1 Jan 2009

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:


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:




Fechner image from:
Fechner, Gustav Theodor. Elemente der Psychophysik. Zeiter Theil. Leibzig: Druck und Verlag von Breitkopf und Härtel, 1860.
Available online at:

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