Understanding Mathematical Proof Techniques / Strategies...

[Stephen Camp]

Note: cross-posting

I am working through several texts on Splines, interpolation, mathematical
approximation on my own. Some of the texts are:
A Practical Guide to Splines by Carl de Boor
Curves and Surfaces for Computer Aided Geometric Design,
A Practical Guide by Gerald Farin
Numerical Analysis by Kincaid & Cheney
etc...

Anyways, I find the reading pretty heavy duty for me, and I have trouble
following the proofs. To really 'get' them, often times I have to do a
lot of work with pencil and paper to satisfy myself that what is 'proved'
in the book is in fact correct. In particular, often times the 'proofs'
are not what I am used to / was taught in high school or college: e.g.
we prove by "induction" the following...

I have a hunch that there are several / many types of proofs or strategies
followed when proving something... but aside from the typical deductive
calculus / geometry proof, I feel lost. For example, on pp.34-35 of
Farin's book on splines (1st ed), he offers an 'inductive' proof of the
equation relating the de Casteljau recursive algorithm for Bezier curves
to a series (summation) of the polygon points multiplied by Bernstein
polynomials. The proof substitued the equation to be proved into the
recursive definition of the de Casteljau algorithm, then massaged the
new equation into a form that allowed one to apply the recursive
property of the Bernstein polynomial to then recover what was being
proved... At first, it seemed like a circular argument to me... but now
I understand it and accept it, but it wasn't the deductive, iron clad
approach I learned in geometry and saw in most of my calculus proofs...

So, I ask: are there any articles/books/texts/anything that teaches one
ABOUT proofs, what kind of proofs exist, proof techniques (or strategies)
and why they are valid. I feel that if I understand the underlying
strategy and why it is valid, then I have a much better chance of following
and accepting proofs as they are offered...

Also, are proofs involving series, and/or recursive formulas or algorithms
more often inductive than deductive?

Thank you very much for any help you can give me. I'm sorry if this has
been long winded.

-steve camp
hob...@spacemanspiff.den.mmc.com

[M. J. Saltzman]

In article <jfrancoe-1...@jfrancoeur.mitre.org> jfra...@mitre.org (Joe Francoeur) writes:
>
>> So, I ask: are there any articles/books/texts/anything that teaches one
>> ABOUT proofs, what kind of proofs exist, proof techniques (or strategies)
>> and why they are valid.
>

>Yes. Sorry I can't give you an exact title, but it's something fairly
>obvious like "How to Do Proofs". It's in paperback - I saw it in Barnes
>and Noble. Incidentally, their math/science book section is quite good,
>considering it's a "general" book store.

Daniel Solow, _How to Read and Do Proofs_(2/e), Wiley (1990).
ISBN 0-471-51004-1.

There's also George Polya's _How to Solve It_, Princeton (1971).
--
Matthew Saltzman
Clemson University Math Sciences
m...@clemson.edu

[Nelson G. Rich <x327>]

I believe the book to which you refer is "How to Read and Do Proofs",
by Daniel Solow, 2nd edition, John Wiley & Sons, Inc., 1990.

--
Nelson G. Rich
Department of Mathematics and Computer Science
Nazareth College
4245 East Avenue