20130523 06 - The mathematics the way algorithms think / the mathematics the way we think

Thomas J McCabe
May 23, 2013
Le Club Qualité Logicielle
Paris, France

 An algorithm as a mathematical object
 Shifts of consciousness as a mathematical object
 History – Where did the math come from?
 Consciousness bandwidth
 The geometric shape of the space of our
consciousness
 ‘Sacred geometry’
 Applications
 Increasing complexity of consciousness
 Computer science applications

 The McCabe Cyclomatic complexity quantifies the fundamental paths in
algorithms. It elaborates the intrinsic paths inherent to any algorithm. It
drives the testing process and predicts the error rate of algorithms.

 The McCabe Dihedral complexity quantifies the shifts of our
consciousness. It elaborates the intrinsic perspectives inherent to any of
our thoughts. It expands our empathy and illustrates all the possibilities
before us.

 This talk will describe the mathematical derivation of both algorithmic
complexity and consciousness complexity. The impact of algorithmic
complexity on the software industry will be discussed. Also the emerging
impact of consciousness complexity will be discussed – – including
several computer science applications.

Graph Theory: Leonhard Euler Group Theory: Évariste Galois
1811 - 1832

 Algorithms as mathematics
 Paths through an algorithm as a mathematical object
 Consciousness as mathematics
 Shifts of consciousness as a mathematical object

 Cyclomatic complexity
 V(G)
 Quantifies and derives
basis paths of an
algorithm
 Comes from Graph
Theory
 Dihedral complexity
 Dih3
 Quantifies and
derives the corollaries
of a statement
 Comes from Group
Theory

 The story of the structure of consciousness with
the illustrated by algorithm complexity

 Nothing to believe
 Rigorous
 Operational
 Conform to our intuition on small things and
extend our intuition on big things beyond
where we can think
 Comes from another place
 A 10x improvement

20130523 06 - The mathematics the way algorithms think / the mathematics the way we think

 Executives try to uncover their ‘blind spots’
 Their unknown unknowns
 Artists, inventors, scientists, entrepreneur's try
to uncover their intuition
 Their unknown knowns
 Jeremy Lin
 Steve Jobs
 By empathy, change in perspective

 An early attempt to explain the concept of extra
dimensions came in 1884 with the publication
of Edwin A. Abbott's Flatland: A Romance of
Many Dimensions. This novel is a "first-person"
account of a two-dimensional square who
comes to appreciate a three-dimensional world.
 See Higher
Copyright c Thomas J McCabe 2012

 Focus on ‘the object’
and it becomes the
subject
 ‘I’ becomes the object
 Focus on ‘know’ and it
becomes ‘knowing’ as
subject
 ‘I’ becomes the predicate
Subject, Predicate, Object
I know the object

 What we have just done is a classical
mathematical group know as
 The Dihredal group D6
 There are 100s of thousands of books about D6
and 10s of thousands of PHd mathenaticans
specializating in group theory
 The theorems of group theory apply to our
shifts consciousness

 e: SPO: I know the object
 a: PSO: Knowing I create the object

 b: SOP: I objectify knowing

 aba: OPS: The object knows me
 ab: OSP: The object I create knowing

 ba: POS: Knowing objectifies me

 This presence of expanded
awareness is exactly
analogous to a light prism
in optics. Each expands
the original into its six
constituent components –
white light beams
expanded into its six
visible colors of the
spectrum, original
thoughts expanded into
its six constituent
colloraries

 Blank tool here
 Then ‘ I know the object’

 Steve Jobs accomplished 10,000 times more
than most of us
 His IQ was not 10,000 time higher – the scale
doesn’t go that high
 He didn’t work 10,000 time more hours – there
are not that many hours in the week
 How did Steve Jobs do it?

 Jobs has a vision of an emergent PC industry
and sees the immediate steps. He articulates a
commanding vision and follows his instincts.
He scales up both the industry and scales up
himself.

