Fechner's Law Simplified

by Corry Shores
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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	0.06367248	3.183624
4	0.06494593	3.247296
5	0.066244848	3.312242
6	0.067569745	3.378487
7	0.06892114	3.446057
8	0.070299563	3.514978
9	0.071705554	3.585278
10	0.073139665	3.656983
11	0.074602459	3.730123
12	0.076094508	3.804725
13	0.077616398	3.88082
14	0.079168726	3.958436
15	0.0807521	4.037605

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.

Available online at:

http://www3.interscience.wiley.com/cgi-bin/fulltext/121634139/PDFSTART

I also modified material from this site:

http://www.diracdelta.co.uk/science/source/w/e/weber-fechner%20law/source.html

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Bergson, Time and Free Will, Chapter 1, §42 "The Underlying Assumptions and the Process by which Fechner's Law is Reached"

by Corry Shores
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[The following is summary; my commentary is in brackets.]

Bergson, Time and Free Will

(Essai sur les données immédiates de la conscience)

Chapter I, "The Intensity of Psychic States"

Part XIV: "Psychophysics"

§42 "The Underlying Assumptions and the Process by which Fechner's Law is Reached"

Bergson will show how Fechner converted Weber's experimental findings into his psychophysical law (61d). [See this entry for more on Weber's Law, and see this page for a version of Fechner's Law that will more easily produce accurate figures for the sake of demonstration. Bergson is not very clear, and he veers from Fechner's rendition. My interpretation is guesswork.]

We noted before that Weber's findings:

1) we can gradually increase a stimulus until a change is first detected, and then take note of the quantity of stimulus needed to induce that least-perceptible-change.

2) Consider we want to compare two levels of stimulus, a lower and a higher one. Then we increase gradually the lower one until we feel a change. We do the same for the higher one. We find that higher starting stimuli always require greater increases to produce a minimal sensation.

3) The proportion of least-perceptible-stimulus-change (ΔE) to starting-stimulus-quantity (E) is always the same ratio for each type of sensation. For example, for the sensation of weight, it is 1/51.

We call the original starting stimulus value 'E.' And the least perceptible-stimulus-change then is 'ΔE.' Their proportion is constant for any any given sense type, so we formulate it as:

Now, we will call any given sensation 'S.' When we increase E by ΔE, we experience the minimal sensation of difference. We will call this minimal sensation ΔS. And we will place them into an equation. But ΔS for weight sensation, for example, does not equal 1/51. Rather, no matter the circumstances, ΔS always equals itself, because it is constant (so it always equals 1). Thus we need to make the other side of the equation equal 1. Recall also that ΔE can be determined by applying a function to the starting E value. In the case of sensing weight, that would be:

ΔE = f (E) = (E/51).

Now, when we divide ΔE by f (E), we obtain 1/1, or 1. So now we can equalize it with ΔS. To express the constancy of Weber's Law, we incorporate the C which is just a constant value, for our purposes it should equal 1 so not to change the proportions.

[Note that Bergson already has veered from Fechner's explication. Fechner does not use the function notation.

So it seems in his case the constant K would stand for 51 when measuring weight sensation.]

Now imagine that we are holding 100 grams of weight. We know how much it will take to obtain the least-perceptible-change: 100/51 or approximately 2. [From now on we will use the ratio 1/50 to simplify the mathematics.] But Fechner wants to be able to determine the amount of sensation we have when just holding 100 grams. Because the least-perceptible-sensation of change is always the same amount of sensation, he will standardize the measurements in terms of the number of ΔS's. He will set as the base-line the least amount of stimulus needed to have a feeling of the stimulus in the first place. Then we will try to find the number of ΔS's it takes to rise from the base-line stimulus to the one we are translating into a sensation value. So if the least we can sense is 1 grams, then we want in our example to determine the number of ΔS's it takes to rise from 1 gram to 100 grams. But we want to determine it mathematically, not experimentally. So we need an equation that allows us to compute the amount of sensation by performing some mathematical operation on a given stimulus quantity.

One problem is that all the ΔS's remain the same homogeneous value, even though the ΔE's vary according to the starting weight. However, they are logarithmically correlated [see the Fechner entry for more on the logarithmic correlation.] But that also means that their correspondences become increasingly disproportional as the increases become larger. However, if we only look at an infinitely small amount of S and an infinitely small amount of E, they will not verge from each other, in fact they will be directly proportional.

Now we may sum all the differentials of S to obtain the total value of S. But we do not know S yet. However, we know E, so we will also sum-up all the differentials from the base-line stimulus to the stimulus quantity we are translating (0 grams to 100 grams, for example.)

Because each differential of S and of E were directly proportional, the sum of all the differentials leading up to E should give us the value for S. [It is still not clear to me how to specifically use this formula to obtain a specific value for S. However, the Fechner entry provides a simpler way to produce a specific accurate value.]

Weber's Law alone is provable, but it does not help us measure sensation. However, Fechner's Law does allow us to measure sensation, but it is not provable:

the transition will thus be made from a proved law, which only concerned the occurrence of a sensation, to an unprovable law which gives its measure. (62cd)

l'on passera ainsi d'une loi vérifiée, où l'apparition de la sensation était seule en cause, à une loi invérifiable, qui en donne la mesure. (46-47)

In the next part, Bergson will explain the ways that Fechner's method succeeds and fails. (62-63)

[Next entry in this series.]

Images from the pages summarized above, in the English Translation [click on the image for an enlargement]:

Images from the pages summarized above, in the original French [click on the image for an enlargement]:

Bergson, Henri. Time and Free Will: An Essay on the Immediate Data of Consciousness, Transl. F. L. Pogson, (New York: Dover Publications, Inc., 2001).

Available online at:

http://www.archive.org/details/timeandfreewill00pogsgoog

Images and French text from:

Bergson, Henri. Essai sur les données immédiates de la conscience. Originally published Paris: Les Presses universitaires de France, 1888.

Available online at:

http://classiques.uqac.ca/classiques/bergson_henri/essai_conscience_immediate/essai_conscience.pdf

and

http://www.archive.org/details/essaisurlesdonn00berguoft

Fechner image from:

Fechner, Gustav Theodor. Elemente der Psychophysik. Zeiter Theil. Leibzig: Druck und Verlag von Breitkopf und Härtel, 1860.

Available online at:

http://www.archive.org/details/elementederpsyc00fechgoog