Abstract
Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of physical geometry (or idealized perceptual space), the space of the mathematical science of physical nature (in which science, not only raw perception has a word) and the abstract spaces of mathematics (free creations of the mathematical mind), each of them with its peculiar geometrical structure. Perceptual space is proto-Euclidean and the space of physical geometry Euclidean, but mathematical physics, Husserl allowed, may find it convenient to represent physical space with a non-Euclidean structure. Mathematical spaces, on their turn, can be endowed, he thinks, with any geometry mathematicians may find interesting. Many other related questions are addressed here, in particular those concerning the a priori or a posteriori character of the many geometric features of perceptual space (bearing in mind that there are at least two different notions of a priori in Husserl, which we may call the conceptual and the transcendental a priori). I conclude with an overview of Weyl’s ideas on the matter, since his philosophical conceptions are often traceable back to his former master, Husserl.
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Notes
Empirical verifications, in this case, involve not only direct perception via the outer senses (vision, tact, etc.), but mathematically informed experiments if necessary.
But can we say from experience that space is tri-dimensional “all over”? As we will see below, Husserl seems to admit that, although we cannot access every corner of reality intuitively, we extend to the space as a whole the structural properties perceived in our limited chunk of space. Space as represented is, for him, a uniform extension of space as directly intuited. There are no reasons, presumably, for our space-constituting functions to work differently if we were placed in another region of space. The fact that physicists conjecture today that space may have more than three dimensions in subatomic scale does not pose any treat to our pre-scientific representation of space; science, Husserl believed, has the right to adopt any formally correct explanation of observed phenomena (even incurring in the risk of formalist alienation, as denounced in Crisis).
There is in Husserl a clear distinction between pure sensorial data (the hyletic data) and percepts (perception is an intentional experience). In his Lectures of 1907, Husserl presents a minute description of the constitution of spatial rigid bodies, and the medium where they are placed, physical space, in which we are not allowed to forget this distinction. Husserl believes there are essentially two systems of sensorial data involved in spatial perception, the visual and the tactile (although, in that work, the visual appears with by far more relevance), which are molded into spatial percepts by a series of intentionally motivated kinesthetic systems working in isolation and cooperatively; there are basically four of these systems (some terms are Husserl’s, some are mine): (1) the oculomotor system, by means of which a non-homogeneous 2-dimensional flat finite space is constituted; (2) the restrict cephalomotor system, by which a non-homogeneous 2-dimensional curved space is constituted, limited “above” and “below” by closed lines, like the section of the earth surface between the tropics; (3) the full cephalomotor system, by which a 2-dimensional spherical space is constituted (which Husserl calls Riemannian space); and, finally, (4) the (full) somatomotor system, by which Euclidean space is constituted. It is worth noticing that, for Husserl, binocularity does originate depth, but depth is not yet, by and in itself, a third dimension comparable to breadth and height; spatiality requires the subject to be able to move freely towards, away from and around the body, and would be constituted independently of binocularity.
The contemporary scientific image of space is that of a system of formal relations induced primarily by physical relations among physical entities (independently of how we happen to perceive or conceive it). It follows that if these entities were separated in clusters, with entities in one cluster having no physical relation to those in other clusters, space could very well be conceived, for scientific purposes, as disconnected into isolated “multiverses”, bearing no spatial relations with one another (logical relations such as that of difference, would, of course, still hold).
We could in principle conceive different organisms, with different space-constituting mechanisms, representing space with a different number of dimensions (for example, bodies incapable of motion might represent physical space with only two dimensions—see Husserl 1997 or note 3).
It is also conceivable, I think, that the psychophysical functions responsible for the representation of physical space do in fact make it impossible for us to perceive spatial discontinuities, even if they existed in transcendent reality. The situation might be analogous to us representing movement in fast running discrete sequences of still pictures: closely packed atoms of space might necessarily be perceived as a continuum. But this seems not to be what Husserl thinks (although I’m not willing to bet on it).
There are other methods, such as trying to draw a square by drawing congruent segments at right angles; only in Euclidian space this will succeed.
