What if the lower part of your thumb hurts? Cut it.

Not the tip? I assume you are talking about the part nearest the arm, correct? Depends--certainly it will hurt for a day or two after the cut. However, if the pain gets worse, if there is increasing redness or drainage, or if the thumb is numb and/or you have trouble moving it, you should have it checked out by a doctor.
Clean it. If you've cut your thumb and it hurts you are normal. Diabetics often cut themselves and because of nerve dysfunction rhey cannot feel it. So if the cut is deep it should be closed with sutures by a doctor. If not cleanse it with soap and water thoroughly dry it with a clean towel or wet wipe and apply some antibiotic ointment and a bandaid. Cleanse and change the dressing twice a day.