Part 10: Seldon on ‘‘‘Dialectical Categorial Progressions’’’ Series. Dialectical Categorial Progressions’ ‘‘‘FINGERPRINTS’’’.

 

                                   
                 

 

 

 

 

 

 

Part 10: Seldon on

 

‘‘‘Dialectical Categorial Progressions’’’ Series.

 

 

Dialectical

 Categorial

Progressions’

‘‘‘Fingerprints’’’.

 

 

 

 

 

 

 

Dear Reader,

 

It is my pleasure, and my honor, as an elected member of the Foundation Encyclopedia Dialectica [F.E.D.] General Council, and as a voting member of F.E.D., to share, with you, from time to time, as they are approved for public release by the F.E.D. General Council, Seldon’s commentaries on key Encyclopedia Dialectica concepts of Seldonian Theory.

 

 

This 10th text in this series is posted herewith, together with supporting text-images and diagrams [Some E.D. standard edits have been applied, in the version presented below, by the editors of the F.E.D. Special Council for the Encyclopedia, to the direct transcript of our co-founder’s discourse].

 

 

 

 

 

 

 

 

 

 

 

Seldon –

“For any, dialectical, ontological categories-progression, whether ‘‘‘historical-dialectical’’’ or ‘‘‘systematic-dialectical’’’, but modeled in the algebra of the _WQ ‘meta-numbers’ for modeling dialectics, and whether that _WQ category-symbols progression is fully-solved, partially-solved, or not solved at all, there is an algorithm that produces a ‘quanto-qualitative’ value that uniquely characterizes per the number of _WQ category-symbol terms in any such _WQ progression or series, summed non-amalgamatively, in that modeled, heterogeneous, dialectical ontological categorial progression.”

 

“We noticed this progression-metric years ago, but I do not believe that we have ever published anything about it here, before this very blog-entry.”

 

“Why do we call it a ‘quanto-qualitative’ value, instead of just a [“purely”-]quantitative value?”

 

“We do so because that value involves a Real number quantifier, yes, but one that ‘‘‘multiplies’’’ a metrological unit ‘qualifier’, e.g., belonging to the –

 

q₇ (---] _Rq_MQN ¶|-º _Rq_MU |-º  _Rm

 

– axioms-system of arithmetic, the 7^th such system in the ideo-[meta-]systematic dialectic of the Encyclopedia Dialectica arithmetics for modeling dialectics.” 

 

“The candidate ‘metrological unit qualifiers’ for this value include grads [or “grades”, “gradians”, or “gons”], radians, and degrees.  The specific quantity of the metrological unit quantifier varies with the metro-logical unit qualifier chosen.” 

 

“Any of the metrological units could be chosen to fill the role of ‘m°_u°₆, Î the _Rq_MU or _Rm space, 

or set, our ‘metrological unit qualifier ‘dialectical meta-numeral’ for angle-measurements.”

 

“But, how is this possible, to quantify category-symbols’ progressions, other than simply by their count of _WQ generic arithmetical categorial-qualifier-symbols itself?”

 

“It is possible because, in the analytical-geometrical representation of the _WQ space, or ‘unit-line-segments’ ‘‘‘set’’’, each generic ‘meta-number’ is represented by a unit-length line segment, which is orthogonal to all of the other generic meta-numbers’ unit line segments, so that every distinct _WQ ‘meta-number’ occupies its own, perpendicular, unit-size dimension, without any 2+-dimensional “volumes”, only such single-dimensional line-segments, being included in the substance or ‘content-structure’ of _WQ ‘meta-number’ space, and with all of these rectilinear dimensions joined and intersection in the q₀ “origin” of this space.”

“That is, the _WQ space is defined as a space, not made up out of any “infinitude” of “infinitesimal points”, but, on the contrary, as a space made up out of an – ‘operatorially’ growing – but always finite number of finite length – unit length – ‘impartible’[‘a-tom-ic’] line-segments.”

“Thus, q₁¹ generates the 1-dimensional space q₁, 

q₁² generates the 2-dimensional space q₁ ^ q₂; 

q₁³ generates the 3-dimensional space q₁ ^ q₂ ^ q₃, 

and so on.”

 

“Thus q₁³ generates a dotted-line cube, whereas q₁² and q₁¹ generate, respectively, 2-dimensional and 1-dimensional dotted-line ‘hypo-cubes’, and q₁⁴ generates an, idealized, dotted-line 4-dimensional “hyper-cube”, etc.”

 

“As its takes two, end-points-sharing line-segments to form an angle, the angle represented by q₁¹ = q₁ is, not 0, i.e., not ‘empty zero’, but ‘full zero’: non-existent.” 

 

The angle represented by q₁² =  q₁ + q₂ is that of the diagonal of a unit-lengths-sided square, of length Ö2, with angle 45°.”

