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


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The mathematics the way algorithms think / the mathematics the way we think

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

  1. 1. Thomas J McCabe May 23, 2013 Le Club Qualité Logicielle Paris, France
  2. 2.  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
  3. 3.  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.
  4. 4. Graph Theory: Leonhard Euler Group Theory: Évariste Galois 1811 - 1832
  5. 5.  Algorithms as mathematics  Paths through an algorithm as a mathematical object  Consciousness as mathematics  Shifts of consciousness as a mathematical object
  6. 6.  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
  7. 7.  The story of the structure of consciousness with the illustrated by algorithm complexity
  8. 8.  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
  9. 9.  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
  10. 10.  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
  11. 11. Copyright c Thomas J McCabe 2012
  12. 12. Copyright c Thomas J McCabe 2012
  13. 13.  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 Copyright c Thomas J McCabe 2012 Subject, Predicate, Object I know the object
  14. 14.  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
  15. 15.  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 Copyright c Thomas J McCabe 2012
  16. 16.  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
  17. 17.  Blank tool here  Then ‘ I know the object’
  18. 18.  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? Copyright c Thomas J McCabe 2012
  19. 19. The many hidden truths
  20. 20.  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. Copyright c Thomas J McCabe 2012
  21. 21.  Steve Jobs tool here  Or testing example
  22. 22. 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) Copyright c Thomas J McCabe 2012
  23. 23. 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 Copyright c Thomas J McCabe 2012
  24. 24. 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 Copyright c Thomas J McCabe 2012
  25. 25.  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. Copyright c Thomas J McCabe 2012
  26. 26.  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. Copyright c Thomas J McCabe 2012
  27. 27. Copyright c Thomas J McCabe 2012 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
  28. 28. S O P S O P S O P S O P S O P S O P Copyright c Thomas J McCabe 2012 S O P O P S P S O P O S S P O P O S
  29. 29. 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.
  30. 30. 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.
  31. 31.  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
  32. 32.  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
  33. 33. 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.
  34. 34. Copyright c Thomas J McCabe 2012
  35. 35.  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 Copyright c Thomas J McCabe 2012
  36. 36.  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!! Copyright c Thomas J McCabe 2012
  37. 37.  Where EC has been applied so far  System requirements validation  Gun control  CEO coaching  Business strategy  Entrepreneur's lifestyle  Cyber security Copyright c Thomas J McCabe 2012
  38. 38. Copyright c Thomas J McCabe 2012
  39. 39. 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. Copyright c Thomas J McCabe 2012
  40. 40. Copyright c Thomas J McCabe 2012
  41. 41.  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.
  42. 42.  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!
  43. 43.  See your biggest opportunity/challenge in it’s full presence – see it ‘six ways from Sunday’  Why would you not?  Use http://www.expanded-  Contact me:  Thomas McCabe   301 775 9454
  44. 44.  Thomas McCabe – 301 775 9454   Web: Copyright c Thomas J McCabe 2012
  45. 45.  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
  46. 46.  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
  47. 47. Copyright c Thomas J McCabe 2012