Binary and Analogical Reasoning

The Sommer Cube (S3) is about Dialectic as Play


                                                       HEGEL FOR TOTS

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


"The Dialectic Incarnate -- Hegel for Tots"

“Froebel developed the concept for the second gift by 1836.  He considered it the most profound of all, claiming that to understand the second gift was to understand kindergarten.  Consisting of a sphere and a cube linked by an elucidating cylinder .... [emphasis added] Alike in their perfection, the sphere and the cube are, in respect to form, pure opposites …. the dialectic incarnate – Hegel for tots – and the clearest expression of Froebel’s law of opposites.”

                                                        Norman Brosterman:  Inventing Kindergarten, 1997                                                                                                                                   


What does Sdo?

In operational terms, Silluminates / integrates diverse problem sets (classes) across a broad range of disciplines  CHALLENGES / GAMES  as algebra (“al-jabr”:  reunion of broken parts) facilitates the solution of geometry problems by other means (Cartesian Plane). 

S3 is transformative, a kick-start, a self-generating, self-sustaining escalating causal loop:  an exploration of deep causal structure (in "hard science”, beyond behaviorist psychology descriptive terms -- think endogeneity); an evolutionary cycle of cognitive development where the manipulator is both agent and object of change. 

It’s about classic toy.


And, it’s about low-tech, high-yield, cognitive tool which can play a critical role for the individual as well as formal education and science disciplines' (STEM) efforts to recognize and develop spatial reasoning -- key to intelligence.

"Recognizing Spatial Intelligence", SCIENTIFIC AMERICAN (2010) 

And it's fun.

Variations on a theme of square and circle -- flip-flop, a counterpoint of logic and intuition.  An interplay of objective and subjective reality.



A tension of switchiness (compound cognitive, perceptual, mechanical flip-flop and schedule of reinforcement — reversal learning":  analogical reasoning, instrumental learning):  switching of directed attention, relative motion and form, and navigational strategies


                                                                              Rolling Ball “Tilt-Switch      

Thus, S3 is about uncertainty of analogical / binary interplay.  A competition.

"… the logic of dichotomies ….  A container shall be provided with holes in such a way that they can be opened and closed.  They are to be open at those places that correspond to a 1 and remain closed at those that correspond to a 0.  Through the opened gates small cubes or marbles are to fall [gravity] into tracks, through the others nothing.  It [the gate array] is to be shifted from column to column as required."

                                                                            Leibniz, G.W., "De Progressione Dyadica -- Pars 1", 1679

It’s about self-regulation of analytic and holistic thinking; a competition of rational (programming desired logic functions) and nonrational.

Compounding logical arguments.

Thus, it’s about grasping everyday complex functional relationships solved and displayed.

Untitled t subscript copy

Changing (24 S3 attitudes), robust (multiple input / output), tunnels (tilt-switch / rule) "function box”. 

(Think of the ball as “ping”:  a network test for reachability.  Part of the methodological process of logical reasoning, the logical argument.)  

A switching of “true" and “false" logical arguments presented visually and tactilely — “sameness" and “differentness” at multiple levels of abstraction in terms of relational complexity.  Causal interaction within dynamic core.  Patern-matching.

Sis a paradigm shift, a hybrid, an orchestration of ideas:  combination of simple puzzle, labyrinth / maze and rolling ball (apparently unrelated, but in equilibrium, exploiting the symmetry of geometry and algebra, to higher and higher levels of abstraction), where intuition, naive commonsense reasoning, are prerequisites -- logic is necessary but not sufficient.

“... a work of instruction and enlightenment, a kit to be assembled ... that looks backward and forward at the same time ...architectural, rotatable, invertible.”   

                                                                                                                    Paul Epstein on J.S. Bach

In other words, the S3 process requires a cycle of intuition and reasoning, naive commonsense reasoning, to: 

Expand fluency with efficient procedures;

Analyze change in both concrete and abstract contexts;

Connect concrete things and events with their abstract representations;

Develop mathematical arguments about geometric relationships;

Reason about / better understand common-sense and formal logic; and 

Structure proof.

The S3 validation is elegant: success, as characterized by manipulator criteria, is a structure, conceptual or physical, which allows the exit of the ball, or give the mathematics solution.  

And it’s fun.

