A Review of a Research Article in the Summer, 2000 Issue of JRME,
(A commentary on the mathematical understanding of those who seek to explain the ways of mathematicians)
by Leone Burton and Candia Morgan, in Journal for Research in Mathematics Education, vol 31 (2000), p429-453.
The first two sentences of the Abstract are:
In this article we report on part of a study of the epistemological perspectives of practicing research mathematicians. We explore the identities that mathematicians present to the world in their writing and the ways in which they represent the nature of mathematical activity.
I read this far because I am interested in mathematical writing, and could not believe that this journal would have anything sensible to say about it. At this point it already appeared that my suspicions were justified. I cannot see that mathematicians' epistemological perspectives, which have been commented on a great deal in recent years by mathematicians themselves, are likely to be illuminated by a study of "the identities that mathematicians present to the world".
Indeed, their research writing is generally not "presented to the world" at all, but to other mathematicians, though some of it, if it is of any value, tends to filter through to other audiences in time. It is notorious that only mathematicians, and very few of those, read or are even able to read, the work of other research mathematicians. Thus "the identities that mathematicians present to the world" through their research writings, while certainly present, are unlikely to be understood except by certain other mathematicians. And what these ("constructed", of course) identities have to do with their epistemology, which is generally quite simple and Platonic, is something worth reading on about, if only for the laughs.
One noticeable feature of educational literature is the abundance of bibliographical reference. The present article contains 23 pages of text and 51 bibliographical references. Unlike references in mathematical research articles, education articles tend not to give page numbers, so if a reference in an educational research paper is to be used the user will have to comb the entire book referred to, and even then (as my own experience has indicated) not necessarily find out just what it was about the book or paper referred to that makes its lessons a basis for, or apposite to, the text that has invoked it. The opening sentence of the present paper of Burton and Morgan is as follows:
In recent years, there has been increasing recognition in the mathematics education community of the social nature of mathematical activity and of the importance of communication within the practices of doing, teaching, and learning mathematics (e.g., Boasler, 1997; Burton, 1999b; Resnick, Levine, & Teasley, 1991; Steffe & Gale, 1995; Stephens, Waywood, Calrke, & Izard, 1993; and with a particular emphasis on equity, Secada, Fennema, & Adajian, 1995).
"Within the practices of" is an ill phrase, I thought, and I didn't much like that bit about "equity", but I haven't time to consult the work of these fourteen cited authors just to find out the justification for the remark that the mathematics education community has (in recent years, anyhow) become cognizant of the social nature of mathematical activity, whether within or without the practices of the doing, the teaching or the learning. It seems much too much in the way of heavy guns for so bland a statement. I believe it, I believe it! -- even the equity part.
The authors then go on about various sorts of things that can be considered "mathematical writing", and include brief mention of things more popular than current research literature. They mention, too, some of the commentary on the writing of mathematics, some of it written by mathematicians. The 1973 AMS report, How to Write Mathematics, especially its long segment written by Paul Halmos, but not ignoring Steenrod, Schiffer and Dieudonne, is a superb guide to the best of present practice, and I hope it is still known to many mathematicians. That report is referred to in this article, as is Leonard Gillman's Writing Mathematics Well (MAA, 1987). But the authors of the present article have a deeper goal in mind than the mere analysis of good mathematical writing, or the distiguishing of the good from the bad or indifferent in today's mathematical research literature. It is hard to say, actually, why they make such references to mathematicians' own announced criteria for good mathematical style.
Their analysis proceeds, they write, "from two major perspectives: the interpersonal and the ideational (Halliday, 1973)." Halliday, they explain in a footnote (p435) "also identified a textual function -- making the text into a meaningful message -- that we do not consider in this article." This footnote shocked me, even though up to that point I had already expected to be displeased by the paper. Goodness, I have spent my entire professional life thinking the "textual function" of a research paper or book, or even textbook, to be the basic one, almost the only decent one; and since Dieudonné, Gillman, Halmos, Schiffer and Steenrod had all written as if conveying the overt mathematical message were the sole purpose of learning to write one well, I could imagine what the rest of this particular article would say about the deeper motives of our arrogant tribe (and I turned out to be right, too). But that this paper would expect to analyze the writing without any regard for the announced purpose of the text struck me as plain impossible.
Of course we all take pride in our successes, and of course I (among others!) have always been self-conscious to some degree in writing as elegantly as I can, but such motives underlie all human activity, even the most beneficent. To say of a good scientist's science that he only did it for personal satisfaction is to say nothing important. Nobody could have been more arrogant than Isaac Newton, from what I have read, but that could not make much difference in the way his contemporaries, the competent ones, must have read his works. Newton's works reflected his milieu as well as a general 17th Century philosophy, and certainly something of Newton's personality as well as his unique mathematical genius must be visible to today's scientific historian, but it seemed to me that the textual message of Newton's Principia was what made it important and valuable. How can one analyze the style apart from the substance? The words "textual function" now joining a few other warning markers at the back of my mind, I read on in this paper of Burton and Morgan fearing the worst. Part of the evidence for my prejudiced conclusion was already in the footnote: They weren't going to bother with the merely textual function of mathematical writing.
