Just a suggestion. Get an aerorol of some kind, and blast the gum with the contents (that is, take the spray button off, apply direct to the gum and press the can down). The gas in the can should freeze the gum making it brittle, smash it with a hammer, then hoover up.
If all that fails, cutting it out may be the the only choice (that is citting the pile, not cutting a hole in the carpet)