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: e + 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.

Essay 43: Knowledge Puzzles of “Far-Fetched Questions”

Heidegger (died in 1976), the German thinker (and Hannah Arendt‘s lifelong boyfriend) is walking along somewhere in France with Jean Beaufret, the French poet-philosopher, and wants to “delimit” what topics should be admitted and discussed and manage to dismiss other kinds of topics.  Heidegger says, “we do not need to ask what the connection is between Newton’s laws and the French national anthem, ‘La Marseillaise’ or between Carnot’s Principle and the sign on the shop across the street, ‘This Store is Now Shuttered.’”

In Gulliver’s Travels, the satirical masterpiece, we find a scene where the Academy of Projectors (mad scientists profs.) are trying to make cucumbers out of moonbeams and have other crazy projects.  The Academy is described in the Laputa/Lagado “flying islands” section of the satire.  Again, we grin when we read these lines in Jonathan Swift and marvel at his inventive genius. It’s not quite as simple to pin down exactly why Heidegger’s or Swift’s examples of “crazy questions or projects” are so comically nutty.  Clearly, there are experiences we all agree on as being indicative of insanity or are at the outer limits, perhaps, of Quixotism (Don Quixote).  If a person tells you he or she plans to go to the roof and reach up and put the moon in their pocket and then go the county Registrar of Deeds and declare it their property, we see multiple impossibilities and figure the person is joking, drunk or insane.

On the other hand, many questions or projects that would seem silly at one point seem less silly now: an example is, say, bringing “dinosaurs” back via DNA “resurrections.”

Thus the “knowledge quest” and its parameters is evolving in strange ways, on top of all the other uncertainties.

The Heidegger/Beaufret dialogue, mentioned above, occurs in the following book:

Dialogue with Heidegger: Greek Philosophy
Jean Beaufret
Series: Studies in Continental Thought
Publication date: 07/06/2006
ISBN: 978-0-253-34730-5