Our intuition is a wonderful source of ideas for innovations, general judgments and quick assessments. However, what it provides us in fast and frugal modes of thinking, it lacks in precision and scope because it is compiled from our internalized experiences. Consequently, it is often incorrect, especially when dealing with realms like quantum physics where we lack direct experience. But even in issues of everyday experience, our intuition can often lead us astray. Consider the following problem.
Peter is looking at Mary, while Mary is looking at Paul. Peter is married. Paul is unmarried.
Is a married person looking at an unmarried person?
Answer: a) yes b) no c) cannot be determined
Most people get the wrong answer to this question because they make an intuitive judgment and don’t reason it through in a logically deductive fashion.
Or consider the following problem.
It is given that everyone is prejudiced against prejudiced people. Mary is prejudiced against Peter. Is Paul prejudiced against Peter?
Answer: a) yes b) no c) cannot be determined
Again, this problem requires a slow logically reasoned approach, but most people get it wrong.
Let’s consider the famous Coincident Birthday Problem: What is the probability that in a randomly chosen group of 24 people, there are at least 2 people with the same birthday? (i.e., born on the same day and month, but not necessarily the same year)
Few people realize that the probability is about 50%, because it is counterintuitive. With 365 days in a year, who would expect two people in a group of 24 to have the same birthday? We can prove this is true mathematically or demonstrate its plausibility by computer simulation. In this problem, intuition is a detriment.
In a recent post, I presented the Monty Hall Dilemma with a solution showing that the intuitive answer to that proposition is incorrect. Even when the correct solution is presented, many people refuse to believe it, although simulations verify the mathematical solution.
From the time of the ancient Greeks, philosophers have inquired about the basis of our certainty about what is true and what is not. Out of this search for truth evolved Euclidean geometry, in which truths were deduced from simple axioms that we could all agree are true. In the millennia that followed, the entire field of mathematics evolved as a way of avoiding the rabbit holes created by flaws in our intuition. When Isaac Newton put science on a foundation of mathematics with his Law of Universal Gravitation, he gave us a science with predictive power. Using mathematics, we could not only express relationships, but we could also predict events like eclipses, conjunctions and a variety of other natural phenomena.
Today, physicists have been able to predict phenomena like dark matter, the Higgs boson and gravitational waves by merely analyzing equations describing entities that lie outside our experiential world and our intuition. Mathematics describes patterns that transcend our experiences and enable us to probe areas outside our intuition and correct perceptions when our intuition is flawed.
Henri Poincaré, one of the greatest mathematicians of the 19th century and quintessential spokesman for the role of intuition in mathematics observed:*
Intuition cannot give us rigor, nor even certainty; this has been recognized more and more… This is why the evolution [of rigorous proof] had to happen; let us now see how it happened.
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*Henri Poincaré. Intuition and Logic in Mathematics in Swetz, Frank. 1994. From Five Fingers to Infinity: A Journey through the History of Mathematics. Chicago, IL: Open Court Publishing. pp. 691–2.