 Steve Jobs tool here
 Or testing example

Steve Jobs is an
example par excellence
of expanded
consciousness and
thinking beyond the
obvious.
The six permutations of the
statement ‘Steve Jobs seizes the PC
industry’ were generated by the
following transformations:
(SPO)
(OPS)
(PSO)
(OSP)
(POS)
(SOP)

SIX SHIFTS OF CONSCIOUSNESS:
 e: SPO → SPO
 a: SPO → PSO switch 1st two
 b: SPO → SOP switch last two
 aba: SPO → OPS switch 1st and last
 ab: SPO → OSP rotate right
 ba: SPO → POS rotate left
SHORTHAND:
 Two switches, two rotations,
empathy
 sserr
 ssrre
 s2er2

the 'e-ba-ab' loop reflects the fact that (ba)2=ab and (ba)3=e, as well as the fact
that (ab)2=ba and (ab)3=e The other "loops" are roots of unity so that, for
example a2=e

 applying transformation 'aba' repetitively just keeps repeating
switches from SPO to OPS back to SPO again and again. So our
empathetic mediator never becomes conscious of the process itself; he
is blind to ‘the knowing.’
 Likewise the transformation ‘a,’ when repetitively applied, just keeps
switching back and forth between SPO and PSO. A consciousness
limited to this transformation gets into a restrictive infinite loop and
never becomes conscious or empathetic with the object.
 On the other hand, the transformations 'ba' and 'ab' are more robust –
– since (ba)2=ab and (ab)2=ba, a person repeating the transformation
'ba' will see three states of consciousness: SPO, POS, and OSP.
 This shows the bandwidth resulting from focusing on a singular
transformation – – some are better than others. Given such a singular
switching mentality, it also shows the resulting repeating cycle. The
cycle graph clearly illustrates how a more robust switching
mechanism, involving two or more transformations, will yield more
states of consciousness.

 The mathematical dihedral group of order six is also
known as the group of symmetries of an equilateral
triangle. The structure of permuting three objects as
compared to rotating and flipping an equilateral
triangle, are the same structure, called isomorphic.
 Intuitively you can see what's going on.
When we permute states of consciousness
we rotate the triangle – – when we rotate
the triangle we permute states of
consciousness. In math speak, they are the same.

S O
P
O P
S
P S
O
P
O S S P
O
P O
S
Original(1)
Rotate (1) right = Rotate (1) left =
Flip (1) at P Flip (1) at S Flip (1) at O

S O
P
S O
P
S O
P
S O
P
S O
P
S O
P
S O
P
O P
S
P S
O
P
O S S P
O
P O
S

The space of the original statement is a point, which is one dimension.
The space generated by first corollary together with the original statement is a line, which
connects the two points of the two statements.
With the second corollary we have added a third statement to our geometric space – – we
have added the point of the third statement to the line of the first two -- this yields a triangle.
With the third corollary we add a point to a triangle which yields a square.

 With the fourth corollary we add a point to the pyramid’s base which yields a pyramid
with a four sided base.


 The fifth corollary adds a point yielding the final octahedron space of our consciousness.

Enlarge the original point, to a line of
reasoning, to triangulating three
statements, to thinking in the four
corner box, to thinking outside the box
(pyramid), to seeing total octahedral
awareness.

 For centuries man has been fascinated with geometry explaining
nature – ‘sacred geometry’ as borderline religion --- for good
reason, it’s the shape of how we think
 The shifts in our consciousness is a geometry
 Stunning?
 What else would you expect?
 Once you see the geometry you can use it
 We will have a complete view replacing a singular perspective
with a 6 fold awareness
 Riding the math leads to generalization, abstraction, centuries
of rigorous theorems, all applying to consciousness

Artists get
there first
 This painting captured the
same truth years ago, well
before this discovery in May
2013.
 The genre is ‘sacred
geometry’ – mystics thought
the heavens must have a
geometric shape. The instinct
was well placed but
understated, their very
thoughts had geometry, and
it was indeed sacred.

