From the symposium volume, "The role of axiomatics and problem solving in mathematics", The Conference Board of the Mathematical Sciences, 1966, being a report of a sub- conference at the quadrennial International Congress of Mathematicians in Moscow.

Peter Lax p114

"I will take my illustrations from the lower end of the high school curriculum:  the multiplication of fractions and the multiplication of negative numbers, two somewhat dry subjects...."

He multiplies a/b by c/d by partitioning a rectangle into suitable subrectangles, counting them, and seeing how many are needed to represent the product of a/b and c/d; he thus "proves" the definition ac/bd to be the right one.  As for the arithmetic of possibly negative numbers, he imagines a train on a track, on some point of which a controller takes measurements of distances and times from the control station at a certain moment, using ∆x/∆t as a measure of average velocity; to avoid using four different formulas, as might be required to avoid the use of negative numbers, he measures time from 0 and distance from some point along the route (+ or -) and gets a single formula that serves the purpose no matter where he stands relative to the two positions and the two times.

To Lax these two problems are what dictate the algebraic rules concerning fractions and negatives, and not the fortuitous fact that the definitions preserve some of the algebraic laws already known to be valid for whole numbers, and positive numbers.

“Had we defined multiplication or division otherwise than we did these operations would be useless for these particular applications; on the other hand if these new operations did not share the usual properties of multiplication and division, then we would be unable to manipulate them [conveniently].”

p115:  "In contrast to this approach through problems, the current trend in new texts in the United States is to introduce operations with fractions and negative numbers solely as algebraic processes.  The motto is: Preserve the Structure of the Number System.  I find this a very poor educational device: how can one expect students to look upon the structure of the number system as an ultimate good of society?"

In another place (p. 115) Lax writes,

"Of course I agree that many of the traditional problems of high school algebra are just as artificial [as the logical exercises leading from given hypotheses to predictable conclusions, usually of trivial interest to students], such as the ones involving perverse children who, instead of disclosing their age as asked, relate it in obscure ways to that of their brothers, sisters, parents, etc.  The remedy is to stick to problems which arise naturally; to find a sufficient supply of these, covering a wide range, on the appropriate level is one of the most challenging problems for curriculum reformers.  My view of structure is this:  it is far better to relegate the structure of the number system to the humbler but more appropriate role of a device for economizing on the number of facts which have to be remembered."

And in conclusion (pages 115, 116):

"What motivates textbook writers not to motivate?  Some, those with narrow mathematical experiences, no doubt believe those who, in their exuberance and justified pride in recent beautiful achievements in very abstract parts of mathematics, declare that in the future most problems of mathematics will be generated internally.  Taking such a program seriously would be disastrous for mathematics itself, as Von Neumann points out in an article[1] on the nature of mathematics – the most perceptive ever written on the subject -- it would eventually lead to rococo mathematics.  As philosophy it is repulsive, since it degrades mathematics to a mere game.  And as guiding principle to education it will produce pedantics, pompous texts, dry as dust, exasperating to those involved in teaching the sciences.  If pushed to the extreme it may even cause the disappearance of mathematics from the high school curriculum along with Latin and the buffalo."

Lax was and is not alone in his view that presenting mathematics in strictly axiomatic form is poor pedagogy at grade school levels, for all that it is philosophically necessary at some point, and enables some part of our profession, maybe even all of us, to pay close attention to the details of reasoning behind all putatively correct mathematical conclusions.  Lax had already seen some of the “dry as dust” textbooks produced by some of the folks “with narrow mathematical experiences.”

Me, too.  I once inadvertently conducted a small experiment by asking a number of acquaintances the question, “How do you tell students why the product of two negative numbers is positive?”  I wasn’t looking to verify some hypothesis; I was just interested.

From Charles Rickart, a Yale mathematician, I got this answer (this was in about 1956):  “I tell them that if you have two dollars deposited in each of three banks, you are ahead by six dollars; if you owe two dollars to each of three banks, you are behind by six dollars; but if there are not three banks to each of which you owe two dollars, why, then you are ahead by six dollars, aren’t you?”

