Often we are confused that others are able to hold contradictory ideas at the same time. The examples are as familiar as they are banal: "how can you be for/against the death penalty when you are also for/against the state intervening in women's reproductive rights. Both sides clearly see how the other side is hypocritical but claim that their side is actually not a contradiction at all, but rather it's just a matter of understanding some additional information.
A few things need to be unpacked:
1) We tend to understand ourselves as static with moments of sadness or depression or anxiety or joy or confusion. In every case what is assumed is that variation is the exception. I believe this is exactly backwards. I am always in a state of change. I'm always excited by this or that thing I'm reading, but then I get bored and so I turn on the tv, and then I flip channels, but then I'm hungry, so I make food, but maybe I ate too much, so now I'm tired and then maybe I wake up and have to use the bathroom. I'm always changing to make things restabalize for brief moments. This is the idea of the gyroscope. I appear stable but in reality that apparent stability is maintained only because of many moving parts working simultaneously.
2) In his work Hypermodern Times Gilles Lipovetsky discusses that the hypermodern is characterized by an embracing of opposites - the world is both hyperconnected and yet people feel very lonely - we are both close and far away and so on. I think the problem with this sort of model is that it's not ambivalence - there's multiple flows intersecting; it's not simply a matter of opposites. Also, the reason the world is both intimate and distant is not because of some contradiction in logic; it's simply contextual, i.e., in what sense do you mean "intimate." The reason it feels like opposites is because Westerners have been trained to think in dichotomies.So I don't think people are holding two ideas in different quadrants of their brain because you can point out these contradictions and often people understand - they see both compartments at the same time, and then they usually start attempting to explain away contradictions. I'm not suggesting we all don't hold competing and contradictory ideas - what I'm suggesting is that depending on different scenarios and variables, one idea usually becomes focused while others recede. And I'm arguing this happens for stability - we want to feel at home in the world. We want to avoid doubt when possible.
Another important event in the emergence of identity is the notion of narrative. Because we believe we need to be a self, we need to understand our self in terms of a story. We do this by taking where we currently and tracing backwards, often coming up with insights like "If you think about it, this is what my life has always been leading to." Of course, this is simply a matter of the narrative moments we freeze and constitute as "significant" at the expense of other events we consider tangential. It would be nihilistic to suggest some events aren't more significant. But I believe that in reality lots of events from lots of possible narrative configurations would also be meaningful - essentially it is not that this or that narrative we construct is inaccurate; rather, it's one out of many, many different possibilities. This is why if one narrative falls apart, often people have some version of "Well I know it seemed like this is what I was supposed to be doing, but actually, now that I've thought about it..." And so on.
To recap: Stability is not the norm. We seek "stability" by isolating moments and "freezing" them in order to create a narrative whereby our personality and behavior feel logical and predictable. We want to make sense. Also, the instability we feel is not simply because of opposites playing in some kind of Hegelian synthesis that won't resolve; rather, there there are multiple flows that do not resolve into each other - we are always teetering this way and that and trying to feel "at home" in the world. This process over enough time produces this thing we refer to when we say "I."