic style can be a disarming and effective mode of political persuasion. Huemer, an analytic philosopher, acknowledges that he is making arguments, but classic style generally does not. As another example of political speech that is partly but not wholly classic, Thomas and Turner cite the Declaration of Independence. The authors of this document refuse to admit that they are making an argument. Truth is “self-evident.” 

  20. Thomas & Turner, 2011, p. 60. 

  21. Strunk & White, 1999, p. 21. 

  22. Thomas & Turner, 2011, p. 58. 

  23. One exception is the genre of field guides. Fittingly, Thomas and Turner devote signficant time to studying field guides as prototypes of classic style. 

  24. Ibid., p. 59.