“There is no doubt that the conviction which Euclidean geometry carries for us is essentially due to our familiarity with the handling of that sort of bodies which we call rigid and of which it can be said that they remain the same under varying conditions” (Weyl 1963, p. 78).
Helmholtz, before Einstein, told us that the metric is not a formal aspect of space, but depends on its material content. So, even if space were not homogeneous and its metric not constant, free mobility would still be valid, since, in Weyl’s words, “a body in motion will ‘take along’ the metric field that is generated or deformed by it” (Weyl 1963, p. 87).
But what sense could be attributed to the hypothesis that the world and everything in it change dimensions in such a way that no change can in principle be noticed? As Weyl tells us (1963, p. 118), a metaphysically real difference that cannot as a matter of principle be detected is non-existent.
It is tempting to read this analogy in terms of the notion of homeomorphism. Represented space may be only a homeomorphic copy of transcendent space; i.e. only partially and, even so, only formally identical to it.
A manifold is no more, no less than a structured multiplicity of things; we would call it today a structured system of entities.
This is the transcendental synthetic a priori in Husserlian version.
In answering this question Husserl seems to be answering Helmholtz, who argued against the non-intuitability of a non-Euclidian space. For Helmholtz, to intuit a non-Euclidian physical space means to imagine spatial sense impressions, captured by our sense organs according to the known laws, which would force a non-Euclidian character on space. As I show in the main text, Husserl explicitly denied this possibility.
All the definitions (and subsequent assertions) are to a large extent Husserl’s own; I only tried to give the ensemble a more coherent (although not logically flawless) presentation, remaining as close as possible to Husserl’s original ideas. Although Husserl’s approach to the axiomatics of geometry is remarkably similar to his colleague Hilbert’s, in his famous axiomatization of 1899, it was for the most part developed before they became colleagues in Göttingen, in 1901. But this can be explained: both were buds of the same Paschian branch. Recall that also in Hilbert’s system the basic elements are point, line and plane, and the fundamental relations those of incidence (point lies on line or plane, line lies on plane), order (betweenness) and congruence. The notion of continuity (which in Hilbert’s system is given by the axiom of completeness and the Archimedean axiom) does not appear in Husserl’s sketchy system explicitly.
I didn’t find this definition in Husserl, but I think he wouldn’t object to it. It may also be that he thought sameness of direction was a primitive notion.
Discrete (finite or infinite) manifolds (nets) have a natural notion of distance: we can define the distance between two points as the smallest number of points one has to go through to reach one from the other. Non-discrete manifolds, continuous ones in particular, on the other hand, have no “natural” notion of distance and can accommodate various.
I have Bergson’s critique of Einstein’s conception of time in mind.
References
Husserl E (1931) Ideas: a general introduction to pure phenomenology. Allen and Unwin, London
Husserl E (1970a) The crisis of European sciences and transcendental phenomenology. Northwestern University Press, Evanston
Husserl E (1970b) Philosophie der Arithmetik, mit ergänzenden Texten (1890–1901). Husserliana XII M. Nijhoff, The Hague
Husserl E (1983) Studien zur arithmetik und geometrie (1886–1901). Husserliana XXI. M. Nijhoff, The Hague
Husserl E (1994) Early writings in the philosophy of logic and mathematics. Kluwer, Dordrecht
Husserl E (1997) Thing and space: lectures of 1907. Kluwer, Dordrecht
Husserl E (2001) Logical investigations. Routledge, London
Von Helmholtz H (1866) On the factual foundations of geometry. In: Pesic 2007
Weyl H (1952) Space, time, matter. Dover, New York
Weyl H (1963) Philosophy of mathematics and natural science. Atheneum, New York
Weyl H (1994) The continuum. Dover, New York. Trans. of Das Kontinnum, 1918
Acknowledgments
I also want to thank another friend, Guillermo E. Rosado Haddock, the editor of this special issue, for having suggested and introduced me to the topic of this paper.
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This paper is dedicated to my friend Claire Ortiz Hill for her sixtieth birthday.
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da Silva, J.J. Husserl on Geometry and Spatial Representation. Axiomathes 22, 5–30 (2012). https://doi.org/10.1007/s10516-011-9161-0
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DOI: https://doi.org/10.1007/s10516-011-9161-0