It’s easy to neglect the detail of one person’s, now historic, philosophical discussion of math and science. But there is a moment, in Henri Poincare’s well known text Science and Hypothesis, that I would like to shine a light on today. The first English translation of the book was published in 1905. Chapter IV is a discussion of space and geometry and in it Poincare takes the time to make an artful distinction between our physical sense of space and mathematical space. He’s writing at a time when mathematics has successfully transgressed the restrictions of Euclidean geometry, once thought to simply be geometry. I’ve written about Riemann’s definitive contribution to the mathematical notion of ‘space’ in an earlier post. In his chapter on Space and Geometry Poincare begins with a simple statement of what is not true:
It is often said that the images we form of external objects are localized in space, and even that they can only be formed on this condition. It is also said that this space, which thus serves as a kind of framework ready prepared for our sensations and representations, is identical with the space of the geometers, having all the properties of that space.
He then summarizes the properties of a Euclidean geometrical space – that it is continuous, of three dimensions, homogeneous (i.e. all the points are identical to one another) and isotropic (identical in all directions). The interesting part of the discussion begins when he contrasts this with the properties of our sense of the space around us. He refers to this as representative space, built from visual, tactile and motor experience.
Poincare points first to the retina, a two dimensional surface and a limited framework where all of the points are not identical – all points on the retina do not play the same role. He takes particular note of the difference between points at the retina’s center and those at the edges. While this visual space may be said to be continuous, it is neither homogeneous nor isotropic. He moves away from the retina’s surface to describe our perception of a third dimension. This “reduces to a sense of the effort of accommodation which must be made, and to a sense of the convergence of the two eyes, that must take place in order to perceive an object distinctly.” Since these are muscular sensations, they are experienced differently. A complete visual space is, therefore, neither infinite, nor homogeneous nor isotropic. The question then arises: Is it, in fact, three dimensional? The mathematician takes over this argument. Since our sensation is built on at least four independent variables, two from the two-dimensional retina, and two from two kinds of distance cues, then our visual space is determined by four independently varying parameters and the complete space may be said to have four dimensions. Modern theories of the visual brain would likely say the dimension is higher. But Poincare also takes note of the fact that movement contributes to the genesis of the concept of space and argues that the sensations which correspond to movements in the same direction are only connected by an association of ideas in our minds. The aggregate of muscular sensations actually depends on as many variables as we have muscles.
Poincare summarizes his analysis in this way:
Thus we do not represent to ourselves external bodies in geometrical space, but we reason about these bodies as if they were situated in geometric space.
For Poincare, our idea of space is secondary to our experience of space which is based entirely on correlative events in our bodies.
This was a time when a mathematician would not ignore the way mathematics transgressed the boundaries of common reason and experience or would not wonder about its nature. As Poincare once said, “Though the source be obscure, still the stream flows on.“ Revolutionary and far-reaching insights in both mathematics and physics were brought to the fore in the early twentieth century. And these were problematic for some. Being an early participant in the transforming ideas of modern physics Poincare writes later in the book:
When a physicist finds a contradiction between two theories which are equally dear to him, he sometimes says: “Let us not be troubled , but let us hold fast to the two ends of the chain, lest we lost the intermediate links….It is quite possible that they both express true relations, and that the contradictions only exist in the images we have formed to ourselves of our reality