A major objection to laying down criteria for the excellence of an exposition is that the effectiveness of an expository effect depends so heavily on the knowledge and experience of the reader. A clean and exquisitely precise demonstration to one reader is a bore to another who has seen the like elsewhere. The same reader can find one part tediously clear and another part mystifying even though the author believed he gave both parts equally detailed treatment. (Steenrod, 1973)
He advises the division of the formal/logical mathematical part, and informal/introductory/metamathematical part. Some, he says, might say, “Show me the mathematics, I’ll supply my own philosophy” p. 2 but he advises supplying some gloss nonetheless.
“He must strive throughout to describe his own attitudes towards the various parts of the subject, and also such other views as he regards valid, but all such material must be labelled as distinct from the formal structure so that a reader can omit and skim such parts as are not to his taste. P. 2
Expects formal to come before informal.
Case for introduction:
“The fact that a reader forgets the introduction is no objection if the introduction helps him grasp the formal structure more quickly. At stake here is the question of how a student learns best. The first of two contending procedures is to ask him to examine first the lumber, bricks, and small structural members out of which the building is to be made, then to make subassemblies, and finally to erect the building from these. The second procedure is first to describe the building roughly but globally and provide a framework for viewing it, and then examine the construction of the building in detail. The first procedure would appeal to a student with a leisurely attitude who enjoys successive revelations. The second procedure, which I espouse, has the advantage that motivation is present at every stage; the student knows where each item belongs when he examines it. The second procedure can be elaborated by inserting between the first rough scan and the final detailed examination a series of scannings revealing successively finer details. P. 11-2
For example, Dunford and Schwartz make no attempt to entice readers to study their book, they do not say at the start what linear operators are about nor why they are important. The reasons for this omission are undoubtedly that their book is a reference work and text for a well-known standard field, and every mathematical education already includes much about linear operators; a sales-promotion job is unnecessary. One consequence of this omission is that they give no overall picture of the results that they obtain; I would like to have had such a review for study, and I suspect that some students of their book would have found it useful. P. 11-2
Writing mathematics is like“writing biology, writing a novel, or writing directions for assembling a harpsichord” p. 20
Halmos advises to write as though addressing someone and mentions mind reading p. 22-3 He estimates the longevity of a paper vs. a book: book 25 years, paper 5. p. 22
Bad notation can make good exposition bad and bad exposition worse; ad hoc decisions about notation, made mid-sentence in the heat of composition, are almost certain to result in bad notation.
Good notation has a kind of alphabetical harmony and avoids dissonance. Example: either ax+by or a1x1 +a2x2 if preferable to ax1 +bx2. Or: if you must use Σ for an index set, make sure you don’t run into Pσ∈Σ aσ. Along the same lines: perhaps most readers wouldn’t notice that you used |z| < ε at the top of the page and z ε U at the bottom, but that’s the sort of near dissonance that causes a vague non-localized feeling of malaise. The remedy is easy and is getting more and more nearly universally accepted: ∈ is reserved for membership and ε for ad hoc use.
Mathematics has access to potentially infinite alphabet (e.g. x, x’, x’’, x’’’, . . .), but, in practice, only a small finite fragment of it is usable. One reason is that a human being’s ability to distinguish between symbols is very much more limited than his ability to conceive of new ones; another reason is the bad habit of freezing letters. Some old-fashioned analysts would speak of “xyz-space”, meaning, I think, 3-dimensional Euclidean space, plus the convention that a point of that space shall always be denoted by “(x, y, z)”. This is bad: if “freezes” x, and y, and z, i.e., prohibits their use in another context, and, at the same time, it makes it impossible (or, in any case, inconsistent) to use, say, “(a, b, c)” when “(x, y, z)” has been temporarily exhausted. Modern versions of the custom exist, and are no better. Examples: matrices with “property L” — a frozen and unsuggestive designation.
There are other awkward and unhelpful ways to use letters: “CW complexes” and “CCR groups” are examples. A related curiosity that is probably the upper bound of using letters in an unusable way occurs in Lefschetz . There xpi is a chain of dimension p (the subscript is just and index), whereas xip is a co-chain of dimension p (and the superscript is an index). Question: what is x23?
As history progresses, more and more symbols get frozen. The standard examples are e, i and π, and, of course, 0, 1, 2, 3, . . . (Who would dare write “Let 6 be a group.”?) A few other letters are almost frozen: many readers would fell offended if “n” were used for a complex number, “ε” for a positive integer, and “z” for a topological space. (A mathematician’s nightmare is a sequence nε that tends to 0 as ε becomes infinite.
Sometimes a proposition can be so obvious that it needn’t even be called obvious and still the sentence that announces it is bad exposition, bad because it makes for confusion, misdirection, delay. I mean something like this: “If R is a commutative semisimple ring with unit and if x and y are in R, then x2−y2 = (x−y)(x+y).” The alert reader will ask himself what semisimplicity and a unit have to do with what he had always thought was obvious. Irrelevant assumptions wantonly dragged in, incorrect emphasis, or even just the absence of correct emphasis can wreak havoc. P. 34
p. 34-5 advice on stating theorems—briefly as possible, irrelevancies excluded.
If you have defined something, or stated something, or proved something in Chapter 1, and if in Chapter 2 you want to treat a parallel theory or a more general one, it is a big help to the reader if you use the same words in the same order for as long as possible, and then, with a proper roll of drums, emphasize the difference. P. 35
The symbolism of formal logic is indispensable in the discussion of the logic of mathematics, but used as a means of transmitting ideas from one mortal to another it becomes a cumbersome code. P. 38 Generally avoid symbols. P. 40
He addresses himself to colleagues and coworkers whose knowledge of the subject and interest in his contribution can be taken for granted. He may be as brief and concise as he wishes and omit history, background and motivation for his work. However, even here it might be worthwhile to consider that by adding a little background information one might widen the audience from the close circle of specialists on the subject to a much more extended group of interested mathematicians…. While writing the paper, the author should envisage the reader who has taken the paper to a place without a library and who is willing to believe a few facts on the say-so of the author, but also wishes to understand what he means. P. 50