## Education and “Intuition Pumps”

Professor Daniel Dennett of Tufts uses the word “intuition pumps” in discussing intuitive understanding and its tweaking.

Let’s do a simple example, avoiding as always “rocket science,” where the intricacies weigh you down in advance. We make a U-turn and go back by choice to elementary notions and examples.

Think of the basic statistics curve. It’s called the Bell Curve, the Gaussian, the Normal Curve.

The first name is sort of intuitive based on appearance unless of course it’s shifted or squeezed and then it’s less obvious. The second name must be based on either the discoverer or the “name-giver” or both, if the same person. The third is a bit vague.

Already one’s intuitions and hunches are not fool-proof.

The formula for the Bell Curve is:

$$y = \frac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{2}}$$

We immediately see the two key constants: π (pi) and e. These are: 22/7 and 2.71823 (base of natural logs).

The first captures something about circularity, the second continuous growth as in continuous compounding of interest.

You would not necessarily anticipate seeing these two “irrational numbers” (they “go on” forever) in a statistics graph. Does that mean your intuition is poor or untutored or does it mean that “mathworld” is surprising?

It’s far from obvious.

For openers, why should π (pi) be everywhere in math and physics?

Remember Euler’s identity: e + 1 = 0

That the two key integers (1 and 0) should relate to π (pi), e, and i ($\sqrt{\mathrm{-1}}$) is completely unexpected and exotic.

Our relationship to “mathworld” is quite enigmatic and this raises the question whether Professor Max Tegmark of MIT who proposes to explain “ultimate reality” through the “math fabric” of all reality might be combining undoubted brilliance with quixotism. We don’t know.

## Education and the Question of Intuition

An intuition pump is a thought experiment structured to allow the thinker to use his or her intuition to develop an answer to a problem. The phrase was popularized in the 1991 book Consciousness Explained by Tufts philosophy and neuroscience professor, Daniel Dennett.

We argue in this education-completing book, that our intuitions are puzzling in a way that “intuition pump” talk does not cope with at all.

Let’s go immediately to the example of simple versus compound interest in basic finance.

You borrow $100.00 for a year at an annual interest of 100%, without compounding and hence simple. A year passes and you owe the lender the initial$100 plus one hundred percent of this amount (i.e., another hundred). In a year, you owe $200.00, and every year thereafter, if the lender is willing to extend the loan, you owe another hundred to “rent” the initial hundred. This is written as A+iA, where A is the initial amount (i.e.,$100.00) and i is the interest. This can be re-written as A(1+i)n where n is the number of years. Thus, if n=1, you owe: A(1+i), which is 100×2 (i.e., the $200 we just saw). There’s nothing tricky in this. You then are introduced to compound interest (i.e., where the interest accumulates interest). You can see where compounding by 6 months (semi-annually, or half a year) or 12 months involves dividing the n (the exponent over 1+i) by 12 months, two half-years or 365 days. You could routinely go to days and hours and minutes and seconds and nanoseconds and you could calculate interest payments compounding for each case. But here is where your intuition falters and fails: suppose you compound continuously? You get to the number e as growth factor where e=2.71823 Simple algebra does show that at 100% interest,$100 of a loan becomes $100 multiplied by e1 (hundred percent=1) or just e (i.e., you owe$100e).

This gives you $271.82. So what has happened? At one hundred percent simple interest you owe$200.00 to the lender. Continuous compounding means you owe $271.82. Instead of owing$100 in interest, you owe $171.82. Your interest bill has gone up by$71.82 or about 72 percent.

Does that seem intuitive? Probably not.

How could one ever apply an “intuition pump” to this arithmetic? We get to the 72% increase in interest by using e which has nothing very intuitive about it. Thus it’s not clear that “intuition pumps” will work here.

You use compound interest arithmetic to get a number which you would never have been able to estimate based on standard intuition since like the 22/7 or 3.14 for π (pi), there’s nothing to “recommend” 2.71823 in and of itself. This means that the link between computational arithmetic understanding and your “gut” or “sixth sense” is feeble at best.

By exploring this way of thinking you could deepen your “meta-intelligence” (i.e., perspective-enhancement). The British economist Pigou (Keynes’s teacher) says that people have a “defective telescopic facility” (i.e., have a poor or even erroneous sense of time-distance).

How one might strengthen one’s sense of time-distance or “far horizons” is not clear.