Learn to Understand.

1. Observation as the first learning process.

When a living organism developed senses it could ‘sense’ some objects present in reality without knowing anything about reality and objects in it, just like a thermometer can sense temperature. To actually observe something it was necessary to have available some brain activity and intelligence, then the organism could analyse its sensory activity using some abstract machinery thinking allowed it to construct. We will not go into detail about how this evolution took place but we will put ourselves in the actual situation of a human observing things today. In our education system there is a difference between acquiring knowledge and understanding the knowledge and often critical thinking is referred to as a source of understanding yet it is usually not explained what “critical” means. Critical of what? The government, the teachers, the self? Perhaps it is necessary to provide a formalization of the learning process in order to fix concepts with a fixed meaning here. First remark is that thinking is a biological activity mainly situated in the brain a bio-electro-chemical process of neurons and synapses, this activity produces ideas which are codes for the contents of the process and these are stocked in the memory as abstract (non-existing in the material universe!) things. We give meanings to these codes and these meanings are remembered with the code in the memory. The very first thing the memory remembered was the “identity” of some exterior interaction, like the drop in temperature at night,day after day for thousands of years. Even if the interaction was not really identical it was similar enough to create a first memory and it was an approximation, a class of events ,a series of similar interactions which was encoded as the ‘type of interaction’. From the very beginning the memory had the defect of viewing identity as a type or class without knowing these concepts then. But then evolution went swiftly and humans constructed abstract notions which could be used to compare different sensed events, we called them properties and used these to describe sensed effects of the events.Such properties did not exist and there was no way to decide whether these actually belonged to the event or to our sensing of the event, that is it could be a property of the interaction between us and the real event. So immediately at this elementary level we meet a problem which is also present in modern quantum theory i.e. the role of the observer in the observed structure! So abstract properties did not necessarily really express some fact about the observed event yet it presented a unique way to describe abstractly the real event we assumed to exist in reality. Step by step, often by repeated sensing and acting upon the object ,for example rotating it to have another view while assuming this rotation did not change the object in reality, we create an ‘abstract object’ consisting of many layers of properties, we call observed properties, converging to a more elaborate ‘image’ in our mind and put in the memory as the object X, where X was a name, another code different from the biological sensing activity, for the abstract onion representing the real object. The memory did a fantastic job because by just thinking X we get the whole idea of the object and its properties, even plus other objects in the memory for which we had observed ‘relations’ with the new object . Again relations could be based on other sensatory registrations of interactions between real objects or also on logical relations between abstract properties already formulated. At this point we meet the necessity of having a logic in making an observation from a set of sensory registrations. This logic was also necessary for the development of a language and language and observation skills developed more or less simultaneously in people. The logic follows from the chronological ordering of events in reality which gave rise to observations of cause and consequence, at least these were interactions which we saw and interpreted as such while being unaware of the situation in reality, so just judging on the basis of ‘observed reality’. The method of developing an abstract model of a real event by constructing properties and linking them by logical implications is the learning process of observation. It gives an approximation of the real object in our abstract world of concepts and logical connections between them. We can keep on looking for new properties by interacting with the object physically,that is in reality. Such experiments can make our image of the object much more precise in the sense we will know more about it in terms of abstract properties. Yet we do not worry whether the experiment has changed the identity of the object in reality, and whether the more complex experiments yield properties of the real object or properties of the experiment! Whatever one would think of these fundamental problems,we accept at some moment that the object is ‘described’ well enough to be identified by the image, that does not mean we know everything about it and certainly not that we understand anything even.

