Logic and Intuition

                       Cognitive Payoff Under Uncertainty

The Sis about navigating three-dimensional problem space, with Hands and Mind:

A simple block and ball networking system which leverages full-spectrum cognitive / perceptual processing, learning algorithms, and the art of design, with an emphasis on advanced thinking and intuition -- and its reflection upon itself (not "mere facts", but principles):  functional relations, particularly goals and feedback (What are my assumptions? Are they justified?)

                            “Logic is for demonstration, intuition for invention.”  

                                                                                            H. Poincare, L’Enseignement Mathematique (1899)


The S3 is about pattern,


the rule which governs a system or phenomenon, exactly like numeric, musical, or visual relationships -- and its simultaneous mental and manual rotation. 

S3 is a self-generating, self-sustaining, escalating causal loop of pattern, a cycle of cognitive development where the manipulator is both agent and object of change.


"From the perspective of a user, the crucial factor is not whether the problem is intrinsically complex (suitable software can make even complex problems seem simple), but rather the cognitive resources that the user is prepared to devote to solving the problem."

                                                     Alan F. Blackwell, ˆWhat is Programming?” (2002)

What does S3 have to do with our new world?  It develops the cognitive resources necessary to program a computer (0/1) -- but without the computer.  There is too much computer, not enough peopleness.  That is, unlike the iPhone, etc., the S3 allows us to hold on to some of the things we are losing … (see also Norbert Wiener’s, The Human Use Of Human Beings)

S3 is a hybrid:  combined puzzle and maze and rolling ball device where intuition is a prerequisite (logic is necessary but not sufficient -- note the Quad Problem). While Rubik’s Cube exercises logic, S3 demands logic and intuition.  As the Pythagoreans subordinated prevailing logic by demonstrating the existence of the irrational (not logical or reasonable) number , with their right angle triangle, when everyone believed that the integer, and its logic, was inviolate (necessary and sufficient), S3 demands the manipulator ask, continuously:  What are my assumptions? Are they justified?

In Buckminster Fuller’s  Tensegrity (tension / compression) terms, S3 is about  Synergy -- a "coordination of thought and physical action, the genesis of geometry, system, and structure.”   

Sis about awareness of environment, modification of self-behavior on the basis of experience (feedback), through the optimization of multiple simultaneous paths, in a constant stream of problems, within a topology of paradox (Watzlawick, Paul et al., Pragmatics of Human Communication: A Study of Interactional Patterns, Pathologies, and Paradoxes, 1967.)

In other words, S3 is about continuous synchronous switching, a “strange loop”:  nested, dynamic, switching of cognitive and perceptual and mathematical dichotomies, where the manipulator must constantly re-examine reigning assumptions, transcend and control nested and evolving recursion and contradiction, by inventing new propositions.  

In Cybernetic (regulatory / transdisciplinary) terms, S3 is a tangible, lucid demonstration of basic behaviors of the brain by mechanical concepts (switches); a cubical maze module (four tunnels = four binary (0/1) switches = gate array) offering a development of choices (control flow) to create linearly independent / dependent paths, using a ball, or symmetry in mathematics. 




                                                                                                                 Binary Processor (0/1)

Thus, Sis a router, which generates oscillating, intermittent restriction of cognition and perception within a dynamic logic hierarchy, a topology of paradox (properties and relations unaffected by continuous change -- symmetry under transformation) which employs the cognitive mechanics of Gregory Bateson's Double Bind paradox (Zen koans employ the same mechanics):  contradictory communication at different levels, where acknowledgement of that contradiction is forbidden) and Wieners’s oscillatory response, "pathological oscillations”, to paradox (Russell’s).

In Zen terms:  embracing the contradiction (instead of accepting the Law of Non-Contradiction) creates the necessary tension to escape from the underpinning concepts -- to see them as dynamic patterns, part of a process of transformation.

The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced and combined... this combinatory play [emphasis mine] seems to be the essential feature in productive thought before there is any connection with logical construction in words or other kinds of signs which can be communicated to others."

                                           Albert Einstein:  letter to Jacques Hadamard,

                                                     The Psychology of Invention in the Mathematical Field, 1945 

S"combinatory play" simultaneously requires a spiral of pattern recognition (inductive - abductive / deductive flip-flop) while impeding it (gestalt figure / ground illusion flip-flop) introducing active resistance, error (in “new” Cybernetic terms, overcoming entropy by using noise as positive feedback), exciting curiosity:  topology of paradox. 

