Tag: π (pi)

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:

\begin{equation} y = \frac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{2}} \end{equation}

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: eiπ + 1 = 0

That the two key integers (1 and 0) should relate to π (pi), e, and i (-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.

Essay 1: Unnoticed Dimensions of Knowledge

Let’s “get down to cases” right now:

  1. You learn decimals and fractions in school. You see that 1/2 can be written as 0.5 or 0.50 or with as many zeros as you like. That seems “clean.”

But 1/3 is equal to something more complex (i.e., 0.3 recurring or repeating, like 0.3333 and so on infinitely).  If you divide 1 by three you keep getting three.

Imagine you want to experiment a bit, and multiply the fraction 1/3 by three and the 0.3 recurring by three, thus not affecting things since you’re doing the same thing to both sides of the equation.

You get:  1 = 0.9 recurring or repeating.

You’re suddenly puzzled: How can 1 be obtained by adding “slices of 9 fractions” (i.e., 9/10 + 9/100 + 9/1000) to infinity. How do you get to the end? What end? 

It turns out that it’s not that simple to get a grip on all this.  A person who allowed themselves to become fascinated by this specific conundrum would enter a “beautiful ocean” of mathematics beginning with so elementary a phenomenon.

This shows you a deep connection between a part (e.g., the fraction and decimal 1/3 and 0.3 recurring) and the wider world or domain or universe of numbers.

How can it be that such a simple elementary “thing” becomes so intricate, deep and elusive?

  1. Let’s jump over to an entirely different kind of example. Think of Dinesen’s novel Out of Africa. Remember the movie with Meryl Streep and Robert Redford.

Suppose you turn the movie “inside out” and “upside down” and ask: is this movie about coffee and coffee bushes, coffee markets and coffee growing, in a colonial context?  The coffee plantation is near Nairobi (today’s Kenya) and involves plantation economics, colonial relations with Kikuyu peoples, German-British colonial tensions around World War I.

Suppose I take the “backstory” and make that the “frontstory”.

The story of “economic botany” (coffee growing is one case) and colonial tensions between and among Europeans as well as Europeans and Africans is the deeper and larger story while the “musical beds” of the Westerners is a colorful footnote.   

We have the perennial question of “parts and wholes” which is one theme of this book.       

  1. Why does science “orbit” some numbers such as π (pi) (i.e., 22/7)?

You learn in school that there’s a ratio called π (pi) which is 22/7. Think of π (pi) as some kind of essence of circularity. Remember πr2 and 2πr in grade school.

Why does it keep appearing in almost every equation of physics? Why would “circleness” “haunt” science and math? Probability and statistical theory are dependent on π (pi) as a variable. Why?

You could peruse:

A History of Pi is a 1970 non-fiction book by Petr Beckmann that presents a layman’s introduction to the concept of the mathematical constant π (pi). 

Why does science “orbit” some numbers such as π (pi)?

This is an example of this quest for connectedness.