2. The formal learning process and its refinements.

Starting with the observation process we can immediately generalize the structure of such process to a system with a set of ingredients, these may be real events,concepts or idea clusters, and implications between them; the links could be a causal relation or logical implications or measured phenomena defining the relation. Formally a relation from a set A to B is just a collection of ordered pairs (a, b) where a is in A and b is in B, you may represent this by an arrow a →b and then draw the diagram of the relation as the set of all arrows corresponding to the pairs. This diagram provides a visual interpretation of the logical structure of the relation considered. If we look at Just one set of ingredients, A say, then a relation R on A is reflexive if for every a in A we have aRa, thus a →a, it is anti-symmetric if a →b and b →a then b=a and r is transitive if a →b and b →c yields a →c . A relation with all three properties is a partial order of A with a<b if and only if a →b. Every total ordering is a partial order , a partial order is said to be total when for every two different elements ,say a and b , in A we have that either a<b or b<a. Causality is a partial order which need not be a total order;typical examples of totally ordered sets are sets of real numbers. The chronological order on the time as a set of moments is a total order but the chronological order induced on a set of events in reality may not even be a partial order because two different events may happen at the same time. In fact there is much ore to say here because existing events or objects in reality do exist in a time interval, not in a moment in time, but we do not go into that here.

Now a learning process has a study object,that can be something we observed as happening in reality or it can be a completely abstract series of ‘related’ ideas, the ingredients of the study object are related by a relation, very often a causal or logical implication ordering.The description of the study object a well as any communication or thinking about it is linearly ordered by the irreversible time passing by in observed reality, while the structure of the relation diagram for the logical structure, that is the partial order diagram, is a complex structure far from being linear. The knowledge hidden in the process is exactly reflected in the shape of the diagram while the learning happens in time depends on some ‘linear’ way one follows through the non-linear diagram. There are many ways to explain the structure of the diagram,one can start at an arbitrary element a of A and start to build the diagram from arrows ending or starting at a, the after no more such arrows can be found take a new element b not connect by any arrow to a and then repeat the construction with arrows involving b. These ‘stars’ attached to given elements a in A, ordered in some arbitrary way, provide a linear picture of the complex diagram structure. Here we see a fundamental problem in the learning process at this elementary level! The “learning uncertainty property”. In a study, either by reading, listening or conversing, one may be in different linear accounts of the complex diagram and so there will probably result confusion about the logic of the relation (implications)with the complex diagram. Teachers (or text books etc…), do usually not show the structure of the study object before explaining the contents chronologically linearly ordered by time, and that causes all learning problems by a form of non-synchronization. From this one may derive some interesting didactic guide lines, let me just refer to two publications.

1. Ma Min, Fred Van Oystaeyen, A Measurable Mathematical Model for Processes, J. of Mathematical Sciences, vol.34, 2015.

2. Ma Min, Fred Van Oystaeyen, A Measurable Model for the Creative Process in the Context of a Learning Process, J. of Education and Training Studies,4(1), 2015.

After a first learning process one obtains some ‘knowledge’ in the memory but the knowledge cannot be total as it depends on the description of the ingredients in the process, just like some physics experiment depends heavily on the construction of the experiment. The solution is to do further experiments and connect the results. So in a learning process, the studying does not necessarily stop after the first ‘reading’ of the study object and a ‘finer’ approach may be necessary in order to obtain more knowledge and in fact knowledge about the knowledge obtained in the first process. Here comes the ‘critical’ thinking; critical of what,well of the structure of the learning process and the ‘correctness ‘ of the description of the ingredients. A ‘refinement deformation’ of the process is obtained by providing for each ingredient A a finite set of aspects of A, say A={a(1),a(2),… ,a(n)} and on the aspect set of all aspects of all ingredients of the process one defines a new relation r which is reflexive and no two aspects of the same ingredient are related by r if they are different. In the papers mentioned one studies a numerical weighing so that r on aspects induces R on ingredients in some nice way and A,r is the aspect deformation of A,R. In physical processes the aspects arte usually obtained by decomposing an ingredient in substructures,e.g. a molecule in atoms or further in sub-atomic particles,in studying the pollution of a river the aspects can be measured quantities of heavy metals or some probabilities about the debite, in mathematics aspects can be earlier results used in a new phrase. We have a very general method of deepening the study in the deformed process and reaching more understanding about the first process in the knowledge of the second! The critical thinking is structural! The contents of the initial learning process is of secondary importance, the structure of the implication diagram or the general relation diagram is now scrutinzed. For example the star of an ingredient A,star(A) say, may be looked at by taking a B in the star and looking at star(B) then a C in star(B) need not be in star(A),because we may have A →B and C →B and no relation between A and C. Putting all star(B) for all B in Star(A) together we obtain the A-component of the process. One can now do similar for all aspects a of A with respect to the relation r and study the stars for r trying to see the relation with the cluster structure for R. Inclusion defines a partial order of the stars for R as well as for r and we can define cluster processes for both R in the learning process and r in the aspect process. Without going further I hope this convinces you that a lot of deeper structure of the process does now reveal itself and that is ‘understanding’, so the aspect process is called the “micro-process”. In Physics quantization may be seen as an aspect deformation of mechanics. In philosophy it is a refinement ,usually of linguistic nature, of the description of concepts ‘and their relations(!)’.In the lifting from R to r some interesting phenomena related to fantasy and creativity will appear later.

