The Neutralizing of Mathematics



            I recently reviewed a book for the McGraw-Hill company. It was the third edition of a well-known calculus book by Sherman Stein, and McGraw-hill was considering a revision.  My job was to suggest improvements. 


            The text, being mathematics, was (naturally) impersonal, full of things about graphs and exponential functions, and their derivatives and integrals, things which called for the pronoun "it" whenever the need for a pronoun arose.  But not entirely, at least in the exercises at the ends of the chapters.  Even calculus problems have their human element, and in fact the author strove to place his abstractions firmly in the real world as much as is possible, for calculus is a very practical branch of mathematics and should be appreciated as much for its usefulness as for its intrinsic beauty.


            Some problems concerned balloons :  "A large spherical balloon is being inflated at the rate of 100 cubic feet per minute.  At what rate is the radius increasing when the radius is 10 feet?"  Some concerned fish: "What is the acceleration of the fish described in Example 1 when the length of line is 300 feet?"  Some concerned women:  "A woman is walking on a bridge that is 20 feet above a river as a boat passes directly under the center of the bridge..."  Some concerned (inter alia) men:  "A man in a hot-air balloon is ascending at the rate of 10 feet per second..."  Honest, these are all from page 273.


            Mr. Stein, it may be seen, has been assiduous in spreading his examples around so as not to leave anybody (or any thing, almost) out.  Older books, it is said, tended to render women invisible, by using the pronoun "he" whenever a person of undetermined sex was mentioned, and by using male examples almost as a matter of course when the sex of the person running, flying, or walking on a bridge was for mathematical purposes a matter of indifference.  Stein's publisher was in fact a pioneer in the rectification of this affront, having promulgated as early as 1974 a manual called "Guidelines for The Equal Treatment of the Sexes in McGraw-Hill Book Company Publications." (This, for those curious about such things, may be found reprinted in all the recent editions of The Norton Reader, an anthology much used in freshman English courses.)  And so Professor Stein has striven with the demon of sexism, and won.


            Pronouns are no problem either. Or are they?  Page S-31, Problem 40:  "A flea, to amuse itself, jumps 1/2 meter to the right, then 1/4 meter to the left, then 1/8 meter to the right, then 1/16 meter to the left, and so on, as shown in Fig. C.1.  On the nth jump the flea travels 1/2n meter, but continues alternating right and left.  The flea starts at the number 0 on the number line. (a)  Show that after n jumps, where n is odd, the flea is at [a certain place].  (b)  Show that after n jumps, when n is even, the flea is at [a certain other place].  (c) As the flea keeps on jumping, what single number does it keep jumping over?


            One senses a little anxiety here concerning the sex of the flea.  Having pictured the flea as a creature capable of a desire for amusement, it is a bit harsh to keep calling it "it," as if it were an ash tray.  In older editions one could have said "he," of course.  Here Stein gets around the indignity of using "it" five times by using it only twice and repeating "the flea" rather than its pronoun the other three times.  This makes the prose a bit bumpy.  It is an instructive exercise to read the paragraph aloud, once as printed, once with

"itself" and "it" throughout, and once with "himself" and "he" throughout.  The last sounds best, most human, and presents the picture to the mind's eye more clearly than the others.


            The only objection to what would have been done by every writer until very recently is that "he" is now averred to convey the information that a necessarily male flea is being spoken of.  The instruction now being given in all our schools and colleges is in fact rapidly bringing about this apprehension, which when applied to all English literature written before 1970 (and much since) is in fact a misapprehension.  Our children will have as hard a time, once the McGraw-Hill guidelines become universal, reading James Thurber and Virginia Woolfe, at least in their use of pronouns, as they now have with Chaucer.


            The most important reason not to use "it" throughout, and a reason doubtless recognized by Stein as he wrote, is that "it" could have other antecedents:  "Show that after n jumps, when n is even, it is at..." can momentarily deceive the reader into believing that "it" refers to the integer n, the most recently mentioned neuter noun.  This is in fact one reason European languages distinguish genders where sex is not in question, and even where it is:  to make back-references (whether via pronouns or other devices) less ambiguous.  The system is not perfect, but it helps.  In the case of the flea it would completely settle the matter:  "He" must be the flea, not the integer n.


            Having been intimidated by the McGraw-Hill guidelines, Stein could, of course, have used the device favored by most schoolteachers in recent times: "his or her."  Innumerable letters-to-the-editor are still being written on this matter.  An op-ed piece by one Linda Cotton, reprinted in my local paper from The Baltimore Sun just last December 31 (1988), was devoted to stamping out the last vestiges of linguistic sexism.


