Minimum numbers are always being asked for, so various individuals come up with the "minimum" number of a species. With most shoaling fish, and certainly with all species of Corydorus, the more the better for the fish. A group of 10-12 of a species does have proven (by scientific study) benefits to shoaling fish.
Cories are a bit different, in that most species will happily chum around with other species. This does not apply to most shoaling fish, such as the characins, cyprinids, etc., which need sufficient of their individual species to be well-adjusted and healthier. There are however some issues with multiple species of cory.
We must first recognize that the genus Corydoras is polyphyletic; this means that it contains descendants of more than one ancestor. Eventually this will get sorted out, and we will have nine genera in place of the one. Species within the same lineage, which would normally be separated into genera, interact a bit differently, and they can hybridize if there is not a group of males and females of each species. This is one problem, polluting the gene pool, something we should always avoid.
A second issue here is that males of the species in lineage 1 become quite territorial especially when they spawn, and spawning cories will be the norm in any healthy tank with several of them. Fortunately, species in lineage 1 are much less often seen in the hobby that are species in some of the other lineages. The species in the genus Scleromystax, which used to be in Corydoras, can be very rough with one another. But aside from these specifics, multiple species are basically OK, but try to have a few of each species if at all possible.