As I began my studies into atheism I found that many people used the phrase "square the circle" (forcing incompatible things to fit together) to belittle those who struggle with reconciling their religious views with science. Many people see the apparent conflicts between science and religion as mere gaps in our understanding as to "how it all fits together." I remember my dad saying once that, although there appears to be a conflict, someday god will reveal the truth and we will see how science confirms the teachings of the Mormon Church. And as a believing Mormon teenager, that was good enough for me. I believed in the church, so reality must fit within its teachings, even if we don't know how yet.
This was more or less the attitude I had while attending BYU. Many of my science professors (and I had a lot) also viewed science in this way. Even on the first day of Biology 101, the professor handed out a sheet of paper with quotes from various church leaders saying things in favor of, and in staunch opposition to evolution (the very basis of modern biology). I recall him saying that one can accept or reject evolution and still be a "good Mormon" (emphasis, of course, being placed the "good Mormon" part). He also pointed out that the church has yet to reconcile the traditional creation story (as taught in certain "sacred places") with aboriginals, who have lived in Australia for over 40,000 years.
As a teenager, I believed in the traditional creation myth, as taught by my seminary and Sunday School teachers. Then I learned more about evolution and talked with my father (a physician) about it. He said, "We don't know exactly how god created everything, but evolution may have been the mechanism by which god made the animals." This sounded reasonable to me, so I adopted the view that evolution only applied to animals, and not humans (this was the most common view I came across at BYU).
Then, again in Biology, I heard an alternate explanation of the creation myth which allows for human evolution. In Genesis (Gen. 2:7) it refers to the "Breath of Life." In Hebrew, "breath" can be interpreted as "spirit." Therefore, god could have had humans evolve like animals and then used the "Breath of Life" (or spirit) to give them a conscience or soul, thus making them human. This sounded the most reasonable to me and I tried explaining it to my family. This was met with hostility and arguments, such as "I believe in evolution for animals, but not humans" and "prophet so-and-so said we didn't come from monkeys." Things got emotional and dinner was needlessly uncomfortable.
Later, I realized why it was so uncomfortable. We were all using emotional appeals, arguments from authority and unsupported scientific claims to build our arguments. Not one of us used reason and evidence. In other words, none of us, including myself, had any validation or rational reasons for our opposing cases, and we were simply making assertions. We were all trying desperately to reconcile our beliefs with science. This is the essence of squaring the circle. It doesn't matter how you rationalize your beliefs; unless you support it with reason and evidence, it is no better than any other unsupported claim, and you have no argument. This applies to any conflict between science and religion.
On some level, when you reconcile science with religion, either religion or science has to give something up or concede some point. Because science is based on evidence and reason, it is very difficult to justify conceding anything to an assertion of beliefs. On the other hand, the more ground religion gives, the less relevant or useful it becomes. Once I realized this was the case (several years after that uncomfortable dinner, mind you), I came to the conclusion that the only way to rationally square the circle of religion and science was to throw out religion completely. And, having thrown out religion completely, I consider my circle squared.
Here is Christopher Hitchens showing how hard (and, at times, ridiculous) it is to square the circle of human evolution with creationism: