"I do not believe that there exists anything in external bodies for exciting tastes, smells, and sounds, etc. except size, shape, quantity, and motion." (Galileo Galilei, On Motion, p.48) When Galileo proposed his doctrine of subject and object and the distinction between primary and secondary qualities, he established the scientific prejudgement that the concept of space is of something geometrical and not differentiated qualitatively.
Newtonian "absolute space" was based on a realist conception of mathematics (see Jammer p. 95) To Newton, mathematics, particularly geometry, is not a purely hypothetical system of propositions...instead geometry is nothing but a special branch of mechanics. Newton's first law of motion, which links change in motion with force requires an absolute (or inertial?) framework. It requires a distinction between absolute motion and relative motion and links force to a change in absolute motion. For example, as the train pulls away from the station, the station may appear to be moving and it can be said that the station is in relative motion to the train, but the force is acting upon the train, and it is the train that is accelerating absolutely. Newton tried to establish an absolute frame of reference for the universe defined in relation to its center of gravity. (not necessarily identical with the sun) Absolute spatial movement and position could then be measured in relation to that point.
But is geometry an empirical or ideal activity? For Cassirer, the most radical removal of geometry from experience had already occurred with Euclid, which was already based on figures that are removed from all possibility of experiment. Not only the idealizations of point, line, and plane, but the idea of similar triangles, whose differences are considered inconsequential or fortuitous, and that become identified as "the same" mark an immense step away form ordinary perception.
The mathematization of space and its representations in Cartesian grids allowed space to become more abstract and less tied to a specific set of conditions. If the axes of the grid could stand for any set of variables, then a proliferation of types could take place. But even as Descartes' discovery of analytic geometry gave the problem of space an entirely new orientation, his own metaphysics describes space as some sort of absolute thing in the form of an extended substance, not simply a certain pattern of order.
"In all the history of mathematics there are few events of such immediate and decisive importance for the shaping and development of the problem of knowledge as the discovery of the various forms of non-Euclidean geometry." (Ernst Cassirer, The Problem of Knowledge) In Euclidean geometry, the axiom of the parallels states that through a given point there is one and only one parallel to a given straight line that does not go through the given point. Non-Euclidean geometry starts with the opposite axiom... When Riemann published "On the Hypotheses Underlying Geometry" (1868) the axioms of Euclid, which had been regarded for centuries as the supreme example of eternal truth, now seemed to belong to an entirely different kind of knowledge. For Cassirer, "the whole problem of the truth of mathematics, even of the meaning of truth itself, was placed in an entirely new light. Until that time, both rationalist and empricist philosophers had agreed that the relations of mathematical ideas were rigorously necessary and unalterable. How could entirely different and wholly incongrous systems of geometry uphold the claims of truth? "To recognize a plurality of geometries seemed to mean renouncing the unity of reason, which is its intrinsic and distinguishing feature." (Cassirer, p.24)
"Mathematicians appropriated space and time, and made them part of their domain, yet they did so in a rather paradoxical way. They invented spaces: non-Euclidean spaces, curved spaces, n-dimensional spaces, abstract spaces (such as phase space), and so on. (see Lefebvre , p.2 for a social critique of abstract space) For example, Gerald Edelman uses the concept of an n-dimensionsal neural space of all potential qualia, that includes every possible discrimination between states of consciousness. For Edelman, the dimensions of this space are given by the activity of actual groups of neurons in the brain.
Henri Poincaré
In this way, space became a "mental thing" Physicists, according to Rudolf Carnap are free to choose among spatial systems according to their own requirements. He quotes Henri Poincaré 's observation that no matter what observational facts are found, the physicist is free to ascribe to physical space any one of the mathematically possible geometrical structures, provided that he makes suitable adjustments in the laws of mechanics and optics and consequently in the rules for measuring. For Poincaré , "The object of geometry is the study of a definite group, but the general idea of the group preexists, at least potentially, in our mind, having forced itself not as a form of sensibility but as a form of our understanding. All we have to do is choose among all possible groups the one that will constitute a standard for us, as it were, to which natural phenomena are referred. Experience guides us in this choice but does not dictate it; nor does it permit us to know which geometry is truer but only which is more 'useful.' " (H. Poincaré , La Science et l'hypoth se, quoted in E. Cassirer, The Problem of Knowledge, p. 43)
Rudolf Carnap rejects Kant's claim that geometry is a priori and synthetic. He splits geometry into mathematical geometry which is a priori because analytic and physical geometry which is synthetic and not a priori. In physics the choice of geometries becomes a pragmatic one.In his Philosophy of Space and Time, Hans Reichenbach develops this empiricist conception of geometry. (see visualization)
Ernst Cassirer shows Poincaré's assessment of the impact of non-Euclidean geometry as a shift in the meaning of mathematical axioms. For Cassirer, the theory of sets had shown that the different geometries were all equally true in an ideal and mathematical sense. Geometry could be defined as a theory of invariants in respect to a certain group -- only properties that are characterized by an invariance with respect to certain transformations can be called "geometrical." While Euclidean geometry applies to a "basic set" of rigid bodies that are freely movable in space without changing form, different transformations can be applied to different sets of objects (defined as "same" with respect to a particular criterion) For Cassirer, the modern sense of axioms differs from the ancient. Axioms are no longer assertions about content that have absolute certainty. Rather they arre proposals of thought that make it ready for action. (p.45)