11.03.09
Attempts to resolve the issue of space/time in Kant
Comment on Space/time and idealism/realism
Stephen P. Smith said,
November 3, 2009 at 10:44 am
One approach is to construct a space-time geometry within the intuitionist paradigm, and then relate this geometry to concrete space-time. I think this is doable, and I start this effort in my book on the topic of general relativity. The construction becomes dialectical. For example, Euclidean geometry is found passing over into Riemannian geometry, and invariant geometric constructs are found being held together by a middle-term that unites the covariant and contravariant. Kant`s synthetics can be collected and used to induces a geometry (built up from noted equations), but for abstract geometry to approach real space-time it must be that the synthetics are refined and purified by the dialectical program. That the synthetics are real in the limit means that the essence of mind sources the space-time fabric, agreeing with transcendental idealism.I will note that our macro-dimensions that are apparent are very telling, removed from the quantum scale of the uncertainty principle and the galactic sale of black holes, of three spatial dimension and one temporal dimension. I believe we see the our space-time dimensions because it is necessary for us to perform error recognition. We have the necessary dimensions that are needed to correct all our mistakes, confirming a transcendental idealism. We have the needed dimensions to distinguish constructed distance from constructed straightness, and we can distinguish intrinsic straightness that is limited to a surface manifold from extrinsic curvature that embeds the manifold. This is a three fold passage again, where the impetus to construct distance, straightness, and curvature is found held together by a middle term that hints of the beyond.
Stephen P. Smith said,
November 3, 2009 at 6:28 pm
Here is a re-write of the above, and my take on space-time as represented by geometry.
An approach that offers interesting results is the method of construction as professed by intuitionist mathematics (e.g., Brouwer`s mathematics). Construction may require the activity of measurement, and what is generally required is a time dimension where watching an internal clock permits measuring the duration of activity. With the addition of one spatial dimension it becomes possible to measure distance and distinguish distance from duration. Distance is measured by laying a rigid body end to end. With the addition of a second spatial dimension it becomes possible to distinguish constructed distance from constructed straightness. Straightness becomes the shortest distance between two points. With the addition of the third spatial dimension it becomes possible to distinguish intrinsic straightness that is limited to a surface manifold from extrinsic curvature that embeds the manifold.
Awareness of three external spatial dimensions and one internal temporal dimension is supported by a three-fold passage: the impetus to construct distance, straightness, and curvature. The impetus that measures distance is negated when straightness is discovered in the making, this is related to Hegel`s first negation if our geometrical construction purports to be dialectical. The impetus that measures straightness is negated when extrinsic curvature is discovered in the making. In abstract geometry many more spatial dimensions are possible, but in concrete space-time our macro dimensions (that are removed from the quantum scale and galactic scale) are strangely limited to three spatial dimensions and one temporal dimension. As far as our ordinary space-time is considered, the making that gave awareness to duration, distance, straightness and curvature ends with three spatial dimensions and one temporal dimensions. The blunt ending implies that the impetus carried by all activity ends, or reaches its visible limit, within realized space-time. After the second negation there is no more dimensionality to be discovered within the macro state, and whatever impetus that remains must be redirected internally. It is redirected to what holds the first negation to its second negation, a middle terms that implies that space-time is two-sided.
Abstract geometry is also apparently two-sided: there is the covariant that is offset by the contravariant, but with the contraction of the covariant and contravariant returns the invariant presented to awareness; there is the intrinsic that is offset by the extrinsic that offers a logical map to locate (by the activity of measurement) the intrinsic. Abstract geometry is found agreeing with realized space-time, agreeing with Kant and his view of intuitions that are a-priori and synthetic. However, we have now reached beyond Euclidean geometry that Kant knew. Euclidean geometry has passed over into Riemannian geometry following the requirements of constructed activity that is rediscovered to be dialectical, as noted above.
Kant`s views are saved, even with Riemannian geometry, an observation that is not widely known.