Thus, S3 is about human control of complex systems:  dynamic decision making.  Uncertainty as tool.

Mental and manual rotation in the face of contradiction of action and environment:  an illogical universe, and its implications.  A topology of paradox, Watzlawick’s "contradiction that follows correct deduction from consistent premises”.


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



The S3 process requires a cycle of intuition and reasoning, naive commonsense reasoning, to: 

Expand fluency with efficient procedures;

Analyze change in both concrete and abstract contexts;

Connect concrete things and events with their abstract representations;

Develop mathematical arguments about geometric relationships;

Reason about / better understand common-sense and formal logic; and 

Structure proof.

The S3 validation is elegant: success, as characterized by manipulator criteria, is a structure, conceptual or physical, which allows the exit of the ball, or give the mathematics solution.  

And it’s fun.

                        Operation and Activity of Abstraction

                               (Think "Pre-K math" -- keep it simple.)


"[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word." 

                                                                                                      Galileo Galilei, Opere Il Saggiatore, 1623


Q.  What is a number?

A.  An abstract idea, used in measurement and counting.

Q.  What is a numeral?

A.  A symbol, representing a number.

Q.  Where to begin?

A.   Albert Einstein's “Combinatory Play”, the genesis of mathematics.


“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 

                       S3:  Programming Binary Logic Gates

             Discovering and Representing A System Of Rules


                                                                                                      Asynchronous Analog-Binary Processor (0/1)

Each S3 reorientation simultaneously reprograms the four "gravity feed” tunnels differentially, nonlinearly; each acts as a binary (0/1) logic gate (rolling ball "tilt-switch") to impede (0) / allow (1) ball flow.






Michael’s little idea:

Symmetry under transformation.

And feedback (Ashby’s double loop):  exploration of network topology, physical and logical -- with hands and Mind; Apex reasoning, patterns of thought exceeding the parameters normally experienced in logical operations:  where the strategies and tactics are evolutionary.

Thus, S3 is about uncertainty as tool.

Switchiness (compound cognitive, perceptual, mechanical flip-flop and schedule of reinforcement — "reversal learning"): switching of relative motion and form. 


Juxtaposition of square and circle across three dimensions. 

Patterns of thought (and their systematic breakdown -- learning requires negative knowledge), intuition, under simultaneous mental and manual rotation.  


Buckminster Fuller's "coordination of thought and physical action, the genesis of geometry, system, and structure.


While S3 demands recognition of sequence / pattern of repeating events formed in accordance with a definite rule(s), different individuals can perceive the same pattern differently, reach different generalizations.  

It’s mysterious.

In Cybernetic terms, S3 is about demonstration of functional relations, particularly goals and feedback, of Norbert Wiener’s ANIMAL  /  MACHINE cooperation in DYNAMIC SITUATIONS -- fields which can change (or seem to change) autonomously.

S3 fields are "pattern blocks", which create porus polyhedron "chains" formed by distinct collections of figurative elements:  circles and squares.

Thus, Sis a conspiracy of pattern, a dynamic router, a system of programmable binary (0/1) switches, a “flying junction” (an over-and-under junction of tunnels which weave past each other) housed within a cubical exoskeleton with multiple axes of rotation and alternative, manipulator-controlled ball trajectories, offering exit in all orientations.  

A distributed processor (S3) programming network.



(Remember, think of the ball as “ping”:  a network test for reachability.  Part of the methodological process of logical reasoning, the logical argument.)  



With each additional S3 (transparent or opaque) linked tunnels offer paths of  nonlinearly 

a)  more combinations (complex binary sets and patterns and groups), but 

b)  fewer permutations (possible successful paths).

That is, the closer to the desired end, the more high level reasoning is required to proceed without disrupting the entire Snetwork.  And with each additional path the other path(s) / algorithm(s) must be redone in concert with the new interrelationships to optimize multiple simultaneous paths:  (ike the Frame Problem:  no boundary for what information is relevant for each action, what will change, what will remain the same), the problem of delimiting the consequences and non-consequences of actions.  

"'solutions to the narrowly conceived problem have the effect of driving a (deeper) difficulty into some other quarter of the broad problem ” 

                                                   Daniel Dennett  (Hayes):  Cognitive Wheels: The Frame Problem Of AI

      Approaching Complexity (Something You Cannot Predict)

                                      Unconscious Inference

"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)

The developing S3 network requires constant revision of operational sets and simultaneous consideration of many possibilities from many perspectives: the order of steps is not fixed, there is no algorithm.

