Bergson, Time and Free Will, Chapter 2, §53 "We Cannot Form an Image or Idea of Number without the Accompanying Intuition of Space"

by Corry Shores
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Bergson, Time and Free Will

Chapter II, "The Multiplicity of Conscious States," "The Idea of Duration"

Part XVI: Numerical Multiplicity and Space

§53 "We Cannot Form an Image or Idea of Number without the Accompanying Intuition of Space"

Previously Bergson wondered if number and space had anything to do with each other. To answer this we consider the ways that we conceived numbers since our childhood.

Recall that first we see a row of balls, for example.

Then, we imagine that row of balls.

Then, the balls become just simple points.

Finally, the points disappear, leaving only an "abstract" number.

However, the abstract number that remains is really just a symbol. It allows us to remember the concrete images. And, it provides a conventional way of expressing number.

Consider for example that we learn basic principles of arithmetic well enough that we know 12 to be half 24, even though we do not need to conceive either the number 12 or the number 24. But, if we want to "picture" number itself rather than merely figures or words, then we have to produce an "extended image."

However, we might think that we are able to conceive the number 50, for example, without an image of fifty things. For, we might begin with "unity" and count-up one-by-one until we arrive at 50. In this way, it seems to us that we are not imagining things in space, but rather an abstract number existing in duration. Moreover, when we are counting durations rather than points in space, we also use a temporal succession. This again leads us to believe that we consider number in abstraction from space. (78d)

But Bergson wants to know if what we really do in these cases is count moments by means of spatial points. Imagine that we are adding things, but as we add new things to the old ones, the old ones disappear. If the things we are adding disappear, we could not add them. Thus to sum things, we must take them together, and they all must remain together. Now, when we counted up to 50, we thought that we did not give any spatial place to the figures in the succession. Rather, we believed that they were only separated by duration, and hence were non-spatial. But, for us to go from 12 to 13, for example, we cannot erase the previous 12. They must also remain in our consciousness, which means they must be given an ideal spatial place in our imagination. (79a)

We involuntarily fix at a point in space each of the moments which we count, and it is only on this condition that the abstract units come to form a sum. (79ab)

Involontairement, nous fixons en un point de l'espace chacun des moments que nous comptons, et c'est à cette condition seulement que les unités abstraites forment une somme. (60b)

So we conceive number by means of space. However, later we will see that we can conceive time independently of space. But when we count, add, or imagine a number through duration, we are not dealing with each durational moment as a pure duration. For, these moments themselves have vanished forever. Rather, we are dealing with "the lasting traces which they seem to have left in space on their passage through it." (79c)

Although, when we count through duration, first we begin with mental images and continue with them for the first two or three numbers. Then, we just presuppose that the rest could be mentally pictured if we needed to grasp them. But the very fact that we must start with images means that "every clear idea of number implies a visual image in space." Also, when we conceive of a discrete multiplicity, it is made-up of the same sort of imagined units that we use also to conceive number. Hence as well, any clear idea of a discrete multiplicity also requires such spatialized images. (79d)

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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

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