Thus, S3 is a cognitive tool offering the benefits of simultaneous high-level reasoning and manual experience (note manual / mental rotation studies) through a powerful exercise of recursion and contradiction, requiring self-examination of processes and management:  reconciling feedback loops to accomplish a goal.  

In other words, in spatial processing switching terms, when you brush your teeth do you watch the mirror image for reference?  Or, do you avoid watching the reflection and use egocentric reference (location with respect to somatic perspective of perceiver)?  Or do you ever brush, switching frames of reference back and forth, and experience visuospatial disorientation?  

That's how Sworks, switching from one mode to the other -- confusing the mind with alternative frames of reference, crossing levels of abstractionover and over again.


Sis a holon (at once whole and part), an architecture of nested uncertainty, an oscillating, intermittent restriction of cognition and perception within a recursive, dynamic logic hierarchy.

Almost a “purposeful” entity (think like a child, or a Cybernetician).

"Here is how it works: first you decide to treat the object whose behavior is to be predicted as a rational agent; then you figure out what beliefs that agent ought to have, given its place in the world and its purpose. Then you figure out what desires it ought to have, on the same considerations, and finally you predict that this rational agent will act to further its goals in the light of its beliefs. A little practical reasoning from the chosen set of beliefs and desires will in most instances yield a decision about what the agent ought to do; that is what you predict the agent will do.

                                                                             Daniel C. Dennett, The Intentional Stance, (1987)


Sis about reference frame: allocentric (location with respect to perspective of perceiver vs. egocentric (location within frames external to and independent of perceiver) -- spatial processing switching.  Think hand-flying (in aviation terms, the pilot’s cognitive workload of controlling the demands of a reference frame (airplane attitude) roll / pitch / yaw, within another frame of roll / pitch / yaw, within yet another frame of roll / pitch / yaw).    

While the S3 exterior has six faces (three sets of opposing faces), each "quartered" face with one or two holes (mismatched hole quadrants block path exit between S3s), the S3 interior has four intertwined chiral tunnels (each tunnel with independent  “logic gate” orientations) with eight entrance / exits:

A three axis rotation problem space (physical and mental) of nested uncertainty (uncertainty of environment, and cognitive flip-flop and perceptual flip-flop -- "unknown unknowns"), of constant and unexpected contradiction, recursive abstractions and relationships;  opposing systems, domains of expectation / reference frames (which exhibit completely different properties and changing order of operations).

AHegelian logic hierarchy (holistic -- Thesis / Antithesis / Synthesis). 

To optimize multiple simultaneous paths the manipulator must simulate the future, conceptually "drill down” a hierarchy of nested coordinate systems.  

In Michio Kaku’s, The Future of the Mind terms, “self awareness; the ability to put one’s self inside a model of the environment.” 


                                                                          Binary Processor (0/1)

In other words, with each rotation (precession of the rotational axis of each tunnel), and each additional S3, the manipulator must repeatedly focus on the binary (0/1) issues of a given tunnel (part), proceed up the logic hierarchy to the S3 (whole), then to the expanding universe of S3 (overarching whole / system of systems), while coping with contradiction of action and environment.  

A dynamic cycle of recalibration and reflection. 

CUBICAL exoskeleton (six faces)  >>  

HOLES quadrant (eight holes)  >>  

TUNNELS gate array (four binary switches)  >> 

TUNNEL (binary switch).

Thus, Sis about the exploration of symmetry, influences among parts and whole, square and circle --  adaptation to accelerating rates of change and complexity of system and environment (cognitive development, unlike Rubik’s Cube cognitive “exercise” without paradox).

A robust exploration of abstract concepts in visuospatial terms:  transformative play through nested dichotomies, logic and intuition, recursion of reason and perception.



Play with this:  study the square and circle patterns below and find the next shape.  


Inductive (& abductive) reasoning allows us to make a generalization based on pattern and relationship (arguably, we have been exercising this reasoning from birth).


The Sarchitecture, its patterns and relationships of square and circle, capitalizes on this inductive (& abductive) reasoning and its opposite, deductive reasoning, as well as our dependency on the perceptual polarity of Gestalt figure / ground Illusion,  


to demand seemingly paradoxical chains of reasoning: 

1)  Illusion; 

2)  Reality; 

3)  Meta-level of the two.  

A mixing of different levels of abstraction, self-reference.

Just like the viewer of Escher’s paradoxical drawing must rise to another level and reflect.

