Just one assumption: if a private organization provides grants for mathematical research, presumably it admits that mathematical research by itself does make sense.

This granted, if you manage to explain how results of your research might be useful for some other researchers (either in mathematics or in any other area that the organization under consideration might find equally sensible) whose results in turn would not be entirely isolated from the outside world, then this should suffice for justification.

There can be of course cases when grants are provided under certain conditions, to investigate this and that particular problem. But then trivially either you do not have anything to do with it or you show evidence that your previous experience gives you chance to tackle this particular problem, and do not need any further justification either.