In Question 74707, we ask what mathematical habits of thoughts are useful in other areas. It seems only fair to ask also what we can learn from them. It is also fair to ask what they should not learn from us.
The first question is
What habit of thoughts in other areas can be of use in mathematics.
Here are a few suggestions to the first question:
In many areas the quality of exposition is very important.
In some areas simplicity is considered as an advantage. In mathematics to some extent difficulty is a criterion for quality.
Other areas give more weight to heuristic and non rigourous arguments compared to pure mathematics.
In other areas there is much heavier use of computers.
In some areas discussions and debates are basic part of the academic discipline. This is not the case in mathematics.
The second question is:
What habit of thoughts in mathematics should be avoided (even by mathematicians) outside mathematics.
Here are a few suggestions (to the second question) for starters.
Mathematicians (as a rule) avoid ambiguity. Nonmathematicians recognize the value (in appropriate circumstances) of ambiguity.
Mathematicians don't care what anybody thinks. We know when our assertions are facts, because we can prove them, and we know how important they are; we don't need anyone's opinion on that. There is a danger that this attitude carries over to our everyday lives. Nonmathematicians recognize the value (in appropriate circumstances) of the opinions of others.
Mathematicians, with all due respect to Godel, think simple declarative sentences are either true or false. Nonmathematicians are better able to deal with shades of gray.
Those of us who teach are constantly judging the mathematical abilities of others. If we are not careful, we start to judge the worth of others by their mathematical abilities. Nonmathematicians know that some of the best people alive can't add fractions.
Mathematicians (and theoretical physicists) consider a spherical cow. Nonmathematicians understand that conclusions based on unrealistically oversimplified models are untenable.