"… a kind of Strange Loop, an interaction between levels in which the top level reaches back down towards the bottom level and influences it, while at the same time being itself determined by the bottom level.”    

                                                                                                       I Am A Strange Loop  /  Gödel, Escher, Bach 



In operational terms, the S3 architecture creates nested, dynamic, cognitive and perceptual loops / dichotomies / switches:

a)  Flip-flop between inductive (& abductive) reasoning (discovery driven / bottom up); 

and deductive reasoning (rule driven / top down); 

b) Flip-flop of figure / ground (distinctive-feature information);

which dictate the manipulator employ analogical mapping and perceptual processing --visuospatial strengths -- "combinatory play", to retrace and reinforce associations, to substitute and manipulate Piaget's "pure proposition" for possible action, within a three-dimensional information (manual / mental rotation) framework. 

"Whenever one encounters a new situation (or makes a substantial change in one's viewpoint [S3 rotation] he selects from memory a structure called a frame:  a remembered framework to be adapted to fit reality by changing details as necessary....  Attached to each frame are several kinds of information… how to use the frame… what one can expect to happen next… what to do if these expectations are not confirmed …. a simplified frame-system to represent the perspective appearances of a cube." 

                                                                   Minsky, Marvin, The Psychology of Computer Vision

                                                                   McGrawHill, 1975

Thus, it is transformative, a kick-start, a self-generating, self-sustaining, escalating causal loop, a cycle of cognitive development where the manipulator is both agent and object of change.

In behaviorist operant conditioning terms, the S3 manipulator is both experimenter and experimental subject (rewarded with a falling ball), while the pigeon manipulator in Skinner’s Box (a topology of axioms) is simply experimental subject (rewarded with a falling ball of food).  Note that the S3 flip-flops guarantee intermittent reinforcement (a seminal issue), even for a world-class mathematician.  

Decision under uncertainty of environment and cognitive flip-flop and perceptual flip-flop.


Thus, the Smanipulator has the additional satisfaction / cognitive development / fun of the working scientist creating, continuously reformulating / recalibrating dynamic systems problems with changing ground rules, and then solving them -- no small task.

"The S3 readily engages and captivates children, sustains their attention, and challenges and stretches them cognitively.  Moreover, it enables children to calibrate their own complexity of play …

                                                                        Laura E. Berk, Child Development, 8th Edition (May 13, 2008, cc mss)

                                 Continuous Synchronous Switching

                                            Symmetry And Dialectic 

“Conflict is the gadfly of thought. It stirs us to observation and memory. It instigates invention. It shocks us out of sheep-like passivity, and sets us at noting and contriving…conflict is a sine qua non of reflection and ingenuity.”

                           Dewey, John, Human Nature and Conduct: An Introduction to Social Psychology, 1930

The Sis a non-linear, recursive binary problem space of abstractions, patterns and relationships in conflict:  independent of disciplines -- dynamic nested figure / ground illusion (ambiguous distinctive-feature contours:  squares / cubes -- circles / tunnels -- illusory and real).




                                                              "trompe l'oeil / trompe l'esprit"

A topology of paradox -- an environment of contradictory cognitive and perceptual signals / messages at different levels, with a negative injunction forbidding acknowledgement of that contradiction.  (Paradoxical intervention in behavior modification is a classic tactic which appears to contradict the desired purpose of the intervention.)

That is, Sis an engine of paradox which generates continuous ambiguous, bi-stable stimuli, cognitive and perceptual, such that the brain cannot commit to just one interpretation, but must switch back and forth, providing best guesses, unconscious inference ("unbewusster Schlussin Helmholtz's 1867 terms. 

To be precise:  a pair of contradictory signals (cognitive) which contradict another pair of contradictory signals (perceptual), where those higher level pairs themselves contradict each other, with a negative injunction forbidding acknowledgement of that contradiction. (Note, the brain does not broadcast the operation of its own processes.)  

The only way out of the paradox is for the manipulator to shift point of view to third person perspective (just like the viewer of Escher’s paradoxical drawing must rise to another level and reflect on the paradox). 

In other words, escape a recursive, nested, back and forth of cognitive and perceptual cycles which impede / allow cognition and perception, simultaneous covert contradictions, thereby introducing active resistance, error (in the “New Cybernetic" terms, overcome entropy by using noise as positive feedback); 

in effect, a mandatory focus on process / frame -- not just outcome.

