## The Natural Log

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What is the natural way to place numbers on a number line?

Apparently it’s not with even spaces. And no, I’m not referring to drawing ability, but to what seems to be a tendency to put smaller numbers farther apart, and larger ones closer together.

Some psychology folk visited the Mundurucu (an indigenous culture in the Amazon) and gave them a number line that had 1 at one end and 10 at the other (or, in some cases, 10 at one end and 100 at the other). Then a number word was spoken, or a bunch of dots shown or sounds heard, and the people had to place the number somewhere on the number line. In general, the numbers were placed like the logarithmic scale above rather than equally spaced, at least most of the time

Those with more than three years of education tended to place numbers indicated by Portuguese spoken words at equal intervals on the line. However, those same individuals showed a compressed mapping for arrays of dots and for spoken Munduruku words, as did all of the other Munduruku participants. (from Science Daily)

Earlier studies have shown that preschoolers do the same thing until they’ve had some formal schooling, as do adults in Boston, at least sometimes:

The Boston-area participants showed linear or nearly linear mappings in all the conditions of the study when they were presented with dot arrays that were small enough to count or with number words. Nevertheless, adults in Boston also showed a compressed mapping when presented with sound sequences or with arrays of dots too large to count (from Science Daily)

The conclusion in the abstract of the study (in Science) is that “[t]he concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education.”

If I’m remembering my physics correctly (which is not a sure thing by any means), there are many things that humans perceive logarithmically: sound, light, and the weight of objects come to mind (where we can hear/see/feel both very small amounts and very large amounts). Indeed, sound (and possibly light?) are measured using an actual formal logarithmic scale. But I find myself wondering if the logarithmic scale described in this study is actually precisely logarithmic or simply compressed. Because when I have to write something in a limited space, like a cake, even though I know where it’s supposed to begin and end and I know how many letters I’m going to write down, it ends up looking like the following:

Which is certainly compressed, but probably not an actual logarithmic scale. Unless I’m a heck of a lot more talented at cake-writing than I thought.

(But I haven’t seen the actual paper, and in fact the abstract uses “logarithm” or “logarithmic” three times, so they probably do mean an actual logarithmic scale. Forgive me for doubting.)

The log scale was created by this site at incompetech.com, which looks like incompetent.com but is actually a very competent and very useful site for free online graph paper, including dots and triangle and rectangles and even circles.