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

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