Thus, the S3 demands, gently at first, the manipulator simultaneously plan and direct multiple lines of thought, rotate mentally and physically, differentiate and process visual parts and wholes, separate evolving tasks into manageable, self-referential subsequences (each step is specified in terms of previous steps that the same function has already calculated -- like the Tower of Hanoi classic recursive programming problem).


And escalates from there.

A cognitive development tool which demands the manipulator integrate opposites, by linear and non-linear reasoning with hand and mind; 

Transformative, a kick-start, where voluntary behavior is modified and maintained by its consequences (intermittent reinforcement) -- Ratcheting. 

A cycle of concrete to abstract which preserves within it each of the ideas or stages which it has subsumed; a totality, a fractal architecture.  

A simple block and ball networking system which leverages full-spectrum cognitive / perceptual processing, learning algorithms, and the art of design, with an emphasis on advanced thinking and intuition -- and its reflection upon itself:  functional relations, particularly goals and feedback (What are my assumptions? Are they justified?)

Ssynergy:  “operationalizing” the collapse of our expectation of a logical universe, and how we adapt by advanced thinking and intuition -- and reflect upon those processes -- in an eminently researchable and simple form. 

A demonstration of Hegelian dialectics of pattern and of process, a mixing of different levels of abstraction, self-reference:

                            Thesis,                  Antithesis,                     Synthesis; 


In Hegel’s terms (best seen / experienced in practice)

Synthesis: Integration of a set of contradictions to a higher level, transcending the polarity and forming a new proposition.

In Piaget’s terms:

Assimilation:  a new network scheme into the old network scheme, to accommodate the new scheme  (re-examines and adapts to the new information) to produce a synthesis of the two: an equilibrium, a higher level network scheme

In Bateson's terms: 

“Double Bind” paradox:  contradictory signals / messages at different levels, where acknowledgement of that contradiction is forbidden. 

(In Zen Koan Master tells student: "If you tell me this stick is real, I will beat you. If you tell me  this stick is not real, I will beat you. If you say nothing, I will beat you." Student escapes the paradox by changing the "level of communication”.  She takes the stick from the teacher, and breaks it.  Thus,  Zen mechanics of enlightenment, with an implicit injunction forbidding discussion of the contradiction -- the sanity / wisdom of the Master.)

In Russell’s / Cantor's (logic / set theory) terms: 

Reconcilement (manual and mental) of the paradox of sets / groups, symmetry under transformation (transformation geometry -- rotation [turns], reflection [flips], translation [slides], etc.) as learning medium:  evolving reflection symmetry > group theory > analytic geometry > linear algebra, etc. (higher dimensional geometry, topology).

Thus, a topology of reciprocal influence of Hands and Mind. 

In Spenser’s terms: 

the S3  demands the manipulator program: 

Propositions, not just objects [simple blocks];

Ideas within a hierarchical structure; and 

Alternative possibilities within a combinational array. 

Or, from Bruner’s perspective:

the S3 is a manual cognitive development tool which spirals learning:

Enactive:  how to do things -- actions which achieve results;

Iconic:  images as surogates for concepts; and 

Symbolic:  reflect on propositions rather than objects. 

Or in plain english, the S3 demands a conscious, self-reflective process of: 

Reorganizing, again and again, orientations and evolving networks;

Analyzing and intuiting evolving, dynamic patterns and connections;

Manipulating according to rules (deductive reasoning) or inferring new rules from particular instances (inductive / abductive reasoning);

Switching / jumping back and forth from internal to external reference frames, survey perspective (bird’s eye view) and route perspective (environment tour); 

Coping with tricky Gestalt figure / ground illusion and structure (relationship among the parts and whole) and function (step-by-step operation), etc.    


                                                                (checkered tunnels show paths)

In a nutshell, S3 demands the manipulator constantly re-examine reigning assumptions and embrace a new paradigm: a spiral of self-reinforcing change of cognitive structure; a balance of evolving recursion, dichotomies, pattern matching and creation; transcend and control nested and evolving recursion and contradictions by inventing new propositions.


                                                    (checkered paths allow exit)

For infant, child or world-class mathematician, from manipulation to study, from intuition to reasoning, from object to proposition, from developing intellect to powerful mind, each at his / her own pace — testing hypotheses, cycling between the concrete and the abstract.

Low-tech cognitive tool.  Intelligent toy.

Reciprocity of Hands and mind. 

S3 Play.  


© Michael S. Sommer, Ph.D. 2015