[longshadow]

In it, formal proof would not be the only acceptable way to establish mathematical knowledge."

Slippery slope (no pun intended), IMHO.

[Doctor Stochastic]

One of the new uses of computers is in functional iteration. This is rather hard by hand. (Poincaré did seem to do it though.)

One can study things like the fixed points of Ax(1-x) for example.

Of course, proofs are necessary at some point.