Historical Phenomenology of Mathematics

One of these days, I'll have to pick up the book. For now, I'll write about the ideas, using incomplete information.

First, a definition from the man himself, the late Hans Freudenthal. This is his description of historical phenomenology as it concerns mathematics:

Our mathematical concepts, structures, ideas have been invented as tools to organise the phenomena of the physical, social and mental world. Phenomenology of a mathematical concept, structure, or idea means describing it in its relation to the phenomena for which it was created.

In a previous post, I shared an example of what seems to be an early conception of the median, quoted in a paper by Bakker and Gravemeijer titled An Historical Phenomenology of Mean and Median. The example comes from a book by Edward Wright, cartographer and mathematician, circa 1599:

Neither if there be disagreement betwixt observations, are they all by and by to be rejected; but as when many arrows are shot at a mark, and the mark afterwards away, he may be thought to work according to reason, who to find out the place where the mark stood, shall seek out the middle place amongst all the arrows: so amongst many different observations, the middlemost is likest to come nearest the truth.

For a more accessible example of "Wright's median," one can think of a guess-the-number contest at a fair in which participants are invited to publicly guess (write down) the number of, say, jellybeans in a jar to win a prize. Here, each guess is like an arrow, and the exact number of jellybeans like the missing bullseye. Assuming that every participant had access to the same information (they all saw the same jar of jellybeans) and that the distribution of their "errors" was relatively normal, the median (or mean) of all the guesses would likely fall close to the exact number of jellybeans in the jar:

Treynor, former editor of Financial Analysts Journal, told us that when he taught finance, he would pass a jar of beans among his students and have them guess the number. As I wrote: "The guesses would vary wildly, but always, when the number guessed in total was divided by the number of students guessing, the result was within 3% of the correct number, he said. As there were 52 of us assembled, and a bowl of peppermint candies on the table, we tried the experiment. A low guess of 32 was recorded, a high of 71. The median guess was 46, the mean was 45. The correct total was 46, a number only one of the 52 had guessed."

More later. Bye!

Reference: Bakker & Gravemeijer. 2006. An Historical Phenomenology of Mean and Median Educational Studies in Mathematics. 62: 149-168.