Adjective

1. not having normal physical feeling
   • 1897, Bram Stoker, Dracula

     It is something like the way dame Nature gathers round a foreign body an envelope of some insensitive tissue which can protect from evil that which it would otherwise harm by contact.

2. not having normal emotional feelings, cold, tactless, undiplomatic
   • 1895, Grant Allen, The British Barbarians

     Somehow, when Bertram Ingledew let it once be felt he did not wish to be questioned on any particular point, even women managed to restrain their curiosity: and he would have been either a very bold or a very insensitive man who would have ventured to continue questioning him any further.

   • 1994, Jann Arden, "Insensitive" (song)

     Oh I really should have known by the time you drove me home, / By the vagueness in your eyes, your casual good-byes, / By the chill in your embrace and the expression on your face, / That told me you might have some advice to give / On how to be insensitive.