Having had waaaaayyyyy too many dog bites, I can vouch for the need to see a doc right away on these. It does go beyond this though, any puncture wound introduces infection deep into the tissue and the closes over it effectively sealing it in. Very difficult to clean (been there) even with a irrigation syringe. This is particularly important say for example when building a shelter and put a nice long splinter in your palm, grabbing a rusty nail by accident (tetnus), etc. Having to lance in the field may be the difference between good and very bad. Hope your finger gets better quick.

Bill