As it turned out, Burton and Morgan managed their analysis of the deeper properties of mathematical writing not only apart from the substance, but without understanding the substance. A tour de force. To say someone does or does not "understand" something is to make a disputable statement, since we do not inhabit one another's minds, but within the mathematical community there are certain indicators of ignorance that are never denied. Without them we could not grade our students' examinations, after all. I have on my office wall a framed letter from a man in Alberta who believed he had trisected the angle. It is very large and elegantly typed, with figures drawn by genuine ruler and compasses in two colors. It is of course quite wrong, and if I say that the author did not understand the mathematics by which it is proved that what he thought he was doing was not so, there is not a mathematician in the world who would deny my thesis. That single paper, which I have framed for the pleasure of visitors to my office, is sufficient evidence for me, and for most of them.
But for the man who writes such nonsense, things can appear different. While my correspondence with this Canadian was pleasant, and ran to three exchanges before I gave up further explanation, I have read some of the letters and manifestos of other mathematical and scientific "cranks", people who have invented perpetual-motion machines and disproved the existence of the electron, or have squared the circle, and it is common among such people to be angry. They say -- and appear to believe - - that the world of mathematics, or physics, is a closed guild, so anxious to preserve its authority, its hegemony as it were, that it will forever deny the validity and importance of that crank's discoveries. We (mathematicians or scientists) refuse to print their discoveries, they complain, and often even refuse to discuss them, or show them to their satisfaction how they are wrong. The circle-squarer's analysis of professional mathematical research writing is not the same as that of Halmos or Steenrod. So with the authors of the paper I am discussing here, and of course I cannot impose my judgment on them, but I can at least exhibit for the possible persuasion of others some paragraphs of what Burton and Morgan have written on pages 439 and 440 of their paper, and claim it to be evidence of mathematical ignorance akin to the ignorance demonstrated beyond reasonable doubt, on a single page, by the genial angle-trisector from Canada. Beginning on page 439:
Authority -- Positive and negative: The first aspect of identity. One important aspect of the identities of authors as projected by their texts is the extent to which and the manner in which they claim to be authoritative within their community. If an author appears too tentative in his or her claims, less value might be placed on the results, whereas if one appears inappropriately self-assured, a reader might question the author's right to be so certain and even dismiss the work. Terms such as "clearly" and "obvious" are relative to the individuals using them (implying that this information is obvious to me but may not be so obvious to you). Our interpretation that the use of such terms serves as a claim to authority on the part of the writer (implying that this derivation is clear to me and I do not need to explain it further because if it is not clear to you, that is your fault not mine) is reinforced by the interview data, as demonstrated in some of the earlier quotes. We believe that, whatever the author's intent, the extent or absence of such words is one of the interpersonal aspects of the writing that will influence the ways in which the readers of the text will construct an image of the author and will consequently judge the worth of the text itself.
This section exhibits a curious switch that makes it hard to argue with. It begins by attributing an arrogant attitude to the user of "clearly", and describes that attitude twice, in parenthetical imagined monologue, but ends only by saying that "whatever the author's intent" might be, the use of such words will generate an image of some sort in the reader, only an image of the author, not necessarily the real thing. Is the author who uses "clearly" parading his authority or is he merely subjecting his prose to misinterpretation of his motives? Burton and Morgan get away with claiming both, I suppose.
I cannot deny the latter reading, any more than I can deny that some circle-squarers consider my denial of their discovery a defensive act of authority. But I will nonetheless deny that "clearly" is used to put down the readers. I ask my own readers here to find, if they can, samples of articles in the Proceedings of the American Mathematical Society (say) exhibiting such behavior. It is hard for me to believe that the writers of this analysis of the prose of mathematical research have had much experience reading any of it for its mathematical content – or not successfully, at any rate. I must explain to them that "clearly" is an abbreviation for some statements that would take up unnecessary space, and perhaps, even probably, interrupt the exposition in such a way as to make it harder (not easier) to read and understand. Further, that the author who uses "clearly" generally knows the statement is not "clear" to everyone, not to one person in a million actually, for it only needs to be clear to those who have a good chance at understanding his piece at all.
