Pi -- or pesky pi as I like to call it -- is that irrational number that represents the relationship between the circumference of a circle and its diameter. Because pi is irrational (it cannot be expressed as a simple fraction) it is not possible to construct a square with the same area as a given circle. Thus "squaring the circle" is a metaphor for trying to do the impossible.
For centuries mathematicians tried and tried to square the circle, using a ruler and compass. It wasn't until the nineteenth century that they came to the realization that it is impossible. This caused a real shift in mathematical thinking -- namely that it didn't make sense to keep trying to come up with a solution. The conclusion to the problem is that there is no resolution.
It turns out that humans, like pi, are irrational as well. James Clear's post of 7/6/17 states that we are all irrational, and goes on to describe several of the irrational choices (he refers to them as mental errors) we make when faced with a decision. One is known as the Availability Heuristic, which refers to our tendency to overvalue the importance of things we remember, and undervalue the importance of things we hear nothing about. Payola is a good example of this -- people think that because they hear a song everywhere it must be a good song. If we hear it enough, we'll start to like it. Even if it's a really bad song.
Another phenomenon which leads to irrational behavior is Anchoring. Marketers are experts at using this to their advantage. An example of this is putting a limit on something. If you go to the store to buy a beach towel, and there is a big display of beach towels with a sign that says "Limit 5 per customer," you may be swayed to buy more than one beach towel, even though you didn't intend to buy 4 or 5 beach towels.
Okay, so we know that it is impossible to square the circle, because pi is used in formulating the area of a circle, and pi is irrational. We also know that once mathematicians shifted their focus away from trying to solve that problem, they could do a lot of useful calculations using an approximation of pi. They were able to adjust for the irrationality factor.
Likewise, we know that intuitive thinking is not all it's cracked up to be. There are myriad behavioral science research studies that point out the limitations of relying on our gut. Knowing this can actually give us an advantage, and inform our decision-making processes. At every turn, outside forces are pressuring us to succumb to the traps of irrational thinking. But as long as we can adjust for the irrationality factor, and remember that intuition is far from foolproof, we can make really good decisions which will lead to positive outcomes.