Re: Geometry, algebra, and computers

[karshner@stmarytx.edu (Karshner)]

> It seems that prior to the time of Newton all proofs were geometric.
> Newton lived at a time of transition from geometric to algebraic >proofs
and apparantly delayed publication of his papers while he found
> the 'correct' geometric proof for propositions already proven 
> algebraically.  Modern communication may be opening up to us new forms
> of proof.  We may live at a similar time of transition.  If we do, 
> then we will be employing double proofs in the same way that Newton
> did.  An example is the observation of beats between two waves.  These
> are easily seen in a computer simulation but we still feel it 
> necessary to derive the algebraic expression and explain it to the 
> students.
>
> My question is:  Do you employ any 'double' explanations in your >physics
teaching?  One of which is algebraic while the other uses
> modern communications such as videos and computer simulations.

Michael,
        You missed part of Newtons motovation for using geometry. He felt
that classical geometry was a proven system that had stood the test of time
and he wanted to append his theory on to that structure and not the new
fangled Cartesian Geometry with whose followers he had many dissagreements.
It was removing the fluxian theory (his calculus) that caused the delay. His
fluxian theory looks far more like geometry then the Leibnizian calculus we
use today. 
        As to your question, I use all three. I just retruned from a class
where we were solving mechanics using the energy point of veiw. I solved
some examples using algebra and a little calculus. Showed the importance in
finding potential minima for stability using both the calculus and geometry,
and then used Mathcad on the computer to find the mimima of a particularly
pathelogical potential. I view
modern simulations and computer tools as just other tools to add to
my tool bag along with calculus and geometry. Which ever tool gives the
greatest insight is the best one for the job. 
        I do have some reservations about modern computer simulations as
they are only as good as the models behind them. In addition proofs done
using computers are only as good as their programs and I know of no way to
prove a program is solving things correctly, but I know how to prove things
mathematically or geometrically.
Gary
Gary Karshner
St. Mary's University
San Antonio, Texas 78228
karshner@stmarytx.edu