Is 'a diagonal' an objective material fact or a mental construct?

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When I carve a line / 'a diagonal' - do I carve wood or the idea  (die Vorstellung oder den Gegenstand / den vorgestellten Gegenstand) ?

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Two opposing surfaces in a cubic piece of wood are necessarily parallel to each other. “Parallel" applies to each and every part of these surfaces - there are no areas or points which could be excluded from this 'parallel-ness'. A diagonal drawn on one side of this piece of wood is necessarily parallel to the surface on the opposite side. If another diagonal is drawn at a different angle on the opposite surface, then these two diagonals will necessarily not be parallel to each other - despite both diagonals sitting on two parallel surfaces. 

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How can this be understood - two lines which are not parallel, but which clearly and necessarily are parallel to each other at once ? As they each sit on parallel surfaces, each point of which is part of the attribute ‘parallel’ -? This situation necessarily reveals the need for attributes to be applied: the line thus must have a 'front', 'back', 'left' and 'right', defining the seemingly bodiless line as a multi-dimensional body.  Geometry here is produced by the Logic the situation requires. 

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The 'top' and the 'bottom' of the lines must accept the way the surface provides space, but sideways the line has the 'autonomy' to negate the attributes that apply to this surface. Lines on parallel surfaces can be described by attributes which are not totally determined by this surface. Can one say that these lines have a degree of freedom-? How can this divergence of the line from the whole be best described and understood? 'Freedom': the possibility of the detail not being fully determined by the whole, whilst being fundamentally determined by the whole - ?

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The totality of a line is congruent with the surface it sits on. The totality of surface is not part of the totality of the line. The surface can only determine the line in the area the line covers. The surface being parallel to another implies that its attributes apply to the lines it carries. The diagonals on parallel surfaces being not parallel to each other implies that the lines are entities with attributes either relating to the relationship between lines (on surfaces), or to the relationship between surfaces as whole entities . The freedom of one set from another despite being contained in it? The question of the line and surface is a question of the relationship between the part / the detail and the whole. Surfaces and lines are physical and logical sets which partly contain each other. Attributes can satisfy the logic for sub-sets or parts, yet on the whole a paradox is produced.

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Everything we do -and think- is embedded in a ‘geometrical’ physical situation. 

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In Parallel Diagonals (photos above) the four diagonals on four sides of the beam have become edges of  the volume - connected as if by 'minimal surfaces', stretched taut between those diagonal lines on parallel surfaces. The edges of this new volume are the only parts of the original surface of the cubic wooden beam that remain untouched and thus keep the logic of the cubic volume intact. The volume outside of these surfaces has been removed, resulting in a twisted and curved shape of surprising complexity. 

 

Through the midpoint on each of those new and twisted surfaces at the middle of the piece, lines parallel to the diagonal edges are drawn. These parallels touch at the centre and 'move' further apart towards the ends. As the surfaces curve in every direction, no two points next to each other on this curved parallel line could be said to be sitting on a line which could be called 'straight' in any sense. Even in the act of measuring and drawing these parallel lines, they curve away under the ruler which nevertheless enables me to draw this parallel line.

 

The lines parallel to the diagonal edges are only parallel and straight lines in one extreme and singular sense, position and relationship.

 

The straightness and parallel-ness of the parallel line is only a partial aspect of their shape. Their parallel-ness and straightness can only be held for brief moments in the mind, identified and understood only when seen in their parallel relation to the original diagonals-  showing only one 'side' of themselves, one aspect, only one relationship out of countless others it engages in and which are all equally embodied.

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The parallel diagonals embody a fundamental uncertainty and complexity that is much deeper, much 'more physical' than a merely logical paradox.

 

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The lines parallel to the diagonals, which touch at the centre and move apart at the ends cross over at the centre: half of the parallel is parallel to half of one diagonal edge, and then crosses over to being parallel to the next diagonal edge in the other half. Each parallel line is a parallel, and not parallel at once, mirrored in a way that makes the non-parallel parallel et vice versa. 

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No physical foundation for physical change.

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The epistemological problem: To describe our mental activity, we require, on one hand, an objectively given content to be placed in opposition to a perceiving subject, while, on the other hand, as is already implied in such an assertion, no sharp separation between object and subject can be maintained, since the perceiving subject also belongs to our mental content. ...