From Henry Pollak, a Bell Laboratories mathematican I questioned on this and related matters many years later, in a telephone conversation of about 1996, I got two different such stories before I cut him short, since our conversation was really about something else, that I was more anxious to get to in the time we had.  I no longer remember his illustrations in detail but both of them involved such things as the distance advanced by a train unwittingly headed in the wrong direction, but forced to back up for a while.

Neither Pollack nor Rickart mentioned the field axioms!

But when I asked a math teacher of less mathematical experience or insight than Rickart or Pollak, his almost instinctive answer (this was also in 1996) was that the proposition was an easy consequence of the distributive law and the definitions concerning negatives; and he was certainly able to produce a lucid proof on that basis.  Curiously, however, he was unable to understand my next question at all: I asked him how he knew the distributive law applied to negatives, or mixtures of negative and positive numbers.  He became angry, thinking I was mocking him.

I can think of no better illustration than this, of the danger Peter Lax was calling to our attention in the final sentences of his article.  The field axioms are not the cause of ab = (-a)(-b); they are a summary – a miraculous fact – of properties we have other reasons to observe and want to codify for computational purposes.  It is wonderful that so many things fall under the field axioms, but they aren’t the Ten Commandments, after all.  There are rings that don’t happen to be fields, and Gibbs had to give up several properties that would have been handy for cross products, could they have been made consistent with the rest of his system of vector analysis.  He was driven by the geometry, not a preconceived axiom system, to stick with the ungainly product he seemed to have, whether he liked it or not.

It was a bit of a fad in the 1990s, among NCTM devotees, to point out that children who used the mistaken formula a/b + c/d = (a+c)/(b+d) got a result that had meaning in baseball, for example:  If a batter scores a “hits” out of b “at bats” on one day his batting average is a/b, and if he scores c out of d on the following day, his average for the following day is c/d.  To get his average for the two days taken together, the computation is (a+c)/(b+d), and not the average of the two earlier averages (or their sum).  How wonderful, but why call this combination by the name “a/b + c/d”?  In adding fractions “+” already has a meaning, a different meaning, and one of many uses.  It is as idle to call the new operation “addition” as to say “one plus one is eleven” – an old schoolyard joke.

Even if this new operation (of “averaging the averages in baseball”) were of such importance as to warrant a new symbol, and some systematic study of its arithmetic properties, one must recognize that while it says something about fractions, it says nothing sensible about rational numbers.  There is more value in daily life in calling equivalent fractions equal, and making sure that our comments on such numbers are independent of the representative fraction used, than in trying to use every fraction as a separate entity in computation.  I don’t believe Lax would argue with that.  In the daily measurement of lengths, adding ¾ of an inch to ¾ of an inch will simply not give a total length of 6/8 inches, whereas adding 1 inch to 1 inch gives 2, the length we can verify by direct measurement.  We want fraction addition to reflect the additive properties of length, whether we make our measurements in units that produce integral results or by some other units that give fractional values for the same lengths.  Our rules for addition of fractions can be derived from suitable axioms, to be sure, but the source of the rules is in the measurement, not the axioms.  Just as dictionaries are compiled after a language has been in use, and not the other way round, so it is with mathematics and its axioms.  Their main value in mathematics, as elsewhere, is to codify what you have already taken to be true, i.e. taken as a sufficient, though preferably simple, description of what is at issue.  Axioms are a way to keep the basics in mind, and later help understand truths one might not have thought of without the clarity brought on by our codification; but to teach the grammar before looking at any of the literature is a mistake of philosophy and pedagogy alike.

Ralph A. Raimi

15 September 2004

[1] Von Neumann, The Mathematician, reprinted in vol. 4, p.2053ff of The World of Mathematics, an anthology edited by James Newman and published by Simon and Schuster, 1956