Using whole uncored apples in chutney?
Apple cores contain more pectin than the rest of the apple, so including the cores will likely get you a thicker, more gelled chutney. 

With enough cooking they'll indeed soften, as long as it's an apple variety that softens when cooked (as opposed to baking varieties that hold their shape) so it's really just the seeds you'd have to worry about. Sounds like the recipe doesn't really deal with them, so I guess you'd have to pick them out after chopping, or strain or pick them out at the end.