2013 June 10
PART 6: Poincaré's words are in boldface.
A few remarks to end with. There is a striking contrast between the roughness of this primitive geometry, reducible to what I call a table of distribution, and the infinite precision of the geometers' geometry. And yet the latter is born of the former, but not of that alone; it must be made fruitful by the faculty we have of constructing mathematical concepts, such as that of group, for instance. It was needful to seek among the pure concepts that which best adapts itself to this rough space whose genesis I have sought to explain and which is common to us and the higher animals.
The evidence for certain geometric postulates, we have said, is only our repugnance to renouncing very old habits. But these postulates are infinitely precise, while these habits have something about them essentially pliable. When we wish to think, we need postulates that are infinitely precise, since this is the only way to avoid contradiction; but among all the possible systems of postulates there are some we dislike to choose because they are not sufficiently in accord with our habits; however pliable, however elastic they may be, they have a limit of elasticity.
We see that if geometry is not an experimental science, it is a science born à propos of experience; that we have created the space it studies, but adapt it to the world wherein we live. We have selected the most convenient space, but experience has guided our choice. As this choice has been unconscious, we think it has been imposed upon us; some say experience imposes it, others that we are born with our space ready made. We see from the preceding considerations, what in these two opinions is the part of truth, what of error.
In this progressive education whose outcome has been the construction of space, it is very difficult to determine what is the part of the individual, what the part of the race. How far could one of us, transported from birth to an entirely different world, where were dominant, for instance, bodies moving in conformity to the laws of motion of non-Euclidean solids, renounce his ancestral space to build a space completely new?
The race seems indeed to play a preponderant part; yet if to it we owe rough space, the pliable space I have spoken of, the space of the higher animals, is it not to the unconscious experience of the individual we owe the infinitely precise space of the geometer?
This is a question not easy to solve. Yet we cite a fact showing that the space our ancestors have bequeathed us still retains a certain plasticity. Some hunters learn to shoot fish under water, though the image of the fish be displaced by refraction. Besides they do it instinctively. They therefore have learned to modify their old instinct of direction, or, if you choose, to substitute for the association A1, B1 another association A1, B2, because experience showed them the first would not work.
THE RELATIVITY OF SPACE, by Henri Poincaré