the purpose of life and existence
Colin Singleton struggles throughout the novel to find happiness in his identity. His early success as a child prodigy leads him to feel pressured to live an abundant life, one that is undeniably special and outstanding. As a child, he found his identity in his amazing intellect, but he worries that he might not retain his advantage as an adult, which would result in an empty, failed identity.
His desire to be amazing does yield some continued advantages for him, such as his awareness of the lethargic effect of entertainment. Eg. "[Colin thought] What is the point of being alive if you don't at least try to do something remarkable? How very odd, to believe God gave you life, and yet not think that life asks more of you than watching TV" (Ch. 5). He does enjoy a certain advantage over the apatheticness of his peers, even though his awareness causes him desparity and frustration.
After realizing that his value system for his identity has depended on his intellect and his relationships, Colin has an existential crisis, saying, "I feel like I've only ever been two things.... I'm a child prodigy, and I'm dumped by Katherines. But now I'm [neither]" (Ch. 15). This offers the unique challenge of recreating his value system and trying to find joy in his existence through other means.
love and romance
The title of the novel, An Abundance of Katherines is a reference to Singleton's continued failure in his 19 relationships with girls who are all named Katherine, spelled Katherine. The repetition of girls named Katherine is a Kafta-esque detail drawing attention to the utter failure of all his relationships. It might suggest that the problem is not in the girlfriends at all. His frustration with his identity has caused the failure of his relationships, not any specific problems in any one of the relationships.
Adversely, however, his failed relationships only add to his frustration, and early in the novel, we see his cynical view of relationships: "Dating, after all, only ends one way: poorly. If you think about it, and Colin often did, all romantic relationships end in either (1) breakup, (2) divorce, or (3) death" (Ch. 3). This proves to be an inauthentic view informed by anger, and when he meets Lindsey, he realizes that his cynicism is defensive.
It's through his blooming attraction to Lindsey, who has also dated another guy (on and off) named Colin, that he begins to come out of his disenfranshisement. Instead of viewing people as "Dumpers and dumpees," he begins to view people as more complex and complexly motivated. Through his relationship with Lindsey, it becomes evident that his theorum will always be inadequate in judging future relationships because it can only diagnose failed relationships and not all relationships are bound to fail, although they cannot make him feel valued and happy. That is a question of his internal capacity to appreciate himself, and no relationship can supplement the peace he lacks.
the value of knowledge.
Colin's life has always been predicated on his potential as a student. He has a prodigious ability to learn and retain knowledge, as well as to solve problems. But his abilities yield frustrations in his life and he resists the force of his abilities on his life. His father says early in the novel, "It pains me to say this, Colin, but if you wish to continue to grow intellectually, you need to work harder right now than you ever have before. Otherwise you risk wasting your potential" (Ch. 3). Knowledge is portrayed as a double-edged force: It both helps Colin and gives him a high potential for the future, and it also leaves him feeling frustrated through the caustic backlash of his potential failure or inadequacy.
Throughout the novel, knowledge is contrasted with the organic complexity of human life. His relationship with Lindsey is stalled by his desire to finish the perfect math formula to determine how a relationship will fair, instead of just braving the relationship and learning from the other person. Knowledge positions itself as a solitary pursuit and having a high view of knowledge leads Colin to view relationships in an unfair, narcissistic way. Colin's journey through the novel involves relocating his priorities, lowering knowledge and raising up his view of others and their inherent value outside of his knowledge of them.