Living with Paradox

I have heard this term tossed around a few times but I don’t really quite grasp what it means. I heard something in regards to being “self aware” means having to live with paradoxes, but again I am lost. Are there any examples of this? IF at all?

Interesting subject.

Let’s start with some examples to establish some baselines;

Patrick Hughes outlines three laws of the paradox:[15]

Self-reference An example is the statement "This statement is false", a form of the liar paradox. The statement is referring to itself. Another example of self-reference is the question of whether the barber shaves himself in the barber paradox. Yet another example involves the question "Is the answer to this question 'No'?"

“This statement is false”; the statement cannot be false and true at the same time. Another example of contradiction is if a man talking to a genie wishes that wishes couldn’t come true. This contradicts itself because if the genie grants his wish, he did not grant his wish, and if he refuses to grant his wish, then he did indeed grant his wish, therefore making it impossible either to grant or not grant his wish without leading to a contradiction.

Vicious circularity, or infinite regress
“This statement is false”; if the statement is true, then the statement is false, thereby making the statement true. Another example of vicious circularity is the following group of statements:“The following sentence is true.” “The previous sentence is false.”

Other paradoxes involve false statements ("‘impossible’ is not a word in my vocabulary", a simple paradox) or half-truths and the resulting biased assumptions. This form is particularly common in howlers.

As an example, consider a situation in which a father and his son are driving down the road. The car crashes into a tree and the father is killed. The boy is rushed to the nearest hospital where he is prepared for emergency surgery. Upon entering the surgery-suite, the surgeon says, “I can’t operate on this boy. He’s my son.”

The apparent paradox is caused by a hasty generalization, for if the surgeon is the boy’s father, the statement cannot be true. On the other hand, the paradox is resolved if it is revealed that the surgeon is a woman—the boy’s mother.

Seems to me that even a perfectly logical statement can be false if based on a false premise and that the false premise can be discovered through mathematics.

I wasn’t really referring to the logical paradoxes, which seem more like quirky word games to me. More like the everyday sort of stuff I guess, like how if you try to impress people you end up doing the opposite. Or the more you try to hold on to someone the more you push them away. Or “less is more”. That kind of stuff. Also the logical paradoxes, in this example the doctor, are cleared up by mentioning it is the boy’s mother. In that case it is hidden information that changes the nature of what is said. But this is entirely different, I’m also trying to get at this link:

Which seems to imply that they lived in a sort of “tension” of non-resolution, although I have no idea what that means and they are entirely vague on the subject. Which is why I was probing around hoping to find out what was meant by living with paradox.