## Is Euclidean Geometry the ‘Form of Outer Sense’ for Kant?

Abstract:

Kant is often thought to identify the necessary conditions on our “outer in- tuition” with the principles of Euclidean geometry. On this interpretation of his view, it is refuted by 19th and 20th century developments in mathematics and physics. But some commentators observe that the necessary conditions Kant articulates (in the “Metaphysical Exposition” of space), to which he refers as “singularity” and “infinity”, correspond to a broader class of geometries. (It is then suggested that Kant uses Euclidean theorems to illustrate the conditions only because no other geometry was then known.) I wish to develop this insight in terms of more general spatial notions than the metrical ones which feature in recent literature.
Interpreters have typically considered Kant’s view against the back- ground of the emergence of non-Euclidean geometry. If they are sympathetic, they seek to explain how his view could accommodate the other geometries of constant curvature. A prominent example is Michael Friedman’s view that the a priori form of outer intuition makes possible, specifically, the constructive operations of classical Euclidean geometry. Friedman explicitly construes the a priori features of spatial intuition as conditions on measurement. On this interpretation, Hermann von Helmholtz’s view, viz. that the principle of free mobility is a necessary condition for quantitative spatial representation, is a “minimal generalization” of Kant’s (2012, 257).
Following Lisa Shabel, I take Kant to conceive intuition’s distinctive contribution to geometrical reasoning primarily in terms of topological features and mereological relationships, rather than congruence relations. This calls for considering Kant’s view against the background of a different generalizing trend, from metrical geometry to projective geometry and then topology.
It can accommodate these generalizations if, as I suggest, Kant’s notion of space as the form of outer intuition is understood as what Riemann was to call a triply-extended magnitude. Such an identification is licensed by Kant’s conceiving of regions of space (including space itself) as quanta, and (in his metaphysics lectures) regarding them “not as existing independently of position and not as expressible in terms of a unit, but as regions in a manifold”, as Riemann was to put it (transl. Ewald 2005, 653). Hence, Kant could have agreed with Riemann that the “general concept” of extended magnitude does not determine, independently of contingent empirical facts, the metrical geometry of space. (I do not claim, however, that Kant could have anticipated Riemann’s use of differential calculus for assigning metrical properties to space.)
Although this paper is not primarily an account of Riemann’s views, his Habilitationsvortrag is relevant because it supplies an argument (Ewald 2005, 660) for Kant’s view that these concepts do determine that space’s magnitude is unbounded. This doctrinal agreement is particularly striking because Riemann (following Herbart, through whose writings he was exposed to Kant) seems to have rejected what he took to be Kant’s main conclusions.
References:
Ewald, W. From Kant to Hilbert, Vol. 2 (Oxford University Press, 2005). Friedman, M. “Einstein, Kant, and the Relativized A Priori”. In Constituting Objectivity (Springer, 2012), 253-268.