 

The angle represented by the maximum-length diagonal of –

 

 q₁³ =  q₁ + q₂ + q₃ 

 

– is that of a unit-length-sided cube.  The length of that diagonal is, per the Pythagorean theorem –

 

Ö((Ö2)² + (1)²)  = Ö3,

 

– and its angle is –

 

 sin^-1(Ö3)  »  35.26°.” 

 

“One of our interested volunteers consulted an AI about the extension of this pattern to the generic _WQ arithmetic’s generic 4-D categorial progression –

 

q₁⁴  =  q₁ + q₂ + q₃ + q₄

 

– and beyond, and obtained the following advice during that [K.S.-edited] dialogue:

 

 

“[note that the ‘hypo-diagonals’, “diagonals”, and “hyper-diagonals” herein addressed are not IN the _WQ space; they all ‘‘‘transcend’’’ that space, ‘‘‘except’’’ for q₁¹ = q₁.  That is, _WQ space is not “closed” for its operations of categorial ‘‘‘addition’’’, categorial ‘‘‘multiplication’’’, and categorial ‘‘‘exponent-tiation’’’]” 

 

 

 

 

Q1.  I am interested in calculating the angle as well as the length of the maximal DIAGONALS unit-[hyper]cubes of various dimensionalities, and request your help in doing so.  If we consider that a unit length one-dimensional line segment is a 1-D unit ‘hypo-cube’, then call its angle is 0 degrees, its length is 1 unit.  If we also consider a two-dimensional unit square to be a 2-D unit ‘hypo-cube’, then its diagonal’s angle is 45 degrees, or pi/4 radians, and that diagonal’s length is the square-root of 2 units [irrational length].  For a three-dimensional unit cube, the length of its diagonal is the square-root of 3 units [again, an irrational-magnitude].  But what angle does the diagonal make with the base square face of that unit cube?  Likewise, what is the angle and length for the maximal diagonal of a mentally-constructed four-dimensional unit hyper-cube, and for a five dimensional unit hyper-cube?  Do the lengths of those diagonals happen to be the positive square root of four and the positive square root of five, respectively?

 

 

R1.  For a 3-D unit cube, the maximal diagonal makes an angle of approximately 35.26 degrees ( arcsin(sqrt(3)) ) with its base square face.  For 4-D and 5-D unit hypercubes, the maximal diagonal lengths are indeed sqrt(4) = 2 units and sqrt(5) units, making angles of 30 degrees and of approximately 26.57 degrees with their respective orthogonal base hyperfaces.

 

 

1. Generalize diagonal lengths

The length of a maximal diagonal in an n-dimensional unit hypercube is calculated using the Pythagorean theorem extended to n dimensions:

L_n = sqrt(1^2 + 1^2 + … + 1^2) = sqrt(n)

• 3D Cube: L₃ = sqrt(3) units (irrational).
• 4D Hypercube: L₄ = sqrt(4) units (rational).
• 5D Hypercube: L₅ = sqrt(5) units (irrational).

 

2. Calculate the 3D angle

To find the angle q₃ between the main diagonal of a 3D cube and its 2D base face:

• The height perpendicular to the base face is 1 unit in length.
• The length of the diagonal of the 2D base face is sqrt(2).
• The length of the main 3D diagonal is sqrt(3).

q₃ = approximately 35.26°, or 0.6155 radians.

 

3. Calculate the 4D angle

To find the angle q₄ between the main diagonal of a 4D hypercube and its 3D base cube hyperface:

• The length along the new 4th dimension is 1.
• The maximal diagonal of the 3D base cube is sqrt(3).
• The main 4D diagonal is sqrt(4) = 2.

q₄ = approximately 30.00° or p/6 radians.

4. Calculate the 5D angle

To find the angle q₅ between the main diagonal of a 5D hypercube and its 4D base hyperface:

• The length along the new 5th dimension is 1.
• The maximal diagonal of 4D base hypercube is sqrt(4) = 2.
• The main 5D diagonal is sqrt(5).
• q₅ = approximately 26.57°, or 0.4636 radians.

 

Summary Comparison Table

Dimension (n)	Geometric Name	Maximal Diagonal Length	Angle with Base (n-1)D Hyperface
1	Line Segment	1	0° or 0 radians
2	Square	sqrt(2), approx. 1.414	45.00° or p/4 radians
3	Cube	sqrt(3), approx. 1.732	»35.26° or »0.6155 radians
4	Tesseract (4-D hyper-cube)	sqrt(4) = 2	30.00° or p/6 radians
5	Penteract (5-D hyper-cube)	sqrt(5), approx. 2.236	»26.57° or »0.4636 radians.

 

 

✅ Final Answers

The 3D main diagonal forms an angle of approx. 35.26° with its base face. Your hypothesis is correct: the 4D and 5D maximal diagonal lengths are exactly sqrt(4) = 2 and sqrt(5) units, forming angles of 30.00° and approx. 26.57° with their respective underlying base hyperfaces.

