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Replies: 49   Last Post: Jan 13, 2012 2:37 PM

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 Wayne Bishop Posts: 5,465 Registered: 12/6/04
Posted: Dec 23, 2011 7:09 PM

At 10:12 PM 12/22/2011, Dan Christensen wrote:

>The axioms for geometry and real numbers are not
>built into the DC Proof system. The numbers of
>axioms alone would overwhelm the beginner. For
>the axioms of geometry, see the work of Tarski
>or Hilbert. You can find a version of the axioms
>for the real numbers in the DC Proof/Samples directory.
>
>Beware, though. In my experience, even
>elementary results in the Euclidean plane can
>quickly explode into proofs of several hundred,
>even thousands of lines. So, I really can't
>recommend geometry for the beginner just
>learning how to write mathematical proofs.
>Better to stick to logic, set theory, and
>elementary number theory, the axioms of which
>ARE built into DC Proof. (See "To the Educator" at my homepage.)

All true but I come to exactly the opposite
conclusion about beginners just learning how to
write mathematical proofs. I continue to believe
(as with far better mathematicians than either of
us over the last couple millennia) that Euclidean
plane geometry remains a wonderful place to
introduce mathematical proof to
beginners. Moreover, I am absolutely convinced
that much of the reason that US students pursuing
degrees in mathematics have so much difficulty
with mathematical proof is the loss of a good (in
spite of its logical problems) introduction to
proof in the context of the traditional high
school (usually sophomore) course in Euclidean
plane geometry. So-called "modern" introductions
to geometry that have replaced it are so awful as
to be useless if not outright
counterproductive. I have been teaching our
upper division university course entitled "Modern
Geometry" for the past 40 years and the
performance has gotten steadily worse over the
decades. With very few exceptions, students who
Mideast, Far East, Russia, Ethiopia, etc. - are
the best students of the class and that did not
used to be the situation, say, back in the 70s
when we had lots of students from Iran. A nice
example was my first student assistant after I
was left quadriplegic and in need of competent
classroom assistance. She was from Lebanon, a
beginning high school mathematics teacher there
who had lost all of her school records when she
left Lebanon at the height of the conflict there
years before. With her children now in middle
and do whatever she needed to get US credit for a
bachelor's degree in mathematics. At the time, I
had not yet figured out how I could write so she
needed to score my neighbors as well as write
stuff on the board that I had not been able to
anticipate and use Geometer's Sketchpad or
Cinderella (wonderful for examples in the
Poincaré disk model for hyperbolic geometry).

I have always felt the need to start the course
with a quick review of stuff learned in the
traditional high school course but over the years
it has become less and less "review" than initial
introduction and, although they have earned a C
or better in a course entitled "Math Notation and
Proof", they have little idea of what
mathematical proof is really all about. On the
first quiz, she came back from scoring absolutely
amazed, better said, appalled at some students
inability to even regurgitate easy stuff that was
both in the printed material I had distributed
and discussed in class. Her assessment, "We learned that stuff in 7th grade!".

Beginning as you propose strikes me as being
perceived by most students as an exercise in
abstract nonsense, not in providing some level of
understanding of the solid foundation for
mathematics that proof provides. Beyond that,
except for mathematical logic itself, no
professional mathematician appeals to axiomatic
set theory or formalistic logic in their thinking
nor in their publications. More than a few have
no idea what you're even talking about once you
get beyond, say, discussions about the Axiom of Choice.

Wayne

Date Subject Author
12/11/11 kirby urner
12/12/11 Dan Christensen
12/12/11 kirby urner
12/12/11 Dan Christensen
12/13/11 kirby urner
12/13/11 Dan Christensen
12/13/11 kirby urner
12/13/11 Dan Christensen
12/13/11 kirby urner
12/13/11 Joe Niederberger
12/13/11 Dan Christensen
12/13/11 kirby urner
12/13/11 Dan Christensen
12/15/11 Joe Niederberger
12/15/11 kirby urner
12/15/11 Joe Niederberger
12/15/11 kirby urner
12/15/11 Joe Niederberger
12/15/11 kirby urner
12/15/11 Joe Niederberger
12/15/11 kirby urner
12/15/11 Joe Niederberger
12/15/11 kirby urner
12/16/11 Joe Niederberger
12/16/11 kirby urner
12/16/11 Dan Christensen
12/16/11 Joe Niederberger
12/17/11 Joe Niederberger
12/17/11 kirby urner
12/17/11 Dan Christensen
12/18/11 Wayne Bishop
12/18/11 Joe Niederberger
12/18/11 kirby urner
12/23/11 Dan Christensen
12/23/11 Wayne Bishop
12/24/11 Louis Talman
12/23/11 Joe Niederberger
12/23/11 kirby urner
12/23/11 Wayne Bishop
12/24/11 Joe Niederberger
12/24/11 Wayne Bishop
12/24/11 Joe Niederberger
12/24/11 kirby urner
12/24/11 Joe Niederberger
12/24/11 Wayne Bishop
12/24/11 Dan Christensen
12/25/11 Dan Christensen
12/25/11 Dan Christensen
1/13/12 Joe Niederberger
1/13/12 kirby urner