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In speaking of the nature of representative space, Poincaré introduces the idea of a flat surface, or "canvas" if you will, upon which a painter might paint:

"Our representations are only the reproduction of our sensations; they cannot therefore be arranged in the same framework—that is to say, in representative space. It is also just as impossible for us to represent to ourselves external objects in geometrical space, as it is impossible for a painter to paint on a flat surface objects with their three dimensions." (Science and Hypothesis, p. 66)

When he speaks of a "representation of our sensations", the idea seems to hearken back to what he said previously about the retina:

"A cursory analysis shows us this image [formed on the back of the retina] as continuous, but as possessing only two dimensions..." (Science and Hypothesis, p. 61)

None of that is all that surprising, because many people think of the visual field as a two-dimensional plane that could be likened to a canvas. However, what I find surprising is that, in many ways, he denies that the sense of vision is geometric or spatial in nature. In fact, he denies that any of our sensations have a spatial character ("On the Foundations of Geometry", p. 1) or that we could have any sense of direction from vision in isolation from other sensations, especially muscular sensations (Science and Hypothesis, p. 67). In addition to that, he claims that we can could never have any notion of distance in isolation from motion:

"Suppose images formed at four points A, B, C, D of this immovable retina. What ground would the possessor of this retina have for saying that, for example, the distance AB was equal to the distance CD?" ("On the Foundations of Geometry", p. 2)

Supposedly our sense of direction comes from feelings associated with movements that occur in the same direction (Science and Hypothesis, p. 65). It's unclear to me how we could know when movements occur in the same direction prior to a sense of direction. Nor does he seem to allow for any sense of position, which would help with both the problem of direction and relative distance. In fact, he speaks of different points of the retina as supplying a succession of sensations which would be spatially indistinguishable if not for the corrective changes which we can bring about through voluntary motion ("On the Foundations of Geometry", p. 6-7).

Finally, Poincaré asserts that none of our sensations in isolation could impose upon our minds the concept of geometrical space (Science and Hypothesis, p. 67). If that's true, the physiology of my retina shouldn't impose any spatial restrictions with respect to how the world is represented. This seem to contradict his claim that we only have a flat two-dimensional surface to work with. (Besides that, my canvas doesn't appear flat at all.)

How should I understand the limitations of a flat canvas which is purportedly not spatial in nature? Perhaps even more importantly, on what basis can he account for it having any dimension at all?

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