Not sure what you're trying to describe. The wheel gets larger, the tire does not.
Yes, it looks different due to the change in wheel diameter. Gives you a lower profile sidewall, but the overall diameter of the tire is, for example, 32 inches, then it's still 32 inches. Just with more wheel inside the tire and shorter sidewalls.
Most 15's were/are "flotation" sizes. This means that your 32x11.50x15 might turn into a "metric" size when you choose a tire for a 16" wheel. Don't know what that tire size might be, but a 265/70-16 comes to mind.
But there are possible discrepancies here. A "32-inch" tire in flotation sizing probably measures in at 31, or maybe even 30.5 inches tall. Whereas a 265/70 might be called a "32 inch" tire, but it might actually measure out at 32 inches!
So be wary of using dissimilar measuring criteria. Get out a tape measure and verify!
Paul