It is only in Chapter 14 that we find out the true identity of the "Spider Man". It is when David, Rosalind and Petra are captured by the men of the Fringes that David comes face to face with the leader of the Fringes, who he has worked out is his uncle. Note the similarity that is drawn between David's father and his uncle:
A figure seated on a stool just inside the entrance looked up as we approached. The sight of his face jolted me with panic for a moment - it was so like my father's. Then I recognised him - the same "spider-man" I had seen as a captive at Waknuk, seven or eight years before.
During the dialogue that occurs between the spider-man and David, his nephew, the story comes out. David tells him:
"My father had an elder brother," I said. "He was thought to be normal until he was about three or four years old. Then his certificate was revoked, and he was sent away."
Looking back at David's father's actions, this explains a lot. Having had a "mutant" discovered in his own family as a brother, it is understandable why he is so intent on discovering and disposing of other mutants. The fact that his own very brother was a "mutant" who remained undiscovered for four years would have made him all the more desperate and urgent to root out any form of impurity everywhere - including in his own family.