            "It isn't necessary," she wrote, after an apparently dispassionate review of the pitfalls of the current language reforms, "to neuter and contort every noun like a pretzel.  A simple 'he or she' will do."  But it won't.  Not only is it clumsy to be driven to three syllables in a pronoun used so frequently in writing and speech, but "he or she" injects the distracting mention of sex into contexts where it is irrelevant.


            "A flea, to amuse himself or herself, jumps 1/2 meter to the right, then 1/4 meter to the left, then 1/8 meter to the right, then 1/16 meter to the left, and so on, as shown in Fig. C.1.  On the nth jump he or she travels 1/2n meter, but continues alternating right and left.  He or she starts at the number 0 on the number line. (a)  Show that after n jumps, where n is odd, he or she is at [a certain place].  (b)  Show that after n jumps, when n is even, he or she is at [a certain other place].  (c)  As he or she keeps on jumping, what single number

does he or she keep jumping over?"


            This being ludicrous, Stein's compromise, a couple of "it"s and a few repetitions of "the flea", gets through the exercise with a minimum of trouble.  Of course, there are those who will say that this example is badly contrived, since a flea is not human and therefore may as well be treated as neuter anyhow.  Whenever I, or any other defender of the status quo ante of gender-indifferent pronouns, bring up an example to illustrate the awkwardness or perversity of the new linguistics, my opponent is able to point an alternate phrasing that is smooth enough and more or less says the same thing.  In this case, it might be that a flea ought not to be referred to by a personal pronoun.  Yet there are matters of style that will not be denied.  What if I want to personify the flea here because it means something to the quality of the exposition.  Shall this be forbidden?  Yes, it shall.


            It is instructive to contrast Stein's book with another book published by McGraw-Hill, but in 1962:  Calculus, by Ralph Palmer Agnew.  Here on page 300 we have, as Problem 21:  "An observant senator observes that if he hires just one secretary, she will work nearly 30 hours per week but that each additional secretary produces conversations that reduce her effectiveness.  In fact, if there are x secretaries, x not exceeding 30, then each one will work only 30 - (x2/30) hours per week.  Find the number of secretaries that will turn out the most work."


            I do not believe this problem could stand in today's climate, in which calculus-book senators must be as often female as male, and their secretaries as often male as female.  On the same page, Problem 28 begins, "The x axis of Figure 5.296 is the southern shore of a lake containing a little island at the point (a,b), where a > 0.  A man who is at the origin can run r feet per second along the x axis and can swim s feet per second in the water.  He wants to reach the island as quickly as possible..."


            Any modern textbook selection committee would have to pass this book by.  Can only men be senators, and swim and run, while women must serve as their secretaries? There is no doubt that Agnew pictured the senator as male and his secretaries as female, certainly a typical picture even though there were a couple of female senators in his time, and even though the chief secretary of an important executive (male or female) was more likely to have been a man than a woman.  But Agnew's mind was on mathematics, not sex, or justice.


            Had he written Problem 21 to read, "An observant Senator observes that if she hires just one secretary, he will work nearly 30 hours per week...", the reader's attention would certainly have been diverted to thoughts about the probable sex of senators and secretaries, and to curiosity about Mr. Agnew and his purpose in writing so strangely.  Most readers, anyway.  Some, including many of today's college students, have been educated by so many textbooks enforcing the equality of the sexes that they recognize the phenomenon at a glance.


            Reading about a female senator and her male secretaries, today's freshman will immediately recognize the advertising, or consciousness-raising, function of the pronouns used.  They are unrealistic if taken literally as descriptive of a typical case -- even a freshman will recognize that -- but they should be realistic.  It is a crying shame, he will have been taught, that the preponderance of secretaries are female and their employers male; therefore the equalizing language is necessary, to help, by changing attitudes, to bring about a more just future.


            In a calculus book?  Agnew's purpose was to teach about derivative and maxima, and neither to exalt nor to degrade women.  His language was not abusive.  His casual acceptance of the nature of his world was deliberate, designed to keep the student's mind on the structure of the mathematical problem.  If the purpose of a calculus book is to improve society in other ways, to advance the cause of women, say, or of world peace, then there might be some point to the introduction of distracting elements, but this is a political decision that Mr. Agnew was, in 1965, unwilling to make.  Mr. Stein, in 1985, had had it made for him.


