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Updated: May 25, 2025
For if contradictions between theory and experience manifest themselves, we should rather decide to change physical laws than to change axiomatic Euclidean geometry. If we deny the relation between the practically-rigid body and geometry, we shall indeed not easily free ourselves from the convention that Euclidean geometry is to be retained as the simplest.
In this way he arrived at a space-structure which possesses neither the three-dimensionality nor the rectilinear character of so-called Euclidean space a space-picture which, though mathematically consistent, is incomprehensible by the human mind.
In the sense of Euclidean geometry, a plane is the sum-total of innumerable single points. To take up a position in a plane, therefore, means to imagine oneself at one point of the plane, with the latter extending around in all directions to infinity. Hence the journey from any point in space to a plane is along a straight line from one point to another.
For in a space of a kind we are accustomed to conceive, that is, the three-dimensional, Euclidean space, the existence of such a field with its characteristic of increasing in strength in the outward direction is a paradox, contrary to mathematical logic.
For although I suspect that even Einstein is an imperfect relativist, and retains Euclidean space and absolute time at the bottom of his calculation, and recovers them at the end, yet the effort to express the system of nature as it would appear from any station and to any sensorium seems to be eminently enlightening.
But I understand the nature of men better now. What an eye-opener!" Diva by this time was trundling away round the room, and longing to be off in order to tell everybody. She could find no hole in Elizabeth's arguments; it was founded as solidly as a Euclidean proposition. "Ever occurred to you that they drink?" she asked. "Believe in Roman roads and diaries? I don't."
None of these levels can intersect, and they form a family of parallel instantaneous planes in the instantaneous space of moment A. Thus the parallelism of moments in a temporal series begets the parallelism of levels in an instantaneous space, and thence as it is easy to see the parallelism of rects. Accordingly the Euclidean property of space arises from the parabolic property of time.
But always this patterning was ordered, always COHERENT. As though it were a page on which was spelled some untranslatable other world message. Fourth Dimensional revelations by some Euclidean deity! Commandments traced by some mathematical God!
The realization of the reversibility of the relationship between Point and Plane leads to a conception of Space still free from any specific character. By G. Adams this space has been appositely called archetypal space, or ur-space. Both Euclidean and polar-Euclidean space are particular manifestations of it, their mutual relationship being one of metamorphosis in the Goethean sense.
This may suffice to show how realistic are the mathematical concepts which we have here tried to build up. We were thereby led to realize that man's general capacity for thinking mathematically is the outcome of early experiences of this kind. It is evident that geometrical concepts arising in man's mind in this way must be those of Euclidean geometry.
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