3. The organic process of a learning process.

Organic society is defined by viewing the members together with all links between them,both of a physical and an abstract nature. Somebody writing a text and another reading it isa link or abstract nature, talking to somebody can be a physical link but also an abstract because of the impact of the messages delivered. In the same way the organic learning process about a study object can be defined , starting from the learning process on the ingredients and then passing to the linking structure like stars and clusters,then going on to a refinement in some aspect process and the linking structure for the lifted relation r, eventually reiterating this process.

The star of an ingredient or aspect may be see as the formal equivalent of a “context”, that would be even a correct statement if the language in some text is seen as a process of giving (getting) meaning and reaching understanding of the text; the context of ingredient A ,resp. aspect a, is then indeed given by the A-component for R ,or resp.the r-component of aspect a. The understanding of the meaning of the things in the components is then the understanding of A or a in context. If the A-component is the whole set of ingredients of the process then we say that A is a generator for the process,otherwise the A-component is said to be a sub-process of the original one. Very complex processes can be decomposed as the finite union of A-components (similarly on the level of the micro-process) and it may be more simple to study the X-components separately for a chosen finite number of ingredients X, for example a minimal number such that the components cover the whole process. In loc.cit. we study certain extensions of processes and one special type is obtained by extending the set of ingredients of ,say I, to say I’, and the relation R on I to R’ on I’. Typical examples would be to embed the study of the pollution of a river into a study of all water structures in an area, or the development of the behaviour of a person into the development of some group behaviour, or the role of one theorem in a chapter of mathematics into the role of a section in a theory. The nature of theembedding is strictly related to the behaviour of stars (clusters) under the embedding, a star for A in I with respect to R, is embedded in the star of A with respect to R’ in I’. The embedding is “A-thin” if no new ingredient in I’ not in I is connected to A, thus the star of A does not change at all in I or in I’. The embedding is “A-full” if all ingredients in I’ not in I are connected to A. For nice relations,like a partial order like causality, or stronger there are nice structural properties of someextensions whichb may be exploited. A very special extension of a learning process is to include a description of the observation process (the observer or student) to the description of the learning process. Then ingrediehts of the learning process may be connected to emotions or moods of the observer. This is particularly interesting for learning processes of the artistic type, for example listening to music, and now the study object focusses more on the interactions between observer (student)and the studyobject(music) of the initial process. This often is obtained from an evaluation process by a teacher of the interaction of the student with the learning process. This evaluation process is not a formal examination, it is a check on every link of the study process for the student’s reaction to the link, for example testing his understanding of the local structure (star!). Similar on the micro-level where the evaluation will by definition evaluate rigorously the ‘understanding’ of the initial learning process.Ideally such extension process should be A-full for every ingredient A in I, and similarly for a in A. That represents a much more thorough test of all understanding than what isusually understood by an examination. Such extension (evaluation) of the process will be called a test process, these processes may in the weighed version (cf. loc. cit.)be used to calibrate the original weighing of the aspects and ingredients of the processes involved. So even if the original weights for importance given to ingredients and aspects are done by an educated guess the testing processes one may suitably install will allow a more scientific re-calibration of the weights! On the other hand if the original weights have been obtained by scientific measurements the new insight (understanding) stemming from a micro-process may lead to new “experiments”by a new (test-)extension of the process, leading to better measurements of the deeper structure addressed by the new process. In the language of didactics, an organic examination procedure does not aim at an evaluation of the student’s knowledge but at the clarification of his points of misunderstanding plus an indication of where the misunderstanding results from in the process structure. The aim is to help and guide the student not to point out he does not know enough.

