I Like Eq

I grew up in an environment that glamourized mathematical equations. Equations adorned a text like jewelry, set there to dazzle, and often to outshine the text that they were to illuminate. Needless to say, anything I wrote was dense, opaque, and didn’t communicate what it set out to. It was not until I saw a Reference Frame essay by David Mermin on how to write equations (1989, Physics Today, 42, p9) that I realized that equations should be treated as part of the text. You should be able to read them. David Mermin set out 3 rules for writing out equations, which I’ve tried to follow diligently (if not always successfully) since then.

  1. Number or label all displayed equations (Fisher’s Rule):

    The most common violation of Fisher’s rule is the misguided practice of numbering only those displayed equations to which the text subsequently refers back. … it is necessary to state emphatically that Fisher’s rule is for the benefit not of the author, but the reader.
    For although you, dear author, may have no need to refer in your text to the equations you therefore left unnumbered, it is presumptuous to assume the same disposition in your readers. And although you may well have acquired the solipsistic habit of writing under the assumption that you will have no readers at all, you are wrong.

  2. When referring to an equation within the text, identify it by a phrase as well as a number (aka the Good Samaritan Rule):

    A Good Samaritan is compassionate and helpful to one in distress, and there is nothing more distressing than having to hunt your way back in a manuscript in search of Eq. (2.47), not because your subsequent progress requires you to inspect it in detail, but merely to find out what it is about so you may know the principles that go int othe construction of Eq. (7.38).

  3. Punctuate the equation (aka the Math is Prose Rule):

    The equations you display are embedded in your prose and constitute an inseparable part of it.

    Regardless … of how to parse the equation internally, certain things are clear to anyone who understands the equation and the prose in which it is embedded.

    We punctuate equations because they are a form of prose (they can, after all, be read aloud as a sequence of words) and are therefore subject to the same rules as any other prose. … punctuation makes them easier to read and often clarifies the discussion in which they occur. … viewing an equation not as a grammatically irrelevant blob, but as a part of the text … can only improve the fluency and grace of one’s expository mathematical prose.

  1. John Scholes:

    Yes. All useful advice. But the real way to make papers easier to understand is to distil. You have to think really hard about the essence of what you are trying to say. Of course, you have to be a genius to do this really well – a good example is Fred Hoyle’s autobiography, which contains more astrophysics than thousands of pages of the ApJ, all without a single equation. But we can all aspire to do better.

    On a more modest expository level, I am looking for important mathematical arguments which can be simply explained to the lay reader (in a page of A4 or less). I started with the infinitude of primes. Grateful for any ideas.

    08-14-2008, 4:40 am
  2. vlk:

    True that. It takes a great deal of effort to simplify a complex concept into natural language. Math is basically shorthand, I suppose, to describe complicated ideas without ambiguity. Think of how difficult it would be to say what a straight line is in English, as opposed to saying “y=mx+c”! Newton, for instance, in the Principia, writes his equations and does complicated integrals entirely as geometric constructs — an amazing feat, but rather hard to read now without commentary.

    PS: Could you describe what sort of mathematical arguments you are looking to explain? Who are the target audience? One thing that may be useful: take a look at Prof. James Robert Brown’s lecture on “Proofs and Pictures: The Role of Visualization in Mathematical and Scientific Reasoning” at the Perimeter Institute, where he discusses ways to replace induction-based proofs of some well known theorems with geometrical sketch proofs.

    08-14-2008, 6:02 pm
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