 Not where life works - on the many things we do well
– leave your auto pilot doing what it does well
 Where life doesn’t work --On areas where you’re stuck:
 Mental blocks
 Bad Habits
 Where’s you’re stuck
 Biggest challenges
 Relationships you want to change
 Areas where you want a fresh look

 Marriage counseling. Uncovering the blind spots in relationships.
 Problem solving – – permute the triple – – verbalize the relationship with the
problem.
 Imagining new relationships – – with people, with home, with God, with
meditation, with health, with wealth, with problems.
 Legal. Discovering nuances of the application of laws --- discovering blind spots.
 Learning something difficult, when you get to a ‘blind alley’ and have to change
perspective so learning can proceed. You later go back with a larger perspective to
the erstwhile ‘blind alley’ and realize you can see your way through it. Changing
into a new consciousness state and change perspective.
 International negotiation – – negotiation in the arms race, global warming,
international tariffs. … where the stakes are big.
 Game theory and utility theory in economics. This is the mathematics of trade-offs,
mathematical techniques are used to realize the optimal trade-off and highest value
to each party. This theory of consciousness can be used to broaden the set of
considerations.
 Unlocking your own frozen perspective, shedding light on blind spots. For
example, if you are a conservative Republican, start with the statement ‘I mistrust
Democrats’; go through the six states of consciousness and see if it breaks any blind
spot scar tissue – – who knows!!

 Where EC has been applied so far
 System requirements validation
 Gun control
 CEO coaching
 Business strategy
 Entrepreneur's lifestyle
 Cyber security

Dih6 is the key to a
captivating phenomenon as
we move consciousness to
higher dimensions. Dih6 has
12 members. It cycle graph
is very similar to the cycle
graph of Dih3: notice the six
sub groups on the bottom,
instead of three with Dih3,
each representing flipping
from
each of
the six
vertices. Also notice the rotational symmetries
on the top -- in this case we have five instead of
two with Dih3.
It's intriguing to notice that the group Dih6 has
two distinct Dih3 subgroups. In the diagram above you can see
subgroup (S1,P1,O1), in blue, and subgroup (S2,P2,O2), in red.
Intuitively what's happening is the hexagram group Dih6 represents
our two fold consciousness of (S1,P1,O1) and (S2,P2,O2), and the
subgroups show the exact phenomenon we have been discussing.
This suggests Dih3 is a ‘consciousness building block’ that is the
foundation when we are conscious of many things.

 The idea is to put the requirements statement into the Thought Prism
Tool and compare our respective interpretations. If you and I agree on
the interpretation of the original statement then we each write down our
respective interpretations of the first permutation and see if
our interpretations agree. Do this blind to each other, write down the
respective interpretation, and then compare results. Do the same thing
with the 2nd permutation, ... all the way to the 6th.

 A metric s where 1<= s <=6 would express the degree of our concordance.

 The upshot is to test our agreed interpretation --- it makes us each
interpret 6 different views and compare our interpretations.
Differences interpreting the permutations show implicit
misunderstandings of the original statement.

 Most people are in
one dimension
 A few gifted operate
in two dimensions
 The extraordinary will
be in three
 And we can stretch to
six!

 See your biggest
opportunity/challenge
in it’s full presence – see
it ‘six ways from
Sunday’
 Why would you not?
 Use
http://www.expanded-
consciousness.com/
 Contact me:
 Thomas McCabe
 tom@e-c-i.info
 301 775 9454

 Thomas McCabe – 301 775 9454
 Tom@e-c-i.info
 Web: http://www.expanded-consciousness.com/

 To err is human. To loaf is Parisian. -Victor Hugo
 That Paris exists and anyone could choose to live
anywhere else in the world will always be a
mystery to me. -Marion Cotillard as Adriana in
Woody Allen’s Midnight in Paris
 Everything ends this way in France – everything.
Weddings, christenings, duels, burials, swindlings,
diplomatic affairs – everything is a pretext for a
good dinner. -Jean Anouilh
 Lunch kills half of Paris, supper the other half. -
Charles de Montesquieu

 The whole of Paris is a vast university of Art,
Literature and Music… it is worth anyone’s
while to dally here for years. Paris is a seminar,
a post-graduate course in everything. -James
Thurber

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