Comments on: Unnatural logarithms

By: Charles Petzold

Charles Petzold — Sun, 08 Jun 2008 16:03:50 +0000

I think Weber’s Law reveals something very fundamental about human perception, so it’s always interesting to see it show up in unexpected ways.
Obviously the linearity of the “natural” numbers has been so drilled into our educated minds from early childhood that we are mentally forced to use that number sense in evaluating similar numbers or measurements. But I think this linear number sense is restricted to local regions — much like we treat the surface of the earth as a plane when overall its really an oblate spheroid.
For example, you cite our evaluation of a 10 cm line being 10 times the length of a 1 cm line. But measurement involves two objects — a value and a unit. The values (1, 2, 3…) may be linear, but the units (millimeter, centimeter, decimeter, meter…) are logarithmic. The combination is essentially a logarithmic scale. For every 10 values we bump the units up by 1.
In real life, numbers are always attached to things, and are always accompanied by units; these units subsume the relentless linearity of the numbers into a more natural logarithmic scale. The population difference that turns a “hamlet” into a “village” is insignificant for a city.
Similarly with money. In the U.S. we have money units of 0.01, 0.05, 0.10, 0.25, 0.5, 1, 2, 5, 10, 20, 50, and 100 — also a roughly logarithmic scale. We tend to treat money logarithmically. Everything else being equal, we might be inclined to walk an extra block to save 10 cents on a slice of pizza, but not bother driving the extra mile to save 10 dollars on a widescreen TV.
Our sense of time is also logarithmic. When looking into the past, the last year seems full of detail, years earlier less so, but when we turn our gaze back to distant centuries, whole years and decades go by where seemingly “nothing happens.”
Here’s what might be interesting: Ask children to arrange animals (for example, ranging from ants to elephants) by size on a line to reflect how much bigger one animal is than another. I would bet the children to arrange the sizes logarithmically rather than linearly. (Adults, on the other hand, know “too much” to let their Weber’s Law instincts take over in this taks. Yet, ask even educated adults to draw the Solar System to scale without consulting Wikipedia, and it’ll definitely be scaled in a logarithmic factions)
As a wise mathematician once said, “God invented the logarithms; all else is the work of man.”

By: David

David — Fri, 06 Jun 2008 22:01:53 +0000

Caleb,

Yes, roughly speaking, humans perceive sound and light intensity on a logarithmic scale. So, a ten-fold increase in sound intensity is perceived as a two-fold increase in loudness.

From this fact we cannot, however, conclude that our number sense, or our sense of linear geometric magnitude, is also logarithmic. If I were to draw a 1 cm line and then a 10 cm line on a piece of paper, I don’t think you’d tell me that the second line was twice as long as the first.

By: Caleb Eggensperger

Caleb Eggensperger — Fri, 06 Jun 2008 07:46:22 +0000

Neil Tyson’s “Death by Black Hole” has a well written introduction that discusses the fact that human senses are logarithmic by nature. (And the entire introduction is available online, courtesy Amazon: http://www.amazon.com/gp/reader/0393062244/ ).

The idea that human senses are logarithmic is clearly evident in the usefulness of things like the decibel scale or the use of apparent magnitudes to gauge the brightness of stars.

The latter is actually very interesting, and clear support for your claim. The practice comes from the Greeks, who divided the stars up into six categories of brightness, based on their perception of the brightness of the stars. Later, when more modern astronomers used this data, they found that it had a logarithmic correlation with actual brightness of the star.

By: David

David — Tue, 03 Jun 2008 21:53:02 +0000

The report is interesting in and of itself, but unless yours is the Editor-in-Chief of “Mundurucu Scientist”, don’t the findings offer some vindication of *her* intuition regarding your readership’s number sense?

By: rms

rms — Mon, 02 Jun 2008 19:19:53 +0000

living overseas and not enjoying the pleasures of electronic access I still will need to wait for a (linear) couple of days to read the original article before being too much opinionated.
however, I have two non-independent comments:
1. regarding linear or log scales for plots. I agree with you; absolutly. but more drastically: I use a log scale whenever there’s the slightest hint of a non-linear trend in the data AND at least 2 (preferibly 3 or more) orders of magnitude appear in the scale (i.e., we go from 1 to 100, or from 10 to 1000). this leads us to the other comment
2. it seems to me dangerous to extract conclusions on the exponential (or, for that matter, potential or scale-free or fractal..) behavior of things when you have information spanning just two orders of magnitude (let alone just one like here). I would find the case much more compelling if the experiment included numbers say, from 1 to 1000 (or as you suggest, probably not without some irony, from 0.1 to 100)