From this perspective, 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. 

S3 is about continuous synchronous and asynchronous switching:  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, like Boolean algebra -- a way of seeing new structures -- fundamental to the design of binary computer circuits and programming language.  (Individual S3 logic elements do not necessarily have a discrete true / false state at any given time; simple Boolean logic is inadequate for this, thus extensions are required.)  

It demands the manipulator understand and assimilate evolving information about patterns and functional relationships and analyze change in both concrete and hypothetical contexts: a meeting of abstract ideas with the spirit of mathematical rigor.

In other words, the Sis about coordinating mental and manual environment, modifying self-behavior on the basis of experience (feedback), through the optimization of multiple simultaneous paths, in a constant stream of evolving problems -- within a topology of paradox (contradictory cognitive and perceptual signals / messages at different levels, with a negative injunction forbidding acknowledgement of that contradiction:  Double Bind):

A dialectic engine, demanding logic and intuition; a tangible, lucid demonstration of basic behaviors (consciousness) of the brain through 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.


                 Asynchronous Analog-Binary Processor (0/1)  /  three axis rotation problem space   


With each rotation (precession of the rotational axis of each tunnel), and each additional S3, the manipulator must focus on the binary issues (0/1) of a given tunnel (part), proceed up the logic hierarchy to the S3 (whole), then to the ever expanding universe of S3 (overarching whole / system of systems).  Then go back down the hierarchy of hierarchies and repeat the process (while coping with contradiction of action and environment).  A dynamic cycle of nested abstraction, recalibration and reflection.

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

In C.S. Peirce’s Pragmatism terms, A) flip-flops of abduction (intuition) / deduction / induction, a dynamic cycle (or spectrum) of inquiry only understandable as a whole, B) systematically impeded by gestalt figure / ground illusion flip-flop (active resistance, error).

The exploration of symmetry, influences among parts and whole, square and circle --  


adaptation to accelerating rates of change and complexity of system and environment: a cognitive development tool.  While Rubik’s Cube (Machine-think:  about problem solving competence), without that paradox, exercises logic, S(about human learning proficiency) demands logic and intuition.

A topology of paradox.  

"… 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.”    Gödel, Escher, Bach

                          Symmetry Under Transformation

From another perspective, Sis a self-generating, self-sustaining, escalating causal loop, where objects / ideas are taken to pieces, analyzed, and reassembled (an evolutionary cycle of cognitive development where the manipulator / observer is both agent and object of change).

Thus, Sis transformative demonstrating:

Open Systems Thinking as a working epistemological premise: a rotating, cubical field of “forbiddingly complex environmental interactions” (F. Emery, Systems Thinking, 1969). 

Second-order Cybernetics:  "the cybernetician, by entering his own domain, has to account for his or her own activity. (Heinz von Foerster, Cybernetics of Cybernetics)

Artificial Intelligence / Mathematics (the science of patterns) Play (without symbols), Symmetry and Consciousness -- “strange loops”, feedback on higher levels. (Gödel, Escher, Bach / Norbert Wiener / Ross Ashby)

Play of causal inference / adaptation.

A cubical maze module.  Symmetry under transformation.



From yet another perspective, the Sis about active control of thinking while learning -- not just goal-attainment;   partnership of motor and mental rotation (unlike computer-focused learning), essential in the very early years, which facilitates robust exploration of abstract concepts in visuospatial terms. 

        “Logic is for demonstration, intuition for invention.”  

                                                                                      H. Poincare, L’Enseignement Mathematique (1899)


Simple "puzzle solving" logic is not enough -- there is no algorithm with S3.  As combinatorial binary issues (0/1) rapidly become overwhelming a robust flexibility of logic and intuition,"intelligent trial and error", must be exploited to continuously compose / decompose the problem from / into sub-problems and multiple constellations, by continuous analogical mapping and perceptual processing to program dynamic patterns and causality of structure, function, and process of square and circle:  the manipulation of abstractions without symbolic and numeric tools -- core ideas spanning all disciplines.


                                              (four checkered paths cross and exit within a single S3)

© Michael S. Sommer, Ph.D, 2018