## 2 Feb 2009

### Fechner's Law Simplified

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In Weber's Law, the ratio of the smallest-perceptible-stimulus-change to the original-stimulus-value remains constant regardless of the original stimulus value. Hence the smallest corresponding sensation-change is constantly the same ratio, for example, in the case of weight sensation, the constant is always 1/51 [To simply the math we use 1/50]. We will now formalize the way that Fechner described Weber's law. We use a dot "●" to mean "is constant when there is a constancy of":

In the case of weight, the minimal sensation of difference is constantly 1/51. So suppose in one case we begin with 100 kg, then feel a minimal increase. We will find that the minimal-perceptible-increase was 2 kg. Now suppose in another case we begin with 10 kg. Then we will feel a minimal increase of sensation when 0.2 kg is added. So we know that in any case the ratio of smallest-perceptible-change to original-stimulus-value remains the same: 1/51.

Weber's Law tells us that for each sensation, there is a constant ratio like the one above, and also that so long as this ratio holds between any stimulus values, we can expect that we feel the minimal possible sensation. But Fechner wants to be able to say that the sensation of 100 kg is greater than the sensation of 10 kg. Moreover, he wants to be able to say how much more of a sensation we have when holding 100 kg instead of 10.

In the first place, he will have to create an equation with the appropriate variables.

For the weight experiments, our ratios of least-perceptible-change to starting-weight might be:
0.02/1
0.2/10
2/100

In each case, the fraction equals 1/50. Thus, if we multiply each ratio by 50, we have the same constant minimal sensation of weight change:

Thus Fechner uses this pattern to translate Weber's Law into an equation:

In the case of lifting weights, C = 51 [although we use 50]. But for changes of brightness, C = 62. For changes of sound-volume, C = 333. So the constant in each case is consistent for that particular type of sensation, but it varies for the different kinds of sensations.

Now let's presume that we begin with no weight in our hands but we still slowly increase up from zero. Suppose that as soon as we get to three grams, we first feel weight. Prior to that we felt nothing, but at 3 grams we feel something. So 3 grams is the minimal threshold for feeling weight [it could actually be otherwise, but we here suppose it is 3 grams]. Then we keep track of every time we feel another change. We do this until we have felt 15 minimal changes. We find that the weight at the 15th change is slightly above 4 grams. The left column of figures is the number of changes. The middle column are the amounts of weight-increase needed to produce the following sensation of weight change. And the right column is the total amount of weight being held:

 0.06 3 1 0.0612 3.06 2 0.062424 3.1212 3 0.0636725 3.18362 4 0.0649459 3.2473 5 0.0662448 3.31224 6 0.0675697 3.37849 7 0.0689211 3.44606 8 0.0702996 3.51498 9 0.0717056 3.58528 10 0.0731397 3.65698 11 0.0746025 3.73012 12 0.0760945 3.80472 13 0.0776164 3.88082 14 0.0791687 3.95844 15 0.0807521 4.03761

We see that the ratio 1/50 maintains all throughout. Also we see the first weight increase was only 0.06, while the last one was 0.08. We can imagine that if we began at other numbers, like 3.5 grams, our increments would have been slightly different, and the resulting weights slightly different, yet the whole time maintaining the proportion 1/50. So in other words, so long as the proportion continually holds, there is a continuum of corresponding values. Also, because the increases themselves increase, their pattern of increasing can be represented using logarithms.

But what if we wanted to know how many sensation-changes there are when we increase from 3 grams to 4 grams? We know that there must be between 14 and 15. But since there is a scaled correspondence, there should be a way to quantify exactly that number of minimal sensation-changes for any given stimulus.

Fechner uses integral calculus to produce the formula that allows us to determine the number of minimal stimulus changes. For our purpose of understanding Bergson's critique, we want the clearest illustration possible. So we will use a much simpler method that is based on the same principles.

The first thing we do is establish a log value for the weight-sensation constant. We do so by adding 1 to the value of the constant ratio for the given sensation. In the case of weight sensitivity, this ratio is 1/50 or 0.02. When we add that to 1 we get 1.02. Now we take the log value of that, log(1.02), and put it aside for the moment.

Now we will consider the log of the stimulus value when we first feel weight: log(3). We subtract that from the stimulus value of the weight we want to determine: log(4).

log(4)-log(3)

this figure we divide by the standard log value for weight sensation that we previously set aside:

[log(4)-log(3)] / log(1.02)

The number we obtain is 14.527, which is between 14 and 15, as we expected. [Again, Fechner's method is far more complex, but ours does not seem too inaccurate.]

In this way, Fechner believed he found a way to quantify a sensation's intensity, based objectively on our knowledge of the stimulus' extensive measurements.

[The text above modifies material from:

Masin, Zudini, and Antonelli. "Early Alternative Derivations of Fechner's Law." Journal of the History of the Behavioral Sciences. Vol. 45(1), 56-65, Winter 2009.

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