The Dialectic of Euclidean Geometry. Part 8.: Cases of Dialectic Series.

 

 

 

 

 

 

 

 

 

 

 

 

 

      The Dialectic

 

of

 

Euclidean

Geometry.

 

 

 

Part 8.:

 

Cases of Dialectic Series.

 

 

 

 

 

 

 

 

 

 

 

Dear Reader,

 

 

 

In the ‘dialectogram’ posted above, the depicted «arché» category, or ‘premise category’, is the category whose units are the axiomatic systems of the Euclidean Geometries, color-coded ‘red-ish’.

 

For those geometries, based upon Euclid’s fifth axiom, or “5th postulate”, the number of lines parallel to a given line is exactly one, or, in the ‘dialectogram’s short-hand,

#(||) = 1.

 

The depicted ‘first contra-category’ to this «arché» category is the category whose units are the axioms-systems of the standard, classical “Non-Euclidean Geometries”, color-coded orange, and placed at a higher elevation in the ‘dialectogram’, to signify the greater ‘thought-complexity’ of this second category, and of its units.

 

The key characteristic of these classical “Non-Euclidean Geometries” is that the number of lines parallel to a given line is different from one, or, in the dialectogram’s short-hand, #(||) ~= 1. 

 

Two sub-species of this species of “Non-Euclidean Geometries” are also depicted in that ‘dialectogram’. 

 

The left-most sub-species category depiction is that for the axioms-systems units of the “Elliptic” Non-Euclidean Geometries, for which the number of lines parallel to a given line is less than one, or, in the dialectogram’s short-hand, #(||) < 1; in fact, for which #(||) = 0.   

 

The right-most sub-species category depiction is that for the axioms-systems units of the “Hyperbolic” Non-Euclidean Geometries, for which the number of lines parallel to a given line is less than one, or, in that dialectogram’s short-hand, #(||) > 1; in fact, for which #(||) = “¥”.   

 

The depicted third category, the ‘first uni-category’, uniting or unifying the «arché» category and its ‘first contra-category’, is the category whose units are the axioms-systems of the “Riemannian Geometries”, color-coded ‘yellow-ish’, and placed at the highest elevation in that ‘dialectogram’, to signify the greater ‘thought-complexity’ of this third category, and of its units, relative to the other categories depicted in this ‘dialectogram’, and to their units.

 

The “manifolds’ – the spaces of Riemannian Geometries – can combine regions of “flat”, Euclidean-like geometry, with regions of “curved”, Elliptic-like geometry and with other regions of “curved”, Hyperbolic-like geometry, as well as with other kinds of “curved” geometries!

 

Humanity’s presently-known best – most observationally and experimentally corroborated – mathematical model of large-scale physical space[-time]; of the large-scale universe, and of universal gravity, uses the “pseudo-Riemannian” or “semi-Riemannian” geometry, developed by Albert Einstein in his theory of universal gravitation, “The General Theory of Relativity’.

 

The Domain of this case of dialectic, ‘the dialectic of Euclidean versus Non-Euclidean geometries’, we name “Euclidean and Related Geometries-in-General”, or ERG for short – D = ERG.

 

The category-symbol that we have adopted for the «arché» category, of “Euclidean Geometries”, is ‘q_E’.  The category-symbol that we used, above, for the second category, the ‘first contra-category’, of “Non-Euclidean Geometries”, is ‘q_N’, or, equivalently, is ‘q_EE’, connoting the immanent critique, or self-critique, of the “Euclidean Geometries”.  The category-symbol that we used, above, for the “Riemannian Geometries”, is ‘q_R’, or, equivalently, is ‘q_NE’, signifying the mutual critique, mutual conversion, combination or unification of category ‘q_N’ and category ‘q_E’.

 

 

 

 

 

 

 

 

 

 

 

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.

 

 

 

 

 

 

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