It's Day 9 of the Year 2019 CE,
January 9, 2019, FWIW

Ahhh, but not those logs, these logs: log_b(x). It turns out that our perceptions of many things in this world, including aging and duration follow a logarithmic curve of one sort of another. With that said, I'm going to drop a relatively short video on the matter right here:

In summary the presenter, Sarah, uses Brady as a foil and has him try to discriminate between two different weights, one in each hand. He can, for 100 and 120 grams, respectively, but not for 200 and 220. She states that this is Weber's law and that it applies to many things, including our perception of time and demonstrates graphically how this works. On the x axis she has weight (stimulus) and on the y perceived weight (response). At low x values the log curve is nearer to vertical and the perceived difference is great relative to the actual difference, while at higher values the log curve is nearer to horizontal and the perceived difference relative to the actual difference is quite small. So, when you're a kid, a year is a big chunk of your life and lasts a loooong time, but when you're 70 something, it just flies by. She applies this to many things such as marketing, pricing, and jail sentences (an area of concern to her). It is noted that there is more fine granularity in short judicial sentences, 30 days, 6 months, etc. than in long sentences, 20 years, 30 years, etc. Six months is perceived as way more than 3 months, but 20 years and 3 months isn't really perceived as that much more than 20 years. This is reflected in judicial sentences all over the globe. OK, that's all well and good, but the whole presentation is something of a gloss.

Zo, Weber's law is strictly about perception. It was formulated by Gustav Fechner and is part of the foundation of psychophysics, which Fechner also invented. Fechner came up with several formulations of Weber's law, such as:

"Simple differential sensitivity is inversely proportional to the size of the components of the difference; relative differential sensitivity remains the same regardless of size."

This is a more comprehensive version of Sarah's statement that delta I over I = k. That the perceived change is proportional to the initial stimulus can be best illustrated by another aspect of Weber's law relating to the Just Noticeable Difference. In the experiment with Brady, had they continued with ever decreasing differences, they would've hit a point where Brady couldn't detect the difference. At, for example 104.999999, he couldn't and at 105 he could. That would mean that the just noticeable difference, for Brady, for weight, would be 5 grams at the 100 gram level. That would be 5% of the base rate of 100 grams. The 5% proportionality would hold for Brady with respect to weights, for any base weight. Thus, if the initial weight were 400 grams, then the just noticeable difference would be 20 grams, 5% of 400. This just noticeable difference differs for each sense and for each person, but there always is one and the proportionality rule holds. A horizontal line on a monitor or billboard will have to be a certain percentage longer for any of us to notice the difference, and that difference, which varies among individuals, is proportional to the original length.

So, beyond that, perception is proportional to the log of the stimulus. The basic Weber's law and the formulations concerning just noticeable differences, were based on experiments performed by Weber. From those laws, Fechner then derived the law that perception is logarithmically proportional to stimulus. This law is not strictly true, but works over a very broad range. It is only a very useful approximation, as formulated. All that said, however, it is kind of good to know. Also, though it is strictly about "perception" as in sensation, there does seem to be some applicability to the broader sense of "perception" such as our perception of the passage of time and maybe even the value of tangible things.

Of course, the discussion of sentencing sent me straight to "30 days in the workhouse, 6 long months in jail":

Some got 6 months, some got one solid year, but...

Image is: Truck load of ponderosa pine, Edward Hines Lumber Co. operations in Malheur National Forest, Grant County, Oregon (LOC)

Its an open thread so have at it. The floor is yours
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