FOM: Re: Midwest Model Theory Meeting

[Charles Silver]

-----Original Message-----
From: Harvey Friedman <friedman at math.ohio-state.edu>
To: fom at math.psu.edu <fom at math.psu.edu>
Date: Monday, November 08, 1999 8:04 AM
Subject: FOM: Midwest Model Theory Meeting




Harvey wrote:

>7. Baldwin commented that a lot of nice proofs in mathematics avoid
>induction and make good use of intuition. The example he gave was
>1+2+3...+n = n(n+1)/2. You can draw a staircase and notice that you have
>exactly half of the rectangle of points of width n and height n+1. Voila!
>No induction. Baldwin's question: how do we formalize this common approach
>to mathematics.
>
>I commented that indeed most mathematicians avoid induction in favor of
>other techniques. In fact, Rota was fond of saying that if one has to
>resort to induction, then one does not have a good proof.


    Why not just use that old technique supposedly intuited by Little Gauss
(when he was eight?), which can be used to solve all problems of this
general type?   In particular:

          1  +   2   +    3    +  ... +(n-2) +(n-1) +n
   +     n  +(n-1) + (n-2) + ...  +  3    +   2    +1
-----------------------------------------------------------------
      (n+1)+ (n+1)+(n+1)+...+(n+1)+(n+1)+(n+1)

    Since the sum above, n(n+1), represents twice the number we're looking
for, the result is half that, or  n(n+1)/2.


Charlie Silver