Pythagorean Triples & Dialectical Triads – the DIALECTIC of the RIGHT TRIANGLE. Part 1.: Cases of Dialectic Series.

 

 

 

 

 

 

 

 

 

 

 

 

 

Pythagorean Triples

&

Dialectical Triads –

 

the

 

DIALECTIC

 

of the

 

RIGHT TRIANGLE.

 

Part 1.:

 

Cases of Dialectic Series.

 

 

 

 

 

 

 

 

 

 

 

Dear Reader,

 

 

Claim: The category of hypotenuses, within the Domain category, the «genos»-category, i.e., the ‘right triangles-in-«gene»-ral’ category,’ is the ‘dialectical synthesis category’ for this Domain, relative to the two, antithetical categories of the, mutually-perpendicular, contrarily-directed “sides” of the general right triangle.

 

 

As distinct from the transcendental diagonal of the, generic – 

[ q₁ [+] q₂ ] 

– value, derived via the Encyclopedia Dialectica _NQ arithmetic for modeling dialectics, that ‘‘‘transcends’’’ the closure of that arithmetic, the hypotenuse of a right-angled triangle instantiates another kind of ‘‘‘transcendent’’’ diagonal.

 

 

This is because each hypotenuse of a right triangle is a unit of the dialectical synthesis ‘uni-category’, of the categorial triadic dialectic, of the right triangles [sub-]Domain of classical geometry.

 

 

It is so because each such hypotenuse «monad» bridges, connects, and in that sense combines the two, mutually-orthogonal sides «monads»; it mediates the gap between those 2 line-segments «monads», and, in that sense, unifies them; ‘“contains”’ them both, by ‘‘‘tieing’’’ them together.

 

 

A right triangle’s hypotenuse line-segment ‘“contains’’’ its two sides’ line-segments also quantitatively, per the famous, ancient “Pythagorean” Theorem.  If the length of the right triangle’s hypotenuse is represented by c, the length of one of its two sides – arranging the triangle so that we can call that side the “Vertical side” – is represented by a, and the length of the other side – call it the “Horizontal side” – is represented by b, then c, the quantity of spatial extent metrological units “in” the hypotenuse, quantitatively contains the quantity of spatial extent metrological units “in” a, as well as those “in” b, as follows:

 

 

c²     =     a²  +  b²,

 

or,  

 

+Öc²    =   +Ö(a² + b²)    =     c.

 

Thus, the hypotenuse line-segment ‘‘‘contains’’’ its two sides’ line-segments fully ‘qualo-quantitatively’.

 

 

The Vertical side and/versus the Horizontal side antithesis is not a 180 degrees opposition in direction per se, but the two sides are the, ‘categorial antithesis’-forming, opposite ‘dialectical sides’ relative to the context of this Domain – relative to the right triangles Domain.  The thus opposing “sides” of right triangles constitute the ‘intra-duality’ of the right triangle «genos», resolved by its hypotenuse.

 

 

It is not just that the Vertical side is a – determinate negation’ –‘not-Horizontal side’, and that the Horizontal side is, likewise, a – determinate negation – ‘not-Vertical’ side.

 

It is that, within the limits of the Right Triangle «genos», the “linear independence”, the mutual perpendicularity, of the two sides, as two of the three «species» of that «genos», conveys an intuitive sense of antithetical oppositeness in the relation between the two.

 

This dialectic is not one of «aufheben» ‘meta-«monad»-ization’, except, perhaps, a the sense “straight-edge and compass construction” of one side from the other, e.g., the vertical side, grasped as a single line-segment «monad», constructed from the horizontal side; the one «monad» of the horizontal side re-directed into the vertical direction, to form the single «monad» of the vertical side, as a determinate [directional] negation, and conservation, and ‘‘‘elevation’’’ of the horizontal line-segment.

 

Instead, the two sides’ line-segments-units’ categories are co-posited, and co-present, together with the hypotenuse unit’s category – thus constituting a systematic dialectic, not an historical dialectic.

 

The hypotenuse units are neither vertical side units, nor horizontal side units, but, in a sense, are, or partake of, both vertical units and horizontal units, as their interconnectors.

 

 

 

 

 

 

 

 

 

 

 

For more information regarding these Seldonian insights, and to read and/or download, free of charge, PDFs and/or JPGs of Foundation books, other texts, and images, please see:

 

www.dialectics.info

 

 

and

 

 

https://independent.academia.edu/KarlSeldon

 

 

 

 

 

 

 

 

 

 

 

 

For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights -- specimens of ‘dialectical art’ – as well as dialectically-illustrated books published by the F.E.D. Press, see –

 

https://www.etsy.com/shop/DialecticsMATH

 

 

 

 

 

 

 

 

 

 

 

¡ENJOY!

 

 

 

 

 

 

 

 

 

 

 

Regards,

 

 

 

Miguel Detonacciones,

 

Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];

Elected Member, F.E.D. General Council;

Participant, F.E.D. Special Council for Public Liaison;

Officer, F.E.D. Office of Public Liaison.

 

 

 

 

 

 

YOU are invited to post your comments on this blog-entry below!