FOM: Re: Midwest Model Theory Meeting

[Vaughan Pratt]

>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.

The inventor of Boolean logic also invented the finite difference calculus
("A Treatise on the Calculus of Finite Differences", first edition 1860,
second edition substantially reworked by John Moulton 1872), whose toolkit
provides induction-free methods for a much larger class of such problems,
including summing n^i for fixed integer i and much more.

Vaughan Pratt