4. Critical thinking in an organic learning process and fantasy.

The aspect is saturated by its context, (a phrase used in the paper about Wittgenstein’s vision on music and art in general) is a restricted version of what I call the organic process, it fits the notion of natural clustering of concepts and ideas,that involves emotions and moods too when dealing with an extended process including the observer in the process! This may be seen as a quantization of the learning process and its micro-process. This quantization moves the theory out of the more abstract and rational contexts, the construction is a version of testing (evaluation)processes of learning processes. In the case we were dealing with weighed processes the extensions could also be viewed as test-processes for calibration or re-calibration which may be seen as a quantization interpretation of the old measurements, the choice of doing this by introducing probabilities in the measurements was one about the choice of the mathematical model for the usual quantum theory, which is probably not related or induced by any process structural argument! In fact since non-commutative geometry of a dynamic time-space may be seen as a process deformation of the usual manifold structures one may claim that the passage from manifoldstructure to the non-commutative topology is more indicated by the process (dynamical) philosophy.

So in our observations of something we never “see” an object alone,we observe the links with other existing things as well as non-existing abstractions we constructed already and while organically observing we also analyse some emotional connections in us depending on the mood we are in. So the organic structure of something extends in reality as well as in the abstract world we constructed AND in our Self, both on the level of emotions and spiritual properties as well as biological (unaware even) construction properties! So here critical thinking about the process structure will now imply an analysis of the Self and its relation with the original learning process, thus a deeper inclusion of Self-study but focused on the topic of the learning process at hand. On the micro-process level this leads to deeper understanding about the involvement of the person and the study-material, for example a piece of music being analyzed or just …enjoyed ,with stress on the emotion joy!

I do understand Wittgenstein perhaps stayed too close to his mathematical-linguistic background, putting emotions in the “culture” or the context. For me,as defined above, the context of an aspect is an integral part of the organic learning (observing,researching) process. Therefore,on that level I do not make that distinction between emotions and rationality, they are completely intertwined in the Self and even deeper in the unaware structural identity I call the soul (without any a priory religious assumptions on it!). Thus in some sense, while we are observing and analyzing some work of art-or anything in reality-we construct an amalgam of part (only a part!) of our Self and the observed object, we do not become one with the object as claimed in some Zen-type explanations, but we “eat” the object in some philosophical process of ‘digesting’ it. I guess I believe it can indeed be related to the primitive reaction of “eating” just like babies try to eat everything. Like we transform food to part of us we transform abstract observations of reality to part of us, kept in the memory both aware and non-aware. All of this is related to the process we call understanding where the context of aspects is exactly the organic structure of the aspect deformation of the learning process of observations. It is exactly there where the counter-causal relations stemming (or in fact defining) creativity and fantasy appear. So the point of view of Wittgenstein catches only a small part of my interpretation which connects eating and the soul via the tower of micro-processes leading at each step to deeper understanding and deeper also meaning more deeply connected to the social micro-network (abstract and real!).