 

 

“So, the higher the number of distinct category-symbols in a _WQ-language-formulated categorial progression ‘meta-model’ – i.e., the higher the dimensionality of the _WQ space generically undergirding that specific/interpreted categorial progression – the greater the “fingerprint” length of its [hyper-]diagonal, and the smaller the “fingerprint” angle of that [hyper-]diagonal.”

 

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“After the dialogue with the AI about the “fingerprint” lengths and angles of the vector-sum-like [hyper-]diagonals, in _WQ space, uniquely characterizing, ‘quanto-qualitatively’, categorial progressions of any given finite count of category-symbol-terms, and, noticing that the AI’s algorithm, for ‘hyper-diagonal’ fingerprints, rested upon a dimensionality > 3 generalization of the, originally 2-D, Pythagorean theorem, we asked the AI for its proof of that generalized Pythagorean Theorem, and obtained the following from that secondary dialogue.”

 

 

“Summary of the AI’s [edited] proof –”

 

 

 

Theorem-proposition statement [to be proven]: The square of the length of the diagonal of an n-dimensional rectangular cuboid (hyperrectangle), generalized for n > 3, is equal to the sum of the squares of its n orthogonal edge lengths.

 

Inductive Proof of the Theorem-proposition.

1.  Establish the Induction Base Case.

For n = 2, the “hyper”-rectangle is a standard, flat rectangle.  A single diagonal splits it into two right-angled triangles.  By the classical – and abundantly already proven – 2D Pythagorean Theorem, the square of the lengths of the diagonal, D₂, spanning the first two dimensions, is, via the lengths of rectangle’s sides 1 and 2, s₁ and s₂:  

D₂²  =  s₁² + s₂²  

2.  Formulate the Inductive Hypothesis.

Assume that the proposed, generalized Pythagorean Theorem holds for a space of k dimensions, k > 3.  According to this hypothesis, then, the square of the length of the diagonal D_k, spanning across k mutually-perpendicular dimensions, is equal to the sum of the squares of the lengths of those k edges: D_k²  =  s₁² + s₂² + … + s_k²

 

3.  Apply the Inductive Step.

Extend the hyperrectangle into the (k+1)th dimension by adding one new orthogonal edge, of length s_(k+1).

Because this new edge is perpendicular to all k dimensions of previous, mutually-perpendicular, edges, it must also be perpendicular to the existing intermediate diagonal line-segment, of length D_k.

Together, the intermediate diagonal, of length D_k, and the new edge, of length s_(k+1), form the two legs of a new, locally-flat, locally-2D, right-angled triangle.  The hypotenuse of this triangle is the full, dimension-(k+1) diagonal, of length D_(k+1).

Applying the, base-case, 2D Pythagorean Theorem to this right-angled, locally-2D triangle yields: 

D_(k+1)  =  D_k²  +  s_(k+1)²

 

4.  Substitute and Conclude.

Substitute the induction hypothesis expression for D_k² into the new equation: 

D_(k+1)²  =  s₁² + s₂² + … + s_k²  +  s_(k+1)²;

so, D_(k+1)²  =  S_{i = 1 to i = (k+1)}(s_i²)

Thus, by the mathematical induction argument given above, the proposed theorem is proven true for “Natural” Numbers hyper-dimensions n > 3, and, indeed, for all dimensions, 

n ³ 2.  

Q.E.D.

 

Final Algorithm Formulation.

The distance or Diagonal length D, across any n-dimensional Euclidean space figure with orthogonal component lengths, {L_i}, is determined by squaring each individual orthogonal component length, then summing all of those squares, and then computing the positive square-root of their sum: 

D = +Ö(L₁ + L₂ + … + L_n).

‘DIAGONAL TRANSCENDANCE’.  What we call ‘diagonal transcendence’ is actually a widespread phenomenon, even in vastly disparate contexts, where ‘ideo-ontological’ or ‘physio-ontological’ constraints are overcome. 

One can see this phenomenon irrupting ‘ideo-ontologically’ already in ancient times, as when the Pythagoreans recognized that the length of the diagonal of a unit square cannot be expressed by any ratio of two “Natural” numbers, in an early encounter with the conceptual possibility of ‘“ir-ratio-nal”’ number ideas. 

This phenomenon can be seen also in the Cantor “Diagonal Proof”, that the – potential, never constructible or actualizable – infinitude of the “Natural” numbers cannot be put into any one-to-one correspondence with, for example, the potential infinitude of the “Real” numbers. 

This phenomenon, of ‘diagonal transcendence’, can even be seen in the urban planning of certain cities, such as Phoenix, Arizona, where the primary rectangular grid of streets, and the navigational limitations that this imposes, is interrupted, & partially overcome, by a diagonal street, that intersects and short-cuts street travelers to other streets of the rectangular array in complex and convenient ways.

 

 

 

 

 

 

 

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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