Even for that minority of his audience that does not see it clearly, the word “clearly” is not a put-down but an invitation to find out for oneself why the statement follows, something the reader will have to do at some point in any case if he is to keep up with the status of that particular branch of mathematics. I am somewhat embarrassed; actually, at having had to write the paragraphs immediately preceding this one, since in the community of mathematicians this function of "clearly" is well understood without my explaining it. The only criticisms I have heard, or can imagine, about its use concern the size of the fraction of the readership whose understanding is adversely affected by the absence of the material elided by the "clearly", i.e., whether the line was drawn too narrowly or too broadly in a given paper, but never about whether it was used in some authoritarian manner, or whether some part of its audience might "construct" a mistaken image of the personality of the author, or "dismiss the work" in irritation.
We have all known arrogant mathematicians, but that arrogance is never exhibited as a claim of authority overriding the claims of logic, after all. On the following page, a few sentences later,
... Expressing an appropriate and acceptable degree of authority is likely to be important for beginning mathematical writers in order that they may be accepted as (properly positioned) members of the community; Ward (1996) pointed out that "the author must consider the tactics to use in order to convince an audience of his or her truth claims" (p40). We felt, therefore, that the forms and extent of claims to authority were significant enough to warrant examination.
Since such "tactics" did not sound like mathematical behavior to me I looked in the bibliography for the reference to Ward, and found that it was a book, Ward, S.C. (1996), Reconfiguring truth: Postmodernism, science studies, and the search for a new model of knowledge. London: Rowman and Littlefield. And on this warrant (a single postmodernist citation for a questionable claim of "significance", as against the fourteen authors cited early on in support of a platitude) the authors say that "therefore" they will examine the "forms and extent of claims to authority" in mathematical research. Well, in truth they need no warrant at all. Let us consider the examination itself:
Indicators of authority claims from the use of modifiers, such as "clearly", "easily", "of course", and "immediately obvious", and phrases, such as "without loss of generality", "it suffices to consider the case", and "the last stage is trivial", all signal a gap in the argument (implying that there is no accompanying justification for the claim that generality is not lost) and hence signal to the readers that they should accept the author's claim....
There is a great deal more, none of it a model of expository prose by the way, and I refer my reader to the original for further horrors. Probably the most tellingly misconstrued "indicator of authority" is "without loss of generality", which is a technical phrase in such frequent use that its abbreviation, "wolog", is familiar to every graduate student. If Burton and Morgan are to be believed, we are therefore even teaching authoritarian manners to our graduate students.
Rather than interpose here an explanation of the way "wolog" is used in principle, much as I could have written out an outline of the nature of mathematical proof for my Canadian geometrical crank, I shall here give an example: If a mathematician were to write a proof, using the Cartesian coordinate system, of some theorem concerning the a triangle, such as that the medians intersect at a single point, he might begin by saying, "Without loss of generality, assume the triangle is situated with one vertex at the origin and another at the point (a,0) for some a > 0." A sufficiently sophisticated audience will recall that any triangle, wherever situated in the plane to begin with, can be brought to the described position by a rigid motion, with the consequent preservation of all the Euclidean features of the problem; hence proving the theorem for this triangle proves it for all others. This explanation itself is abbreviated, for "Euclidean features", "rigid motion" and the like themselves require definition, so that the "hence" in the preceding sentence also needs proof; but for mathematicians and advanced students this much information about Euclidean geometry from an analytic standpoint is a standard expectation.
Thus, far from trying to depreciate a gap in the argument by "authority", the use of "wolog" even signals that gap to the reader, so that those who for some reason need more explanation will not be trapped into unreasoning belief, but will seek the explanation elsewhere; for if there is anything a mathematician does not want in his audience, it is an appearance of agreement, but without understanding. Meanwhile the audience that does not need the rest of the explanation (and that should be the major part of the audience intended by the writer) will not be fatigued by excessive verbiage. Moreover, "wolog" provides more than just consideration for the time of the reader or the space of the publisher, as does, e.g., the use of "clearly", for "wolog" may also serve a genuinely expository function, directing the reader's attention to an interesting feature of the method employed, rather than burdening him with a tedious lemma tending to obscure it. But I won't go on.
I might summarize by saying that in no way can "wolog" be interpreted as anything but consideration for the reader (and publisher) except by someone who doesn't understand the phrase, or willfully chooses it to make a political point for an audience that doesn't understand it.
Authority has never proved a theorem, or discovered one. Authority sometimes determines what the mathematical world considers important, sometimes does someone out of a job, and sometimes puts people in the Gulag; but Burton and Morgan are looking in the wrong place for the misuses of authority when they are analyzing the language of mathematical research literature. Indeed I believe, though this is probably impossible to prove, that the world of mathematics education is more burdened by authority than are most other domains of human endeavor, while, fortunately, the world of mathematics is afflicted with less.
2 September 2000
revised 25 September 2004