> Happy New Year to all on this list!
I agree.
> Second, concerning synthetic geometry (a term I like and still use),
I agree again, but didn't know the term was on its way out. Is
it? How do you know?
> Resolved: The Greek synthetic
> approach to geometry is outmoded and clumsy and should no longer
> be taught.
In 1959, after "The New Math" was off to a flying start in the USA
there was held (by the OECD -- or OEEC? -- something -- the "European
Union" anyhow) in a place called Royaumont in France a conference of
European mathematicians and math educators (the latter category not as
clearly distinguished from the former then as it is now) to consider
reform of school mathematics. Ed Begle of SMSG was also there, but it was
an essentially European meeting. The Proceedings were published in 1961
under the OECD (OEEC?) imprint.
Jean Dieudonne there gave the speech which set the tone for the
next few years in much of Europe, and reinforced recent developments in
America: he advocated teaching high school mathematics as a deductive
system in the most economical way. Among other things, he said, "Euclid
must go!", meaning that a *sufficient* set of synthetic axioms were much
too cumbersome for rigorous use, while Euclid as taught in the 19th and
20th Century schools was based on too few axioms plus an uncertain visual
intuition. He recommended beginning with R, either axiomatically or built
up from something simpler (Peano's N), and *defining* Euclid's space as
RxR with the Pythagorean metric.
Now, this *will* do the job, even though it would take quite a few
unintuitive theorems and proofs to bring us to recognize this thing as
Euclidean space. No American "New Math" project dared do it this way.
All the ones that tried to be rigorous offered a compromise between
Euclidean phrasing and numerical axioms. All these programs failed,
however, and except for the Bourbakist Grands Lycees in France I don't
believe anyone in Europe really tried to axiomatize Euclid in the analytic
manner alone, for school math purposes.
I agree that the customary use of Euclid's system has logical
flaws, but I cannot see any analytic alternative as being more rigorous at
the K-12 level because any development intended for high school (or
earlier) students will surely leave as many holes in the formation of R as
any 19th Century Euclid has in the traditional postulates for points and
lines in the plane. A rigorous development of Euclid is simply not
possible for children of that age, at least on a mass scale.
On the other hand, nobody has found any false propositions in
Euclid, and once one has got to I;47 the process can become pretty
rigorous and intuitive at the same time. The theory of proportions
presents a problem which can be handled by asserting that the similarity
theorems are as true for incommensurates as for commensurates ("You'll
learn why when you get old enough"), much as we assert the existence of
irrationals on the line in the present analytic dispensation ("You'll
learn why when you get old enough"). One *could* proceed from a rigorous
(e.g. Hilbert) Euclid to show that there is a 1:1 mapping of R (assuming
we know what R is) onto the points of any line, with distances on R
corresponding to congruence of segments on the line, and that identifying
the points of the plane with coordinate pairs defined by distance
involving two such lines, thus arriving at Dieudonne's formulation and
thenceforward using whichever scheme one wants, or a mixture (as indeed
any sensible person today does).
All this was the subject of much pedagogues' debate in the 1960s,
and many textbooks, some of them written by very knowledgeable
mathematicians, and often carefully tested in actual classrooms with
teachers of varying abilities, designed to answer objections to the old
schemes and introduce better ones. The upshot was that the whole
enterprise bombed ("bombed" is a current American idiom meaning "failed
spectacularly"), and most textbooks today have nothing at all that can
bear the name of geometry. I therefore ask: Abolish Euclid? We don't
*have* Euclid to abolish. But I believe we'd be better off if we did.
To professional mathematicians the Dieudonne scheme is clear and
good, and is in fact what we use as a foundation even when we argue
synthetically (secretly drawing little triangles on the blackboard out of
sight of the class), since using RxR + Pythagoras we can easily prove the
first few "Euclidean" theorems and argue from there with or without
numbers (or pictures) as we please; furthermore, we *need* the analytic
scheme if we are to have extensions to manifolds, etc., which Euclid knows
nothing of. Hence Cooke's proposition, echoing Dieudonne, is attractive.
But in SCHOOL? Whatever would we do with the "proofs" of the
Pythagorean Theorem, of which there seem to be hundreds, if we take it as
an axiom? Our whole language would suffer. It is bad enough as it is,
when modern children with the blessing of the National Council of Teachers
of Mathematics learn the sum of the angles of a triangle by cutting the
corners off a paper triangle and laying them out to make a line. This may
be the astronomer's way of checking whether Euclid was right, but that's
science, it is said, not mathematics. Does that make it wrong?
Yes, we have no right to denigrate science as the origin of
mathematics. The Pythagorean formula can be checked by measurement as
easily as the postulate that two points determine a line; why *not* use it
as an axiom? There is nothing against it from a *logical* point of view.
But while Dieudonne has won a *de facto* victory, it has been quite
Pyrrhic, since it has eliminated Euclidean reasoning without substituting
any other sort in fact. Look out there: The kiddies do not understand R
any better than they did before 1959, but have lost all vesatige of
Euclid in the reform process; and mathematical reasoning *as such*
has all but vanished from the schools. I believe the price of Dieudonne's
efficiency is too high, and that Euclid should be retained, for the mental
exercise it affords and its glimpse of beauty bare. Until there is
evidence that such parts of Euclid as were traditional in 1900 are
necessarily a waste of valuable school time and necessarily delay the
mathematical competence of school children, these virtues should be
sufficient.
Ralph A. Raimi Tel. 716 275 4429 or (home) 716 244 9368
Dept. of Mathematics FAX 716 244 6631
University of Rochester Webpage http://www.math.rochester.edu/u/rarm
Rochester, NY 14627 (Webpage contains links to papers)