The Intuition Behind Math

A con may be a photo in our news feed on Facebook. It could be a spam mail that bypassed the spam protector. Cons can even be well researched evidence that look fantastic on face value. Cons fool us because many of the times we aren't able to perceive the intuition behind math and probability.  In his recent book, “How Not to Be Wrong: The Power of Mathematical Thinking”, the mathematician Jordan Ellenberg seeks to unravel some of these cons to make us think better. The message is simple: intuition is more powerful than math.

A horse bookie sends you five correct bets. They're all free. You take the bets and gamble. As you win, you become more and more adventurous. You start raising the bets. The bookie now asks you to gamble your life savings on the sixth horse. This time the bookie asks you for a fee. What will you do? Pause. Why would the bookie want to give you free bets in the first place? That was the intuition. You didn't ask this question when you were winning. Life was on the rise. You never realised that every good song can come to an end. You didn't think you could have been on a rising function that will soon reach a peak and then start to fall like an upside down U. Did it strike your mind for once that this bookie could have sent 10,000 people random bets? With a probability of a success rate of 1/10,000, which isn't unrealistic, you may have been the 'unlucky' gambler who hit the jackpot with five correct bets in a row. This was a real world experiment. Here's another version of the upside down U curve from a political debate.

Sweden is a Welfare State. The state takes care of its citizens from cradle to grave. America is based on free markets. People buy insurance to protect themselves. It therefore seems logical that Americans should advocate for a Welfare State. Does it seem logical that the Swedes want to abandon their welfare model and take the path of free markets like the Americans? It can. Let's avoid math and imagine. That'll make it intuitive.

Imagine a hill. Look vertically up. The top of the hill is the maximum amount of Swedishness a nation can achieve. 'Swedishness' means the beneficial effects of the Welfare State. The bottom of the hill at the horizon (looking from left to right) measures where the nation is at its point of history. Now imagine America, a nation that's to the left of the peak. Surely, Americans would like to become more like the Swedes. Now think about the Swedes. The Welfare State can't increase forever. It has to be like a hill that rises and then starts to fall. The Swedes find themselves to the right of the peak of the hill. They've been to the peak. The more they move now, the more downhill they'll go. Now does it sound logical why the Swedes would want to be like the Americans? Or why the Americans would want to be like the Swedes? Isn't math more intuition than the math itself?

Statistics is obsessed with significance. We're trained to focus attention on only that area that's significant (successful). We also follow this unknowingly. When your professor asks you to read a well established journal article or when you see a photographer posting one of their photos on Facebook, did you ever ask, how may insignificant (unsuccessful) drafts of the journal article or how many clicks the photographer made that he/she didn't show? We see only the success, but never know the long or short list of failures behind that success. Significance is a tiny subset of the entire spectrum of a story.

A mathematician himself, Ellenberg argues knowing math may never make us mathematicians. Knowing math and thinking intuitively can certainly help us make more sense of the world we live in. However, all science (and arts) is approximation only. That's another story for another day.

Source: “How Not to Be Wrong: The Power of Mathematical Thinking” by Jordan Ellenberg. Penguin Press. May 2014.

Asrar Chowdhury teaches economic theory and game theory in the classroom. Outside he listens to music and BBC Radio; follows Test Cricket; and plays the flute. He can be reached at: asrar.chowdhury@facebook.com