            Now the apparently sex-related references in the two examples from Agnew, i.e.  (a) the (male) senator and his (female) secretaries, and (b) the "man" who combines running and swimming to reach an island, are not actually comparable, and here is a most important and misunderstood point in the debate over gender in English.  While Agnew's senator was clearly pictured as male, his use of "man" to describe the swimmer is purely conventional.  It could have been an otter and the problem would have been no different, nor would the use of "otter" have shocked the reader into political musings, as the use of "she" would in the senator problem.  The political purpose served by making imaginary senators as often female as male can be accomplished without corruption of the language; one says "she" when a woman is definitely meant, and "he" when the man is referred to, and if the secretary is to be a man, "he" is merely correct, not a convention.  One can argue about the

advisability of misrepresenting the sexual composition of the Senate in

calculus books, but there is nothing to argue about when, the choice having been made, the pronoun "she" describes that choice.


            On the other hand, the swimmer who wishes to minimize his travel time might be a man, woman, child, otter or water-bug.  I have seen all of them invoked in calculus book minimum problems, and have at all times been indifferent to everything about them except their postulated speeds when running and when swimming.  But the McGraw-Hill Guidelines would require me to have begun the present paragraph, "On the other hand, the swimmer who wishes to minimize his or her (or its) travel time might be a man, woman, child, otter or water-bug."  This is not the same thing as requiring me to depict senators as female, it is a purely linguistic requirement, designed to give the feminine-gender pronoun equal billing with the masculine, as if pronouns could suffer from unfair discrimination, and had votes.


            In Agnew's Problem 26 he postulates a man as the runner-swimmer, and therefore has no need of the purely conventional "he"; his "he" means "he."  That is because Agnew has chosen to make his problem definite, clearly pictured, "true to life".  He could have been less definite, and written only of "a swimmer," as I have just done.  Faced with the McGraw-Hill "he or she" a few times in the next sentence or two, I think he would have changed his mind, and, looking back to see what sex his last human example had, choose a boy, girl, man or woman accordingly.  This is the solution recommended by those who advise textbook writers who protest that "he or she" is awkward or distracting.


            But what advice does McGraw-Hill have for me?  I cannot make my swimmer any one of those things, because my discussion concerns the generic swimmer whose sex is still occupying Mr. Agnew's attention.  We are writing a book here, and we are discussing what is about to become Problem 26.  There is a swimmer in that problem, and the swimmer has a certain speed on land and another in the lake.  We are now discussing whether to make the swimmer a man, a woman, a boy, a girl, an otter, or a water-bug.  The choice has not yet been made, and I'm getting right weary of using the word "swimmer" -- for the sixth time in this paragraph.


            I prefer to use "he" and "him", pronouns invented by our ancestors to simplify and shorten our speech and writing, or "she" when it is known the antecedent is feminine.  A few years ago, I might have written, "There is a swimmer in that problem, and he has a certain speed on land and another in the lake.  We are now discussing whether to make him a man, a woman, a boy, a girl, an otter, or a water-bug." 


      Today's textbook writers are forbidden this locution, and today's schoolchildren are taught it is wicked.  People write letters to the editor of their local newspapers saying that "he" means masculine and should be avoided for clarity.  A few years ago this was disingenuous, but today it is fast becoming true.  Deprived of this convenience, this use of "he" that English has enjoyed for centuries, this usage that was as natural to Virginia Woolf as it was to James Thurber,  our schoolchildren now learn to write "he or she," but in fact will say "they" in these contexts. 


            Not only children, but professors in faculty meetings; and it invades the printed page too, when the minutes are taken verbatim.  From the (confidential) minutes of the Faculty Senate of the University of Rochester, 17 May 1988:  "Steve, I would like to ask another question about the last sentence.  Does that mean if we adopt it that if a student fails to pay his or her tuition, we cannot disenroll them?"  This shift in the number of the antecedent, turning it into plural, has the obvious advantage of avoiding what is declared to be the sex specific pronoun "he" or "she," thereby evading the wrath of McGraw-Hill and its (?) allies.


            One reason the revolution in pronouns has had the success it has in fact had is that it does not inconvenience people whose speech and writing does not run to abstraction; and they are probably a large majority.  If all my conversation is of the form, "He said... and then she said..." I have no need of indefinite or hypothetical antecedents.  My antecedents are all named John or Mary, or the mayor, or Jack's cow; it is clear what pronouns mean to me.  Another form of writing not inconvenienced is scientific writing where the subject matter is not human, where one speaks of molecules, forces, carcinogens or Grassmann algebras.  "It" will do the job here, though sometimes the algebra or molecule is named A, and repetition of its name is as easy as the use of some pronoun.


            The person left out of the feminist calculus is the writer who, like me, writes of people, not John and Mary, but of typical or generic specimens of some sort.  A diabetic comes into my office; do I tell him or her to bare his or her arm while I take his or her blood pressure?  ("A simple 'his or her' will do," wrote Linda Cotton.)  As a real doctor with a real diabetic before me (or him or her) we have no problem;  our patient is named Martha-Jane McDougall, and by golly we tell her to sit down.  As a writer, imagining the generic doctor's behavior when the diabetic comes into their office, or his or her office --- well, you see what I mean.  I'd like to keep my mind on the blood pressure, and not on the war of the sexes.