5. Anti-relation and fantasy in organic processes.

When in A,R we have a relation A →B and in A,r we have a relation b →a for a,b aspects of A, B resp., then we say the b →a is an ‘anti R-relation’ since it goes against the direction of A →B. When R is causality then anti-R is said to be anti-causal. Instead of starting from a relation R on ingredients and constructing some relation r on aspects, and there is some freedom in this, one can start from an aspect relation r and via some weighing of aspects according to their importance in the ingredient they describe-for example if the genetic structure is an aspect of some plant then one can give this a high weight because it is enough to classify the plantspecies it belongs to-one can induce a relation R on ingredients via a simple mathematical counting formula (cf. loc. cit.). If this R is a partial order then r is said to be ingredient ordering. Starting from a partial order R and constructing some r on aspects inducing a relation R’ on ingredients according to the formula mentioned before then R’ needs not be R and it even does not necessarily needs to be a partial ordering! Thus then one has to manipulate the construction of the lifted aspect relation so that it works well. Hence we will usually assume that we start from an ingredient ordering relation. There is however a big advantage in allowing weaker relations if one sets up a learning process in an environment which has an organic description,for example in society ,some group of organisms, some species,Life, Nature. The formal learning processes allow abstract constructions not actually changing in time but processes described in reality almost always have the chronologically induced ordering. An organic process where the links are given by steps in the process is not behaving differently with respect to the appearance of anti-R-relations compared to any learning process, and the same for the use of fantasy in the aspect relation where the causality of the ingredient relation is broken,thus where one acts as if the relation does not represent a reality,which is a possible definition of fantasy. However if the ingredients are viewed in organic society or the organic process of the universe, then the links between ingredients not depending on time or the moment considered define a cluster of ingredients, for example cl(A) the set of all ingredients linked to A in the organic way. The process considered in its organic environment by putting cl(A)Rcl(B) if ARB neglects at first sight some relation CRD for some C in cl(A) and D in cl(B) but this will appear in cl(C)Rcl(D) of course. The organic links not seen as process steps in the time-evolution form symmetric connections between the elements of the cluster, if their selection as organic links in a process in the organic environment has been chose correctly in the philosophy of a process in the organic environment then a C in cl(A) can only appear in CRX for any ingredient X if X is in fact in cl(B), meaning that R is really defined as a relation on the class-level. All of this can be reformulated for the aspect deformation defined in the organic way, this includes that if ARB and a is in A and such that arb with b an a priori arbitrary aspect,then b is an aspect of B’ for some B’ linked to B; since for A(1) in cl(A) we have that cl(A(1))=cl(A) the foregoing says that relations between aspects of any A’ in cl(A) and something in B always end in a fixed B’’ in the class of B.

Now allowing a relation R which is not a partial order but describing a process in time of classes of ingredients,which may be seen as an extension of a causal process (in time) on representatives of classes by adding symmetric links between elements in a class can have anti-R-relations in the aspect sets over some ARB’ even if such do not exist over ARB; note that we cannot have BRA for the chosen representatives A and B of their classes but we can have B’RA ! This means that counter-R-logical deformed relations are now more frequent than in the non-class theory,so more fantasy is used in these processes.

In general a notion of classes may be introduced without the good behavior of class respecting relations as above, the appearance of anti-R-relations (with R not being a partial order relation) is much more chaotic; perhaps one may write computer programs to deal with such situations. For example dividing society in classes like political classes or religious classes and viewing the evolution of organic society as a process with a non-causal relation (one can also go to the first micro-process of a causal one and use this in a second deformation thus starting from a non-causal relation but one may for example use organic causality-which is weaker than causality- as the relation, then the organic chaos should be describable to some extent).

Obviously,the working with classes in the organic setting introduces a higher amount of process uncertainty by the chronological formulation of complex relation diagrams, and these are more complex in the non-ordering case. Thus when communicating properties of the class processesin the organic setting, communication about it should be more structured than in less complex situations and the structure should be made clear before discussing the properties. I hope I made not only this very clear.