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Combining Reasoning and Intuition

It is common for either intuition or reasoning to be mistaken. Much of this website is about ways intuition can be wrong (for example we overestimate the chances of some rare mishap or we believe someone's claim when we should not). If we recognize that we're in a situation where intuition is often faulty, we can apply whatever reasoning skill we can to reduce the chances of making a mistake.

Although we may think of reasoning to be more reliable than intuition, it's not uncommon for that to be faulty as well. That's not because the rules for reasoning are bad, but because we often make mistakes in following the rules. If we do some kind of arithmetical calculation it is pretty common that we will slip up someplace and arrive at the wrong answer. The same can be the case for other kinds of complicated deductions.

I know of a case where a girl who was having difficulty with math was asked how many minutes there were in five hours. Her answer was "twelve." She knew she was supposed to use the number five from the five hours, and that she needed the fact that there were 60 minutes in an hour. However she divided 60 by 5 instead of multiplying. Dividing when you are supposed to multiply or vice versa is a pretty common error, so I can't fault her too much for that. She did the division correctly. She could easily have avoided giving the wrong answer, however, if she had just double checked her reasoning by applying intuition. It should be pretty obvious that five hours should be a lot more than twelve minutes. If she had realized something was wrong and then tried multiplying, she probably would have gotten the problem right.

I know I can easily make a mistake trying to calculate the tip at a restaurant. If the bill is $36 and my attempt at calculating 15% comes out to be $1.80, it seems clear that it is too small.  If I get $15, I know it is too large.  My intuition has told me that I've done something wrong and I should try the calculation again. In cases like this it is almost impossible not to use our intuition, since our own behavior in a somewhat familiar situation is involved. We are more likely to forget to use intuition in classroom situations were we are not thinking of the problem in real life terms.

Good mathematicians often spend a lot of time "playing" with problems. They take calculations or geometrical situations and see what happens when they try certain things. They look for patterns in what they see. In this way they gain a lot of experience with mathematical problems much as the rest of us gain experience with real life situations. Experience gives us good intuition about what works in what situation, and for the mathematician their mathematical intuition can be very helpful in seeing whether their abstract reasoning is on the right track or taking them off into the weeds.

If we want to avoid being wrong about important questions, it helps to develop as good an intuition as possible for the situation in addition to reasoning as carefully as possible about it. If the results of the reasoning and the intuition are incompatible, we need to both check whether our reasoning was correct and whether our intuition could have been fooled. If both are in pretty close agreement, we can feel more confident that we are right.