            In other words, "a simple 'his or her'" will not do, and people are not in fact taking that way out.  They are alternating uneasy uses of 'his or her,' and sometimes 'it,' with ungrammatical use of 'their,' and filling in the cracks with repetitions of the antecedent noun or some synonym if possible.  All of it is being done self consciously, which means that these devices have not yet fully entered the language.  What the future will bring, in the matter of pronouns, is not clear.  The plural "they" seems to be ahead.


            One can make a case for this usage, in fact, by saying, simply enough, that "they" is not ungrammatical, merely sometimes singular and sometimes plural, just like "you."  Context will tell which is meant.  But there are difficulties here which people have noticed in daily life:  Having begun the sentence with a singular noun, shifting to "they" or "their" entrains plural verb-forms from then on, and into succeeding sentences, finally obscuring the situation so badly that a new start is called for.  I could provide an example, but it would be tedious.


            The other aspect of the McGraw-Hill guidelines is quite a different matter.  The movement to picture women in non-traditional roles in our textbooks and our abstract exposition in general, exactly in order to facilitate their entry into these roles in the real world, by making it seem less strange to the next generation, is as well under way as the campaign against smoking.  Women are not the only subject of this propaganda, of course, since now Hispanics (as Spanish speakers are called who do not live in Spain) and blacks are also to be pictured in the textbooks as nuclear physicists and Major Generals as frequently as their fraction of the general population should in "justice" place them there.


            This is merely a political matter, not an assault on language, and no more to be objected to than the imposition, by an important publisher upon the writers it agrees to print, of any other political slant.  It cannot harm literature itself unless it becomes a conspiracy, and even then the bias will probably be replaced by some other in time.  During Victorian days one could not mention anything connected with sex, and the Victorian analogue of the McGraw-Hill

guidelines required the word "limb" rather than "leg," when a woman's body was in question.  A family newspaper or magazine could not print any of the following words:  syphilis, condom, vagina, orgasm.  But with time, things change.  Today's family newspaper or magazine has other taboos, and cannot print any of these words:  chairman, schoolmarm, cripple, idiot.


            In the year 1982 I collected some stories I had been writing over the years to make a book, a book of stories for children, based on tales I used to tell my own children, two daughters, many years earlier.  Naturally they were rather dated, but in searching for a publisher I didn't really count that a disadvantage.  The local color and the way the world looked in 1952 were part of the story.  Alice in Wonderland is quite understandable to today's children, after all, even if there are no references to computers and airplanes.  However, one professor of education to whom I submitted the manuscript for advice pointed out to me that I had pictured the mother of the little boy protagonist as

being mostly in the kitchen, while the father was seldom home during the day.  The father took his son to places like traffic court and the Sears hardware department, whereas the mother showed him how to make frosting for the cake.  This division of labor, said the professor of children's literature, is unacceptable in a children's book today.  The stories are interesting, but really.  One cannot sell stories with such a setting. 


            In fact my stories did not find a publisher, though the reasons for this might have been other than political.  One might suspect that the absence of a daughter in the family described was also a mark against the book, though the professor had not mentioned that.  The editors who rejected the book were noncommittal.   


           A few years later I wrote another book, of a totally different nature.  It was a manual on academic dishonesty among undergraduates in college, and what to do about it.  In this matter I had some expertise, having been occupied with this question for many years at my own university.  One editor to whom I sent it telephoned me with some enthusiasm, asking for "first refusal rights" pending the approval of referees to whom she proposed to send it.  The book itself was great, she said, but before she could send it to referees she needed my agreement in principle to the removal of the unacceptable pronoun "he"

and its relatives in passages dealing with hypothetical people.  It would be  easy, she said, and the publisher would help me.  Hardly any work at all.  When I refused, saying I preferred the traditional usage, which in fact was still to be found five years ago to a degree that is surprising today, she said with regret that in that case the Rutgers University Press could not even consider my manuscript.


            Censorship?  No.  The government is not itself part of this conspiracy, and I can publish anything I want to, myself.  We are in fact better off than in the time of Mark Twain, who chafed at the restrictions he was under in matters concerning sex and religion, no matter who the publisher might be.  And to compare the today's publishing world with what past tyrannies accomplished, the worlds of Stalin and Hitler, or even the much gentler Vatican of the time of Galileo, would be excessive.  Still, it is sad.


                  Ralph A. Raimi